Non-Euclidean Geometry Rick Roesler

I can think of three ways to talk about non-Euclidean geometry. I’m pretty sure they are all equivalent, but I can’t prove it.

1. The Parallel Postulate

Euclidean geometry is called ‚Euclidean‛ because the Greek mathematician Euclid developed a number of postulates about geometry.

One of these was the parallel postulate:

Given a line, L, and a point, P, not on L: there is exactly one line through P that’s parallel to L.

For a really long time (2000 years?) people tried to prove that the parallel postulate could be derived from the other postulates; in fact, it’s independent. And because it’s independent, you can invent a geometry that has a different postulate - these are the non-Euclidean geometries.

Euclidean geometry is the geometry of a ‘flat’ space - like this piece of paper or computer screen (a plane) -- or Newtonian space-time. There are two archetypal non- Euclidean geometries: spherical geometry and hyperbolic geometry. I’ll mostly talk about spherical geometry because it’s easier to picture, and I found some good graphics on the web. In spherical geometry, there are no parallel lines – not even one! So what’s a ‚line‛on the surface of sphere? In Euclidean geometry (a plane), when we have a way to measure distances between points, one way to define a line (segment) is that it’s the curve between two points that has the minimum length. In a plane, the curve isn’t curved, it’s a straight line, but it satisfies the ‘minimum distance’ principle.

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In math, these curves of minimum distance are called geodesics. Get a ball (whose surface is a sphere) and a piece of string; mark two points on the ball and stretch the string tightly along the surface of the ball between the two points. You’ll find that the string is part of a great circle; a great circle is the intersection of the sphere with a plane passing through the center of the sphere. Because we’ve stretched the string as tightly as possible, the curve that the string defines must have the shortest length between the two points; if it didn’t, we could stretch it even tighter. So a great circle path is a geodesic on a sphere.

Figure 1. Three examples of great circles on a sphere. Each great circle is in a plane that includes the center of the sphere. The center of the sphere is also the center of each great circle.

On the earth, the equator is a great circle, as are the lines of constant longitude. (But not the lines of constant latitude. Why?) Now you can convince yourself that on a sphere there are no ‚parallel lines‛. Why?

The surface of a sphere satisfies all the other Euclidean axioms, but not the parallel postulate. So it’s non-Euclidean. By the way, you now understand why a flight from Dallas to Tokyo goes through Alaska. Why? (And this is a great example of an ‘everyday use’ of non-Euclidean geometry. But gravity is better and more exciting, and I’ll get to that later!)

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2. A Triangle’s Angles Don’t Have to Sum to 180˚

In a plane (Euclidean geometry), if you draw a triangle and measure the three included angles, you’ll find that the sum always add up to exactly 180˚. Now draw a triangle on a globe (spherical, non-Euclidean, geometry). The first side goes from the North Pole to the equator via the prime meridian (0˚ longitude, near London). For the second side, start at the North Pole and go to the equator via New Orleans (90˚W longitude). The third side goes along the equator, connecting the first two sides. Now look at the angles - what’s the measure of each one? What’s the total?

You should have found that the total is greater than 180˚; in fact, for this example, it should have been 270˚. This will always be true in a spherical geometry, but how far it deviates from 180˚ depends on the area of the triangle. (In hyperbolic geometry, by the way, the sum of the angles is always less than 180˚.) So the second definition of non- Euclidean geometry is something like: ‚if you draw a triangle, the sum of the three included angles will not equal 180˚.‛

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Figure 2. Example of a spherical triangle. In this illustration the angle at the North Pole is 50˚ rather than the 90˚ angle we constructed in the text; here the sum of the angles is 230˚. When we construct smaller triangles on the sphere (see the inset), we end up with angles that sum to almost exactly 180˚.

Remember, I said that the deviation from 180˚ depended on the size of the triangle. In the picture above, the inset shows that if you draw a triangle in your back yard and measure the angles, you’ll get something very, very close to 180˚; your back yard is locally flat and approximates a Euclidean plane - even though your back yard is really curved because it’s part of the Earth’s surface1. That’s why for a very, very long time, people thought the earth was flat. Because it is – locally.

1 The spherical triangle that we constructed with three 90˚ angles has an area 1/8 that of the earth’s surface 14 2 (Aearth ~ 5 x 10 m ). If you drew an isosceles right triangle in your back yard with each side being 1m, the triangle’s area would be 0.5 m2 which is 1015 times smaller – a million billion times smaller -- than the surface of the earth. And that’s why the earth seems flat to us.

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3. PARALLEL TRANSPORT - THE KEY TO UNDERSTANDING EINSTEIN’S GENERAL THEORY OF RELATIVITY (GRAVITY)!

Imagine that you’re at the North Pole, and you’re carrying a spear. Hold the spear horizontally and point it south (hint: every direction is south!). Start walking along a constant-longitude line until you reach the equator. Note that as you move, the spear never changes direction, it’s always pointing south; each position of the spear is parallel to its previous position. Now, when you’ve reached the equator, keep facing south, but move sideways toward the east along the equator. Your spear is still pointing south; it’s still being parallel transported. After you’ve gone a fair way, maybe 1/8 the circumference, around the earth along the equator, start walking backwards toward the North Pole along a line of constant longitude. Your spear is still facing south.

When you reach the North Pole (your starting point), you’ll find that the spear is NOT pointing in the same direction it started in, even though we never changed its direction while we walked! The result is a global rotation without local rotation.

Figure 3. Parallel transport of a spear. The red arrows represent the spear; the closed path that the spear traverses is in blue. Note that the spear never changes direction; it always points South. Yet when we return it to the North Pole, it’s been rotated from its initial direction.

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You should take some time now to convince yourself that if you’re in a Euclidean (flat) space - like a flat piece of paper, a plane – a parallel transport of a spear will not change its direction when you go in a closed loop; it always ends up pointing in the same direction it started. So another definition of non-Euclidean geometry is something like: if I parallel-transport a spear around a closed loop, it will end up pointing in a different direction than it started.

Now you’re ready to understand gravity! Imagine that you and a friend are ants that are standing at two different locations on the equator. Since you’re ants, you don’t understand ‚up‛ and ‚down‛; you only understand two dimensions: ‚forward and backward‛ and ‚left and right‛. Begin by measuring the distance between you - you’re very sophisticated ants. Now you both start walking north – your initial directions are parallel – toward the North Pole. As you walk, you continue to measure the distance between you, and you find that it’s getting less and less!

You know that Newton, a very famous ant-physicist, said that objects will move in a straight line with constant velocity unless acted upon by an external force; therefore, two objects moving parallel to each other will always remain the same distance apart unless acted upon by an external force. Newton was a Euclidean thinker. And in Euclidean geometry if you and your friend start off walking parallel to each other, the distance between you should always be constant - that’s what parallel lines do in Euclidean space! But your relative distance is decreasing! Therefore, there must be some force pulling you together because you eventually collide once you reach the North Pole. Being good students of Newtonian physics, you decide to call that force ‚gravity‛.2

Einstein modified Newton and said that, instead of moving in straight lines, objects move along geodesics (shortest path curves) in curved (non-Euclidean) space-times unless acted on by an external force. In addition, Einstein said that it’s mass that curves space-time. So gravity really isn’t a force at all. What we call gravity is simply the result of us (or the earth) trying to move along the shortest space-time path when the earth (or the sun) is curving the space-time we’re moving in. Mass here curves

2 It’s interesting that the ants can’t feel anything pulling them together. They just drift closer and closer together as they move North. They infer the existence of the force because they think they’re in a flat space and the distance between parallel lines in a flat space should always remain the same. If you go skydiving or if you’re orbiting the earth, you don’t feel anything pulling you down either - you’re just in free fall. Remember, it’s not the fall that hurts; it’s the sudden stop when you hit the ground!

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And, in fact, gravity is mostly due to the curving of time. Here’s a simple demonstration you can do at home. The last page of this article is a picture that you can print out. It’s what I call ‘cone world’. Cut the page along the line marked ‚floor‛. Notice that, because the paper is flat, the shortest distance between two points is a straight line. Roll the paper into a cone and tape the sides together; space (distance from the floor) goes up and down the cone in straight lines, while time goes around the cone in a circle. In cone world, space is ‘straight’, but time is ‘curved’. Near the vertex of the cone is a tabletop three feet off the ground (the base of the cone). Now imagine that a cup of coffee is knocked off the table. There is no ‚gravity‛ in cone world; objects just move along the straightest-possible path (a geodesic) per Einstein.

I’ve drawn the path of the falling cup. (Remember, it’s a straight line when the paper is unrolled laying flat on the table; you can convince yourself that this is still the shortest possible path by taking a string and stretching it tightly between two points on the cone.) So what happens? Well, one second after it leaves the table, the coffee cup is about 2.75 feet above the floor; after another second, it’s about 2.25 feet above the floor; after another second, it’s about 1 foot above the floor; and it finally hits the floor after about 3.4 seconds. This is really important: the coffee cup ‚fell‛ to the floor NOT because there’s gravity in cone world, but because, in cone world, time is curved!

That’s all we did: we curved time, and, voila, we have gravity!

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3.5

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Height(feet) 1

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0 0 1 2 3 4 Time (sec)

Figure 4. Path of a falling coffee cup in Cone World. The cup ‘accelerates’ as it approaches the floor, just as if it were under the influence of gravity. Yet there is no gravity in Cone World. Like the ants on the sphere, the cup is simply following the shortest path in this world where time is curved. Note: this is not a parabola like a real-world path would be, but it looks similar.

Now because matter curves spacetime, it turns out that clocks run faster at higher altitudes. Your GPS (global positioning system) uses a system of satellites to triangulate your position. To do this, they need to synchronize the time between the satellite and your navigation system; but the satellite clocks tick faster than the clock in your navigation system. So the GPS system has to compensate for this asymmetry which is a direct result of the non-Euclidean geometry of spacetime. If we didn’t correct for this, your navigation system would be off by hundreds of miles after just a couple of days.

That’s it: three different ways to think about non-Euclidean geometry. One really important example: gravity! And another important example: the Global Positioning System.

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Figure 5. Cone World. Cut this out, roll it into a cone, and tape the edges together. Convince yourself that a geodesic (shortest distance curve) on the cone is the same as a straight line on this piece of paper before it’s rolled up. The coffee cup follows the shortest path in Cone World; the result is that it ‘accelerates’ toward the floor.

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