The ÔCoreÕ Concept and the Mathematical Mind
Chris King
Mathematics Department
University of Auckland Feb 2007
Abstract:
Abstract pure
mathematics is often seen as an Ôinverted pyramidÕ, in which algebra and analysis
stand at the focal point, without which students could not possibly have a firm
grounding for graduate studies. This paper examines a variety of evidence from
brain studies of mathematical cognition, from mathematics in early child
development, from studies of the gathererhunter mind, from a variety of
puzzles, games and other human activities, from theories emerging from physical
cosmology, and from burgeoning mathematical resources on the internet that
suggest, to the contrary, that mathematics is a cultural language more akin to
a maze than a focallybased hierarchy; that topology, geometry and dynamics are
fundamental to the human mathematical mind; and that an exclusive focus on
algebra and analysis may rather explain an increasing rift between modern
mathematics and the Ôreal worldÕ of the human population.
2: Landmarks from Early Childhood and the Noosphere
3: The Game, Topology and Two Small Clouds in
Classical Analysis
4: Puzzles and Games as an Expression of Human Mathematical
Imagination
5: State Space Graphs and Strategic Topologies
6: The BrainÕs Eye View of Mathematics
7: The Fractal Topology of Cosmology
8: References
1:
Introduction
The idea of a
fundamental duality between analysis as Ôthe science of continuityÕ and algebra
as the Ôart of discrete operationsÕ, forming the two pillars of conceptual
mathematics raises a number of deep questions. At first sight it provides an
appealing analogy to the idea of complementary left and right hemisphere brain
activity, spanning order and chaos, typified by the specialization for
descriptive language in the left hemisphere, particularly in males, and
creative poetry and music in the right. However, rather than the ÔAdam and EveÕ
primality of algebra and classical analysis, it is the disquieting
entanglements of topology, like a jilted Lilith, which have been seen as
ultimately standing in natural complementation to the ÔdivineÕ order of
algebraic structures in the split brain. Neither does the algebraanalysis
partnership stand on a solid axiomatic foundation, undercut, like the house
built on sand, by the common, yet often discarded, bedrock of mathematical
logic and set and category theory.
However, the argument for algebra and analysis is not built so much on
axiomatic primality as perceived Ôheuristic necessityÕ  the idea that, without
a firm grounding in these two abstract areas, a prospective graduate will lack
essential knowledge and skills. Is this so, or is it a classical misconception?
Figure 1.1: Hmong appliquŽ illustrates a cultural
tradition intuitively based on topology which illustrates
the nested Jordan curve two
colouring theorem of section 2 used to effect in early child maths readers.
The left figure has 7 blue oceans and 19 white islands,
the right has 7 red oceans and 11 white islands.
At stake is a
model of ÔrealityÕ of cosmological proportions. Central to the Ôbelief systemÕ of modern pure mathematics is
the notion that mathematics is itself a cosmology of abstract archetypes,
exemplified by Karl PopperÕs ÔtrinityÕ[1]
in which the brain in the objective physical universe and the subjective
conscious mind are complemented by a third Ôdark forceÕ  a ÔcosmosÕ of
conceivable abstract structures waiting to be ÔdiscoveredÕ by theoretical
research. Nevertheless many of
these structures, stemming from the very notion of an infinitesimal point
singularity, abhorred by the waveparticle complementarity of the quantum
universe, already have a slightly archaic classical tarnish in a world composed
of entangled quantum waves, so are these archetypes really ÔuniversalÕ?
Comparison with cosmological theories of everything provides suggestive clues.
In contrast to the
cosmological view of mathematics, is the ÔculturalÕ view that mathematics is a
human, relatively Ôculture fairÕ language, crafted to enable the development of
concepts, the discussion of hypotheses and the proof of theorems that would
otherwise be impossible without it. Clearly mathematics, and with it the areas
of algebra and analysis, are developed as linguistic branches of mathematics,
containing their own definitions and propositions, which can logically be
proved using the language of the objects, functions and operators in each of
the fields. But if mathematics is primarily a humaninvented language, the
question arises as to whether a given branch of it has any unique claim to
fundamentality and whether another quite different linguistic description might
also lay equal claim to the territory, and whether, in the near future, or at
the restaurant at the end of the universe, the claims algebra and analysis
might make, even to heuristic fundamentality might appear quaint, myopic and
archaic artifacts of a culture doomed to attrition by its own lack of
adaptability to the circumstances around it.
We can see signs
of this dilemma in the relation between analysis and topology, where competing
ideas of continuity, which in analysis are based on measurement and an almost
irresolvable strategic standoff between epsilon and delta, meet their nemesis
in the fall of the metric space empire to the superior topological concept of
continuity, based on the apparently elusive notion of ÔopennessÕ opening the
PandoraÕs box of all the tortuous knotted worms of continuity and connectedness
and the many worm holes back to geometry writhing in the topological category
and in the futility of measurement in a landscape permeated by nonmeasurable
sets. While analytic epsilondelta
is still considered the foundation concept, this is done with the knowledge
that, lurking in the back closet, hopefully kicked upstairs to the graduate
program, is a more fearsome concept, coming right out of left field, or more
likely the enigmatic right cortex, that lays waste to all ideas of measure.
In presenting
these ideas to a human population there is a contest of credibility gaps. While mathematicians rail at the
futility of the na•ve concept of limit in first year courses, performed merely by
making spot checks at a few neighbouring points getting closer to the value
concerned, the epsilondelta game has proven to be a standoff so counter
intuitive that it has, by degrees, been shuffled into its own cubby hole in
specialized theoretical courses, only to reappear as the required phoenix of a
ÔcoreÕ major. By contrast, the topological idea of continuity and
connectedness, while remaining mathematically ÔfringeÕ does have immense
immediate and practical appeal to the human consciousness and imagination. The
alternative na•ve idea of continuity  that which one can execute while drawing
without lifting a pencil off the paper  goes right to the heart of topological
ideas of continuity, and path connectedness, and is the foundation of both
algebraic homotopy and manifold topology and is thus the basis of the PoincarŽ
conjecture.
Figure 1.2: Left: Oldest example of Celtic frieze and
knots from Durham Cathedral (7^{th} cent). Center: Maori string figure
Tahitinui[2], has been found to be topologically
identical to the Native American Osage diamonds, illustrating the deep
crosscultural awareness of topology,
as an intellectual skill associated with textiles, nets and knotmaking.
Chinese children also play string games. Top right: Neolithic meander maze
resembles that on the coinage of Knossos Crete (lower inset) suggestive of the
MinotaurÕs Labyrinth. DuraEuropos ÔArranÕstyle knitting (300BC256AD)[3]
and Andean woven textile (upper inset) illustrate the intrinsic topological
complexity of all textiles made by weaving, knitting netting other methods.
Topological
continuity is also a directly perceivable conservation concept that, in
contrast to its relegation to the ivory towers of graduate mathematics, is
directly understandable by children still learning to read, as successfully
purveyed in some of the best mathematics education books for young children.
For this reason topology courses have been a favorite with training teachers
and those returning for additional mathematical education, for the very reason
that they provide access to a fundamental form of mathematical reasoning wholly
neglected in our emphasis firstly on statistics as an alternative to
mathematics and secondly on algebra and calculus to the exclusion of topology
and geometry in secondary mathematics and undergraduate core papers at the
tertiary level, now prospectively becoming filtered by prerequisites even at
the graduate level.
Topological
continuity also has very plausible foundations in gathererhunter skills, honed
over evolutionary time scales, in terms of tracing tortuous connected paths in
the wilderness, between raging torrents, lairs of lions and dangerous
precipices, to apply oneÕs skills of geometry and dynamics to outwit the prey
and retrace oneÕs steps through the topological maze to finally bring home the
meat, to trade for sexual favours in the Ôamazing raceÕ of genetic survival.
With the foundation of Neolithic cultures, weaving and netmaking added
tremendously to the topological complexity of the human imagination, leading
later to knitting and ultimately, with the flying jenny, to the industrial
revolution.
Figure 1.3: Map of
the Kaueranga Valley illustrates how wilderness terrains give rise to a complex
natural topology of pathconnected routes (tracks and trails) partially
obstructed by other topologicallyÐconnected features, such as water
courses, ridges and escarpments, and patches of dense forest with steep
impenetrable valleys. Learning the
pathconnected topologies of the wilderness is a gathererhunter task essential
for survival.
To support this
topological and discrete dynamic thesis, a stimulating investigation has been
made of a variety of human puzzles and games, as a cultural ÔimprintÕ of human
mathematical imagination, including obviously topological wire, string and loop
puzzles, Rubik cubes and their algebraic and geometrical variants, including
irregular forms such as ÒSquare1Ó, Erno RubikÕs 3ring puzzle, geometrical
tiling puzzles, including the Soma cube and variants such as the ÒLonpos PyramidÓ
and ÒHappy CubeÓ, peg solitaire, the five peg puzzle, numeric puzzles such as
magic squares, squaring the square and Sudoku, logical puzzles such as the
ÔEinsteinÕ Zebra puzzle, and ruleguessing puzzles such as ÒPetals Around the
RoseÓ. In addition are included games of topological strategy, such as Go and
the more recent fractal geometry variant ÒBlokusÓ, based on pentaminoes. and
topological tiling games such as Tantrix and Trax.
A disproportionate
number of puzzles and games have a geometrical and/or topological basis, which
belies the abstractness attributed to topology by mathematicians, and
emphasizes the central place geometry and topology have in the human
imagination. All puzzles and games, whether they are geometrical, logical or
conceptual are fundamentally topological in nature, because they possess a
pathconnected route from an initial state to a final solution, or endgame. A
puzzle is solvable if such a pathconnected route exists, and humans, whether
they are solving a logical puzzle, or a geometrical one, do so by having an
intuitive, or deductive idea of territory within the ÔstatespaceÕ of
possibilities, and how to construct a connected path from beginning to end
along the connected graph of nodes in this space, in which contingencies Ôif É
thenÕ correspond to branches. All
proving of mathematical theorems likewise has a basis in such Ôtopological
logicÕ. In this sense a link is made between the ostensibly continuous subject
of topology and the discrete area of graph theory, but regardless of this, a
human is using, even in a highly abstract situation, the same topological
skills that humans have relied upon for survival in the wild over millennia. In
some puzzles and games the state space is surprisingly small, because the node
transitions are complex, but in others the state space consists of a huge graph
containing up to 10^{50} vertices, comparable even with the 10^{500}
ÔmultiversesÕ now suggested by string theory. Nevertheless a human successfully solves these puzzles,
which closely reflect the superexponentiating npcomplete traveling
salesman[4]
contingencies of n real
connected paths in the world at large, by developing a sense of connectivity
within the territory.
An investigation has also been made of
the current state of research into mathematical reasoning in brain studies,
using functional magnetic resonance imaging and the electroencephalogram. To a
certain extent these are limited by the relatively simplistic arithmetic and
mental rotation tasks frequently assigned to subjects confined in a narrow
noisy detector, but they do provide an informative view of the broader
biological basis of mathematical reasoning lying beyond the axiomatic paradigm
and show in a startling way how language and culture can affect mathematical
processing in the brain, supporting the ÔculturalÕ view that mathematics is a
cultural adaption.
Models of human cognition, imagination
and creativity go even further and are sometimes likened, not just to a
generalized language but more to a Swiss army knife of evolutionary skills
cobbled together, not from a universal foundation but in the subtle adaptive
ingenuity of parallel genetic
algorithms. In terms of brain science, another version of this fortuitous
convergence of parallels is the idea that numerical reasoning is based on a
convergence between three neural ÔsensoryÕ codes of numerical processing, involving analog
magnitude, auditory verbal, and visual Arabic codes of representation. Unlike
mental rotation and targeting, and the need to trace topological paths in the
wilderness, numeracy itself has been found not to be a human universal, with
the Amazonian Pirah‹ becoming
famous for their lack of linguistic categories more specific than ÔfewÕ,
resulting in a cultural inability to distinguish even small numbers of objects
one could count on oneÕs fingers. In a way this is unsurprising since even
ÔadvancedÕ cultures have trouble dealing with a digit span of over seven.
Cosmology also carries with it fundamental
aspects of both topology and fractal dynamics, which we shall investigate to
complete the perspective. To celebrate the new idea of intrinsic duality
between theories of everything, in which supposed fundamental particles, such
as the quark and lepton, exchange roles with perceived composites such as the
magnetic monopole, in the next section we will make a case based on early
learning mathematical texts, that mathematics could as well be founded, in
human evolution and development, on twin concepts of the topology of
connectedness and discrete dynamics.
Figure 2.1: Clues for the ZOO islandslakes puzzles.
2: Landmarks from Early Childhood
and the Noosphere
Two mathematical education books, which made a lifetime
impression on my own children, for their stimulating mathematical
imaginativeness, are the Zoo series[5]
and Inner Ring Maths[6]. These contain a variety of mathematical
puzzles and problems involving sets, classification, sequence completion simple
arithmetic numeracy, rotational and reflective symmetries and geometrical
shapes.
Figure 2.2: The initial topological puzzles have both
animal clues and a two colour coding of the regions.
(a) Topology of Lakes and Islands
The ZOO books are set out with a minimum of language
statements so that even a child who cannot read can identify the tasks visually
and solve the mathematical problems. Most interesting of these to my children
were the two volumes dealing with topology and connected pathways.
The reader is first given the nonverbal clues shown
in figure 2.1, which give instances indicating the problem is not how many
rabbits or fish, but how many (possibly nested) islands and lakes there are.
The reader is then led on an adventure into increasingly abstract and complex
instances of the problem as shown in figures 2.2 and 2.3.
In a mathematical sense, the twin count of islands and
lakes constitutes a topological invariant of a set of nonintersecting Jordan
curves in the plane, each of which divides the region containing it in two in a
way which always permits a twocolour coding of the whole rectangle. If the
enveloping region is considered to be the ocean as is the case for (a flat)
Earth, this results in an unambiguous classification into land and water, even
when the only clues are the curves themselves, as in figure 2.3.
Figure 2.3: The puzzles have now become an example of
using a topological invariant to make a two colour mapping of any rectangle
containing a set of nonintersecting simple closed (connected) curves.
Several points follow. Firstly this is a concept of continuity topologically
dividing a higher dimensional region that is immediately appreciated and
understood by children young enough to not have proper reading skills or a
verballybased machinery to handle abstract concepts. Secondly is provides a
mathematically meaningful expression of a theorem in abstract topology not
generally taught until the third year of an undergraduate degree or graduate
level. Thirdly it provides an example of a numeric topological invariant
spanning algebra (arithmetic) and topology on a par with the Euler
characteristic, Jones polynomial and homotopy group.
Likewise, as noted in the introduction, topological
spaces and their knottings from the Moibius band to the Klein bottle and worm
holes in spacetime provide a rich and stimulating menagerie of mathematical
challenges to our ideas of spatial connectivity and dimension which are
wellknown by many secondary students as the stuff of comic book and space
fantasy on television largely because they can be appreciated directly by the
human imagination even when they cannot be embedded in 3D space.
Specific cultural traditions such as the appliquŽ work
of the Hmong of Thailand, Laos, Vietnam and China figure 1.1 are direct
intuitive expressions of the same nested Jordan curve two colouring property,
showing in another way an ÔinnateÕ appreciation for the topological nature of
these relationships between curves and surfaces.
Figure 2.4: The ÔCollatzÕ problem presented as a number
cruncher for children in ÒInner Ring MathsÓ.
(b) The Discrete Dynamics of Choice
A second problem made the Ôcover pieceÕ of Inner Ring
Mathematics the flow chart number cruncher illustrated in figure 2.5. Once again, this is a problem which can
be appreciated as a recursive arithmetic decisionmaking process and even more
so for its surprising variety and unpredictability by young children who have
no knowledge of abstract algebra, yet, far from being trivial, it remains an
unsolved problem in mathematics whether all numbers generate a sequence forming
a discrete orbit, which is eventually periodic to the portrayed cyclic sequence
. Also called
the ÔCollatzÕ conjecture after Lothar Collatz, it is discussed in detail in
WolframÕs Mathworld[7], in Wikipedia[8] and forms a key text
editing example in Matlab as illustrated in figure 2.6. Paul
Erdős said of the Collatz conjecture: "Mathematics is not
yet ready for such problems" yet it is understood and appreciated by
children learning to read. One clear thing this example illustrates is that
discrete dynamics is as central to human mathematics as algebra and analysis
are.
Figure 2.5: The Collatz problem iconic in MatlabÕs text
editing tutorials
Matlab illustrates the nature of the
problem well as a key example of a discrete dynamical system. Computations using the routine of
figure 2.6 all display eventually periodic iterations to the 4>2>1 cycle,
but with vastly varying orbit lengths and maxima.
Figure 2.6: Matlab simulation of the ÔCollatzÕ problem
shows that although it trends to a logarithmic convergence, individual starting
values have huge variation in the eventuallyperiodic orbit lengths of the iteration,
rendering the problem unsolved to date, despite being able to be appreciated by
young children.
The iteration is poised unstably between a decreasing
and an increasing rule. The fact that some systems can tend to a variety of
asymptotic limits is demonstrated by replacing the 3x+1 by 5x+1. Three distinct
sequences cover the numbers 17, including two different eventual periodicities
 of 7 and 12 and an unbounded orbit.
4 2 1 6 3 16 8 
5 26 13 66 33
166 83 416 208
104 52 26 
73
366 183 916 458 229
1146 573 2866 1433 7166
3583 17916 É 
Table 1: Three orbit types of (5x+1)OR(x/2):
Top period 7, middle period 12, bottom unbounded.
Figure 2.7: Divergences of the
(3x+1)OR(x/2) iteration from [2.1] () for successive path
records.
Computational paths records have been
established for this iteration[9]. A current highest known,
the 88^{th} path record is 1,980976,057694,848447 which reached
64,024667,322193,133530,165877,294264,738020 before eventually entering the
4>2>1 cycle. The successive
path records 2(2) 3(16), 7(52). 15(160) 27(9232) etc. vary erratically along:
[2.1]
Although the Collatz is really a
problem in discrete dynamics it is still using arithmetic binary operations and
is hence algebraic in nature. It also involves theoretical logic of computation
in performing the iteration numerically by mental arithmetic in a manner
immediately apparent to a child. It thus presents what is an unsolved abstract
algebraic problem in immediately accessible terms of repeated discrete
decisionmaking.
In evolutionary terms many social
interactions are molded by a discrete sequence of prisonerÕs dilemma moves of
cooperation and betrayal, which build up our understanding of trust and good
character. The Wason test is a logical puzzle, in which a person is given a
conditional ÔruleÕ, if P, then Q, together with four twosided cards displaying
information of the form P, notP, Q, and notQ. Subjects are instructed to turn
over the cards necessary to determine whether the rule holds. The correct
solution is to turn over the cards displaying P and notQ to see whether their
other sides contain notQ and P respectively, because those, and only those,
cards can violate the rule. is
much better solved by humans when framed in terms of breaking a social
contract, such as detecting cheating (NOT of legal age but IS drinking).
The case could thus be made that, in
evolutionary terms, three of the most the most acutely important mathematical
skills are topological and geometrical skills to do with hunting, generic
classificatory skills to do with gathering and discrete dynamics to handle the Machiavellian
intelligence of human groups.
Figure 5.11a Section of
Wikipedia on the Collatz conjecture, showing its fractal extension to complex
numbers.
(c) Mathematics as a Cultural Maze
Ian Stewart in ÒThe Magical MazeÓ[10]
has portrayed mathematics, not as a pipeline or axiomatic hierarchy, but as a
maze. In his opening words: ÒWelcome to the maze. A logical maze, a magical
maze. A maze of the mind.Ó The maze is mathematics. The mind is yours. LetÕs
see what happens when we put them togetherÓ.
In contrast to the classical view of
mathematics as an axiomatic and heuristic hierarchy in which algebra and
analysis stand as a dual core, a new view of mathematics is emerging as a
result of richer resources, on the world wide web and through packages such as
Mathematica, Matlab and Maple that mathematics is more like a tangled bank, a
maze or an interconnected graph of concepts and states, in which topics
relevant to a particular quest can be rapidly explored and understood in a way
which would require a long bureaucratic journey up an undergraduate pipeline
and major finally passing specialized graduate courses to access. This view is
emphasized by the fact that Mathworld and Wikipedia are themselves, unlike
conventional text books and more like fullyreferenced research articles,
themselves conceptual mazes in which multiple live links carry us from topic to
topic on the wings of research discovery, personal interest or sheer
imagination, healing a rift typified by the comment Ð ÒSo you are a
mathematician! I gave that up in the 4^{th} form!Ó
To illustrate how effective this
mazebased view of mathematics is, let us go back to the Collatz problem that
first appeared in a childrenÕs book and pick out two gems from Wikipedia
(figure 5.11a) and Mathworld (figure 5.11b) concerning the nature of this
problem. The two figures show different perspectives on the orbit of values
from a given stating number. Neither of these connections would be expected in
a mathematics text book, but are immediate extensions of the core concept which
can be explored further by live links and linked references to the original
research. Teilhard de ChardinÕs
ÔnoosphereÕ[11] has thus
become realized mathematically as a networked wiki. The Wikipedia entry is
poignant, not just because it is edited and maintained realtime by the
viewers, but because it carries the problem right into discrete dynamics and
chaos theory, where the intractability of the problem naturally lies. It is
fundamentally the maze properties of mathematics in terms of its logical
relatedness connecting diverse areas, which give mathematics its power of
explanation. The central properties of a language are not hierarchical
definition, but maximal capacity to be used freely to articulate, in an optimal
way, abstract semantic ideas.
Effectively each language generates a maze, not only of words and
phrases, but of expressible conjectures, comments and arguments, as well as
metaphors and tales of the intrigues of human character, which it is up to our
Machiavellian intelligence to put to the most flexible and advantageous use
possible.
Figure 5.11b Section of WolframÕs
Mathworld on the Collatz conjecture.
To make this point even clearer,
mathematical research, to be efficient and innovative, by necessity, has to
link often unrelated areas, such as group theory, graph theory and
computational science, to solve real problems of complexity. In doing so it
also has to find the most direct route to the kind of articulated expression we
call a mathematics research paper. Despite the hierarchical nature of tertiary
mathematics education, at the research level this necessarily means finding the
shortest span across a the graph of mathematical ideas to get us from the
problem to the solution, in much the same way a person solving the Rubik revenge
vanquishes a state space containing 10^{50} vertices in a path to the
solution of only perhaps 50 steps. It is thus a serious dichotomy that to
achieve this level of networking efficiency, a heuristic assumption is made
that we must teach mathematics in a bureaucratic hierarchy, even though this is
largely in conflict with that very fascination with novelty and exploration as
a discovery process the nature of mathematics as a conceptual maze provides.
3: The Game, Topology and
Two Small Clouds in Classical Analysis
The foundation of analytic continuity
is the cryptic statement:
[3.1]
Figure 3.1: Solving the  game requires finding a for each . If this is not to become a
standoff , we need to know the local slope to fit the sandwiches.
This means that no matter how close we
choose to be, there has to be a sandwich in the domain
corresponding to it that will map into the range inside the sandwich.
The counterintuitive aspect of this to a student used to dealing with practical
problem solving is that unless we indulge a subtle form of ÔcheatingÕ this
looks like an unresolvable standoff, because no matter how many values they pick demonstrating convergence, as we tend to x, an opponent can still pick
an ever smaller and more meticulous demand, leading to perpetual impasse.
What we are using here is equivalent to
local Lipschitz continuity[12].
A function is Lipschitz continuous if
and is called a contraction if K<1. [3.2a]
Lipschitz continuity can also be
defined locally f Lipschitz on . [3.2b]
In effect  continuity is a twoplayer
game, in which, to establish continuity, the domain player has to be always
able to outmaneuver the range player, by finding a for each , effectively finding a
functional relationship through a local inverse. As pictured in figure 3.1, to
all intents and purposes, this depends not just on continuity but gauging the
local slope of the function in the neighbourhood of x and hence using the local
derivative . But this is really a form
of cheating at the game, because it really only works when the function is not
just continuous, but differentiable, at least locally, and we are all taught
that differentiability implies continuity, but not vice versa.
Example 3.1: Proving the continuity of , involves discovering the
following functional relationship: [3.3]
What is actually happening here is that
we are estimating a by comparing the ratio of the range and domain gaps, effectively using the derivative , which correctly gives the
varying :ratio for differentiable
functions, which powers of x clearly are. A quick check
of the formula for gives:
[3.4]
The same argument can be extended to
all standard functions, which are generally differentiable on their domains
through their power series representation, which is effectively a fractal
polynomial. This is particularly true for the
trigonometric, hyperbolic and log functions, all of which are derived from the
(complex) exponential:
[3.5]
The question then naturally arises as
to what we do when we come up against proving a nondifferentiable function is
continuous. The classic such function is the broad class of Weierstrass
functions[13] typified by
[3.6]
[3.6]
Figure 3.2 Weierstrass functions are
fractal functions (Matlab simulation).
Weierstrass functions are fractals as
illustrated in figure 3.2, fractality arising directly from their Fourier
series representation. Their Hausdorff dimensions are closely bounded by [14]
[3.7]
Weierstrass functions can be proved to
be nowhere differentiable, which is relatively obvious, since formal
differentiation leads to divergence: [3.8]
The proof that they are continuous
everywhere is immediate. Since the terms of the Fourier series are bounded by and this has finite sum for
0 < a < 1, convergence of the
partial sums to the function is uniform. The uniform limit of continuous
functions is continuous, and each partial sum is continuous:
[3.9]
Thus once we reach,we can switch from worrying about the derivative, or the
local Lipschitz constant, because, no matter how fractally precipitous
successive terms become, in generating the nowheredifferentiable uniform
limit, having used the continuity of the differentiable partial sum , we can now ignore the graph of the function because of the
everdiminishing bound on the divergence caused by the additional fractal
terms.
Uniform convergence guarantees that convergence
is independent of x i.e. , [3.10]
By contrast with Lipschitz continuity
for real functions on [0,1], which are differentiable almost everywhere (i.e.
except on a set of measure zero as defined below), in a topological sense, the
set of nowheredifferentiable realvalued functions on [0,1] is dense in the
vector space of all continuous realvalued functions on [0,1] with the topology
U of
uniform convergence derived from of [3.10] that is. The uniform norm and
uniform convergence are here functioning in much the same way as the Hausdorff
metric (the maximum distance either of two compact sets extend beyond the
other) does for iterated function systems[15]
based on (Lipschitz) contraction mappings forming a sequence in the space of
compact sets in the plane, which converges uniformly to the fractal attractor
of the system.
If we consider the Fourier series for a
real function f(x) on the interval:
[3.11]
since both sin and cos are bounded by
±1, all that is required for uniform convergence is that the coefficients
diminish in such a way that their successive partial sums are convergent. It is
thus possible to construct a variety of Weierstrass functions, for example the
polynomial type in WolframÕs Mathworld[16]:
[3.12]
Demonstrating fractal
nondifferentiable functions form a dense subset is also in principle
straightforward, by forming a sequence using a Weierstrass function:
[3.13]
The are nowhere differentiable, because they are a superposition
of f and
a rescaled Weierstrass function, and tend to f uniformly, just as the partial sums tend to w, as each differs from its limit by the same set of
terms.
Hence the space of continuous functions
on [0,1] is densely permeated by fractal nondifferentiable functions, and we
would be better off working with the uniform topology and teaching students
about topological continuity in a way which admits all the contortions it
provides, including interesting functions in the real world, such as the
fractal waves on the ocean, rather than limiting the arena of interest to the
ideal archetypes we ÔcheatÕ on proving continuity for, by relying on their
differentiability to find .
In a metric space (X,d) we replace the standard
distance in with any real function on pairs of points which obeys
nonnegativity, symmetry and the triangle inequality. The statement of
continuity for then becomes .
[3.14]
If we define an open ball as the
analogue of an open interval i.e. , we can then restate
continuity in terms of open balls: , or
.
[3.15]
We can then define an open
set O as
one where every point has an open ball neighbourhood around it: and can straightforwardly
prove that any open set is a union of open balls and hence any union of open
sets is open. However simple examples such as demonstrate that an intersection of open sets is necessarily
open only if the intersection is over a finite collection. We thus have the
basis for a topological space
 a pair where X is a set and is a collection of open subsets of X defined by:
(i) (ii)
[3.16]
that is a collection of open subsets of X is ÔclosedÕ under arbitrary
unions but only finite intersections and their complements, closed sets are
ÔclosedÕ under arbitrary intersections and finite unions.
This is a primary example
of symmetrybreaking, the open sets throwing off their boundary points, and
their complements, the closed sets, retaining them. Topology breaks the
symmetry between union and intersection which characterizes set theory and
logic in the form of a Boolean algebra in which we have two binary operations
with laws which are mutually commutative, associative and distributive and have
additional laws of absorption (and its dual) and
complements (and its dual) in which we
can exchange and (or and in logic) to gain dual statements like
De Morgan's laws: . [3.17
a,b,c]
When Lord Kelvin said there were two
small clouds on the horizon of classical physics Ð namely the
MichelsonMorley experiment confirming the invariance of the speed of light
foreshadowing relativity and black body radiation foreshadowing quantization
Ð these two statements harbingered a cultural revolution which spelt the
end of classical physics. Analysis likewise has two small dark clouds, which in
a similar sense may spell the nemesis of the classical paradigm.
The first is the nonequivalence of
metric spaces under homeomorphism, the natural definition of continuous mapping
equivalence. A homeomorphism is a function between spaces that is 11, onto,
and continuous in both directions.
However there are simple examples of spaces that are not metrically
equivalent but are homeomorphic. Thus metric spaces cannot be the natural
vehicle for continuity, but topological spaces are.
Example 3.2: The two metric
spaces and are not metrically equivalent. We can see this at once,
because many of the points in are distance apart much greater than 1 tending to infinity,
while all pairs in are closer then 1 apart. There can thus be no rescaling of the finite metric to
contain all the balls in the unbounded metric and hence no metric equivalence
is possible between them.
Moreover is not complete. It does not contain all
its limit points, since the sequence is Cauchy (pairs of
points in the sequence become arbitrarily close to one another), but the limit
0 is outside the space (0,1]. By contrast in , this sequence is not
Cauchy, since does not tend to 0 for all , and this metric space is
complete, being metrically isomorphic to , as noted below.
However the two spaces are
homeomorphic. is a metric identity between and with the standard metric and this is an equivalence which is also necessarily a
homeomorphism. But is also a continuous bijection on using only the standard metric and is also its own continuous
inverse so is also homeomorphic with . Hence and are homeomorphic.
It is thus natural to move
from the metric  definition of continuity to
the topological one:
Theorem 3.1: is a continuous function of metric
spaces .
proof:
() Suppose f is continuous and open. If then it is open, so assume the contrary.
open so . Hence by continuity or . But then so we have found an open ball around any
making it open.
() Suppose open. Consider . Since this is open, is also open and contains x. Hence there exists an open
neighbourhood of x in i.e. .
But this is the same
thing as saying so f is continuous.
This definition becomes the
basis of topological continuity. It might appear even more inaccessible than
the  game because we have to
deal with arbitrary open sets, but this is not so for elementary real
functions, because it is sufficient to show inverse images of open intervals
are open by the above theorem and to do this is no more difficult than the
original Lipschitz type use of the derivative as a scaling factor in the  game, but it also has
manifest advantages in making real the sense of topological discontinuity
directly appreciated in breaking of a continuously drawn curve in space and the
need to assign the boundary points of the breakage.
The second small cloud on
the horizon of analysis comes not from the concept of metric, but of measure.
Although the rationals are spread out densely on the number line so that
between any two irrationals is a rational and vice versa, the rationals are
countable, while the reals and hence the irrationals, have a strictly higher
cardinality. For those interested, you can count the rationals by making a 2D
grid of all fractions and scanning the diagonals etc. This is redundant, but shows we can put all rationals
into a list.
To prove the reals are not
countable, suppose you have a list of all elements of (0,1) as decimals We can always find an element not on the list simply by making . This also shows in
principle a way to make an identification between elements of (0,1) and subsets
of the natural numbers , by identifying the binary
representation where with the subset . Since the number of
subsets of a set containing n elements is , this gives us the famous
relation , between the countable
cardinality of and , and the uncountable
cardinality c of .
Since the development of the
Riemann integral, there has been a love affair with the idea of taming the
rationals sufficiently to prove that a more general notion of integral (known
as the Lebesgue integral) should successfully show that .
[3.18]
Based on the Riemann idea of
a limit of rectangles (figure 3.3), this integral does not exist because the
maximum height of each rectangle is 1 and the minimum is 0, so no limit exists.
The solution to this dilemma
comes in a different nonsymmetrybreaking modification of Boolean algebras in
which we consider instead a collection of subsets closed under countable union
and complement, called a algebra. By De MorganÕs laws, this
is also closed under countable intersection. The smallest algebra over the reals
containing the intervals is the algebra of Borel sets. Naturally it includes
open and closed sets and the additional sets we get forming countable
intersections and unions of these.
If we now expand to consider
any set which differs from a Borel set by a null set, one which can be covered
by a countable union of intervals the sum of whose lengths is less than any , we arrive at Lebesgue
measure[17].
In particular, we can define
the outer measure of any subset B of :
where M is a countable union of
intervals, the sum of whose lengths is . [3.19]
A is then Lebesgue measurable if
[3.19]
i.e. if the measure of the
inside and the outside add to that of the whole set in each case.
This makes things very easy
for an integral over the rationals, because is a null set, since it is countable, and we can cover each
enumerated rational by an interval of length whose sum is .
Figure 3.3: Rather than
partitioning the domain, as in the Riemann integral (blue), the Lebesgue
integral (red) works its limit by partitioning the range and adding the areas
gained from multiplying the measure of the set for which the function is higher
than a given level by that level height.
For the above function
[3.18], which is either 1 or zero, this will just be the measure of , which is 0.
However, while solving the
measure problem for functions over Ôvirtually every set of interest we might
encounterÕ there remains an infinite collection of residual nonmeasurable
sets, called Vitali sets, which cast a pall shadow over the ideal of real
numbers.
Example 3.3: Consider the
rational equivalence class of real x: . This set of equivalence
partitions into disjoint subsets. We construct a Vitali set[18]
by choosing one
representative from each class, via the axiom of choice, an independent axiom,
which allows such choices from arbitrary collections.
Now consider an enumeration of . From the definition of V the sets are pairwise disjoint since
numbers differing by a rational are in the same equivalence class and only one
representative of this was chosen. We can also show . The first inclusion is because, for each x in [0,1] if v is the representative for then , some l and so . The second inclusion is
clear from the maximum divergences from [0,1] of 1.
This gives rise to a
paradox, because Lebesgue measure is countably additive and translation
invariant, so , but each of the are identical by translation
invariance, so we have a countable sum equals a finite number which is
impossible since any such sum must be 0 if the terms are 0 or if the terms are finite and equal.
This is a problem that goes
right to the heart of mathematics as a cultural language, which might have a
very different description on another planet harbouring sentient life.
Mathematicians have of course made a menagerie of number systems incorporating
infinitesimals smaller than any real, including the hyperreals[19],
the long line[20] and others,
but nevertheless there is a serious problem about measure based on countability
when we try to consider Ôinverse countabilityÕ  factoring the reals into
equivalence classes each containing a countable number of members, which makes
the quest of measure a will oÕ the wisp of the mathematical will to order.
Figure 3.4: (a) Open and
closed intervals and their variants are easily appreciated by school children.
(b) A discontinuous mapping on the number line immediately reveals the problem
of assigning boundary points. The inverse image of the open interval (red) has
a boundary point as a result of the discontinuity. This topological idea of
continuity based on openness and boundaries extends naturally to mappings of
curves (c) and regions (d). The topological definition of continuity works as
well as the  game for proving a function
is continuous, as it can access the same Lipschitz arguments in the light of
theorem 3.1, however it has an advantage in searching for discontinuities,
because astute choices of open intervals in the range can highlight points of
discontinuity as boundary points in the inverse image. The function in (e) has
discontinuities at a sequence of values tending to zero, but is continuous at 0. This is confirmed in , consisting of a finite
union of closed intervals, demonstrating discontinuities at the end points,
arbitrarily close to, but not including 0. The function in (f) is continuous on
its domain , but if f(0) is assigned to be 0, consists of a countable union of open intervals limiting to a
single boundary point at 0 which is in the inverse image and is the one point
of discontinuity on .
Figure 3.5 Knots of order 9
illustrate the realizable complexity of knot theory
An alternative to the
credibility gaps of classical analysis is starting from realizable examples of
continuity and path connectedness in which the continuity of curves is broken,
requiring the assignment of the boundary points, leading into knots, manifolds
and fractal topologies and how the idea of open set and topological space
transcends the limitations of measurement in metric spaces, keeping the
topological emphasis, while specializing to Lebesgue measure as particular
studies require, rather than centering on classical analysis to the exclusion
of both topology and geometry, when it is these latter areas that are most
breathtaking to the imagination and still at the cutting edge of analysis, as
the PoincarŽ conjecture[21]
demonstrates.
4: Puzzles and Games as
an Expression of Human Mathematical Imagination
One of the most obvious expressions of
the human mathematical imagination in human culture is its presence in puzzles
and games. Many of these and possibly
the majority are geometrical and topological. There are of course reasons for this in that puzzles are
frequently physical objects, but even when they are logical, conceptual,
abstract or computational they still frequently use geometrical and topological
ideas.
There is also a tendency for a given
puzzle to bring together disparate areas of mathematics, implying that
mathematics is best described as a tangled web, rather than a bureaucratic
hierarchy of axiomatic systems. This is consistent with a new and very
different view of mathematics as presented on the web in sites such as Math
world and Wikipedia, in which mathematics is literally a maze of concepts
related both by natural and logical affinity and by association, generalization
and disparate linkages across widely differing fields to present complementary
vies of a given phenomenon.
We present representative examples of
puzzles and games to illustrate the diverse mathematical areas they bring into
play and the types of mathematical reasoning in humans they highlight. Firstly
let us examine three types of puzzle and game that specifically involve
topological reasoning sometimes associated with geometrical thinking.
Figure 4.1: Five types of topological
ring, wire and string puzzle
Example 4.1 a,b,c,d: Topological wire,
loop and string puzzles.
A large class of puzzles use wire loops
strings and rings to set up situations where the system is not in fact knotted
or linked, which would make the puzzle impossible, but figuring out the unknotting
moves is geometrically and/or topologically challenging.
The twin flight of loops (top left) is
clearly unknotted, as each of the wire loops could be shrunk through those
ÔlinkedÕ over it. The string is
thus not knotted and the puzzle is solvable. However visualizing the deformations of the string required
is complex and involves an exponentiating number of topological moves, doubling
for each additional step in the stairway, in the manner of a Towers of Hanoi
problem. The lower right puzzle
likewise has two nested pairs of unlinked loops, again requiring a recursive
solution. The lower left puzzle
consists of n linked rings and a long loop, which is initially linked over only the left
hand nth
post. The (k+1)th ring can be slid on and off the long loop only if only the kth ring is linked over the
loop. This give rise to the recursive relation for the number of moves to get
to the kth stage: with solution , giving 255 moves for this
8stage puzzle. The top righthand puzzle ÒSquaring OffÓ requires only four
moves, corresponding to the four rings and the successive loops in the square,
but the third is so counterintuitive that many respondents have to ask for the
solution. The centre top puzzle requires three simple moves to integrate the centre
holes in the two pieces of wood and slip the string loop to the front. These
puzzles thus combine three areas of mathematics, topology, geometry associated
with the moves required to respect the fixed dimensions of the rings and wires,
and recursive iterations governing the number of moves.
Figure 4.2: Left Trax has a qualifying
white line and black loop. Centre: Tantrix pieces, including the forbidden
pieces. Right: A Tantrix game in which the forbidden pieces are allowed for
forced moves.
Example 4.2 a,b: Tantrix and Trax
Tantrix and Trax are games using
regular square and hexagonal tilings but the strategy of both games depends on using the tilings to
create topological loops and curves of maximum length. The two games share topological strategic
curve building through a discrete process. Trax using invertible squares with
crossed and uncrossed pathways plays cutthroat race for the first person to
get a loop or line covering 8 rows or columns. Tantrix is more complex, having all
combinations of three of four colours forming curves not crossing in a
cartwheel. Cartwheels are omitted, because they provide sparse rearrangements
being unchanged by a rotation of 180^{o}. Each player chooses a colour
and tries to build the longest line or loop before play runs out. Before and after each turn players fill
forced moves resulting from hollows in the tiling. Tantrix records exist for
the longest ÔlinesÕ and loops, including a computer solution to the Òfour
longest linesÓ puzzle by Paul Martinsen & Jamie Sneddon, April 1998
totaling 146 34 red, 40 green, 35 blue and 37 yellow links.
The 56 Tantrix pieces plus 8
forbidden ones can be deduced easily on a combinatoric basis from all possible
three colour curves on the hexagon.
Pieces can have short curves joining adjacent faces, long curves
spanning a face, and diametric straights. There are Ôtriple shortsÕ 4 combinations of
colours, each in 2 orientations.
The same applies to the forbidden Ôtriple straightÕ cartwheels. The Ôstraights
with long or short curvesÕ each have 4 ways to pick the straight colour and 3
ways to eliminate the fourth colour, or
each. They do not have orientations as a
180^{o} rotation has reverses colour orientation, due to their internal
symmetry. Finally the Òtwo longs and a shortÓ pieces have two orientations as
well, so have 4.3.2 = 24 pieces.
Figure 4.3a: Left: Go is based on
capture by topological enclosure of regions following a discrete 4cell von
Neumann neighbourhood rule based on a player (black) holding the immediately
adjacent squares (see insets). Centre: Blokus uses all geometrical
combinations of piece up to pentaminos to dominate space by building interpenetrating fractal
trees, connectivity being maintained through adjacent corners.
Right: Blokus Trigon uses triangular
pieces up to hexaminos and has different rules because triangular symmetry
allows corner vertices to meet face vertices.
Examples 4.3: Go and Blokus, Dots and
Lines
Go is based on capture by
topological enclosure of regions following a 4cell von Neumann neighbourhood
rule based on a player (black) enclosing a region the immediately adjacent
squares above and below and to either side (see insets). Topological connectedness thus becomes
quantized and discrete. The winner is the player who has captured most of the
board once the uncontested sites where a player dominates have been filled
in. Critical is the idea that the
game depends on topological reasoning although the moves are discrete on a
discrete grid. The Go state space is huge. There are an estimated possible positions on a 19x19 board [22].
Blokus uses all geometrical
combinations of four colours of piece up to pentaminos to dominate the board
by building interpenetrating fractal
trees, connectivity being maintained through adjacent corners. A given player
playing in pieces of a given colour first builds from a corner of the board.
Pieces of the same colour can only be placed corner to corner, so the corners
of each piece constitute future sites of fractal growth. All the pieces played
by a given player thus form a connected graph. Pieces of differing colours can
meet on edges, so a player can fill spaces left by other colours. The graphs of
two players can also cross one another and both be connected, an odd but
obvious property of the discrete connectivity rule based on touching corners,
which parallels the difference between discrete dynamics and continuous vector
fields. The game begins with players building towards the centre and
endeavouring to build fractal dendrites, which will permeate as many regions as
possible in the face of blocking moves by the opponents, using the power of
corners and the capacity to block opposing players vacant options.
As the endgame unfolds, skill moves from strategies of fractal growth, in
which the fractalforming power of corners and the defensive blocking potential
of edges is key, towards careful geometrical placing of the remaining pieces to
play out. The game thus combines
the fractal topology of discrete path connectedness with a geometrical tiling
finale.
Both games illustrate the subtleties
of the way discrete board games can give rise to implied topologies, despite
appearing to be purely geometrical, or abstract strategic in nature and give a
conceptual illustration of how quantization can affect classical properties of
continua in a way which hints at similar properties of quantum transformations.
Fig 4.3b Dots and Lines final
configuration
Dots and Lines, while an apparently
simple fillingin game undergoes a complex phase transition from a ÔgasÕ to a ÔsolidÕ
similar to percolation[23].
Each player is allowed another turn each time they claim a square by completing
the fourth side. At first players make defensive moves avoiding the opponent
gaining squares, but this results in a crystallization of many edges to a point
of selforganized criticality, when all moves will result in escalating
cascades of claimed squares.
The Rubik cube is the most popular
puzzle of al time having absorbed 1/8 of the worldÕs population. Rubik type
puzzles, stemming from the initial 3x3x3 Rubik cube (centre left) now come in a
wide variety of geometrical forms including cubes of various types, pyramids,
stellated dodecahedron (Alexander star right left) cubeoctahedron, truncated rhombic dodecahedron and planar
configurations. All of them
involve skill with mental rotation and keeping track of interacting rotation
processes geometrically and each has a unique ingenious mechanical construction
supporting its rotation set.
However, all these puzzles are subject to a single basic algebraic strategy
to complete the solution Ð examining the symmetries possessed by the
commutators of two noncommuting rotations, which are the compound movements
naturally closest to the identity I.
Figure 4.4a Left: Variants of the
Rubik cube have a variety of geometrical and planar shapes, although all depend
for their solution on the common algebraic method of examining the symmetries
of a commutator of two rotations. Right:
The 4x4x4 Rubik revenge cube has three sets of symmetries generated by
commutators.
Example 4.4: Rubik type
AlgebraicGeometrical Puzzles
For the simpler puzzles there is
only one commutator type , which in the case of
the original Rubik cube permutes 3 edges and exchanges two pairs of corners, at
the same time rotating the corners. then permutes only edges and only corners. A succession of moves of
the type , where T is a transformation
moving the required edges, or corners, to the appropriate positions for C can then solve each of
the puzzles straightforwardly. The
4x4x4 Rubik revenge cube has three possible types of commutator involving inner
and outer rotations, some of which can generate odd permutations. These noncommuting operators give an
everyday insight into the more mysterious noncommuting processes mediating
quantum uncertainty of spin angular momentum in different directions, which
form one basis for quantum uncertainty.
Although a puzzle like
the revenge cube takes only 50 or so moves to reach the solution from an
arbitrary state, the total number of states is huge: There are 8 corner pieces
with 3 orientations each, 24 edge pieces with 2 orientations each, 24 centre
pieces, giving a maximum of 8!á24!á24!á3^{8}á2^{24} positions.
This limit is not reached because: (a) The total twist of the corners is fixed
[3] (b) The edge orientation is dependent on its position [2^{24}] (c)
There are indistinguishable face centres [4!^{6}] (d) The orientation
of the puzzle does not matter [24]. This leaves 7!á24!á24!á3^{6}/4!^{6}=
7,401,196,841,564,901,869,874,093,974,498,574,336,000,000,000 or 7.4á10^{45}
positions, illustrating that the graph of states of realizable puzzles can be
huge, comparable with the number of electrons in the universe or even the
number of potential string theory candidates, thus forming an oracle for
complex systems at the frontier of human knowledge.
Erno Rubik was initially interested
simply in the mechanics of how to construct a cube in which the ÔsubcubesÕ
would rotate, but having done so he discovered the implications: Ò"It was
wonderful, to see how, after only a few turns, the colors became mixed,
apparently in random fashion. It was tremendously satisfying to watch this
color parade. Like after a nice walk when you have seen many lovely sights you
decide to go home, after a while I decided it was time to go home, let us put
the cubes back in order. And it was at that moment that I came face to face
with the Big Challenge: What is the way home?"
Despite their many forms, most of
these puzzles are regular in the sense that every position permits the same
moves. An intriguing exception is the Square1 puzzle, which admits a variety
of irregular conformations, which have varying sets of moves into and out of
these states, some of which permit odd permutations. Thus although the
transformations form a group in which every element has an inverse, the group
and the ensuing graph of puzzle states is highly irregular.
Figure 4.4b: The orbit of states of
(t) repeated 82 times on Square1
The repeated operation
{(tÐ )(82)}, (where ÔtÕ rotates the top clockwise to the next flip
position and ÔÕ is a flip of the right hand side of the puzzle) visits many
such states before returning to the cube.
The full periodicity back to the completed cube is 4 x 82 = 328 since
the permutation of the corners (18) and edges (ah) is (1728)(ag)(cd). The
corresponding sequence
{(tbÐ)(8)} returns to the cube permuted by
(148)(263)(57)(afbecgdh), having an orbit length of 3x8x8=192.
5: State Space Graphs and Strategic
Topologies
Virtually every puzzle, whether
logical, conceptual, arithmetic, geometric, topological or strategic is
navigated by a human subject in an abstract journey from beginning state to
solution, through many possible culdesacs in a journey which takes the form
of a connected path along the nodes of a graph of states which constitutes a
maze of intermediate positions. This is a process akin to a journey through the
wilderness in which various conceptual attributes essential for solving the
puzzle can point the way to the solution much as topographical signposts or at
least sensibly reduce the huge space of possibilities to a feasible number of
options.
Although every solvable puzzle is path
connected, the form and size of the state graphs can vary extremely. A regular
graph with a standard set of moves, such as the Rubik revenge cube, can have a
huge state space. By contrast state spaces in which the transitions are
complex, irregular may have a much smaller state space, despite being of
nontrivial difficulty. We now examine several different types of puzzle to
investigate the common topological thread involved in navigating a connected
path from starting point to solution.
Example 5.1: Who Own the Zebra?
This logical puzzle sometimes
incorrectly attributed to Einstein consists of a series of logical statements
associated with five colours of house, five nationalities, five drinks, five
pets and five brands of cigarette.
The
solution to the puzzle is most easily performed by making a table of the items,
and then analyzing the logical statements, to specify successive entries of the
table, branching to deal with contingencies as little as possible.
Given the statements
listed below, we are asked: ÒWho owns the zebra?Ó and ÒWho drinks water?
1.
There are five houses.
2.
The Englishman lives in the red house.
3.
The Spaniard owns the dog.
4.
Coffee is drunk in the green house.
5.
The Ukrainian drinks tea.
6.
The green house is immediately to the right of the indigo house.
7.
The Old Gold smoker owns snails.
8.
Kools are smoked in the yellow house.
9.
Milk is drunk in the middle house.
10.
The Norwegian lives in the first house.
11.
The man who smokes Chesterfields lives in the house next to the man with
the fox.
12.
Kools are smoked in the house next to the house where the horse is kept.
13.
The Lucky Strike smoker drinks orange juice.
14.
The Japanese smokes Parliaments.
15.
The Norwegian lives next to the blue house.
Figure 5.1: ÒWho Owns the Zebra?Ó portrayed
as a strategic maze of puzzle states.
Figure 5.1 shows a decisionmaking tree
maze for the puzzle, which is conveniently tabulated in the same way as the
solution for ease of reading.
Initially all but two of the statements are processed and incorporated
into the table in terms of links between categories which determine the
relative positions of the linked items.
Although the puzzle is nontrivial its decision making state graph is a
tree with only a few nodes once the logical statements, which
can be processed simultaneously are grouped into one step.
Because there are many ways of
prioritizing the statements and in which order to deal with the categories, a
human subject will frequently navigate a version of the tree, adding one or two
extra assumptions, only to find they have reached an intractable position,
returning to the trunk of the tree, or a variant of it, to try again, releasing
some or all of the assumptions which led to intractability. In doing so, they are navigating a
logical space, abstractly akin to making a pathconnected journey in a
topographical landscape.
Figure 5.2: Maze Presentation of example
5.2
Example 5.2 The Five Peg Puzzle
Figure 5.2 shows the maze for a puzzle
in which one or two top rings can be moved, but only on to a ring, or empty peg
of the same colour as the lower one. Again there are only a limited number of
states because many moves rapidly lead to intractability. The graph now has
trivial loops but is unidirectional upward because the moves are not reversible.
Again the subject is traversing a conceptual territory, which can be described
as a pathconnected region.
Figure 5.4 Elementary Sudoku (left) has
no numeric operations, being based only on each row, column and subsquare
having distinct entries, as the colourcoded version (centre) shows. Black:
initial puzzle. Blue: clues from horizontal and vertical lines. Purple: clues
using subsquares. Red: final solution. This puzzle can be solved without contingencies and thus has a state
space consisting of a meander maze  the unique path from start to solution
(right). Similar colour codings
are used to depict Cayley tables[24],
which also have distinct entries in every row and column.
Example 5.4 Elementary Sudoku
Elementary Sudoku illustrates
the ultimate simplicity of state space structure. Although it presents as a
numeracy puzzle, it is simply a categorymatching puzzle, as illustrated by
colour coding, requiring only that every row, column and subsquare has nine
distinct entries. Because all the numbers can be found simply by filling in
numbers determined in sequence from the provided clues, the state graph is just
a line, as in a meander maze figure 1.2, as illustrated right in figure 5.4. Advanced Sudoku however introduces fewer
clues, requiring testing contingencies, and hence has a simplyconnected tree
maze as in example 5.1.
Figure 5.5a: Four sequences of
geometric moves in the Rubik 3rings puzzle.
Example 5.5 The Rubik Three Rings
Puzzle
The Rubik 3rings puzzle consists of a
set of eight diagonally grooved plates held together by nylon strings woven
over three successive plates in a circuit in overlapping succession, so that
the plates can be folded along certain axes joining the plates, changing the
way the strings link the plates and creating new puzzle geometries. The aim of
the puzzle is to fold the plates into a new arrangement where the three
unlinked rings have seemingly impossibly become linked. This is possible
because the reverse sides of the plates have pieces of a second image of the
three rings linked through one another as shown in the heart shape in the
centre of figure 5.5c, associated with an Lshaped geometry differing from the
rectangular starting position.
The puzzle presents an intriguing mix
of geometrical and topological constraints, the weaving of the strings fixing
the geometry of the hinged shapes, within the basic topology of a ring of
plates in which some, but never all, of the plates are able to be hinged out of
the loop, at least temporarily.
The weaving of the strings itself
presents an interesting topological puzzle, which enables the eight rings in
the rectangular configuration to be transformed in every possible way that
retains their overall ring structure. There are 8 clockwise permutations of the
plates, two directions of orientation around the ring and four orientations the
square plates can adopt collectively. This gives 8 x 2 x 4 = 64 possible states
of the rectangle, however we need to divide this figure by 2, since moving four
steps round the ring rotates the whole rectangle through 180^{o,} if
the top row is coded abcd and the bottom row is an inverted efgh. The way the strings are
woven enables all of the 32 possible states to be reached, although this might
seem impossible from the way they are woven.
Figure 5.5b Colourcoded weaving of the
8 closed loop string pairs holding the 8 3ring puzzle plates in a loop.
In figure 5.5b is shown the weaving of
the 8 looped string pairs in a 4x2 arrangement, which remain unknotted throughout,
although alternate plates have a double winding, with strings linking to 2
adjacent plates, spanning 3 in all. There are no vertical connections in the
centre four plates, so the 8 plates form a ring.
To solve the puzzle requires
negotiating a series of geometrical transformations, some of which lead to
culdesacs, however there are four sequences of transformations illustrated in
figure 5.5a, which lead to a rearrangement of the rectangular arrangement. In
(a) the ring is folded and becomes a literal ring of 8 plates, which can be
unfolded to form four rectangular states involving 3 transformations from the
identity. In (b) the plates can be folded together above and then unfolded in
the vertical direction from below, effectively rotating the plates through 90^{o}. In (c) a sequence of moves takes the
rectangle to a scrambled form of the L or heartshape of the final solution,
which can then be refolded from the other side of the L to gain a different
transformation of the rectangle. There is a mirror image of this entire
sequence, which forms an inverse transformation. Finally in (d) there is another move, which results in a new
set of configurations, resulting in 7 transformations in all.
Figure 5.5c: The state space graph of
the rectangular configurations presented as a noncommutative graph assembled
on the 4D hypercube of 32 states.
Because the geometry of these moves is
complex, we have a nontrivial puzzle which has a state space graph which has
only 32 nodes corresponding to the 32 possible transformations of the 8 plates
above, so we see another example of the trade off between individual transition
complexity and state graph size.
To analyze the state space graph, the
seven possible transformations of the rectangular configuration, a Matlab
simulation was made of each of the geometrical transformations and this was
used to check the definition of each of the 7 transformations arising from each
node. The result is shown in figure 5.5c.
Although this is a richly interconnected
graph with a large number of loops, navigating from one position to another is
still difficult because several of the operations fail to commute in diverse
ways, causing operations performed out of order to arrive at unfamiliar
destinations.
The seven transformations are
colourcoded and the state of each node is illustrated and coded using the
abcdefgh notation above. Each of the pairs of edges forming a parallelogram in
the graph commute while the others do not. Each of the transformations are
selfinverses, except for redyellow shaded ones passing through the
heartshape intermediate, whose two forms are mutual inverses. There is a
corresponding 32 node state graph for the heartshapes, each of which is
connected to two rectangles through inverse transformations, two of which
emerge from each rectangle.
Figure 5.5d Stages in disentangling
the transformation group. (a) Graph of the 7 geometric operations t, o, p, v, r, l and op. (b) Reduced graph with 3 generators
o, p, v and defining new generators n, q and e. (c) Rearrangement of vertices using
the new generators results in a noncommutative hypercubic Cayley graph.
To decode the actual 32member group[25],
first we eliminate redundant operations.
Tracing the connections in figure 5.5d(a) we can see immediately that
three of the key geometrical operations including the simplest (t) and the ones that pass
through the heart solution (l and r) are composites of the others: t=opop, l=povp, r=opvp.
Removing these yields the graph in
(b) and a group with a presentation[26]:
[5.5.1]
In so doing, we have eliminated the
very geometrical transformations that enabled us to get to the heart shaped
solution. We should note that a
similar description could be mounted of all the transformations of the 32
hearts.
Examining the symmetries of the
plates in figure 5.5c however, we can easily see that more natural operations
are available which are composites of o, p, and v but represent
fundamental symmetries of the rectangle.
We define three new transformations
of the rectangle (RH composition):
1. n=op Moves all the plates cyclically
right by one step,
rotating plates 180^{o }when
they move around the end of the ring.
Four
such moves rotate the rectangle through 180^{o} leaving the
puzzle unchanged.
2. q=pv Rotates alternate plates 90^{o}
clockwise and anticlockwise.
3. e=pvpvo Reverses the orientation of the
ring of 8 plates
leaving the top left and bottom right
plates unchanged.
Given these generators, we can
present the group G, with center[27]
, as:
[5.5.2]
We can then rearrange the vertices
to reflect the symmetries and arrive at a noncommutative, hypercubic Cayley
graph[28]
for the transformation group as in figure 5.5d(c). If we use the notation for a
semidirect product[29]
action by inverting elements: , then, from the relations, G can be characterized by: since and , where is the dihedral
group of transformations of the square and is the cyclic group of order n (e.g. of integers modulo n under addition).
Figure 5.6 A sample of die throws from
ÒPetals Around the RoseÓ
Example 5.6 Petals Around the Rose
ÒPetals Around the RoseÓ[30]
is a puzzle that is famous for its account of Paul Allen and Bill Gates introduction
to it in a crowd returning from a computing conference in 1977, in which Bill
was the last active player in the group to discover the rule. The game has only
two clues. One is that the answers are all even, which becomes obvious after a
few throws, and the other is ÒPetals around the RoseÓ, which is significant. No
one is supposed to reveal anything more than the answer to a throw Ð
never the rule itself.
The problem to be solved, rather than
one of deductive thinking as in the zebra puzzle is one of lateral thinking,
faced with a seemingly irregular rule.
The state space of the puzzle now consists of all the lateral shifts of
thinking the subject might imagine, so it cannot be defined precisely in the
way the previous examples were.
There are a great variety of rules which could be applied, some
involving adding or multiplying the values on the faces, others counting how
many die of a given value appear, others dealing with the geometry of the
faces, the way the dice fall or the order of them in sequence, but each of
these conjectures form part of the topography of the state space which the
subject explores till they see a contradiction, until eventually they discover
the rule, which for convenience I will print upside down in light grey below,
so you can read it only if you canÕt deduce it from the instances in figure
5.6.
Critical to the irregularity is that
the rule uses only partial information from the dice. This information is
highlighted both by the high scores and the very low scores, which are
overrepresented in the list in the figure in the interests of quick analysis.
Figure 5.7a The unique simple perfect square of
order 21 (the lowest possible order).
Example 5.7 Squaring the square and
Magic Squares
Not all puzzles involve a state
space. Some are better solved in
one step, or a single defined process, e.g. by defining a system of
equations. One such example is
squaring the square[31]
[32],
where we are asked to find the relative dimensions of the tiled unequal squares
fitting into a single large square in figure 5.7.
This is an ideal candidate for using
symbolic manipulation to take the boredom out of the algebra. We first investigate the geometry and
compare a series of vertical and horizontal side lengths until we have generated
enough equations for a unique solution, and set the smallest square to a
suitable base number.
The Matlab symbolic toolbox provides an
ideal solution platform:
syms
a b c d e f g h i j k l m n o p q r s t u
S=solve('l=k+u,f=k+l,g=f+l,h=g+i,c=h+i,b+g=c+h,o=p+u,o=j+n,a=d+e,b+c=f+g+h,s=m+r,t=r+s,q=p+t,a+b+c=d+e+f+g+h,a+b+c=s+t+q,a+b+c=m+n+o+p+q,a+d+m+s=a+e+j+n+t,a+d+m+s=c+h+q,c+h+q=a+e+o+t,a+e+o=b+f+l+p,b+i=f+g,e+k=j+o+u,o+e=d+n,d+j=m+n');
C=struct2cell(S);
u=2;
for
i=1:21
fprintf('%c=%2.0f ',char(96+i),eval(C{i}));
end
a=42
b=37 c=33 d=24 e=18 f=16 g=25 h=29 i=4 j=6 k=7 l=9
m=19
n=11 o=17 p=15 q=50 r= 8 s=27 t=35 u=2
However
such puzzles are neither common, nor as popular as those which require a conceptual
hunt through a space of possibilities, and in this case the problem is unique,
being the only simple perfect square of order 21 (the lowest possible
order), discovered in 1978 by A. J. W. Duijvestijn[33].
Figure
5.7b Lo Shu the unique 3x3 magic square is associative and generated by the
Siamese method..
To
explore the problem of puzzle generation in numeric puzzles we can explore the
problem of magic squares[34].
A magic square is a square array of numbers, consisting of the distinct positive
integers 1, 2, ...,
arranged such that the sum of the
numbers in any horizontal, vertical, or main diagonal line is always
the same number, known as the magic constant . The unique 3x3 square was
known to the ancient Chinese as Lo Shu. This is also associative if pairs of
numbers symmetrically opposite the centre sum to . If all diagonals (including
those obtained by wrapping around) of a magic square sum to the magic constant, the square
is said to be a panmagic
square also called a diabolic square.
It is an unsolved problem
to determine the number of magic squares of an arbitrary order, but the number
of distinct magic squares (excluding those obtained by rotation and reflection)
of order 15 are 1, 0, 1, 880, 275305224, and an estimate of order 6 is using Monte Carlo simulation and methods
from statistical mechanics. The number of distinct diabolic squares of order
15 are 1, 0, 0, 48, 3600.
Given
the unbounded number of solutions one would expect there exists simple regular
algorithms for generating magic squares and this is the case. The Siamese method consists of placing
a 1 anywhere and placing 2, 3 etc. successively up the right hand diagonal
(vector (1,1)) moving one down (break vector (0,1) if we hit a filled square.
Lo Shu in figure 5.7b can be seen to be generated in this way. The Siamese
method will also generate diabolic magic squares of order 6k±1 with vector (2,1) and
break vector (1,1).
Figure
5.7c A sample 4x4 square puzzle made by removing magic square entries, has a simple tree maze with
two branch points, corresponding to contingencies in the bottom left and top
right corners.
Magic
squares can be used to generate Sudokulike puzzles with state space tree mazes
of varying complexity. In figure 5.7c is shown a sample 4x4 diabolic magic
square in which over half the entries have been omitted. The entries outside
the square give the remainder when the existing entries are subtracted from the
magic constant of 34. In the first stage the bottom left entry is used to
compare information from its row and column. This implies a corresponding set of
contingencies in linked rows and columns leading to an impasse for one (the 9
in position (1,2). This
information can now be used to perform the same analysis for the topright
entry leading to the solution. Once again the numeric puzzle leads to a pathconnected
graph, in this case a tree with two branch points, giving the puzzle an
underlying topological basis.
Because
the number of possible magic squares grows so rapidly, increasing the size of the
square and reducing the number of given entries can rapidly lead to too many
contingencies to make an interesting and ÔdoableÕ puzzle because of the load of
multiplying contingences and the repetitious simple arithmetic involved.
Example
5.8 2D and 3D Tiling with Polyminoes
While some puzzles have
one solution, which might be solved, like squaring the square, by a system of
equations, an abstract proof, or a single algorithm, others have many possible
solutions, often with their own internal irregularities, which require a brute
force computational approach to find all the variations. One of the most
persistent and intriguing types of puzzle to many people are geometrical tiling
puzzles constructed out of systematic geometric variants, such as pentaminoes
(all 12 configurations of 5 attached cubes in 2D), the pieces of the soma cube
(all 7 nonlinear pieces of composed of 3 or 4 cubes in 3D) and the Kwazy
quilt made of all combinations of circles stellated with up to six regular
apices.
Figure
5.8a Anticlockwise from top: 6 variants of the soma cube, viewed front and
back, 6 variants of the ÔLonpos PyramidÕ, one of only 2 possible 3x20 pentamino
solutions, ÔKwazy QuiltÕ, and compound happy cube and hypercube illustrate
tiling puzzles with multiple solutions.
The
soma cube was invented by Piet Hein[35]
the scientist, artist, poet and inventor of games such as hex, during a lecture
on quantum mechanics by Werner Heisenberg. There are 240 essentially distinct
ways of doing so, as reputedly first enumerated one rainy afternoon in 1961 by
John Conway and Mike Guy.
However,
if we count the internal symmetries of individual pieces within themselves,
i.e. and the symmetries of the whole cube we arrive
at . This might be compared
with the maximum number of distinct, possibly nontiling arrangements of the
pieces in space . Because a subject
assembles a cube using less tractable pieces first, it is relatively easy to
find a solution, and to navigate in the maze of solution space using
geometrical intuition using as many back step as necessary to retreat from
culdesacs near completion. A variety of other geometrical shapes can also be
made with the soma pieces, having varying degrees of constraint and hence
difficulty.
Likewise
the ÔLonpos PyramidÕ uses a subset of sphericallybased 2D polyminoes of sizes
3, 4 and 5 to build a pyramid, as well as rectangular solutions. Although the pieces are planar, the
pyramidal solutions involve interlocking pieces aligned horizontally,
vertically and obliquely. Since most are horizontal it is generally easier to
solve from the apex of the pyramid, which places strong local constraints on
the pieces to be used.
The
12 2D pentaminoes, known from the 19^{th} century, are capable of
tiling several rectangles of area 60 units, as well as other shapes, such as
using 9 to tile versions of the individual pieces expanded 3 times in size (45
units area). The number of rectangular solutions are: . This might be compared
with something like independent orderings and orientations of
the 12 pieces. The rectangular puzzles each have similar difficulty, despite
the varying number of solutions, because the narrower rectangles place more
constraints on the feasible partial tilings.
Figure
5.8b Four wooden interlocking puzzles.
The
popularity of such puzzles with both adults and children, including their
variants in wood puzzles (left) that generally have only one way of being
assembled, illustrates a strong theme involving the geometry of mental
rotation, the topology of navigating a path in abstract solution space, and a
preference for dealing with mathematical problems which have a strong sensory
basis, are capable of direct manipulation and promote lateral thinking, to open
unperceived avenues and avoid tunnel vision, as well as deductive
reasoning. These themes all
support a linkage between puzzles and gatherer hunter skills, which have
evolved over long epochs and stand diametrically opposed to the dominance
abstract linguisticbased axiomatic manipulations have in proofs in classical
theoretical mathematics. The gulf between these perspectives becomes ever more
acute in an era when pocket calculators and laptop computers are making
redundant many of the arithmetic skills of mental calculation we have come to
assume go hand in hand with civilization.
Example
5.9 Peg Solitaire as a large State Space with Internal Symmetries
Holes 
Moves 
Positions 
Winning 
Terminal 
Tot Positions 
Dead Ends 
1 
0 
1 
1 

1 
0 
2 
1 
1 
1 
0 
4 
0 
3 
2 
2 
2 
0 
12 
0 
4 
3 
8 
8 
0 
60 
0 
5 
4 
39 
38 
0 
296 
0 
6 
5 
171 
164 
0 
1,338 
0 
7 
6 
719 
635 
1 
5,648 
32 
8 
7 
2,757 
2,089 
0 
21,842 
0 
9 
8 
9,751 
6,174 
0 
77,559 
0 
10 
9 
31,312 
16,020 
0 
249,690 
0 
11 
10 
89,927 
35,749 
1 
717,788 
280 
12 
11 
229,614 
68,326 
1 
1,834,379 
31,920 
13 
12 
517,854 
112,788 
0 
4,138,302 
0 
14 
13 
1,022,224 
162,319 
5 
8,171,208 
386,416 
15 
14 
1,753,737 
204,992 
10 
14,020,166 
1.82E+07 
16 
15 
2,598,215 
230,230 
7 
20,773,236 
5.24E+07 
17 
16 
3,312,423 
230,230 
27 
26,482,824 
5.69E+08 
18 
17 
3,626,632 
204,992 
47 
28,994,876 
3.64E+10 
19 
18 
3,413,313 
162,319 
121 
27,286,330 
3.80E+11 
20 
19 
2,765,623 
112,788 
373 
22,106,348 
8.52E+12 
21 
20 
1,930,324 
68,326 
925 
15,425,572 
1.96E+14 
22 
21 
1,160,977 
35,749 
1,972 
9,274,496 
3.72E+15 
23 
22 
600,372 
16,020 
3,346 
4,792,664 
5.31E+16 
24 
23 
265,865 
6,174 
4,356 
2,120,101 
6.05E+17 
25 
24 
100,565 
2,089 
4,256 
800,152 
4.41E+18 
26 
25 
32,250 
635 
3,054 
255,544 
2.16E+19 
27 
26 
8,688 
164 
1,715 
68,236 
8.25E+19 
28 
27 
1,917 
38 
665 
14,727 
1.36E+20 
29 
28 
348 
8 
182 
2,529 
2.11E+20 
30 
29 
50 
2 
39 
334 
1.05E+20 
31 
30 
7 
1 
6 
32 
1.63E+19 
32 
31 
2 
1 
2 
5 
8.17E+16 


23475688 
1679072 
21111 
187636299 
5.77117E+20 
Table
5.9 Successive board positions in peg solitaire[36][37]
Peg
solitaire has a long and colourful history, being spuriously attributed both to
native Americans and to a French aristocrat imprisoned in the Bastille, but can
be specifically traced back to the court of Louis XIV in 1697, from when its
repeated representation in art shows it had wide popularity. In the classical game, the board is filled
with pegs except for the central position, and the aim is by jumping over and
removing successive pieces, to end with a single peg remaining in the centre.
The English board forms a cross comprising 33 holes, as shown in figure 5.9 and
admits multiple solutions, but the European version with four extra pegs does
not admit a classical solution, so we shall consider the English game, although
there are also many puzzle variants.
A
brute force attack on the possible number of positions in n moves gives the sequence
in table 5.9. The total number of reachable board positions is the sum
23,475,688, while the total number of possible board positions is when symmetry is taken into account. So
only about 2.2% of all possible board positions can be reached starting with
the center vacant. ÔTot PositionsÕ ignores the symmetries of board rotations
and reflections which are factored out in ÔPositionsÕ. Counting successive
board positions into a cumulative set of plays, there are
577,116,156,815,309,849,672 or different complete game sequences, of
which 40,861,647,040,079,968 or are solutions. Thus
although there are theoretically a huge number of solutions, the probability of
finding one at random is about 1 in 10,000. Until a player finds a winning
strategy, they tend to initially move in a haphazard way, hoping to arrive
fortuitously at an endgame they can resolve more easily, and are thus unlikely
to find a solution.
Since
any jump exchanges 2 pegs and a hole with 2 holes and a peg and the start
position exchanges holes and pegs as well, there is a symmetry between start
and finish, which means that exchanging pegs and holes and playing backwards
from the finish will provide a complementary strategy to the original. This can
be seen from the symmetry in the winning positions in table 5.9. One appealing
winning sequence first collapses the cross to one move off a smaller central
diamond game before closing in with a grand circuit. The complement to this
game counterintuitively removes the centre diamond before the arms of the
cross arriving back at the centre.
Figure 5.9 Five stages of a winning
game of peg solitaire which first reduces the game to one move off a smaller
diamondshaped version of the game before making a grand tour leaving a single T
which collapses to the solution. The reverse of this game with pegs exchanged
for holes gives a second counterintuitive solution in which the central
diamond is first removed leaving the peripheral parts of the cross, finishing
with a move to the centre. Other games win by an amorphous strategy.
Figure 5.10 Cover maze from
Supermazes[38]
Example
5.10 Mazes as Topological Puzzles
Finally
we return to mazes, which, in addition to underlying the solution space of
every puzzle, constitute the most ancient and intrinsically topological form of
puzzle known. The state space of
the maze is precisely the set of positions negotiated in traversing it.
Although, as in the example of figure 5.8 they may have a complex topology of
overpasses and underpasses in the manner of knot theory, from the subjects
point of view this is secondary to the path connecting the start and finish, so
the structure of a maze is determined by its pathconnected graph, which is
trivially a line for a meander maze (figure 1.2), a tree for a simplyconnected
maze, which can then be traversed however laboriously by a systematic right
hand rule following all culdesacs to exhaustion, however in a maze with loops
although there may be more than one path, the strategy needs to avoid becoming
locked in cycles.
While
we are told Theseus had to follow AriadneÕs thread to return from slaying the
Minotaur, this may have been merely to avoid becoming disoriented in the dark
winding passage of the labyrinth, because the image of coins from Knossos from
figure 1.2 suggests this, like Roman and floor mazes in many cathedrals was a
simple meander maze requiring no choices, but just a long tortuous walk, in
stark contrast to the duplicitous topological paths in the wilderness humanity
has negotiated, since the dawn of history and the equally elaborate paths in
state space we have discovered in analyzing the above puzzles.
Some
of theses state spaces like the Rubik revenge with and even Solitaire with are huge, but Go with board positions and Chess
with an estimated possible games[39]
and between and board positions at the 40^{th}
move surely take the breath away and make one realize the Machiavellian theory
of the evolution of intelligence, based on social strategic bluffing for sexual
favours and personal power in a complex human society of many players has an
invincible and convincing ring to it!
Example 5.11 ScissorsPaperStone Topological bifurcation as a
basis for a complementary strategy space.
Scissorspaperstone is a game consisting of an apparently
irresolvable cyclic transitive relationship of dominance. There is thus no specific winning set
of moves and winning play depends on a bifurcation between two complementary
strategies of defense and attack.
The defensive strategy is to randomize your moves as completely as
possible so the opponent has no pattern they can fix on to take advantage. The
complementary attacking strategy is to deduce the opponentÕs pattern and choose
the move that will capture the move anticipated by the pattern. Various
statistical deviations in human behavior can also be capitalized on. The
choices are commonly skewed rock gaining 36% paper 30% and scissors 34% so a
player can take advantage of the skew. Players also tend to pick moves that
would have beaten their previous move, so choosing a move which your opponent
would have just defeated is a paradoxically winning strategy[40].
6: The BrainÕs Eye View of Mathematics
Despite the strides of such techniques
as the electroencephalogram and functional magnetic resonance imaging,
research into how mathematics is processed in the brain is still in its
infancy. Evidence from cultural and development studies and the effects of
brain injury, are rapidly being complemented by research to elucidate the localization
in the brain of various aspects of mathematical reasoning, however these have
so far dealt mainly with basic level mathematical skills such as raw numeracy
Ð e.g. comparing numbers and tasks such as mental rotation, which are
already the fare of psychological experiment.
Figure 6.1 Sex differences
in mathematical performance tests are not paralleled in verbal performance
tests[41].
Views of the basis of mathematical
reasoning in the brain vary widely. At one extreme is the notion that numeracy is
a hardwired genetically based trait[42]
located in the left parietal lobe (related to finger counting) and even more
basic than language. On a somewhat different tack, Stanislas Dehaene[43]
the founder of the triplecode model discussed below, sees both hemispheres
being involved in manipulating Arabic numerals and numerical quantities, but
only the left hemisphere having access to the linguistic representation of numerals and to a verbal
memory of arithmetic tables.
There is some evidence for a genetic basis
in mathematical ability, in subtle gender differences in performance at
mathematical tasks[44],
which is not reflected in language acquisition (figure 6.1) despite the
significantly different degree of language lateralization in male and female
brains (figure 6.2).
Figure 6.2 Sexual differences in
language processing[45].
These mathematics skill differences
appear to be real and not just based on differences of educational opportunity.
The most comprehensive study published in Science in 1995 found that in maths
and science in the top ten percent, boys outnumbered girls three to one. In the
top one percent there were seven boys to each girl. By contrast in language
skills there were twice as many boys at the bottom and twice as many girls at
the top. In writing skills girls were so much better, boys were considered Ôat
a rather profound disadvantageÕ[46].
Contrasting a biologicallybased
view of numeracy are studies which demonstrate cultural differences in the way
the same numeracy problem is presented, such as those comparing Chinese and
English speakers (figure 6.3). Whereas in both groups the inferior parietal
cortex was activated by a task for numerical quantity comparison, such as a
simple addition task, English speakers, largely employ a language process that
relies on the left perisylvian cortices for mental calculation, while native
Chinese speakers, instead, engage a visuopremotor association network for the
same task. Also raising doubts
about the genetic basis of numeracy is the discovery of the Amazonian Pirah‹ [47]
who live without any notions of numbers more specific than ÔsomeÕ and cannot
count. This is consistent with the
fact that apart from some savantÕs and geniuses such as Ramanujan[48],
most people have a digit span of only seven, and a mental calculation capacity
vastly inferior to a simple pocket calculator.
Figure 6.3 Languagebased differences
in mathematical processing[49].
While some people from Noam
ChomskyÕs generative grammar[50]
to Stephen PinkerÕs ÒLanguage InstinctÓ[51]
contend that language is a genetically based evolutionary trait, other models
of language[52] see the
genetic basis as more generalized and that spoken languages have ÔtakenoverÕ,
as increasingly efficient systems more in the manner of a computer virus
through their cultural evolution by colloquial use. This view has support in
the much more rapid evolution of languages and the fact that, while we do not
know how long ago people first began speaking, written language has only a
short human history, consistent with our reading skills being an adaption of
more generalized visual pattern recognition systems. Since numeracy and
mathematics depend prominently on Arabic numerals, although having a basis in
analog comparison and finger counting, the visual symbolic basis of mathematics
is likewise likely to be a cultural adaption.
The brain consists of two hemispheres
connected by a bunch of white matter called the Corpus callosum. Ever since
splitbrain experiments on monkeys there has been a fascination with the idea
that the two hemispheres in humans may have different or complementary
functions, stemming partly from the knowledge coming originally from war
injuries and strokes that injury to the ÔdominantÕ left hemisphere which is
usually contralaterally connected to the use of the right hand, is selectively
devoted to language typified in BrocaÕs area of the frontal cortex which
facilitates fluent speech and WernikeÕs area of the temporal cortex, which
mediates meaningful semantic constructions. Although this result came predominantly
from men and brain scans on both sexes have subsequently showed that language
acquisition in women is more bilateral than in men, the idea that the two
hemispheres had complementary functions captured the imagination of
neuroscientists.
There is some evidence generally for
this idea with music and creative language use having a partially complementary
modularization to structured language. This in turn led to the idea that the
more structured aspects of mathematics, such as algebra, and the more amorphous
entangled aspects such as topology might be processed in different ways in
complementary hemispheres. While
this idea is appealing, there are few actual experiments that have tested the
idea, and brain scan studies have tended to concentrate on elementary
mathematical skills, which psychologists and neuroscientists can test on a wide
variety of subjects researching basic brain skills, such as mental arithmetic
and mental rotation, rather than complex abstract procedures.
Theories about how mathematical
reasoning is processed gravitate on common sense ideas linking specific sensory
modalities, known linguistic capabilities and general principles of frontal
cognitive processing to generate parallel processing models of brainrelated
modalities having a natural affinity with mathematical reasoning.
Figure 6.4 Triplecode model[53]
of numerical process in has support from independent component analysis of fMRI
scans of mental addition and subtraction revealing four components. (a) bilateral inferior parietal
component may reflect abstract representations of numerical quantity (analog
code) (b) left perisylvian network including BrocaÕs and WernikeÕs areas and basal ganglia reflecting language
functions. (c) ventral occipitotemporal regions belonging to the ventral visual
pathway and (d) secondary visual areas consistent with a visual Arabic code.
For example the triplecode model[54]
(figure 6.4) of numerical processing proposes that numbers are represented in
three codes that serve different functions, have distinct functional
neuroarchitectures, and are related to performance on specific tasks. The
analog magnitude code represents numerical quantities on a mental number line,
includes semantic knowledge regarding proximity (e.g., 5 is close to 6) and relative
size (e.g., 5 is smaller than 6), is used in magnitude comparison and
approximation tasks, among others, and is predicted to engage the bilateral
inferior parietal regions. The auditory verbal code (or word frame) manipulates
sequences of number words, is used for retrieving welllearned, rote,
arithmetic facts such as addition and multiplication tables, and is predicted
to engage generalpurpose language modules, associated with memory and sequence
execution. The visual Arabic code (or number form) represents and spatially
manipulates numbers in Arabic format, is used for multidigit calculation and
parity judgments, and is predicted to engage bilateral inferior ventral
occipitotemporal regions belonging to the ventral visual pathway, with the left
used for visual identification of words and digits, and the right used only for
simple Arabic numbers.
In research focusing on the
intraparietal regions contrasting number comparison with other spatial tasks[55],
numberspecific activation was revealed in left IPS and right temporal regions,
whereas when numbers were presented with other spatial stimuli the activation
was bilateral[56].
Figure 6.5 Unpracticed and learned
tasks in multiplication and subtraction are contrasted[57].
Further support for the triplecode
model comes from studies of learning complex arithmetic (multiplication)[58],
where left hemispheric activations were dominant in the two contrasts between untrained
and trained condition, suggesting that learning processes in arithmetic are
predominantly supported by the left hemisphere. Activity in the left inferior
frontal gyrus may accompany higher working memory demands in the untrained as
compared to the trained condition. Contrasting trained versus untrained
condition a significant focus of activation was found in the left angular
gyrus. Following the triplecode model, the shift of activation within the
parietal lobe from the intraparietal sulcus to the left angular gyrus suggests
a modification from quantitybased processing to more automatic retrieval. A second study involving learning
multiplication and subtraction supports similar conclusions (figure 6.5). This
trend suggests that learned mathematical tasks of this kind become committed to
linguistic memorization, once they are mastered.
In contrast with this, an experiment
where subjects were asked to analyze a simple mathematical relationship[59],
e.g. x
= A, B = A + 6, C = A + 8 by either forming a number
line picture, or constructing the left side of a solving equation e.g. , in both cases visual
processing areas were activated and there were no significant differences in
processing in language areas. This
suggests visual processing areas are involved in forming equations, at least
unfamiliar newly presented ones.
An intriguing study, which has more
implications for advanced mathematics, where real conjectures are examined and
proved, or found false, examined brain areas activated when true and false
equations were presented to the subject[60]. This study found greater activation to
incorrect, compared to correct equations, in the left dorsolateral prefrontal
left ventrolateral prefrontal cortex, overlapping with brain areas known to be
involved in working memory and interference processing.
Figure 6.6 Prefrontal
areas activated differentially when incorrect mathematical equations are
presented.
Extending this into the geometrical
area and specifically with gifted adolescents is a study of mental rotation
(figure 6.8) involving images such as 3D polyminoes. In contrast to many
neuroimaging studies, which have demonstrated mental rotation to be mediated
primarily by the right parietal lobes, when performing 3dimensional mental
rotations, mathematically gifted male adolescents engage a qualitatively
different brain network than those of average math ability, one that involves
bilateral activation of the parietal lobes and frontal cortex, along with
heightened activation of the anterior cingulate.
Figure 6.7 Differential activation of
the medial prefrontal cortex can predict a personÕs intention to add or
subtract two numbers[61].
It has also become possible to teach
a computer to distinguish subjectsÕ intention to add or subtract two numbers,
using analysis of detailed differential activation of the medial prefrontal
cortex, giving predictions which are 70% accurate (figure 6.7).
A brain imaging study of children
learning algebra (simple linear equations)[62],
shows that the same regions are active in children solving equations as are
active in experienced adults solving equations, however practice has a more
striking adaptive response in children. As with adults, practice in symbol
manipulation produces a reduced activation in prefrontal cortex area. However,
unlike adults, practice seems also to produce a
decrease in a parietal area that is holding an image of the equation. This
finding suggests that adolescentsÕ brain responses are more plastic and change
more with practice.
Figure 6.8 Mental rotation: Above
average subjects, middle gifted subjects, below the difference in activation
between the groups[63].
Other theorists have proposed
differing models to the triplecode, in which there are modules for
comprehension, calculation, and number production. The comprehension module
translates word and Arabic numbers into abstract internal representations of
numbers, calculations are performed on these representations, and then the
abstract representations are converted to verbal or Arabic numbers using specific
number production modules. Here amodal abstract internal representations of
numbers are operated on, rather than numbers represented in specific codes
(i.e., quantity, verbal, or Arabic).
The differences between these models
are great. For example damage in the first would give rise to failures of one
modality of processing or another, while in the second particular abstract
operations would be impaired.
Functional activation would be different in the two cases when stimuli
involving mathematical processing are presented to the subject.
What do all these brain studies add
up to and what bearing do they have on the sort of processes that go on in
advanced mathematics? Although the
subject trials rarely engage anything resembling the sort of advanced mathematics
performed at the graduate level, they do suggest that a broad spectrum of brain
areas are involved in mathematical reasoning, involving spatial
transformations, visual representation of closeness and relative position on
the number line, recognition of numbers and algebraic expressions, making
strategic and semantic decisions and transforming many of these processes into
coded linguistic transformations as they become familiar and memorized. They also suggest that much of the
basis for the richness of mathematics as a palpable reality come from sensory
and spatial processes in contrast to the emphasis placed on formal linguistic
logic in advanced mathematics.
To ensure mathematics continues to be
a real part of human culture and doesnÕt suffer the same fate as classical
languages such as Latin in a world of pocket calculators and laptop computers
which obviate the need for mathematical expertise in much of the population,
mathematicians need to stay in touch with the perceivable richness of science
and artistry and imaginative challenge many directly perceivable areas of
mathematics do provide without consigning all such problems to the trash can of
triviality in an era when new classical results at the research level can only
be produced in esoteric spaces through formal processes that stretch far beyond
the rich landscapes human imagination into the ivory towers of formalism.
7: The Fractal Topology of
Cosmology
An acid test of abstract mathematics
as a description of reality is how well it fits naturally with the emerging
cosmological description of reality we are in the process of discovering. While physics had to face the demise of
the classical paradigm forewarned in KelvinÕs two small dark clouds of quantum
theory and relativity, classical mathematics has not yet come to terms with
these changes to its singular foundations.
Figure 7.1 Top: Quantum interference
invokes waveparticle complementarity. Bottom: Wheeler delayed choice experiment.
Quantum reality and cosmological
relativity display troubling features which raise questions about the classical
model of mathematics based on pointlike singular elements in a space whose
geometry is independent of its components. Rather than contrasting the discrete
and continuous, quantum theory is indivisibly composed of complementary
entities which posses both features through waveparticle complementarity, as
illustrated in the interference experiment, figure 7.1, in which quanta
released as localized ÔparticlesÕ from individual atoms traverse a double slit
as waves, only to be reabsorbed by individual atoms on a photographic plate in
the interference fringes. Such
complementarity arises from a feedback process between dynamical energy and
wave geometry, as expressed in EinsteinÕs law:
[7.1]
However the spacetime properties of
these quanta are counterintuitive, as can be seen from the Wheeler
delayedchoice experiment, where changing the absorbing detector system, from
interference detection to individual particle detection can change the apparent
path taken by the quanta, long before they arrived.
Worse still, in contrast to quantum
theory, which is usually couched within spacetime, general relativity applies
a second feedback between energy and geometry, in the form of curvature of
spacetime, so that the geometry and topology of space is also a function of
the dynamics. This makes
integrating quantum theory and relativity a conceptual nightmare, because, in
the event virtual black holes can be created by quantum uncertainty, spacetime
is locally a seething foam of wormholes, resulting in contradictory
descriptions.
Figure 7:2 The redshifted cosmic
fireball (a) has fluctuations consistent with being inflated quanta. Fractal inflation (b) provides a
topological model of how the largescale structure of the universe might expand
forever. Whether or not it does is also a topological question between a closed
and open manifold structure.
An oracle for the fit of classical
mathematics with reality is the elusive TOE, or theory of everything, which has
remained just around the corner since Einstein made inroads into both quantum
theory and relativity. In every
respect, the search for a unified cosmological theory fundamentally brings
topology into the picture and lays siege to classical notions such as point
singularities.
Inflation, as a key candidate theory
of cosmic emergence, links events at the quantum and cosmological levels.
Symmetrybreaking between the forces of nature at the quantum level is coupled
to a switching from a phase of cosmic inflation in which an ÔantigravityÕ
causing an exponential decline in the curvature of the universe switches to
attractive gravity, the kinetic energy thus equaling the gravitational potential
energy, enabling the universe to be born out of almost nothing.
Figure 7.3 The standard model of
physics involves a symmetrybreaking between electromagnetism and the weak and
colour nuclear forces. A deeper symmetrybreaking is believed to unite gravity
with the others.
At the quantum level, theories
uniting gravity and the other forces are based on a variety of forms of
symmetrybreaking, in which the differences between the two nuclear forces, electromagnetism
and gravity arise from symmetrybreaking transformations of a superforce.
In the standard model of particle
physics, the divergence, first of the weak force from electromagnetism, and
then the color force of the quarks and strong nuclear force are mediated by
forms of symmetrybreaking in which the bosons carrying the weak force take up
a scalar HiggsÕ particle and thus gain nonzero rest mass, at the same time
quenching the inflationary antigravity effect of the HiggsÕ field. The latent
energy released by this process gives rise to the hot shower of particles in
the bigbangÕs aftermath. A similar but slightly different symmetrybreaking
applies to the colour force that binds quarks, involving massless bound gluons.
Figure 7.4 Feynman
diagrams (a) 2^{nd} order and (b) sample 4^{th} order terms in
the infinite series determining the scattering interaction of two electrons.
(c) The full set of 4^{th} order terms. (d) The weak W particles act as heavy charged
photons indicating symmetrybreaking. (e) Timereversed electron scattering is
positronelectron creation annihilation, showing virtual particles are time
reversible[64].
Quantum field theories are fractal
theories, because they define the force, say the electromagnetic scattering
between two electrons, in terms of a power series of terms, mediated by virtual
photons, summing every possible virtual particle interaction permitted by
uncertainty, each of which corresponds to an increasingly elaborate Feynman
spacetime interaction diagram (figure 7.4 ac). The series is convergent in
the case of electromagnetism because the terms diminish by a factor [7.2]
the socalled finestructure
constant. A major quest of all theories is such convergence, to avoid infinite
energies or probabilities.
Figure 7.5 Top: Mtheory can unite
several 10D string theories and 11D
supergravity through dualities[65].
The holographic principle allows an nD theory to be represented on an (n1)D surface. Lower left:
dualities between theories can exchange vibration and topological winding modes
of strings on the compactified dimensions[66]. The algebra of the groups may invoke
the octonians, lower right. String excitations, bottom right, avoid point
singularities, but result in topological connections when strings meet.
Attempts to unite gravity with the
other forces have proved more difficult, with a series of theories striving to
hold the centre ground, from supergravity, through superstring[67]
to higher dimensional (mem)brane Mtheories[68]. All these theories have topological
features attempting to get at the root of the singularities associated with the
classical notion of point singularity and its infinite energy. They are broadly
based on supersymmetry[69]
Ð the idea that every force carrying boson of integer spin is matched by
a matterforming fermion of halfinteger spin to ensure their independent
contributions balance to give rise to a convergent theory. All string and brane
theories are founded on removing the infinite energy of a point singularity by
invoking the quantum vibrations of a topological loop or string, or membrane
for small distance scales, resulting in a series of excited quantum
states. Connecting several of these
theories are principles of duality in which two theories with differing
convergence properties can be seen to be dual, so that a nonconvergent
description in one corresponds to a convergent description in the other. This can result in dual descriptions of
reality in which supposed fundamental particles, like quarks and neutrinos
exchange roles with supposed composites of exotic particles like the magnetic
monopole singularities of symmetrybreaking. These theories also share a basis
in invoking a higher dimensional space, usually of 10 to 12 dimensions to make
the theories convergent. This in turn raises the notorious compactification
problem of how some of these dimensions can be topologically Ôrolled upÕ into
closed loops forming internal spaces representing the 1012 internal symmetries
of the twisted form of the forces of nature we experience as well as the four
dimension of spacetime. These theories involve topological orbifolds[70]
Ð orbit generalizations of manifolds factored by a finite group of
isometries, CalabiYau manifolds[71],
topological bifurcations, and potentially up to candidate string theories[72]
hypothetically representing multiverses with differing properties, only a
vanishing few of which would support life and sentient observers, thus invoking
the Anthropic principle[73],
rather than cosmologically unique laws of symmetry and symmetrybreaking.
Figure 7.6 How the lightest family of
particles in the standard model appear as braids[74]^{,
[75],
[76]}.
Each complete twist corresponds to a third unit of electric charge depending on
the direction of the twist. (a) Electron neutrino and antineutrino correspond
to mirrorimage braidings. (b) Four states corresponding to the electron and
positron with charge depending on the orientations of the twists. (c) Three
colours of up quark and antidown quark.
An alternative to string and brane
theory is loop quantum gravity[77]
and topological quantum gravity based on braided preons (figure 7.6). Here
again we have a topological basis, in which the fundamental particles are
braids in spacetime, consisting of more fundamental units called preons, three
of which make up each quark and each lepton. The orientation of the twists in
these braids determine a fractional electric charge of which can sum in
differing ways to the charge on the electron and positron or up and down
quarks. The theory predicts many features of the standard model including the
relationship between quarks and leptons, the charges of the two flavours of
quark Ð up and down and the fact that each of these come in three colours
corresponding to the combinations of one and two twist braids on the triplet
and can model particle interactions through concatenation and splitting of
braids.
More recently Garrett LisiÕs
ÒExceptionally simple theory of EverythingÓ [78]^{,
[79]}
attempts to integrate all the forces including gravity and interactions of both
fermions and bosons in terms of the root vector system generating E8, with its
subalgebras such as G2 and F4 representing subinteractions, such as the colour
force. The connection he uses
again represents the curvature and action on a fourdimensional topological
manifold
We thus find that all the candidate
theories of reality have an intrinsic topological as well as an algebraic basis
and all lead to situations in which the classical view of mathematical spaces
is replaced by quantized versions, which fundamentally alter the founding
assumptions. One can then ask
whether the difficulty at arriving at a theory of everything results from the
obtuseness of physicists, or the inadequacy of abstract mathematics as a
cultural language of ideals to come to terms with the actual nature of the
universe we find ourselves within.
8: References
In the interests of the mazelike nature
of mathematics, these references have an emphasis on interlinked internet
resources, particularly those from Mathworld and Wikipedia, which themselves
provide direct access to the magical maze in the noosphere, which mathematics
has become.
[2] Andersen Johannes 1927 Maori
String Figures, Steele Roberts Ltd., Wellington ISBN 1877228389
[3] Walker, Barbara 1971 A
Second Treasury of Knitting Patterns, Pitman ISBN 0273360736 p139
[5]Dienes Zoltan, Holt
Michael 1972 ZOO, Longman Group ISBN 0 582 18449
[6] Holt Michael 1973 Inner
Ring Maths II, Ernest Benn
ISBN 0 510 07872 9
[10]Stewart Ian1997 The
Magical Maze: Seeing the world through the mathematical eye, Weidenfield
& Nicholson, London
[11] De Chardin Teilhard 1955
The Phenomenon of Man William Collins Sons & Coy. Ltd., London.
[14] K. Falconer 1984 The
Geometry of Fractal Sets, Oxford.
[15] Barnsley, Michael 1988 Fractals
Everywhere, Academic Press, New York.
[23] Schroeder M. 1993
Fractals, Chaos and Power Laws ISBN 0716721368.
[33] Bouwkamp, C. J. and
Duijvestijn, A. J. W. Catalogue of Simple Perfect Squared Squares of Orders
21 Through 25,
Eindhoven
Univ. Technology, Dept. Math, Report 92WSK03, Nov. 1992.
[38] Myers Bernard 1979 Supermazes
No1, Fredrick Muller Ltd., London ISBN 0584102968
[40] New
Scientist 25 December 2007, 6667
[41] Benbow Camilla 1988 Sex
differences in mathematical reasoning ability in intellectually talented
preadolescents: Their
nature,
effects and possible causes, Behavioral and Brain Sciences 11 169232.
[42] Butterworth, Brian 1999 What
Counts: How every brain is hardwired for math, Free Press NY ISBN
0684854171
[43] Dehaene, Stanislas 1997 Number
Sense: How the mind creates mathematics, Oxford University Press NY.
[44] Kimura, Doreen 1992 Sex
Differences in the Brain, Scientific American, Sept 81.
[45] Shaywitz, B. and S. et.
al. 1995 Sex differences in the functional organization of the brain for
language.
Nature 373, 6079.
[46] Blum Deborah 1997 Sex
on the Brain, Penguin, N.Y.
[47] von Bredow, Rafaela 2006 Living
without Numbers or Time, Nature news May 3
[49] Tang, Yiyuan et. al. 2006
Arithmetic processing in the brain shaped by cultures, Proc. Nat. Acad. Sci. 103
1077510780
[52] Christiansen Morten,
Kirby Simon (ed.) 2003 Language Evolution, Oxford University Press
[53] Schmithorst
VJ. Brown RD. 2004 Empirical validation of the triplecode model of
numerical processing for complex
math
operations using functional MRI and group Independent Component Analysis of the
mental addition and
subtraction
of fractions, Neuroimage
22(3) 141420.
[54] Dehaene S et. al. 1999 Sources of Mathematical Thinking: Behavioral and
BrainImaging Evidence Science 284
970974.
[55] Cohen Kadosh R et. al. 2005
Are numbers special? The comparison systems of the human brain investigated
by fMRI
Neuropsychologia
43 1238Ð1248.
[56] Eger E et. al. 2003 A
Supramodal Number Representation in Human Intraparietal Cortex Neuron, 37,
719Ð725.
[57] Ischebeck A 2006 How specifically
do we learn? Imaging the learning of multiplication and subtraction NeuroImage
30
1365Ð1375.
[58] Delazer M et. al. 2003 Learning
complex arithmeticÑan fMRI study Cognitive Brain Research 18 7688.
[59] Terao A et. al. An
fMRI study of the Interplay of Symbolic and Visuospatial Systems in
Mathematical Reasoning
http://actr.psy.cmu.edu/papers/679/paper507.pdf
[60] Menon V et. al. 2002 Prefrontal
Cortex Involvement in Processing Incorrect Arithmetic Equations: Evidence From
EventRelated
fMRI Human Brain Mapping 16:119Ð130.
[61] Haynes, John Dylan 2007 Current
Biology
DOI: 10.1016/j.cub.2006.11.072.
[62] Qin Y et. al. 2004 The
change of the brain activation patterns as children learn algebra equation
solving
Proc. Nat.
Acad.
Sci. 101 5686 Ð5691.
[63] Boyle M. et. al. 2005 Mathematically
gifted male adolescents activate a unique brain network during mental rotation
Cognitive
Brain Research 25 583Ð587
[64] King Chris 2006 Quantum
Cosmology and the Hard Problem of the Conscious Brain in "The Emerging
Physics of
Consciousness"
Springer (Ed.) Jack Tuszynski 407456. http://www.math.auckland.ac.nz/~king/Preprints/pdf/tuz6.pdf
[65] Hawking Stephen 2001 Universe in a Nutshell
Bantam Books, NY
[66] Duff, Michael 2003 The
theory formerly known as strings (2nd ed.) in The Edge of Physics, Scientific American
[72] Brumfiel Geoff 2006 Outrageous
Fortune
Nature 439 5 Jan p10.
[74] Castelvecchi D 2006 You
are made of spacetime New Scientist 12 Aug.
[75] BilsonThompson Sundance
2006 A topological model of composite preons
[76] BilsonThompson S,
Markopoulou F, Smolin L 2006 Quantum Gravity and the Standard Model