To Maggie, Peter, Marijo and May, and all the children whom I have taught whose interest, enthusiasm and enquiring minds made me want to write this book.
But by the beauty of shape I want you to understand not what the multitude generally means by this expression, like the beauty of living beings or of paintings representing them, but something alternatively rectilinear and circular, and the surfaces and solids which one can produce from the rectilinear and circular with compass, set square and rule. For these things are not like the others, conditionally beautiful, but are beautiful in themselves.
~ Plato
Contents
Title Page
Dedication
Epigraph
Prologue
The Beginnings – Geometry and Nature
1. A Sixth Sense
2. The Art Gallery of the Universe
3. Secrets of the Stone Age
The Ancient Middle East – Geometry and Daily Life
4. Reading the Shadows
5. The Rope-Stretchers
6. The Stargazers
The Ionian Greeks – Geometry and Thought
7. The Sixth Century BC
8. Thales at Home and Abroad
9. How High is the Pyramid?
10. Abstract Thinking
The Secret Order – Geometry, Mathematics and Magic
11. Pythagoras and his Followers
12. A Famous Theorem
13. Dice of the Gods
14. An Unspeakable Tragedy
From the Academy to the Museum – Geometry, Art and Science
15. The Golden Age and the Golden Mean
16. Applied Geometry
17. The Whole, Round Earth
Copyright
Prologue
Three Wonderful Tools
The string, the straight-edge and the shadow – they are easy to find almost anywhere. A string can usually be found in a child’s pocket; a straight-edge in a desk drawer. Shadows are constant companions on a sunny day.
Yet these are also three wonderful tools. Using them alone, the ancients discovered the ideas and constructions of elementary geometry more than twenty centuries ago. It was all done with just string, straight-edge and shadows. And that is the subject of this story.
Behind the modern wonders
Nowadays, people build bridges that span the Golden Gate or the Firth of Forth. They make planes that go six times faster than the speed of sound, submarines that circle the globe without surfacing, missiles that reach outer space. They drill oil wells deeper than the height of Mount Everest. They have harnessed the power of the tiny atom, and placed people in orbit around the earth. But behind these modern wonders lies a long history packed with beauty, adventure and struggle.
Through the ages, people have searched to find the secrets of the universe. As these secrets were learned, they were written down in mathematical symbols. Today the search for secrets goes on: the mystery of the universe is still unfolding. Even the huge telescope at Mount Palomar is only a tiny window into a vast unknown, and when we finally travel to the moon and planets, there will be other worlds circling other stars for us to visit. In this unending search, mathematics has been a key from the very start.
A long, long time ago primitive humans observed the lines and curves and other forms of nature. They marvelled at them, and copied them as best they could.
Then, as need arose at the dawn of history, people learned to construct these lines, curves and forms accurately. They used the string to trace a circle, to lay off a right angle, to stretch a straight line. They used as a straight-edge anything with which they could draw a straight line. They came to realise that shadows are the sun’s handwriting upon the earth to tell the secrets of order in the universe.
By using these easy-to-find tools, the early civilisations learned how to tell time and direction. They designed their dwellings, temples and tombs, laid out their fields, and constructed irrigation ditches. They started to measure and record the apparent travels of the sun, moon and stars. They found ways of guiding their own travels across the sea and the trackless plains.
So, during thousands of years, the ancient peoples of the Near East built up a practical art that came to be known as geometry – from geo, ‘earth,’ and metria, ‘measurement’.
And then other peoples – the Greeks – turned from the tools to the rules.
Thinking people began to question and wonder why. They were still practical, but they were also interested in abstract rules about lines and curves and angles.
For centuries, many worked on these rules. Some studied beautiful geometric forms and tried to combine them with numbers into a pattern of the universe. One solved useful mechanical problems, such as how to raise water out of the hold of a leaky vessel. Others worked on ‘useless’ puzzles, real brain-twisters. Out of their work came theoretical geometry.
The Greek geometers developed logical ways of thought. They discovered that the earth is a sphere, and they measured the girth of this sphere and the tilt of its axis. They discovered the properties of curves which they called the ellipse, the parabola, the hyperbola and the spiral – curves that many centuries later were recognised as the paths of motion for bodies in space. They helped lay the foundations of modern science and invention.
But all that took an unbelievably long time and happened very, very slowly. Nowadays we live in an age of speed: new models of cars and mobile phones are perfected before the old ones are worn out or even paid for. So it may be hard to realise just how long it took to make those ancient discoveries.
For thousands of years, billions of humans have lived on the earth, and only a limited number of them have contributed to the development of geometry. Many of the ‘facts’ are missing from the early part of their story. For the most faraway period, the records are gone and sometimes we have only a name or a legend.
So in this book, we shall try to tell a story instead – only the highlights, the best characters, the dramatic incidents.
This will be the thrilling story of geometry in the ancient world, from the earliest prehistoric humans to the best-selling text in the history of mathematics, the Elements of Euclid. The story really is a thriller, with wonder and adventure and magic, and even a murder mystery!
As you read it, you will also see that it is a timeless tale, for the old discoveries are very much alive today. And all of them were made with just three wonderful tools: the string, the straight-edge and the shadow.
The Beginnings
Geometry and Nature
1. A Sixth Sense
Geometry itself – if we trace its deepest roots – goes back long before the discovery of its tools. It goes back even before the first observations of primitive man. In fact it goes way back to nature and life and a ‘sixth sense’ that is in every one of us. This mysterious faculty can be called an inborn sense of mathematics.
Because we are all part of this immense universe, and are bound by its laws, we have a natural sensitivity to its order and beauty. All people are thinking parts of the universe, so they have used their sensitivity to translate these laws of order and beauty into mathematical language.
You can understand this sensitivity from your own personal experience. When you shook a rattle and recognised rhythm, or rolled a ball in your playpen and noticed the characteristics of a geometric form, your study of mathematics began.
If you ever have entered a room and adjusted the curtains or straightened a picture on the wall, you have used an innate sense of measurement. If you have stopped your ears to shut out strident or confusing sounds, you have sensed a desire for h
armony.
When you first realised security in the fact that daylight and darkness follow a constant pattern, you experienced a sense of order in the universe.
All living things proclaim this order and have a sense of mathematics that comes naturally.
Birds, bees, whales and seals have a natural sense of direction and distance. Why does a flock of birds fly in angular formation and a family of ducks swim along the creek in the same angular form? How do birds fly back and forth with the seasons, finding their way to the same resting place? How do bees direct one another to the source of honey? They never studied the science of navigation – yet they seem to know how to navigate.
Building in nature
In their building, animals show a sense of form. Bees construct their six-sided cells in the most efficient method of space-filling. But they never studied architecture. Spiders spin an almost perfect spiral web, yet they never studied engineering. Few birds fail to observe the principles of symmetry in the structure of a nest. And all animals seem to know that a straight line is the shortest distance between two points. It is almost as though they had instruments built into them.
Within ourselves also there seem to be built-in measuring instruments. A natural compass helps our sense of direction. A sense of mass and weight keeps us from trying to lift too heavy an object. A sense of symmetry guides us in hanging a picture, or spreading a tablecloth, or adjusting a bedspread. A natural sense of rhythm makes us tap a foot or want to dance when we hear music.
And some of us have developed this special sense in our work.
Artists, for instance, are natural mathematicians, for the secret of beauty is order. Artists must continually compare sizes and distances in their relation to one another in the composition of a picture. They have to put form and atmosphere on a flat rectangular canvas. So they have to judge the values of tones from light to dark, and the force of contrast, and the intensity of colour.
Musicians use intuitive mathematics, too. They must keep time, or have a conductor do it for them. They must know the value of notes and rests within a measure. They have to judge the loudness or softness of a tone, and how long to sustain a note. With their fingers, pianists have to measure the speed and strength of their touch.
And what about dancers? The whole plan and execution of a ballet is based upon an amazing precision of time, motion and pattern, all combined with the rhythm of the orchestra.
The maths of poetry
Poets have to measure the time-beats in a line, and the related beats from line to line. They have to judge words not only for meaning and rhyme, but for the number of syllables and the strength of the accents inside the words.
Some people, as you see, have developed a ‘sixth sense’. But within all of us there is a natural mathematical sense tuned to the natural order of the universe. We all like order and harmony. We like things in proportion; when they seem to be out of proportion, we try to correct them.
It was from this inner sense – our sensitivity to the order and harmony of the universe – that geometry really began.
2. The Art Gallery of the Universe
When and how did geometry begin? Who first discovered the lines, curves and shapes that we call simple geometric forms?
These forms were discovered by the earliest humans who wandered on planet earth, for they were forms found everywhere in nature – in the vast art gallery of the universe.
Let us return in our imaginations through tens of thousands of years, to the time when the first humans wandered alone or in small groups over the earth. Future masters of the earth, they were still cringing in fear. All the great secrets and wonderful resources were locked away from them, awaiting the key of discovery.
They hid from the lightning. They were severely frightened by the seemingly blind and ruthless forces of nature. They thought that, as the days grew shorter and the sun sank lower, daylight would disappear forever and leave them in the chilly darkness. So they huddled by their precious fires.
Fire – that was the first great secret wrested from nature. Prehistoric people tended the fires that started when lightning struck trees, and learned to make their own fires. But this did not dispel their dread that the sun was dying. All primitive peoples have shared the fear, and rites and chants and sacrifices were developed to help the sun’s return.
Gradually, however, the returning warmth and light raised their spirits. This cycle had to occur again and again through centuries before they became confident in its dependable pattern. And slowly they began to absorb a feeling of rhythm and harmony and order in nature. Terror gave way to wonder and fresh discovery.
They began to listen to the music of the wind, the beat of the rain. On warm nights they heard insects’ song and the symphony of frogs. They noticed the time-beat of their hearts and the rhythm of breathing.
They began to observe lines. The jagged lightning was terrifying. But they felt the tranquil peace in the line of the far horizon, like the position of their own bodies when they lay down to sleep. They admired the strong permanence in the line of a tall, straight tree, like their own positions when they stood up. They saw the sadness in the curve of a wilted stem, and the airy lightness in the same curve on a soaring cloud.
To the early humans who trudged in fear through the wonderful art gallery of the universe, everything was a new discovery and an amazing surprise.
Exploring the art gallery
Before we learn how geometry came out of these discoveries, you may want to tour the same art gallery for yourself. Just look around and you will find circles, right angles, triangles, squares, five-sided pentagons, six-sided hexagons and spirals, beautifully revealed by nature in sky, earth and sea.
Gaze at the immensity of the night sky! The greatest natural law that we all feel and appreciate is that of order over chaos: it can be seen in the patterns of the starry firmament – not just those visible to the naked eye, but the ones disclosed by telescopes and calculations. In outer space, distant galaxies – immense clusters of stars – unfold in gigantic spirals. Closer to us, planets and comets swing around the sun in elliptical orbits. Meteors blaze into our own atmosphere in parabolic curves.
Or look through a microscope at the intricate beauty of crystals! For centuries down under the earth, under its great heat and pressure, minerals have been solidifying into crystalline form. Crystals are the flowers of the mineral world: they ‘grow’ by building onto themselves the same materials that they are made of. One mineral can be identified from another by its crystal, for that is the geometric form a chemical substance assumes when changing freely from liquid or gas to a solid. Large or small, regular or irregular, all crystals of a given mineral have the same inner lattice structure and the same relations between faces and axes.
Quartz crystals
Have you seen quartz crystals? They are six-sided (or hexagonal) prisms capped by a six-sided pyramid. If you pound one to powder and put the powder in a solution to recrystallise, the resulting crystal takes the same shape: a hexagonal prism capped by a hexagonal pyramid!
Nature always tends toward simple geometrical shapes. These are the result of universal laws of inner structure and outer symmetry: the same forces that mould a teardrop mould a star. And these forces are at work everywhere in the geometry of living things.
Did you ever wander through forests and fields in the early spring? The woodland bursts into bloom with the tiny triangles of the fresh little three-petaled trillium, the square form of the chalk-white, four-petaled dogwood and the pentagonal blossom of the mountain laurel. The fruit trees swell out with five-petaled flowers. Bend close to the ground and see tiny ferns opening, topped with delicate spirals, and notice the spiralling tendrils on a vine.
Stroll across a meadow covered with five-petaled buttercups and the circular faces of daisies and dandelions. And if you pick a dandelion, stop and consider its beauty, from the spiral growth of the plant to the gossamer sphere that holds the seeds.
Take a n
ew look at the flower garden. The lily, iris and jonquil buds open into hexagonal, six-petaled blossoms. Have you noticed that the green ‘petals’ – the sepals – at the base of a rosebud spread into five-pointed stars as the coloured petals unfold spirally? Press a blossom of Queen Anne’s lace in a book if you want to see a design for lace as beautiful as any a queen might wear.
The shape of vegetables
Study fruits and vegetables! Slice a cucumber, and you will find three divisions of seeds. Slice a green pepper, and you will find four divisions. Slice an onion, and the slices will fall apart in circles. Cut an apple crosswise, and you will find the seeds lodged in a five-pointed star.
Best of all, view the seashore with new eyes. If you are collecting shells, examine the spiral on the shell of the tiny sea snail – a spiral that seems to have been drawn there by the huge twirling wave that rolled it up on the beach!
Have you ever seen a sand dollar? It is a very delicate white circle. Look at it carefully, and you will see a little five-petaled flower etched on one side and a larger one extending to the edges of the circle on the other side.
If you have ever come across starfish in the shallow wash of a wave, you will have surely noticed that most of them are five-pointed. Have you ever seen a sea urchin clinging to the rocks with little feelers like pine needles? Dry it and brush the needles off, and you will find that the small domed shell is marked with a five-divided pattern. When you are about to dive off a dock into the warm August water of the bay, the sight of a sea nettle or jellyfish is not welcome. But pause for a moment and study the lacework and structure of its design before you find a new place to swim.
String, Straightedge, and Shadow the Story of Geometry Page 1
