IMELDA ALMQVIST ART: JOURNEYS TO INNER WORLDS, OTHER WORLDS AND AROUND THE WORLD IN PAINTINGS

 Newton, William Blake, Blaise Pascal, Robert & Ellen Kaplan, The Infinite, The Art of the Infinite, Baudelaire, patterns, connections, numbers, Heraclitus, Greek Philosophers, Imaginary Numbers, Plato, Eternal Ideas, The Inner Oracle of Intuition, The Lazy 8 Bus to Infinity, The Poetry of Numbers, Counting Numbers, Cardinal Numbers, Integers, Real Numbers, Surreal Numbers, Primes, Rational Numbers, Irrational Numbers, Complex Numbers, Transcendental Numbers, Imaginary Numbers, Perfect Numbers, Superabundant Numbers and Deficient Numbers, Hippasus, The Square Root of 2, 'Alogos', Pythagoras, Mathematics As The Key To The Universe, Pascal's Triangle, Sierpinsky Gasket, Gottfried Leibniz, The Leibniz Harmonic Triangle, Diophantine Equations, Arithmetica,  'FLT' : Fermat's Last Theorem, Diophantus of Alexandria, Andrew Wiles, Proof of Fermat's Last Theorem In 1995, Fibonacci Numbers, The Fibonacci Sequence In Nature: Sunflowers, Pine Cones, Spirals, Family Tree Of The Male Drone Honeybee, Bartok's Dance Suite, Monsters In Mathematics, Georg Cantor, Continuum Hypothesis, Higher Orders of Infinity, Critical Mass, Critical Moment, Group Mind In Culture, Evolution, Ralph Abraham, Twilight of the Clockwork God, Paradigm Shift, Goethe, Lynn Margulis, Genetics & Cell Evolution, Microbiology, Neologisms, Zeno, Zeno's Paradoxes, The Dichotomy, The Reality of Plurality and Continuity, The Racetrack, Achilles Versus The Tortoise, Regressus In Infinitum, The Vicious Infinite Regress or VIR, An Infinitely Large Circle, Straight Lines & Curved Lines, Infinite Diameter, Chaos Theory, Fractals, Marquis Pierre-Simon de Laplace, The Deterministic Universe, Dynamic Systems, Attractors, Strange Attractor, Finding Patterns In Patterns, Dynamical Systems Theory, States Of Turbulence, The Self Organising Properties Of Chaos, Set Theory, One-To-One Correspondence Between Sets, An Infinity of Infinities, Transfinite Cardinals, First Inaccessible Cardinal, Hyper-Inaccessibles, the Mahlo Cardinal, Cardinals Indescribable, Huge, Supercompact, Rowbottom, Extendible and Ineffible Cardinals, Inexpressible Cardinals, Alephs, Frontier Of Mathematics, Invisible Giants, Quintillion, al-Haytham, Brunelleschi, Vanishing Point, Durer, Maurits Escher

WHERE NEWTON MEETS BLAKE   (size?)   £125

Late in life, Newton said: 'I seem to have been only like a boy playing on the seashore, and diverting myself in now and then finding a smoother pebble or a prettier shell than ordinary, while the great ocean of truth lay all undiscovered before me'.

William Blake took the mystic approach to the same concept: 'To see a World in a Grain of Sand// And Heaven in a Wild Flower// Hold Infinity in the palm of your hand// And Eternity in an hour'

 Blaise Pascal expressed fear: 'The eternal silence of these infinite spaces terrifies me!'

 MATHEMATICS

 I hasten to explain that I am no mathematician and so this page comes with a health warning: visit this page at your own peril to see paintings inspired by mathematical concepts!

About two years ago I was working on the LANGUAGE series, a series of paintings inspired by the languages of our world and concepts unique to those languages. While preparing a talk on the subject, I picked up a mathematical textbook. I wanted to illustrate the point that mathematics is not a language, just as musical notation is not a language. I decided to paint a circle but then messed things up by using the formula for a half circle. The mathematician Robin Whitty was kind enough to give my paintings about mathematics a mention on his website  ( www.theoremoftheday.com and in truth he did warn me I might have some 'stray mathematicians coming my way...')

 Before we get to the paintings, I would like to thank the following mathematicians for taking the time to answer questions and engage with me on the most basic level (so I would have a hope of understanding!):  Robin Whitty (in London), Gerard Michon (in Los Angeles), Evelien Bus and Bob Rink (both in the Netherlands). Obviously any errors and dodgy bits are entirely of my own making! I would also like to say thank you to Judith Bogner (a TV presenter, not a mathematician!) for coming up with the concept of the Lazy 8 London Bus to Infinity!

ALGEBRAIC FORMULA FOR A CIRCLE, ANY CIRCLE   £125

The errors in my paintings were pointed out to me recently by the mathematician Gerard Michon www.numericana.com . While corresponding with him I got intrigued enough to 'enter a time travel machine' (my last brush with mathematics was doing an 'A' level in the subject) and to start reading about mathematics all over again. I was struck by the beauty of the mathematical landscape, the poetry hidden behind seemingly incomprehensive formulae and most of all the way mathematics deals with things that remain invisible to the everyday eye, yet exist...

  It is interesting to note that Robert and Ellen Kaplan start off their book (The Art of the Infinite) with a quote by the French poet Baudelaire, who speaks of 'cradling our infinite on the finite seas'. This immediately poses some questions: does infinity live in our mind? Does it live in our language or in poetry? Or was it out there long before human beings conceived of it? Did it exist in the era of the Dinosaurs?

 DINOSAURS & 'THE IMPLIED SPIDER'    £135

Did Infinity Exist In The Era Of The Dinosaurs?!

 This image appeared before my eyes while waiting for a train recently, while mulling over concepts from mathematics that lend themselves to painting. The 'Implied Spider' from Wendy Doniger's book (or that is my guess! - see bibliography at the bottom of this page) crept in while working on this painting!

 Mathematics is the science of searching for patterns to patterns and for connections between numbers. The Greek philosopher Heraclitus said: 'A hidden connection is stronger than one we can see'.  Could this be true?!

The Kaplans say that mathematicians working right at the frontier of the subject must make a 'leap from seeing with the outer eye to the inner eye'. This is the leap from mathematics to the infinite: this must always be so. As an artist, used to working very intuitively, I find this a fascinating statement. For me painting is a marriage between seeing with the outer eye and the inner eye: I paint things I see with my inner eye, but base them on things I see with my outer eye, to take others on a visual journey that, ultimately, makes them look inside themselves.

David Foster Wallace (in this book 'Everything And More' ) makes an extremely interesting point about the concept 'abstract'. He uses a basic definition for 'abstract': 'removed from or transcending concrete particularity, sensuous experience'. He says that 'used in this way, 'abstract' is a term from metaphysics. Implicit in all mathematical theories, in fact, is some sort of metaphysical position. The father of abstraction in mathematics: Pythagoras. The father of abstraction in metaphysics: Plato'. (Mental note: now find some willing mathematicians to grill on this one!)

THE EXPANSION OF e   £199

 Here follows a definition of 'e' from Wikipedia:

 e is a real number constant that appears in some kinds of mathematics problems. Examples of such problems are those involving growth or decay (including compound interest), the statistical "bell curve," the shape of a hanging cable (or the Gateway Arch in St. Louis), some problems of probability, some counting problems, and even the study of the distribution of prime numbers. It appears in Stirling's Formula for approximating factorials. It also shows up in calculus quite often, wherever you are dealing with either logarithmic or exponential functions. There is also a connection between e and complex numbers, via Euler's Equation.

e is usually defined by the following equation:

e = lim_n->infinity (1 + 1/n)ⁿ.

Its value is approximately 2.718281828459045... and has been calculated to 869,894,101 decimal places by Sebastian Wedeniwski (you'll find the first 50 digits in a Table of constants with 50 decimal places, from the Numbers, constants and computation site, by Xavier Gourdon and Pascal Sebah).

An effective way to calculate the value of e is not to use the defining equation above, but to use the following infinite sum:

e = 1/0! + 1/1! + 1/2! + 1/3! + 1/4! + ...

THE HIGHLANDS OF IMAGINARY NUMBERS   £135

( A phrase coined by Robert and Ellen Kaplan)

 Imaginary numbers are based around the mathematical number i which is the square root of -1. In other words: i = the square root of -1, so i^2 = -1.

Here mathematics touches philosophy: the Greek philosopher Plato came up with the theory that we recognise truths for what they are because the soul had seen them directly in its abstract state, among the Eternal Ideas, before we were born. Is there such a world of Eternal Ideas? At the time of writing this page I am compiling a LONG list of questions to put to willing mathematicians...

Personally I love the way the Kaplans speak of  'consulting the inner oracle of your intuition' ...

Here is a painting inspired by intuition:

THE LAZY 8 BUS TO INFINITY!   (£188)

This is the London Bus my friend Judith takes to work every morning!

Thank you for a great concept Judith! Judith is accompanied by her cat Jingle.

And the 'Lazy 8 Bus to Infinity' takes me to numbers...

One thing that struck me while reading my way through books on mathematics was the 'poetry of numbers', or maybe I should say: 'the poetry evoked by the names of numbers':

Counting Numbers, Cardinal Numbers, Integers, Real Numbers, Surreal Numbers, Primes, Rational Numbers, Irrational Numbers, Complex Numbers, Transcendental Numbers, Imaginary Numbers, Perfect Numbers, Superabundant Numbers and Deficient Numbers...

This list (by no means complete) evokes all sorts of images! Particularly Surreal Numbers, Imaginary Numbers and Transcendental Numbers! What might they be?!

Transcendental Numbers are numbers that cannot be the result of an algebraic equation (i.e. an equation which only uses the power of x ).

 The label 'Imaginary' in Imaginary Numbers is thought to come from Descartes, he used the phrase dismissively. Newton thought that they were impossible. It must be pointed out though that to mathematicians they definitely exist!! Apparently they are of vital importance to aircraft designers and electrical engineers. Imaginary numbers are based around the mathematical number i which is the square root of -1. In other words: i = the square root of -1, so i^2 = -1.

The Surreal Number system is an arithmatic continuum containing the real numbers as well as infinite and infinitesimal numbers. Real numbers include both Rational Numbers and Irrational Numbers (such as pi and the square root of 2) and decimals - they may be thought of as points on an infinitely long number line...

It is like walking a labyrinth, isn't it? Every explanation leads to more questions at best, to feeling more lost  in the worst case scenario...

ROCKING THE BOAT: HIPPASUS   £125

The pythagorean world was shattered by Hippasus's proof that there were numbers, such as the square root of 2, which weren't the ratio of whole numbers. (I.e. you can't arrive at the square root of two by dividing two whole numbers).

Hippasus concluded that the square root of 2 must be an Irrational Number - or as the Greeks called it 'alogos': a nameless number. Pythagoras didn't want to go there, he feared this way of reasoning led to 'monsters'. The story has it that Pythagoras and his followers were at sea when Hippasus came up with irrefutable proof and they, or the gods, drowned him for his impiety...

Anyone who has every studied mathematics at secondary school will have heard of Pythagoras. He was born on the Greek Island of Samos (5th century BC). Many of our most important philosophical and mathematical concepts were first expressed by Pythagoras. He is best remembered for 'that theorem' about the sides of a triangle: a^2 + b^2 = c^2. A less known concept he came up with was that of the Perfect Numbers. (Numbers are said to be Perfect when the addition of the divisors of a number equals the number itself. A good example is 6 : 1 + 2 + 3 = 6)

Pythagoreans believed that mathematics held the key to the Universe! 

SMALL NUMBERS, BIG PERSONALITIES!   £75

 Pythagoreans believed that numbers had personalities.

They believed that odd numbers are male and even numbers female.

 PASCAL'S PYRAMIDS   £125

In truth there is no such thing as Pascal's Pyramid - this is a concept of my invention! However there is such a thing as 'Pascal's Triangle'- and it looks like a pyramid. In the painting it is the smallest pyramid. Pascal's Triangle is famous for its symmetry and hidden relationships. Pascal didn't actually invent it, it was already known to Chinese scholars in the 13th century.

Here I will just add a comment made by Robin Whitty: 'if you print out Pascal's Triangle beyond perhaps 12 or 13 rows it begins to look less and less like a pyramid and more and more like a Chinese hat! To keep it pyramid-shaped you would have to usa smaller and smaller font size'. (See www.theoremoftheday.org/Theorems.html#39). It was back to the easel on this one: please see the section just below this one to see the result.

In Pascal's Triangle each number is the sum of the two numbers immediately above (working from the top down). In Pascal's Triangle the inner numbers form a pattern depending on whether those numbers are even or odd. If we substitute 1 for the odd numbers and 0 for the even numbers, we get a representation which is the same pattern as the remarkable fractal known as the Sierpinski gasket. (In my painting this is the medium size pyramid).

The German polymath Gottfried Leibniz discovered a remarkable set of of numbers in the form of a triangle. The Leibniz numbers have a symmetry relation about the vertical line. However, unlike Pascal's Triangle, the number in one row is obtained by adding the two numbers below it. For example: 1/30 + 1/20 = 1/12. In my painting the biggest pyramid shows the Leibniz Harmonic Triangle.

 THE MATHEMAGICIAN'S HAT   (SOLD)

I.e. The True Shape Of Pascal's Rule

This painting shows the true shape of 'Pascal's Rule': more like an Egyptian hat than an Egyptian Pyramid! I have called this painting 'The Mathemagician's Hat' as an ode to Martin Gardner who coined this phrase and made mathematics accessible to a large audience.

  Gerard Michon sent me this link to a great video featuring Martin Gardner, courtesy of Encyclopedia Britannica: http://concise.britannica.com/new-multimedia/wmv/newdim025.wmv

I showed this video to my children and they loved it, got busy doing magic tricks inspired by mathematics! 

A MAMMOTH CHALLENGE

DIOPHANTINE EQUATIONS & FERMAT'S LAST THEOREM

When first reading about Diophantine Equations: an elephant came to my mind. Proving Fermat's last theorem proved a 'mammoth task' for mathematicians. Gerard Michon then pointed out that not all Diophantine equations are elephantine - the linear ones are not. Having acknowledged his point here, this painting is still about an elephant. Something is the matter with this elephant... This painting was inspired by an image called 'L'Egistential Elephant' in a book about mathematical patterns.

It seems that there will be 'A Mammoth Challenge i' and 'A Mammoth Challenge II': Robin Whitty has drawn my attention to the fact that if I want to 'impress mathematicians' I ought to raise the power in A Mammoth Challenge to 37, rather than 3. He says it wasn't so hard to prove Fermat's Last Theorem for n=3, the real breakthrough was proving FLT for irregular primes, of which 37 is the first. So... back to the easel on this one!

Diophantine equations demand that their solution be in whole numbers. They are named after Diophantus of Alexandria whose Arithmetica became a milestone in the theory of numbers.

Pierre de Fermat was a 17th century lawyer and government official in Toulouse in France. He was a versatile mathematician and enjoyed a high reputation in the theory of numbers. He is most notably remembered for his las theorem, his final contribution to mathematics. (An explanation follows just below).

 It is possible to add two square numbers together to make a third square. For instance: 5^2 + 12^2 = 13^2. Can we add two cubed numbers together to make another cube? And what about even higher powers than 3?

The answer is that we cannot do that. Fermat's last theorem says that for any whole numbers x,y, and z, there are no solutions to the equation x^n + y^n=z^n, when n is bigger than 2. Fermat's theorem is affectionately known as 'FLT' (or so Gerard Michon tells me).

Fermat claimed that he had found a 'wonderful proof', tantalising generations of mathematicians that followed! However Andrew Wiles proved this theorem in 1995.

A MAMMOTH CHALLENGE II £137

(Dressed to Impress!)

FIBONACCI NUMBERS

 The Fibonacci sequence of number is: 1,1,2,3,5,8,13,21,34,55,89,144,233,377,610,987,1597...

Have you worked out the pattern? When I was a child I liked playing with series of numbers in my head, before falling asleep in the evenings. The Fibonacci series is familiar to me because of those nocturnal counting sessions: every number is the addition of the two previous numbers!

The Fibonacci sequence is found in nature in the formation of seashells, the number of spirals in the seed pots of sunflowers, in pine cones and just about every plant and vegetable! Interestingly it is also found in the family of the Honeybee: to be precise the family tree of the male drone bee!

FAMILY TREE OF THE MALE DRONE BEE

following the Fibonacci Numbers! £155

One male drone Honeybee (1) has one parent: the Queen (1), two grandparents (2), three great grandparents (3)... and so forth!

In terms of human achievement, the Fibonacci numbers are found in room proportions and building proportions used by architects. Classical music composers have used these numbers as inspiration: Bartok's Dance Suite is believed to be connected to the sequence. (Mental note: ask mathematician Robin Whitty about this, as he is an accomplished musician and composer!)

HERE BE MONSTERS...! £110

Nested Roots & Other Terrors...

As a complete amateur when it comes to mathematics, I remember how certain algebraic equations I encountered at secondary school made me break out in a sweat. They scared the 'living daylights out of me', they looked like monsters to me!

For a long time I thought that this was the typical reaction of someone who is clueless. Therefore I recently read with great interest about the history of mathematics. And discovered that mathematical concepts and conclusions have inspired fear even in brilliant mathematicians!

We have already looked at a painting of Hippasus. His discovery frightened Pythagoras and his followers! There is a number that is the root of 2 and it is not a whole number...

In the 19th century the brilliant mathematician Georg Cantor was working hard on his Continuum Hypothesis. In his discovery that there is not just such a thing as infinity, but that there are higher orders of infinity, he was ahead of his times. Most of his colleagues and fellow mathematicians did not want to go there, or feel able to go there, so he ended up struggling in isolation. So actually, mathematical concepts have even frightened brilliant mathematicians on more than one occasion.
 

 This painting is about certain mathematical concepts or notations turning into 'Frankenstein's monster' and frightening all who come face to face with it!

Postscript: Robin Whitty points out that 'nested roots',  as the phenomon depicted in this picture is called, are tremendously important in the history of geometry because they exactly delimit what constructions are possible with the Greek 'straightedge and compass' method.

A theorem which uses ever more deeply nested square roots to give Pi is Viete's formula: see www.theoremoftheday.org/Theorems.html#102), as Robin told me over dinner last night. So I shall play detective... I feel a painting coming on!

 And here is that painting featuring ever more nested roots:

VIETE'S FORMULA £145

Calculating the Value of Pi

Calculating the value for pi is, in a sense, equivalent to the ancient problem of squaring the circle: for a given circle construct, using a straightedge and compass, a square of equal area. It was finally established in the 19th century that only values which were built up from nested roots were 'constructable'.

Now pi can actually not be constructed by a finite number of ruler and compass operations. Viete's formula shows an approximation. He published this formula in 1593 and apparently it is the earliest known infinite product approximation of a number.

Personally I think this formula is both elegant and mysterious, a living piece of history!

CRITICAL MASS

n ^2 £125

A concept that I personally find very interesting is that of critical mass. The mathematical formula for this is n^2.

Critical mass is the point where a process reaches a point or density, where something shifts. The circumstances are ready for something to happen. A shift occurs. (My own definition...undoubtedly not very scientific...)

I ran a search on the internet today to find definitions of the concept 'critical mass'. It would seem that just about every discipline uses their own definition! Here are a few:

Critical mass is the minimum amount of something required to start or maintain a venture (= general wikipedia definition).

Critical mass is the minimum mass of fissionable material that sustain a chain reaction (and I guess that's chemistry).

Criticall mass is the conditions of temperature and pressure above which no distinction exists between a liquid and its vapour (physics?)

Just today I stumbled across an interesting mention of the concept 'critical moment' and mathematics in one line. This idea has to do with relating the history of mathematics to cultural history:

'As a culture evolves, through the practice of what they already know, people become more and more prepared to know more. So there is eventually a critical moment in which a new idea can come into the group mind of the culture because the culture has in its evolution arrived at the first moment where it is capable of having that idea'.

Thus speaks Ralph Abraham in Twilight of the Clockwork God ( details in bibliography). He talks about a phase change or paradigm shift and says that such transformations happen on 'two sides of an axis', as it were: with the mathematical model on the one side and a cultural manifestation that swings history on its axis, on the other side.

 'Mathematicians are like Frenchmen', Goethe once said, 'whatever you say to them they translate into their own language and forthwith it is something entirely different'.

Lynn Margulis is a scientist who studies the science of genetics and cell evolution. She makes the same point as Goethe did, speaking as a contemporary microbiologist: 'When you are paradigm shifting, you don't have the language. Whatever you say, people take the words in the context of their paradigm, which makes your scientific life difficult. You don't want to create neologisms. But you are not saying what they think you are saying'.

This is all part of the run up to a critical moment...

 Talking about critical moments, let's move on to Zeno's Paradoxes...

ZENO'S PARADOXES & THE DICHOTOMY

Zeno was a fiendishly clever Greek Philosopher. He came up with some of the most mindboggling paradoxes in world history!

His paradoxes challenge the reality of plurality and continuity. His most famous paradox is known as The Dichotomy. It looks very simple and it appears in two of his most famous paradoxes: 'The Racetrack' and 'Achilles versus the Tortoise'.

ZENO'S PUZZLE  (SOLD)

Achilles Versus The Tortoise

Basically he says that you can never cross a road, (or close a door, or win a race...) because before you can get all the way across the street, you have to reach the halfway point. And obviously before you can get halfway across, you first have to get to the halfway point of getting halfway across... and so forth. This is known as the dreaded 'regressus in infinitum', also known as the Vicious Infinite Regress or VIR.

 Now a mathematician can come up with a formula to answer Zeno's dichotomy. (In case you want to know, that formula is a/(1-r)...) Obviously we all know from everyday experience that doors to shut, people cross streets and catch busses every day.

  Postscript: my starting point was to call this painting 'Zeno's Paradox'. However, Robin Whitty says that nowadays mathematicians regard this as a puzzle, rather than a paradox. He says that to regard it as paradoxical is to merely overlook the fact that, as distances get smaller, so do the amounts of time needed to traverse them. Dealing with infinitesimal changes in distance during infinitesimal changes in time is what the differential calculus was invented for in the 2nd half of the 17th century. Zeno hasn't perplexed mathematicians since then.'

AN INFINITELY LARGE CIRCLE...

A very simple mathematical concept that has profound implications, or so it seems to me, is that of an infinitely large circle.

In mathematics there is a clear distinction between straight lines and curved lines. However, if a circle increases in size until it has a infinite diameter, then it's surface would become indistinguishable from an infinite straight line...

When you spend some time contemplating this, you realise how much depends on our particular viewpoint, as human beings of a certain size and with a brain that is wired a certain way on one particular planet called earth...

It raises questions about both 'what we see'  and if we ever can get to the 'truth' behind what we perceive...

Or is it a case of there being many valid viewpoints, depending on who you are, where you are  and what you are looking at?!

IRREVERSIBLE JOURNEY     (30 x 30 cm) £125

This painting was inspired by an illustration in a science book about how light behaves once enters a black hole. Essentially it becomes trapped. The gravity in a black hole is such powerful that even light particles can't escape it. And so the light entering such a black hole is on a labyrinthine journey from which there is no return.

 CHAOS THEORY & FRACTALS

Within mathematics there exists a field known as Chaos Theory. Now, isn't that a contradictio in terminis, if there ever was one?!

In the 19th centure a belief ruled the world that if we knew all positions, velocities and forces of all objects in the universe, then these quantities could be calculated exactly for all future times. This was the theory of a man called Marquis Pierre-Simon de Laplace, who mentioned this in an essay he published on the deterministic universe.

The general belief was that small discrepancies in initial conditions meant small discrepancies in outcomes. Chaos Theory exploded this idea!!

We have all heard about a butterfly flapping its wings in Brazil and this resulting in a storm on the other side of the planet. Now this is Chaos Theory in action!

A characteristic of chaos is that a deterministic system may appear to generate random behaviour.

This is why forecasting the weather is so difficult, even with very powerful computers. The equations that 'govern' the weather are non-linear: they involve the variables multiplied together and not just the variables themselves.

In dynamic systems we speak of 'attractors', the single point at origin that a motion is directed towards. In systems like this a set of points may form a fractal (see next section). A fractal is called a 'strange' attractor that will have a definite mathematical structure...

FRACTALS

We have already come across a fractal on this webpage: the 'Sierpinsky Gasket' in the painting 'Pascal's Pyramids'.

This is the 'Sierpinsky Gasket'. It is a detail of a small painting enlarged and therefore quite crude. In reality every triangle in the image would contain an infinity of smaller triangles. It is never ending and quite mind boggling!

One definition of Mathematics is to say that it is the science of finding patterns to patterns.

Within the concept of patterns, fractals are a very interesting concept. Fractals are patterns built up by means of a repetition that changes the scale at each application. They can be found all  over the natural world, in the patterns of branching trees, plants and clouds. Snowfakes are an excellent example as well. The thing about fractals is that they look the same under the microscope as to a human eye without magnification.

This process of pattern creation points at infinity. By making the boundary line all wriggly and crenellated, the length of that boundary line can become as long as you like. (Because you can always make it even larger by repeating a tinier version of the same pattern...)

As I said in the previous section: 'Chaos Theory' sounds like a contradiction in terms, you would think that Chaos is what happens where all theories fail. This is not quite true. There is a branch of mathematics called 'Dynamical Systems Theory'. It is based upon the creation of mathematical models of dynamical processes in nature. It has been discovered that when systems enter states of turbulence (think of boiling water or epileptic seizures) they are actually entering into states of organisation so infinitely complex, that they only appear to be disordered. This is referred to as 'the self organising properties of chaos'.

Now ask mothers of small children about the Self Organising Properties of Chaos...!

 insert image

ISLANDS OF MATHEMATICAL IDEAS

(This painting isn't finished yet, my apologies for the gap!)

Or in the words of the Kaplans: mathematics lies in an enchanted world somewhere between reality and imagination.

 Contemporary mathematician Don Zagier speaks of  'the feeling of being the presence of one of the inexplicable secrets of creation' (the distribution of primes) 

 And there is such a thing as the 'Frontier of Mathematics', where discoveries are waiting to be made...

TRANSFINITY AND HIGHER ORDERS OF INFINITY: GEORG CANTOR

I started this page with the concept Infinity and I am going to end it with the concept Transfinity and an Ode to Georg Cantor who discovered that there is not just one Infinity, but that there exist Orders of Infinity, and some are, contrary to intuitive expectations, larger than others...

Cantor is best known as the creator of Set Theory. He established the one-to-one correspondence between sets and proved that Real Numbers are more numerous than the Natural Numbers.

  'CANTOR'S LABYRINTH' £125

(CORRESPONDING SETS)

Cantor's theorem implies the existence of not just Infinity, but 'An Infinity of Infinities'...

His most famous piece of work is the Continuum Hypothesis.

The word 'Transfinity' comes from Latin words (trans & finis) that mean, roughly, 'beyond limits'.

The Transfinite Cardinal of the sets that Cantor established is sometimes called E, or the 'smallest' transfinite number.

  I was intrigued by the Kaplans words about Georg Cantor: apparently Cantor spoke throughout his life of a 'secret voice' - within, above, unknown - a 'more powerful energy' that spoke through him. He always looked for the face behind the mask and then for the mask behind that...'

Postscript: Robin Whitty comments that my picture shows the integers being put into one to one correspondence with the nonzero integers (so Lazy 8 + 1 = Lazy 8). Such correspondences are certainly at the heart of what Cantor did but there are examples which are more representative. Perhaps the most famous one is the one to one correspondence between the natural numbers and the positive fractions. Between any two whole numbers there are infinitely many fractions. Indeed, between any two fractions, however small, are infinitely many more fractions. So it is quite astonishing that there should be the same number of fractions as whole numbers! But such is the case and it is quite easy to see how. (For a good explanation please use the following link : www.homeschoolmath.net/teaching/rationalnumbers-countable.php)

A STAIRCASE OF ALEPHS:   

AN INFINITY OF INFINITIES £125

(CANTOR'S CONTINUUM HYPOTHESIS)

 Subsequently active research has been done (and remember we have the use of computers & computer programmes that didn't exist in Cantor's time) and transfinite numbers have been invoked that dwarf Cantor's remoter alephs. I mentioned the 'poetry embedded in the names of numbers' earlier. What follows here is the 'Poetry of Alephs':

First Inaccessible Cardinal, Hyper-Inaccessibles, the Mahlo Cardinal.... Cardinals Indescribable... Huge, Supercompact, Rowbottom...Extendible and Ineffible Cardinals.... not to mention Inexpressible Cardinals...

As we reach the Frontier of Mathematics, we reach the Frontier of Language too: words fail us in naming these invisible Giants...

 I will finish this section with a quote from Nietzsche:

'If you stare too long into an abyss,the abyss will stare back into you'

THE ORANGE SUN & THE BANANA MOON

In the following painting two entirely seperate and unlikely sources of inspiration come togeter: the Banach-Taski Paradox & Well Ordering Theorem  and a comment made by my 4 year old son about preferring the 'Orange Moon' to the 'Banana Moon'...

Brendan (the 4 year old) recently said that he prefers the 'Orange Moon' (i.e. the full moon) to the 'Banana Moon' (i.e. the waxing or waning crescent moon). Around the same time  Mathematician Robin Whitty kindly gave me some feedback on my paintings about mathematical concepts. He mentioned that the Banach-Taski Paradox would be a favourite theorem with many mathematicians. I checked this out and learnt that in mathematical terms it is possible to cut an orange in five pieces which can then be rearranged to form a ball the size of the moon (or sun!). Robin explains on his webpage

theoremoftheday.org/Theorems.html#17

that this notorious paradox is a bit of cheat because the pieces must be very complicated and they will make up a moon with a huge amount of empty space. Having said that: a mathematical orange has an infinity of material and infinity can fill up as much space as you need it to. At least it can if it is well-ordered! And so in this painting my son's comment and Robin's intriguing Well Ordering Theorem come together. A reconstructed orange makes an Orange Sun and a reconstructed pea makes a Banana Moon...

THE ORANGE SUN & THE BANANA MOON    £145

This painting is an unlikely marriage between two entirely seperate sources of information:

Brendan (our 4 year old son) recently said that he prefers the Orange Moon (i.e. the full moon) to the Banana Moon (i.e. the waxing or waning crescent moon).

Then Robin Whitty told me that the so called Banach-Taski Paradox would be a favourite with many mathematicians. So I checked this out on his website and learnt that it is possible in mathematical terms to cut an orange into five pieces which then can be rearranged to form a ball the size of the moon. This notorious paradox is a bit of a cheat because the pieces must be very complicated and they will make up a moon with a huge amount of empty space. Having said that: a mathematical orange has an infinity of material and infinity can fill up as much space as you need it to. At least it can if it is well-ordered. (Don't try this at home!)

And so in this whimsical painting my son's comment and Robin's Well-ordering Theorem come together. A reconstructed orange makes an 'Orange Sun' and a reconstructed pea makes a 'Banana Moon'...

And this brings us to the next item on the agenda:

 THE BIGGEST NUMBER KNOWN TO MAN...

 For those of us who have small children - or remain children at heart! - here follows a list of the names of 'really big numbers'. My three sons often ask me for 'the name of the largest number known to man'. They talk about zillions and gazillions... I called my 8 year old son in right now and asked him for 'the largest number known in the school playground'. His answer was: million-billlion-zillion and hundreds of thousands! So what are the names of really large numbers?!

 million (n=1), billion (n=2), trillion (n=3), quadrillion (n=4), quintillion (n=5), sextillion (n=6), septillion (n=7), octillion (n=8), nonillion (n=9), decillion (n=10), vigintillion (n=20), centillion (n=100)

As my son is called Quinn, I think he might like to get to grips with the the number quintillion!

These names of large numbers were found on Gerard Michon's website, to find out more, visit  www.numericana.com 

QUESTIONS FOR MATHEMATICIANS...

While I was working on these paintings and decided that they were going to be featured on a page of their own, I came up with a (wish) list of questions for mathematicans. Some of these questions were tricky, bordering on the impossible,  I have been told! This section is based on the answers I received, with once again a big thank you to the people who were kind enough to engage with these issues!!

Right now I am reading a book about revelations and moral dilemma's in science: The Second Tree, by Elaine Dewar. She describes something she calls the Culture War between the materialists and the mystics (or vitalists). Materialists believe that the 'vast machine called life' can be reduced by science to its molecular components and be understood. Vitalists believe that there is some spark in living things that will never be pinned down, will never be measurable and countable. I confess I belong to the second group!

The questions reading about Mathematics sparks off in me touch upon this great debate. From the answers you will see that the mathematicians stand divided, as do the scientists...

 The first question I asked was: if you could prove something (a theorem or other) in mathematics, what would it be? Bob Rink said he would look at patterns and equations that describe in mathematical terms the way objects move in certain experiments in physics. Evelien Bus would love to find a connection between the rational mind and feelings. She would like to see the following question answered: can mathematics cast a light on human existence and provide insights that help us live our lives? She says that she senses there might be such a connection but mathematics only ever 'divorced her from herself', it did not bring her insights that she could integrate into her general life philosophy. Robin Whitty says that he thinks a sizeable proportion of mathematicians would name the Banach-Taski paradox as their favourite theorem (see the Orange Sun & The Banana Moon painting on this page!)

I asked mathematicians what they would paint if they were painters. Bob Rink stated very categorically that he would keep both things seperate, he would definitely not paint mathematical concepts! Robin Whitty could see his point and Evelien was mystified initially by my plan to paint mathematical concepts! Only fools rush where angels fear to tread...

Then my questions became more abstract. I asked if mathematicians believe in Plato's concept of a World of Eternal Ideas while everything in our world is just an imperfect reflection of these. (This question was inspired by Gerard Michon calling himself a Platonist). Evelien wasn't so sure, she believes in a world of eternal souls and spirits that encompasses a much greater dimension than mathematics covers. Bob Rink says he does think there is something in this idea, that mathematics essentially exist in a world of its own. That is is to be hoped our world is a reflection of that, if applied mathematics is to be of use.

Then, inspired by the 'Dinosaurs and Infinity' painting, I asked to what extent does mathematics exist in the pure sense, in its own right and we discover things as we go OR is mathematics defined by the way our brains are wired and the dimension we exist in? Bob Rink thought that mathematics is a discipline created entirely by mathematicians, but that mathematics touches upon universal truths (and hence some discoveries were made by two people independently of each other). Evelien thinks that mathematics is a pure discipline that exists seperately from us human beings. She thinks that due to the limitations of our human language and our 'human-ness' we can only ever grasp part of a much larger world or universe that is mathematics.

My last question was (and a charged issue it is too!) if studying and practising mathematics brings any sense of a Creator or Intelligent Higher Power. Bob Rink says that working as a mathematician does not make him spiritual-minded, that would sooner come about through the beauty of the 'real world'. Evelien says that she sees a Higher Power or the hand of a Creator in everything she encounters in life and so it is with mathematics.

 I have been told that more answers might come my way... Check this page for updates from time to time! And please write in if you have comments or reflections to share!

Imelda Almqvist   (Last updated September 2010)

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Bibliography/Reference

50 Mathematical Ideas you really need to know, Tony Crilly, Quercus, ISBN 10: 1-84724-147-6

The Art of the Infinite, The Pleasures of Mathematics, Robert & Ellen Kaplan, Oxford University Press, 2003  ISBN-0-19-514743-X

THE INFINITE BOOK, John D. Barrow, Vintage 2005, ISBN  0 099 44372 4

EVERYTHING AND MORE, A Compact History of Infinity (spelled as 'lazy eight', which I can't do on this computer), David Foster Wallace, 2003, Weidenfeld & Nicolson, London,  ISBN 0 297 64567 6

Twilight of the Clockwork God, Conversations on SCIENCE & SPIRITUALITY at the end of an age, John David Ebert 1999, Council Oak Books, ISBN  1-57178-079-3

THE IMPLIED SPIDER, Politics and Theology in Myth, Wendy Doniger 1998, Columbia University Press, ISBN  0-231-11170-3

'Zwoegen Door De Modder En Zweven Langs De Hemel', Afstudeerscriptie (Final Thesis) of Evelien Bus, Utrecht, The Netherlands

www.numericana.com  Gerard Michon

www.theoremoftheday.com  Robin Whitty

www.homeschoolmath.net  Maria Miller

 Newton, William Blake, Blaise Pascal, Robert & Ellen Kaplan, The Infinite, The Art of the Infinite, Baudelaire, patterns, connections, numbers, Heraclitus, Greek Philosophers, Imaginary Numbers, Plato, Eternal Ideas, The Inner Oracle of Intuition, The Lazy 8 Bus to Infinity, The Poetry of Numbers, Counting Numbers, Cardinal Numbers, Integers, Real Numbers, Surreal Numbers, Primes, Rational Numbers, Irrational Numbers, Complex Numbers, Transcendental Numbers, Imaginary Numbers, Perfect Numbers, Superabundant Numbers and Deficient Numbers, Hippasus, The Square Root of 2, 'Alogos', Pythagoras, Mathematics As The Key To The Universe, Pascal's Triangle, Sierpinsky Gasket, Gottfried Leibniz, The Leibniz Harmonic Triangle, Diophantine Equations, Arithmetica,  'FLT' : Fermat's Last Theorem, Diophantus of Alexandria, Andrew Wiles, Proof of Fermat's Last Theorem In 1995, Fibonacci Numbers, The Fibonacci Sequence In Nature: Sunflowers, Pine Cones, Spirals, Family Tree Of The Male Drone Honeybee, Bartok's Dance Suite, Monsters In Mathematics, Georg Cantor, Continuum Hypothesis, Higher Orders of Infinity, Critical Mass, Critical Moment, Group Mind In Culture, Evolution, Ralph Abraham, Twilight of the Clockwork God, Paradigm Shift, Goethe, Lynn Margulis, Genetics & Cell Evolution, Microbiology, Neologisms, Zeno, Zeno's Paradoxes, The Dichotomy, The Reality of Plurality and Continuity, The Racetrack, Achilles Versus The Tortoise, Regressus In Infinitum, The Vicious Infinite Regress or VIR, An Infinitely Large Circle, Straight Lines & Curved Lines, Infinite Diameter, Chaos Theory, Fractals, Marquis Pierre-Simon de Laplace, The Deterministic Universe, Dynamic Systems, Attractors, Strange Attractor, Finding Patterns In Patterns, Dynamical Systems Theory, States Of Turbulence, The Self Organising Properties Of Chaos, Set Theory, One-To-One Correspondence Between Sets, An Infinity of Infinities, Transfinite Cardinals, First Inaccessible Cardinal, Hyper-Inaccessibles, the Mahlo Cardinal, Cardinals Indescribable, Huge, Supercompact, Rowbottom, Extendible and Ineffible Cardinals, Inexpressible Cardinals, Alephs, Frontier Of Mathematics, Invisible Giants, Quintillion, al-Haytham, Brunelleschi, Vanishing Point, Durer, Maurits Escher

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