This Manifold of Ours

You might have picked up this book and decided to sit down and read it. This book is about the world in which you live. Before reading, it might be a good idea to look around you carefully. Do you notice anything unusual? If not, look again.

Here’s a thought. Look around at the many shapes and structures which form your world. Do miles of sidewalks stretch from uptown to downtown? Or do you see a spacious lawn with unmown grass, hiding in its depths little groves of mushrooms or wildflowers? Do you see mountains, hills, trees, and forests? What is the shape of your world?

Take a good look at everything around you. With every look out your front window, with every walk outside to get the mail, the curtains lift on the shape of your world. The shapes of our world tell a truly exciting story. Only this time, the characters are not people. The characters are the shapes that form your world. It’s quite a story, and I have the honor of telling it to you.

Like most great stories, this one begins very simply. It all began in my high-school days. I was not good at geometry, but during that year I discovered a passionate love for my Dad’s old mathematics textbooks. They seemed infinitely interesting to me. Instead of big yellow boxes denoting Theorem 1 or Corollary 6.1, one of his books, a TimeLife little book on Mathematics, showed exciting pictures of star fish and nautilus shells. Other books, with strange names like Differential Equations or Calculus with Analytic Geometry contained interesting diagrams, curves, and formulas generously given to the reader. I was captivated by all these things.

One spring day, my Dad and I pulled into our driveway. We lived in the heart of the Palouse region of Washington state, a region of fields and hills on which wheat and lentils are grown. The view from our driveway looked out onto fields, and behind them, the hills – layers and layers of hills.

On this particular day, something looked different. Far out over the lentil fields lay a hill, blue with distance. At just the right angle, a power pole nearly bisected that hill into two symmetric halves. And in front of this, another hill. And behind, another. Something about the layers of hills left an impression. “Look! Do you see it? That’s a differential equation!”

I distinctly remember flipping through Differential Equations looking for that shape. I can still recall searching through the pages and being fascinated by images of curves and lines. Long before I studied mathematics in college, an idea took root and held fast. There was a geometry to the world, and numbers, at least in some general way, were the language and the key to understanding the shape of the world.

Our world is full to bursting with strange and wonderful shapes. Consider the sphere, that humble but ubiquitous shape that really is the stuff of the universe. Stars and planets populate the sky, causing it to seethe with light on a dark winter night. In the daytime, the sun, also a sphere, shines brightly – so brightly in fact that it is difficult to see it as a sphere except at sunset, or on the haziest of summer days. There sphere is everywhere in the night sky. It is the shape of the stars and the planets. And yet, in our Solar System and likely in other systems in the universe, there are numerous asteroids and comets. They are not spheres, but oblong and misshapen amalgamations of ices and rocks. Why is this? And why do some of the great spheres sport beautiful rings, while others do not?

Or consider the wonderful triangle. The triangle is a cunning shape consisting of three pointed corners. There is one thing in nature that, while not strictly triangular, certainly tends to look that way. An army of green leaves scrubs out the carbon dioxide from the air every day, giving us oxygen to breathe as well as providing us with vital sources of food. Leaves often appear triangular due to their blades that peak in a corner, or their saw-toothed and lobed margins. And yet, these toothed and bladed shapes are rare in other cases in nature. Cells do not have margined teeth, and most crystals are not triangular. No blood vessel, artery, vein, or river channel has saw-toothed boundaries. They function according to a different law.

There is a law for everything – a law for the shape of stars and planets, and a law for the leaves. Close to them are the petals of the flowers. They attract birds, butterflies, and other insect pollinators to the flower’s centers so that the flower can be fertilized and the cycle of life continue on. And yet, the petal does not operate by the same geometrical form as the leaves. While most leaves taper to a point, most flower petals do not. They serve different purposes than the leaves, and so they must grow and be shaped differently. While generally smoother and rounder compared to leaves, both are symmetrical and occur in specific numbers according to the plant species. This is not an accident, but by design.

Consider the spiral, a wonderful shape that serves great usefulness in the animal world. The elephant bears spiral tusks, the ram is adorned by spiral horns, and even some varieties of horses grow their hair in tight spiral whorls. The nautilus shell is a famous and much photographed spiral. There are even the spiral shaped bacteria, or the spiral flagella that propel rod-shaped bacteria on their journeys through the microcosmos. But even there, the spiral serves a purpose, for spirals, while common in animals, still only occur in specific instances in nature. Underneath the surface of these animals is a biology that unifies structure and function across the kingdoms and domains of life. And according to general rules of nature, the internal anatomy of animals does not contain spirals. There are no spiral hearts or blood vessels. There are no spiral bones or intestines, tightly coiled though they may be. Circular forms exist, like the starfishes, jellyfishes, and anemones. They have many legs and would look the same if you rotated them in your hand, but there are no true spiral animals.

In the world of shape, there is much room for beauty and nuance. Shapes are not applied sloppily to this world. There is a law and design for everything.

Then consider this. Our own planet, stock full as it is with amazing shapes, is not the only source of geometry in our universe. We know that the universe teems with galaxies – at least 200 billion of them. Each galaxy, whether elliptical, spiral, barred spiral, or irregular, is a source of geometric information. Each galaxy contains anywhere from 10 million to 400 billion stars. This does not include black holes, nebulae, pulsars, quasars, etc. And each of these objects has its own form and purpose.

Why do things in nature take the shapes that they do? Why are shapes common in some instances and not in others? It is to answer these questions that this book was written. To everything in nature is a purpose, and to accomplish this purpose, there must be a design. And this idea – that form and shape indicate design – is manifest in all areas of the universe, from the very small to the cosmologically large.

We live in a wild and wonderful universe with many forms and structures. But even in a busy universe full of things, we hold a special place. We can contemplate questions of science, mathematics, and philosophy. We can look at the world and ask questions about how it was formed. We can look under the microscope and probe the mysteries of the cell. We can turn our telescopes to the far regions of the observable universe. We can contemplate the shape of the world, and here, we stand alone and unique among all the creations of this world.

And this contemplation mathematicians and scientists have been doing for millennia. Ancient Hindus, for example, believed that the earth was flat. The earth rested on a great elephant which, in turn, rested on a tortoise. And to the tortoise was given the great cosmic ocean to float in. But not for long. The idea the Christopher Columbus proved the earth was round by sailing to America in 1492 is not true. As early as 600 B.C., the Grecian civilization widely accepted the idea that the earth was a round sphere and other non-Greek cultures followed suit: Rome by the first century AD, India by the 500s, and the Islamic world by the 900s.

Here lies an interesting question. The earth is a very big place, and to us, barring hills, fields, and mountains, it seems relatively flat. How did people know that the earth was round in the absence of satellite photography from space? The earth being a sphere means that if we were to walk around it, we would get back to where we started. Sailing around the globe, or circumnavigation, would not be achieved until Ferdinand Magellan’s famous voyage (1519 – 1522). For the Greeks to understand that the earth was round, they would have needed to use abstract reasoning, apart from any physical experience.

Two main ways informed the Greeks that the earth was round. First, the Greeks were a maritime people. The sight of ships disappearing across the horizon was a familiar sight to them. They knew from much experience that as ships disappeared, the bottoms of the ships would disappear from sight, but the masts would still be visible. This gave them a clue that the earth was not flat.

Aristotle (400 B.C.) took the reasoning a bit further. This time, instead of looking at the horizon of the water, he looked into the sky, specifically during the astronomical event of a lunar eclipse. During a lunar eclipse, the earth passes between the sun and the moon. During this time, the earth casts a shadow into space and darkens the moon, turning it a deep crimson-brown that occasionally flushes to blood red. Aristotle noted that the shadow cast upon the moon by the earth is always round. Regardless of whether the moon is fully cast in shadow or not, the shadow is always part of a circle.

At this point, Aristotle used some mathematical abstraction born from a culture whose greatest intellectual strength was geometry. What shape casts a circle all the time? Wouldn’t a flat disk cast a circular shadow all the time? Actually, it does not. You can experience this yourself if you take a flat dinner plate and hold it to the wall, making a shadow. It’s helpful, in this experiment, to have the wall brightly lit, from a sunny window, bright light, or storm lantern. The shadow of the plate is only circular if you hold the plate straight out, disk in front of you, like a clock. If you tilt the plate, the shadow becomes an ellipse. Of course, the earth couldn’t be in a cube or a box shape, because that would give corners, no matter the angle. But the sphere is a unique shape. A sphere is the same, no matter how you rotate it. It has no edges, no faces, and looks the same from any and every direction. No matter how you hold it or where you look at it, there are no corners or edges, and its shadow will be circular. Thus, the earth must be a sphere.

Long before Ferdinand Magellan undertook his world-changing voyage to circumnavigate the globe, people understood the world was round. But it would take several centuries more, from 400 B.C. to the early 20the century for another voyage to take place. This time, it would be a voyage of the mind. While the earth’s spherical nature was known, it would be a long time before people understood the full implication of this.

In 1777, German mathematician Carl Friedrich Gauss was born to a poor German family in the peasant class. Through government sponsorship, he rose to become a gentleman of legendary mathematical prowess. His papers, discoveries, and theorems, spanned geometry, astronomy, number theory, and graph theory. He was the last mathematician to master every field of mathematics that existed in his day, and his work birthed diverse fields of modern mathematics. It’s because of Gauss that math majors in college take classes on number theory. It’s because of Gauss that we have statistics and ways to handle data uncovered by scientific studies. It’s also because of Gauss that we have the field of mathematical physics, still in use today. But for our purposes, he was one of the first mathematicians to realize that the geometry of surfaces, a sphere for instance, is distinctly different from the geometry of a flat surface, or plane.

In 1827, Carl Friedrich Gauss introduced the idea of Gaussian curvature. What is Gaussian curvature, you will ask. Gaussian curvature has a surprisingly simple explanation. Go outside to your driveway. Look carefully at the ants and roly-polys and wonder what it is like to be them. A major part of being an ant or a roly-poly is the fact that you know about where you are crawling at the moment, and not much about the faraway world. If you could talk to the ants – and I truly wish that I could – I would ask them, “What’s this surface like, right where you are?” That’s it. That’s what Gaussian curvature is – the answer to the question, “How is a surface behaving right where you are?”

Let’s consider three surfaces: the sphere, the flat plane, and saddle shape. We live on a globe ourselves and so it’s hard to appreciate the roundness of our globe, as we are all very small compared to it. However, you could get a spherical marble and imagine an ant on it. “How is the surface of your world, right where you are?” you ask the ant. The ant is small compared to the marble but he’s not so very small that he can’t appreciate the fact that the marble is bulging up. He is on a dome, as far as the marble goes, and as he walks round the marble, he finds it ever the same – he is always perched on the dome of the ball that is his home. “Right where I am, everywhere that I am,” the ant would say to you, “this surface is curving up.” This is called positive Gaussian curvature, and it is a key component of the sphere.

(To be continued)