Box 15: The Game of Life

John Conway playing the Game of Life in 1974.
Kelvin Brodie, The Sun News Syndication

The great 20th century mathematician John von Neumann pondered the idea of a self-reproducing machine.  In the 1940s there was not enough technology available to build such a machine, but his fellow mathematician Stanislaw Ulam suggested he design a mathematical model using a two dimensional grid or cell space in which the logic of self-reproduction could be explored.   The mathematics of cellular automata was born:

You can read through or download John von Neumann's ground-breaking book at the Internet Archives:

Theory of self-reproducing automata

When Cellular Automata were conceived, the computer was in its infancy and not ready to be applied in this area, so any work had to be done laboriously by hand, so to speak, or else only be theorized.  Von Neumann considered this to be characteristic of a larger stalemate in modern mathematics that computers could someday remedy:

By the 1970s the computer was advanced enough to apply to cellular automata and this branch of mathematical research exploded into LIFE!

The mathematician John Horton Conway's cellular automata GAME OF LIFE became famous when an article appeared in the Mathematical Games column of Scientific American, written by Martin Gardner in October of 1970:

Gardner describes the simple rules for the Game of Life:

From the these guidelines Conway developed the laws of birth and death for the game of life:

Gardner presents a few examples, including the configuration which stabilizes after four generations into the form called the Beehive:

The Glider is a small configuration that "moves" across the grid.  It does not really move, of course, since we are seeing successive generations of different cells that live and die, shifting down on the diagonal.  Strictly speaking "movement" is not possible for cells in this sort of cellular automata:

The Glider Gun produces a steady stream of Gliders:

The following video provides an exciting overview of just some of the endlessly fascinating creatures found in the Game of Life:

Amazing Game of Life

In this caricature sketched by his friend Simon J. Fraser at a conference in Toronto, John Conway’s head has grown a “horned sphere,” a topological entity that is counter-intuitive and ill-behaved, much like the ­mathematician himself.  The cartoon came with a dedication:  "In homage to a diabolical mathematician."  

                          - Siobhan Roberts author of Genius at Play, a wonderful biography of Conway

The brilliantly creative  mathematician John Horton Conway provides some fascinating insights into the game that he unleashed upon the world:

Inventing Game of Life - Numberphile

John Conway Talks About the Game of Life Part 1

John Conway Talks About the Game of Life Part 2

In 2010 John von Neumann's dream was realized within the framework of the Game of Life. 
The Canadian Life Enthusiast from Toronto, Andrew Wade was able to develop a Life form called Gemini that replicated itself.  Adam Goucher, on the website Game of Life News noted: “In fact, this is arguably the single most impressive and important pattern ever devised.”

First replicating creature spawned in life simulator 

 New Scientist:  Self-Replicator

Gemini -- Self-replicating oblique spaceship

 in Conway's Game of Life Cellular Automaton

Cellular Automata continue to have many surprising applications in a wide range of scientific areas:

Prof. Gerard 't Hooft, winner of the 1999 Nobel Prize in Physics gave a talk entitled "The Cellular automaton interpretation of quantum mechanics" at Special Lectures of the Tohoku Forum for Creativity

The Cellular Automaton Interpretation of Quantum Mechanics

 Gerard 't Hooft has also written a fascinating book on the subject which the publisher generously provides, free of charge from the link below:

The Cellular Automaton Interpretation of Quantum Mechanics

Exploring the Game of Life on Your Own:

You can play the Game of Life online at several websites.  An excellent one is provided by Bitstorm:

Conway's Game of Life Online

 

  Anyone wishing to explore the Game of Life on their own computer should download the free and amazing open source version called Golly.  This version has a huge area of play and a wide variety of features and yet can run easily on a home computer, laptop or android tablet (Golly 1.2 for Android is now available at Google Play)

Golly:  Game of Life Download Page

Box 14: Napier's Bones

John Napier contemplated the difficulties of mathematical calculations:

One of his first innovations was what came to be known as Napier's Bones:  a set of sticks that simplified multiplication by converting it into addition, a first step on the road to logarithms:

Napier's Bones

A more in depth look at Napier's Bones:  

History of the Computer: Napier's Bones 

By the time of Johannes Kepler (1571–1630), astronomy had been making great advances and a huge amount of data had been accumulated on the motions of the planets.  However, the number crunching required to analyze this data was horrendously time consuming and tedious in the days before calculators and computers.

As Michael J. Bradley notes in his excellent Pioneers of Mathematics series:

Logarithms: Brief History and Brief Math (TANTON Mathematics)

Introduction to Logarithms

Logarithmic Spirals are found in nature in a multitude of fascinating ways:

From the shape of Nautilus Shells:

To the shapes of hurricanes and spiral galaxies:

Logarithmic Spirals Isabel and M51 Credit: Comparison and M51 image copyright Brian Lula; Hurricane Isabel, courtesy GHCC, NASA

Box 13: Gambler's Blues

 

John Tabak offers some insight into the superstitious state of mind that probably delayed the development of  the theory of probability:

Math Antics - Basic Probability

Many consider probability to be just calculations based on common sense, but the Monty Hall problem is a clear example of how even a simple problem can have a counter-intuitive solution:

Game Show or Monty Hall Problem

 Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. He then says to you, "Do you want to pick door No. 2?" Is it to your advantage to switch your choice?

 In 1990 Marilyn Vos Savant had this problem sent in to her "Ask Marilyn" column for Parade magazine.  The problem is easy to state but the answer Marilyn provided was counter-intuitive and unleashed a storm of controversy...

Marilyn Vos Savant and the Game Show Problem

The Numberphile Channel 

provides a good video summary of the problem and its solution:

The Math Forum provides a detailed explanation of the problem as well as information on its history:

The Monty Hall Problem  

Box 12: Descent into Chaos

Portrait by Bill Tavis:   www.fractalforums.com 

Portrait Description by Bill Tavis:  The father of fractal geometry is "drawn" by the set which he discovered. His beautiful mind is illuminated by a minibrot, while a branch also points to his glowing heart. To his side, we see mountains which are not cones, trees which are not smooth, and clouds which are not spheres. Around his head, an aura of psychedelic fractal colors gives a nod to the fact that an entire art-form was spawned as a result of his explorations.

More than a year in the making, this labor of love is a combination of pencil drawing, digital
painting, photography, and a hand-coded (and heavily modified) Mandelbrot Set rendering. 



The first video gives us some background on Benoit Mandelbrot, his set and Fractal patterns:

The second video takes a more in depth look at the construction of the Mandelbrot Set:

The Amazing Mandelbrot Set 

The only way to fully appreciate the endless depth and fantastic detail of the Mandelbrot Set is to zoom in on it in a mind bending video that uses almost 17 million iterations:

Eye of the Universe - Mandelbrot Fractal Zoom

Even a reverse zoom in black and white is quite mesmerizing:

Black and White - Mandelbrot Fractal Zoom (4k 60fps)

 A great documentary that takes its lead from the poet and artist William Blake:

 

Fractals: a world in a grain of sand 

Ben Weiss | TEDxVeniceBeach

And so Chaos Theory was developed and continues to find new and exciting applications:

PBS:  Chaos Theory

A fascinating look at some paths that led to Chaos:

How Was Chaos Theory Discovered?

Box 11: Artists and Mapmakers

Science History of the Universe - Volume 8

But this enchanted realm of Projective Geometry  had a slow and difficult development with many reverses.  In its early stages as linear perspective it was explored by Renaissance artists.  The artists were interested in adding accurate perspective to their pictures and took their lead from the ancient Greeks.  Pappus of Alexandria is usually credited with discovering the first theorem of Projective Geometry:

Pappus's Theorem Proof

Filippo Brunelleschi the Renaissance architect is credited with being among the first pioneers of the mathematical technique of linear perspective and the method he developed was described by the author, artist, architect, poet, priest, linguist, philosopher and cryptographer etc etc Leon Battista Alberti in his treatise on painting entitled:

 De Pictura

How one-point linear perspective works

 Basic Linear Perspective

 

Later the artists Albrecht Dürer and Leonardo da Vinci employed linear perspective in their work and wrote treatises on the subject, though Leonardo's work is lost.

 ALBRECHT DURER (1471-1528) 

A 1525 engraving showing a procedure for the perspective drawing of a lute

Albrecht Durer Geometric and Perspective drawing

Leonardo's Last Supper provides a famous example of one point linear perspective with the vanishing point powerfully located at Christ's head:

 

The Last Supper

Meanwhile mapmakers wanted a way to flatten out our spherical planet with as little distortion as possible, so that their maps could be used for navigation.  They experimented with many kinds of projections, and such projections became an essential part of Projective Geometry, which is often defined as the study of geometric properties that are invariant with respect to projective transformations.

Map Projections Explained - A Beginners Guide

The magnificently logical edifice of Euclidean Geometry seemed all powerful and all inclusive, particularly when combined with algebra as analytical geometry.  What need was there for a young, upstart geometry that seemed to fly in the face of logic and common sense?

Professor Cassius J. Keyser whose rapturous quote began this post offers some insight into why projective geometry was not easily accepted by many mathematicians:

How can we say that parallel lines intersect at infinity?

From:  Bb 7 1 Point Perspective

Where Do Parallel Lines Intersect?

The seventeeth century inventor of projective geometry was the French engineer and mathematician Girard Desargues, who is famous for the second great theorem of projective geometry:

Desargues Theorem

 

Projective Geometry:  Desargues' Theorem Proof

 

Unfortunately, Desargues' work on Projective Geometry was overshadowed by his friend Rene Descartes' powerful Analytical Geometry and found few admirers, though one such admirer was the great mathematician Blaise Pascal who is credited with discovering (when he was sixteen years old!) the third important theorem of projective geometry:

A 16 Year Old Discovered This AMAZING Geometry Hidden Pattern:  Pascal's Theorem

After Desargues and Pascal, projective geometry was all but forgotten.  All copies of Desargues' book entitled:  Rough draft for an essay on the results of taking plane sections of a cone (1639) were lost or destroyed, and it was only in the 19th century that a handwritten manuscript copy was discovered.

The next major work on projective geometry was not until the 19th century, developed by the French engineer and mathematician Jean-Victor Poncelet while a prisoner in Russia with most of Napoleon's Grande Armée:

Poncelet's Theorem or Porism

Overview of the History of Projective Geometry: Geometry of the Straight Edge

"Projective Geometry is All Geometry!"

Projective geometry finally got its revenge for being overshadowed for centuries by Euclidean geometry, as Morris Kline relates in his article Projective Geometry in Scientific American Vol. 192, No. 1 (January 1955), pp. 80-87:

Projective geometry is more fundamental than Euclidean geometry!  This idea and many others are explored in Richard Southwell's excellent and entertaining series of videos: 

Projective Geometry 0 

Why Perspective Drawing Works

Projective Geometry 1 

Without Equations, Conics & Spirals

Projective Geometry 2 

Foundations & Tilings in Perspective

Projective Geometry 3 

When Does A Parabola Look Like An Ellipse ?

Projective Geometry 4 

Desargues' Theorem Proof

Projective Geometry 5 

Axioms, Duality and Projections

Projective Geometry 6 

Conics Made Easily and Beautifully

Projective Geometry 7 

Harmonic Quadrangles & The 13 Configuration

Projective Geometry 8 

The Line Woven Net

 

Projective Geometry 9 

Brianchon's Theorem (Pascal's Dual)

Projective Geometry 10 

Five Points Define A Conic

Projective Geometry 11 

Projective Transformations Of Lines

Projective Geometry 12 

Involutions Of The Line

 

Projective Geometry 13 

Circle Transforms & The Space Of All Conics

Projective Geometry 14 

Pappus's Hexagon Via Circle Projections

Projective Geometry 15 

Conic Involutions, Pascal's Line And Brianchon's Point

Projective Geometry 16 

Finding A Conic (Quadratic Curve) Through 5 Points : Geometric Concstruction

Projective Geometry 17 

Hidden Harmony Of Conic Involutions

Projective Geometry 18 

Homology and Higher Dimensional Projective Space

The above videos use the free, fascinating and powerful GeoGebra online graphing calculator provided along with other excellent online applications:

GeoGebra Math Apps 

Box 10: Heat Wave

 In 1822, the French mathematician Joseph Fourier published his book  Théorie Analytique de la Chaleur (The Analytical Theory of Heat) in which he made use of a bold conjecture that all complex waves can be broken down into simple sine waves:

from Math Is Fun:  Fourier Series

The great 19th Century British scientist William Thomson, Lord Kelvin praised Fourier's book as a "mathematical poem" and paid tribute to it's lasting influence in his Treatise on Natural Philosophy:

The Vibrating String Controversy

Since the time of Pythagoras and the ancient Greeks, mathematicians and scientists have been fascinated with the vibrating string, such as that of a plucked instrument like the harp.  In the 18th century interest in the mathematics of waves increased with the advent of powerful new tools developed from calculus.  

The vibrating string was the elemental problem in wave theory, and by the end of the 18th century it had drawn in many of the greatest names in mathematics, all pitted against one another, each taking a different approach to the problem.

The Language of Physics by Elizabeth Garber offers fascinating insights into what lay behind the mathematical thinking of this era.

It seemed that the vast mathematical powers of analysis that calculus had unleashed made it seem that mathematicians could proceed deductively, in the light of reason alone and without need for scientific experimentation.

And in their ultimate arrogance, great mathematicians like Lagrange believed that the physical science would be absorbed into mathematics:

Lagrange had approached closer than anyone to Fourier Analysis and had even employed it in a few isolated cases, but believed that the mathematics involved was insufficient to yield a general result.  Stronger mathematical tools were needed, he believed and so, by the weight of his reputation, discouraged any further progress in this area, until Fourier arrived on the scene.

Fourier took a different approach, made his bold conjecture and showed time after time that the mathematical results were in accord with the results of experiment.

Mathematicians were dubious at first about Fourier's conjecture but eventually the German mathematician Peter Dirichlet was able to establish Fourier's result as a theorem instead of just a conjecture, and great mathematicians such as Henri Poincare began to sing Fourier's praises:

In the animation below we see sine waves assemble to approach closer and closer to the form of a square wave.

From Wikipedia article:  Fourier Series

And here is a video showing this assembly of sine waves:

Likewise, in the animation below we see sine waves assemble to approach closer and closer to the form of a saw-tooth wave.

From Wikipedia article:  Fourier Series

These wonderful animations were developed by Pierre Guilleminot and can be experimented with at his website: 

Fourier series visualisation

There are some excellent and entertaining videos on the background to these animations and Fourier Series in general:

But what is a Fourier series? From heat flow to circle drawings

 

But what is the Fourier Transform? A visual introduction

 

The following video explains what the Fourier series does, and why it is one of the most surprising results in mathematics:
 

The Fourier transform lets you have your cake and understand it