MATHEMATICS

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

- Goethe

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!'

Shamanic teacher, painter and author in the UK and the world

 About two years ago I was working on the ART OF 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.  

The mathematician Robin Whitty was kind enough to give my paintings about mathematics a mention on his website and  in truth he did warn me I might have some stray mathematicians coming my way...

 

<  ALGEBRAIC FORMULA FOR A CIRCLE, ANY CIRCLE 

An error in one painting was pointed out to me immediately by mathematician Gerard Michon .

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 incomprehensible formulae and most of all the way mathematics deals with things that remain invisible to the everyday eye, yet exist...

 V  THE DANCE OF LIFE

 

 

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?

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?!

 

<  INFINITY IN THE DINOSAUR AGE

 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!

Robert and Ellen Kaplan 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 LAZY 8 BUS TO INFINITY  (for Judith Bogner)

 

 <  THE EXPANSION OF e    (81 x 48cm)   £199

e = 2.71828..., the Base of Natural Logarithms

 

 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 = limn->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' ...

 Shamanic teacher, painter and author in the UK and the world

 

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... Every explanation leads to more questions at best, to feeling more lost  in the worst case scenario...

<  ROCKING THE BOAT:    HIPPASUS   (30 x 30 cm)   £145

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...

 

 PYTHAGORAS

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! 

 Pythagoreans believed that numbers had personalities.

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

 

<  PASCAL'S PYRAMIDS    (30x 30 cm)    £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.

 

 Shamanic teacher, painter and author in the UK and the world

 THE MATHEMAGICIAN'S HAT   (SOLD)

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.

Here I will just add a comment from 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'

>  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 (so I did!) 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.

 Shamanic teacher, painter and author in the UK and the world

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.

Shamanic teacher, painter and author in the UK and the world

THE 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!    ( 30.5 x 40.5 cm)   £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.

 >   HERE BE MONSTERS...!   

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.

 

 PI

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   (30.5 x 40.5 cm)   £165

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  

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 condition of temperature and pressure above which no distinction exists between a liquid and its vapour.

 Shamanic teacher, painter and author in the UK and the world

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. 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.

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...

ZENO'S PUZZLE & 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'.

It is all about crossing a road, or closing a door... or doing anything we do daily...

Shamanic teacher, painter and author in the UK and the world

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 do shut, people cross streets and catch buses 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...

Ever since I have thought of straight lines as small segments of larger curved lines...

 

When you spend some time contemplating this, you realize 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?!

 Shamanic teacher, painter and author in the UK and the world

 

<  IRREVERSIBLE JOURNEY   

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 

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 behavior. 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...

Shamanic teacher, painter and author in the UK and the world

 FRACTALS  

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!

Shamanic teacher, painter and author in the UK and the world

 

 As an aside: the blue pictures above are NOT recent paintings! I took these photographs at sunrise on January 2nd  2017 , at London Stansted Airport. They are frost patterns on parked cars. I was mesmerized by their beauty and did some research. I discovered that frost patterns are essentially fractals. Use this link for more information!

 This process of pattern creation points at infinity. By making the boundary line all wriggly and crenelated, 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...)

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!

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 organizing properties of chaos'.

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

 

 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...

 Shamanic teacher, painter and author in the UK and the world

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 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 limit

 

<   "CANTOR'S LABYRINTH" (Corresponding Sets)  

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...'

Robin Whitty comments that my painting 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

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'

>  A STAIRCASE OF ALEPHS:   AN INFINITY OF INFINITIES   

(CANTOR'S CONTINUUM HYPOTHESIS)

 

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.

Brendan 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  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 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 banana makes a Banana Moon!

 

ACKNOWLEDGMENTS

 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 remaining 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!

THE BIGGEST NUMBER KNOWN TO (WO)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)...