Everywhere in nature are shapes that we call ‘simple geometric forms’. These shapes seem endless. The simple divisions of a circle, by 3, 4, 5 and 6, are infinite in their variety and beauty in nature’s constructions. For within nature’s law of order there is variation, an overall order with uniqueness of detail.
Snowflake forms
These six-petaled ice flowers are hammered and forged in the upper air by the forces of wind and cold. Snowflakes are always six-sided – that is their law – yet each one is different. So there is freedom within the law. And it is the study of variations within the law that makes mathematics so fascinating and by no means easy. For mathematics goes beyond the simple and familiar into the realm of the abstract and the imagination.
These are some of the wonders and laws to observe in nature’s showcase. We need to have them called to our attention because we have partly lost touch with nature.
But primitive people were terribly, even frighteningly close to nature and nature’s forces. They saw and felt her marvels very strongly. So it was from the vast art gallery of the universe that early humans learned the geometry they would use in the Stone Age.
3. Secrets of the Stone Age
Stone Age humans – prehistoric people who chipped and flaked their tools from stone – used geometry in two great ways: in technics, to make life easier, and in art, to make it more pleasant. For both uses, they borrowed geometric secrets from nature. They felt a sense of swift motion from the slanting line of a tree when it crashed to the ground. If it fell aslant another tree, they felt its strength as a brace. They felt the force of their own slanting positions as they pushed a heavy rock; they realised that it was easier to push a rock along level ground than it was to push it uphill, and they felt the ease of pushing it downhill.
They noticed the speed of birds in angular flight formation. They noticed the sturdiness of three sticks tied at the top to make a tripod.
They saw that curves had a use, too. A round log rolled.
Early people were beginning to discover the secrets of line and force. They used these secrets, which they had found in nature, for countless purposes: to brace their homes, to make their pointed arrows and their wedge-shaped axes, to fashion a log roller, and to extend an inclined plank from the ground to the mouth of a high cave.
Geometry was astonishingly useful – and it was ornamental as well. They noticed that the sky was decorated with stars at night and tumbling clouds by day. The earth was decorated with rolling hills, mountain peaks, winding rivers, flowers, trees, reflections in still lakes and moving shadows. At the seashore, foam-capped breakers curled and spread fan-shaped along the beach, coming only so far and no farther, accompanied by a rhythmic roar. Birds, flowers, reptiles, fish, butterflies and insects were patterned with colourful designs. And as they noticed all this, primitive people were discovering the secrets of beauty in the symmetry and harmony and variety of nature’s forms.
Perhaps unconsciously, they began to borrow these designs as decorations of their own. They borrowed the curve of the snail shell, the repeating pattern of wave after wave, the splash of raindrops and the flash of a flame. They drew them as simple geometric designs, circles and dots, crossed and parallel lines.
And through the millennia, they used these geometric patterns to decorate their persons, their homes, their belongings. They smeared them with mud and ochre on their own bodies. They carved and etched and wove them in their ornaments and tools, their bowls and baskets. They used them to adorn their huts and temples.
Perhaps you are familiar with such designs on the tepees and blankets and beadwork of Native Americans. Did you know that they represent natural phenomena – sun, mountains, trees, lightning? People everywhere have used these designs and this symbolism.
But their development was a fantastically slow process. In prehistoric times, centuries might pass after a geometric form was noticed before it was drawn.
Circles
The first geometric form to be admired was probably the circle. Even the earliest people must have observed the circular shape of the huge, red setting sun, the white brilliance of the full moon, the circles in the eyes of their friends and the spreading circles made by raindrops on a puddle of water or by falling leaves on a still pond.
Admiring a circle was one thing. But drawing a circle accurately was quite another story in those faraway times when people lived without using circles.
Today we are surrounded by circles, more than we can possibly count. Look up, look down, look all around you. How many circles can you find in any room? Think of the kitchen, for example. There are circular plates and pans, gas or electric burners on the stove, rims on the glasses. And these are only a start.
How many circles?
During an assembly quiz programme at a certain school, the question was asked, ‘How many circles are there in the auditorium?’
The first guess was one hundred – far too low. Immediately other students with nimble minds calculated about ten thousand in the circle designs on the ceilings and walls. Then others multiplied the number of students by at least three buttons per jacket, one coin per pocket, four eyelets per shoe, two eyes per face. This came to about twelve thousand. But other students knew there were more circles uncounted. They thought of the polka-dot patterns on some of the girls’ dresses, the rings in notebooks and the circular holes on loose-leaf pages, and all the nails and screws with circular heads holding the floor and furniture together. Finally they gave up!
The circle is everywhere in our lives – sometimes for ornament, more often for use. Nowadays we know the value of an accurate circle. Think how important this accuracy is in the tiny mechanism of a wrist watch, or the dials on an aircraft panel, or all the ‘wheels’ of modern industry.
Certainly the discovery of how to draw an accurate circle was a great accomplishment for early humans back in the Stone Age. But how was it first done? No one knows for sure. Just who discovered the secret – whether it was a man, woman or child – was never recorded. We have no record either of when or where the first accurate circle was drawn, with every point on the circle the very same distance from the centre. But we can make an imaginative guess as to how it might have happened.
Possibly the first true circle was traced on the ground by an animal tied to a stake; it might run around again and again to the limit of its freedom, and trample a circle on the ground – a true circle of overlapping footprints, all the same distance from the centre stake. Perhaps in this way the ‘string’ first revealed the secret of the circle.
This circle may have inspired prehistoric children to play a game. It is easy to picture them using a vine for a string, and swinging themselves around, scuffing out with their footprints perfect circles on the ground.
But their parents would probably not be impressed by this early circle game. They were too busy hunting and tending fires. This great discovery may have remained unappreciated by the adults for long centuries.
Yet, in the end, the secret of the string was recognised. It too was added to the useful knowledge of line and force, and the circle was added to the beautiful decorative shapes copied from nature. The perfect circle was one more geometric discovery in the slowly growing store of secrets of the Stone Age.
The Ancient Middle East
Geometry and Daily Life
4. Reading the Shadows
Here our story takes an exciting new turn. The time was just before the beginning of recorded history. The place was the Middle East. And geometry was about to change from its primitive beginnings into the ancient practical art of earth measurement.
Some people of the new Stone Age had been through what scholars call a ‘revolution’, because it made such a drastic change in their lives. They were no longer hunters and fishers, but herders and farmers. And in their new roles they learned a fascinating new kind of secret from nature: how to read shadows, messages from the sun, as a measurement of time.
As usual, the knowledge came g
radually, for it was built upon the distant past. People of the earlier Stone Age had lived through many millennia, hunting wild beasts for food and being hunted in return. In caves and jungles, across steppes and forests, they had hunted saber-toothed tigers and mammoths and boars – and themselves been stalked and trampled and gored.
Then some wandering groups of humans entered the great river valleys of the Nile and of the Tigris and Euphrates. For a long while before this, they had gathered wild berries and fruits and nuts to eat with their meat or fish. But here, by chance, they learned to prepare wild grain which had a pleasant taste. And in time, as the climate grew drier, somebody noticed that from grain scattered on the river mud – by the wind, by human hands – a plant would grow. Thirsty animals, crowding towards the rivers, were trapped and penned. People were starting to produce their own food along the great rivers of the Fertile Crescent.
In these broad, protected valleys, they could water their flocks and cultivate the land. They lived together, season after season, and sometimes they had leisure to plan and dream.
Among those who had most leisure were the shepherds. They watched their flocks under the scorching sun and took shelter in the shade of large rocks. They had to move many times during the day to keep in the cool protection of the shadows.
Perhaps it was a shepherd, bored by the monotony of his long day, who first placed stones at the end of a shadow as its position and length changed throughout the day.
We can think of him and other shepherds as early scientists who discovered how to tell time. If they wandered far from home, they could tell by the shadow when it was time to start back. They also discovered that the shadow indicated directions on the ground. Along with a need to measure time, they had a need to tell direction. This was before they had any real need to measure distance, for in that time, wandering herders and farmers still moved their homes from place to place, following the wild grasses and the streams.
Measuring time
But time was already important to them. As people discussed the usefulness of measuring the time of day, they must have thought about the usefulness of measuring the time of year. At first, the proper seasons to plant, to sow, to reap, to mow, were all guesswork. Many mistakes must have been made and vital crops lost. What was to be done? If by day the sun marked off the time, why not look to the sky for other needed measurements of time?
Trees bloomed, then birds sang among blossoms and pecked at the first fruit, and what was left developed and ripened. The earth dried up and the leaves fell. The flood of the river came at repeated intervals. Surely there was a pattern in the seasons. Was there also a clue in the sky to all this?
Remember that these people had little to distract them from the sunlight and shadows by day, from the brilliant stars at night. So they watched for signs.
As the shepherds sat through the nights, the sky produced lively entertainment. A bright parade of star groups marched across the sky, and certain bright wandering ‘stars’ moved among them. In the night sky the moon appeared and grew and disappeared – and started this interesting cycle all over again. So these early stargazers counted the nights as the moon went through this same routine, and discovered a repeated pattern.
The sun seemed to follow a pattern too. There were intervals when the days grew longer and longer, and the shadows became shorter. Then this would change, and the days would grow shorter and shorter, and the evening shadows would lengthen early. And this routine, too, would start all over again.
These constant cyclic patterns in the sky and on the earth must have some connection.
Conversations around the campfire
In lively conversations around their campfires, the early shepherds must have talked about the constancy of the change in the moon, the shadows, the seasons and the changing length of the day. Surely the moon and stars, the sun and shadows, were ever-present guide posts for their daily work. The shadows changed through the day, the moon changed through the month, the planets through the year.
So these earliest stargazers counted 30 days while the moon changed; and they watched it change 12 times, from the first bird song in a blooming tree, until the tree bloomed again and the birds returned. They made a crude calendar of 360 days: it can be called the first mathematical formula. The inaccuracies in this first calendar kept people busy studying the sky and shadows for centuries to improve it.
Shadows and time
Shadows and how to read them were to be very important in the future science of geometry. But that comes at a later period of our story. For now, you may enjoy reading shadows yourself – understanding their messages about time, just as the people did after the new Stone Age.
It will not be hard, for shadows are pleasant things. They call to mind cool relief from the summer sun. The sun is a master artist. It sketches a moving picture of you as you walk on a sunny day. It sketches the form of a tree and its leafy pattern in dark blues and purples on a bright green lawn.
Furthermore, probably you already know how to read the sun’s messages from the shadows. Have you ever had the experience of lying in the cool shade undera beach umbrella and suddenly realising that the hot sunlight was on your face? You had not moved, but the shade had. Or have you ever wondered if it was time to leave the beach and get home for supper, and then looked at the shadows for a clue?
Shadow reading is still very important today. Trained scientists can read aerial photographs. If they are given the date, time and position of the camera, they can use the shadows to determine, for instance, the heights and intervening distances of the mountains on the moon.
But before the beginning of history, reading the shadows was crucial to human life. For that was the time of the early farmers and herders, when the world’s first calendar measured day and night, the phases of the moon and the returning seasons of the fruitful year.
Telling the time with shadows
Have you ever thought of making yourself a handy little timepiece? Just prop a stick upright in the ground, and watch its shadow change length and direction (it will shrink towards midday and lengthen afterwards) until it finally tells you when to go home.
It is even more fun to study shadows deliberately for their messages of time measurement. Observe their characteristics and make a record of them. As you go out on the playground or the ball field, mark the end of a shadow. Just before you leave, notice the change in its length and direction. Find out if all shadows point in the same direction at the same time. If you can remember, mark a shadow at, say 3:30 pm, and then examine it at the same hour one month later.
Better still, look up pictures of old shadow clocks and sundials. These will show you how shadows were once used.
5. The Rope-Stretchers
Now at last came the dawn of civilisation. And with it, practical geometry got off to a real start in the work of the surveyors or ‘rope-stretchers’.
To build the early civilisations, people had to get along together in settlements. Living in groups meant great advantages and also great responsibilities. The people who lived together in the protected valleys of the Nile and the Tigris and Euphrates recognised this. We have the proof in their ruins.
Each early civilisation, as far back as several thousand years BC, has left on the land the tracings of field markers, irrigation canals and storage basins. The very earliest tracings go back to about 5000 BC, the date that archaeologists give to the village of Jarmo in Iraq, perhaps the oldest known village in the world. And many of these ruins reveal the work of skilled surveyors. They are laid out with straight lines and right angles.
Nowadays we take right angles for granted, but have you thought what a big part they play in our lives? If you have flown over farm lands, particularly in America and Australia, you have probably noticed that the scene below resembled a patchwork quilt, for the fields are laid out in rectangular patterns. Fences divide one piece of property from another in the same rectangular patterns. And the corners of a rectangle are right angles.
Right angles everywhere
As you drive along through the country, have you noticed that the telephone poles, fence posts, trees and houses all stand upright and at right angles to the ground? Can you imagine a landscape without the right angle? Try to picture driving at fifty miles an hour, with poles standing every which way. And suppose houses and windows slanted in any direction, and the walls of a room were closed inward. You could quickly feel quite disoriented.
Today we are all so familiar with the right angle that we don’t even notice it. Look around the room – how long do you think it would take you to count all the right angles you see? Even in this book, right angles form the four corners of each page. They appear in the structure of the letters L, T, E and H. In modern times right angles, like circles, are everywhere.
But ancient civilisations did not take the right angle for granted. It was a significant discovery made with the aid of a string. It filled needs that were important to people living together in communities.
One very big problem of the early farmers was to mark off their fields for property rights. They had to protect their own property and respect the property of others to get along in peace and security.
What was the best shape for their fields? Remember, these were among the first fields ever laid out. People were familiar with the winding shores of lakes, the jagged lines of mountain ranges against the sky, the endlessly varied outlines of the clouds. But irregular boundaries for a field would not be practical.
String, Straightedge, and Shadow the Story of Geometry Page 2
