HPS 0410 Einstein for Everyone

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Spaces of Constant Curvature

John D. Norton
Department of History and Philosophy of Science
University of Pittsburgh

Linked documents:

Euclid's Postulates and Some Non-Euclidean Alternatives

Unfamiliar Geometries Become Familiar

In the last chapter, we explored the geometry induced by the postulate 5NONE by means of the traditional construction techniques of geometry familiar to Euclid. We drew lines and found points only as allowed by the various postulates. The outcome was a laborious construction of circles and triangles with some quite peculiar properties. We constructed a circle with center O and circumference G, G', G'', G'''.

circle

Its circumference is only 4 times is radius (and not the 2π times its radius dictated by Euclid's geometry). Its cirumference is both a circle and a straight line at the same time. Each of its quadrants are triangles with odd properties. The triangle OGG', for example, has three angles, each of one right angle. So the sum of its angles is three right angles (and not the two right angles dictated by Euclid's geometry).

You would be forgiven for thinking that the new geometry of 5NONE is a very peculiar and unfamiliar geometry and that there is no easy way to comprehend it as a whole. The surprising thing is that this is not so. The geometry of 5NONE and the geometry of the other postulate 5MORE turn out to be the geometries that arise naturally in surfaces of constant curvature. Recognizing that fact makes it easy to visualize these new geometries and one rapidly develops a sense of the sorts of results that will be demonstrable in them.

We will see in this chapter how this arises. Indeed it makes the visualization too easy, that danger is that we overlook the fact that we are really dealing with new an different geometries.

The New Geometry of 5NONE is Spherical Geometry

The geometry of 5NONE proves to be very familiar; it is just the geometry that is natural to the surface of a sphere, such as is our own earth, to very good approximation. The surface of a sphere has constant curvature. That just means that the curvature is everywhere the same. To see how the connection to the geometry of 5NONE works, we need only identify the line AGG'G'' with the equator. The perpendiculars we erected to it in the last chapter then just become lines of longitude all of which intersect at the North Pole, that is at, O.

Perp 12

It isn't quite that simple. We do need to adjust our notion of what a straight line is. The essential idea remains the same. A straight line between two points A and B is still the shortest distance between two points. But now we are forced to remain on the surface of the sphere in finding the shortest distance. There is no burrowing into the earth to get a shorter distance between two points. The curve that implements the shortest distance in the surface is known as a "geodesic".

Geodesic 2

There is a simple way of creating geodesics on the surface of a sphere. They are the "great circles." That is, they are the circles produced by the intersection of the sphere with a plane that passes through the center of the sphere.Geodesic 1

In short, the new geometry of 5NONE is just the geometry of of great circles on spheres.

In such a geometry, there are no parallel lines. All pairs of great circles intersect somewhere. That this is so is sometimes overlooked. People sometimes mistake a parallel of latitude for a great circle. In the figure below, points A and B of the same latitude are connected by a parallel of latitude. The parallel of latitude is a parallel to the equator. However it is not the analog of straight line in this geometry, a geodesic. For geodesics are produced by the intersection of the sphere with planes that pass through the center of the sphere. The great circle passing through points A and B is shown in the second figure. It connects A and B by a path that deviates to the North. Since it is the great circle, it is the curve of least distance in the surface of the sphere between A and B.

Geodesic 3

The great circles are the routes taken by ships and airlines over the surface of the earth, whenever possible, since they are the paths of least distance.

great circle navigation

Circles and Triangles

We can now return to the triangles and circles visited earlier. Their properties were radically different from Euclidean triangles and circles. The triangle's angles summed to three right angles and the circle's circumference was only four times the radius. It is now easy to see that these deviations from Euclidean expectations arise only for very large figures on the surface of the sphere. A very small patch of the surface of a sphere is very close to being a Euclidean plane. The calm surface of a small lake on the Earth is very nearly a flat plane; the surface of an ocean is markedly curved. In those very small patches, circles and triangles are very nearly Euclidean in their properties.

The figure below shows a very small equilateral triangle A''B''C''. The sum of its angles will meet Euclidean expectation near enough and be two right angles. As the triangle grows larger, passing through triangle A'B'C' to the huge ABC, the sum of its angles will grow until they are three right angles at ABC.

triangle

The situation is the same with circles. The circle around the North Pole below with very small radius OA will meet Euclidean expectations, near enough, and have its circumference 2π times its radius. As the circle grows with radius increasing through OB to OC, the formula will mutate. When the radius is OC, so the circle now coincides with the equator, the circumference will have dropped to being only four times the radius.

circles

Corrections to the Other Postulates

Now that we have identified our geometry of 5NONE as the geometry of great circles on spheres, two small corrections are needed. The first postulate allows us to draw a straight line between any two points. In the new geometry, there are two ways of connecting any two nearby points by a great circle. One goes the short way; the other goes the long way all around the other side of the sphere.

The second correction is for the second postulate which allows us to produce a straight line indefinitely. That is not possible for great circles. They are already maximally extended. One part of the original notion of the second postulate was that a straight line never really comes to an end. Any point that looks like an end is only a temporary terminus and the line can be extended past it. That lack of a boundary point is all we need for the revised second postulate.

The two modified forms of the first and second postulates that accommodate these two alterations are:

1'. Two distinct points determine at least one straight line.

2'. A straight line is boundless (i.e. has no end).

The modified postulates are illustrated by the geodesic drawn through two points A and B:

geodesic

Is the Geometry of 5NONE Consistent?

Consider the geometry of 5NONE; that is the geometry that is deducible from the the fifth postulate 5NONE and the other four postulates, suitably adjusted. The expectation of the mathematicians of the eighteenth century and earlier had been that one would eventually be able to deduce a contradiction from them. That is, they expected them to be inconsistent. We started deducing consequences from the postulates but found only odd results, not contradictions.

How do we know that a more imaginative, more thorough analysis might not eventually produce a contradiction? That is, how do we know that the new geometry is consistent?

The question could be answered by a proof of the consistency of the geometry. Alas, advances in twentieth century mathematics have shown that proving the consistency of a rich system in mathematics is typically impossible. However the geometers of the nineteenth century had already supplied us with something that, for practical purposes, is good enough.

In showing that the geometry of 5NONE is really the geometry of great circles on spheres, they provided a relative consistency proof. The idea is simple enough. In a three dimensional Euclidean space, we can recreate or simulate, the different geometry of 5NONE by constructing a sphere. Imagine that somehow we could generate a contradiction within the geometry of 5NONE. That would then mean that we could generate a contradiction within the geometry of great circles on spheres. And that would mean that there must be a contradiction recoverable within the geometry of three dimensional Euclidean spaces.

To get a more concrete sense of how this works, imagine that there is a way of deducing an inconsistency in the geometry of 5NONE. A geometer sits down and begins the steps of the construction that leads to a contradiction. Perhaps the geometer draws a straight line AB; and then a perpendicular to it; and so on. Now imagine a second geometer who works in Euclidean space. That geometer clones exactly everything the first geometer does, but now replaces the first geometer's straight line AB by a great circle through AB on some sphere. The two constructions will proceed analogously for the original geometer working the space of 5NONE and the clone geometer working in the Euclidean space.

geometer
Geometer working with straight lines in geometry of 5NONE.
geometer
Geometer working with great circles on spheres in Euclidean geometry.
Select any two points A and B. Select any two points A and B.
Connect them with a straight line. Connect them with a great circle.
...
...
...
...
...
...
Contradiction! Contradiction!

If the first geometer finds the construction leads to a contradicition, then so must the clone geometer. But that clone geometer is working fully within Euclidean geometry. That is, if the first geometer finds a contradiction in the geometry of 5NONE, then the second must find a contradiction in Euclidean geometry.

pantograph

So, if the geometry of 5NONE is inconsistent, then Euclidean geometry must be inconsistent. Or turning it around, if Euclidean geometry is consistent, then so must the geometry of 5NONE. Of course the big catch is that we cannot prove that Euclidean geometry is consistent. However we can take some comfort that millennia of investigations have failed to find an inconsistency in it. The relative consistency proof assures us that we are no worse off in the geometry of 5NONE.

The Geometry of 5MORE

What of the geometry of 5MORE? One might imagine that there are many distinct versions according to how many parallels can be drawn through a point not on the original straight line. One can quickly see, however, that there is only one possibility for this number. Imagine, for example, that the geometry allows two parallels AA' and BB' through the point but no more.

Five many 2

Then we can always bisect AA' and BB' with a third line CC'. Now AA' and BB' are parallel to the original line in the sense that they never intersect it, no matter how far they are projected. Since CC' is sandwiched between AA' and BB', the same must be true of it.

The basic idea generalizes. Any attempt to limit the maximum number of parallels allowed by 5MORE fails; we can always add one more. So the geometry of 5MORE is the geometry that arises when we may draw infinitely many parallels through the point not on the original line.

We could continue the exercise of discovering the geometry 5MORE through step by step inference. Since we've seen it done once for the geometry of 5NONE, let us just skip to the final result. It turns out that the geometry of 5MORE is the geometry of a negatively curved surface of constant curvature like a saddle or potato chip.

Pseudo sphere 1

In this geometry, lines can have infinite length, just as in familiar Euclidean geometry.

However there are differences that are analogous to those of the geometry of a spherical space:

In very small parts of the space, circles and triangles behave like Euclidean circles and triangles, near enough.

As the circles and triangles get larger, deviations from Euclidean behavior emerge. The circumference of circles becomes more than 2π times the radius; and the sum of the angles of a triangle become less than two right angles.

pseudo sphere 2

The perpendiculars to the equator on the surface of a sphere converge to a single point, the North Pole. On this surface of negative curvature, perpendiculars to a straight line diverge.

deviation negative 2

The Geometries Generalized To Three Dimensions of Space

So far, we have explored the geometries of 5NONE and 5MORE for the case of two dimensional spaces. We can also consider each in three dimensional spaces. The results we would arrive at are summarized in the table (duplicated in Euclid's Postulates and Some Non_Euclidean Alternatives).

Spherical Geometry
Positive curvature
Postulate 5NONE
Euclidean Geometry
Flat
Euclid's Postulate 5
Hyperbolic Geometry
Negative Curvature
Postulate 5MORE
Straight lines Finite length; connect
back onto themselves
Infinite length Infinite length
Sum of angles of a triangle More than 2 right angles 2 right angles Less than 2 right angles
Circumference of a circle Less than 2π times radius 2π times radius More than 2π times radius
Area of a circle Less than π(radius)2 π(radius)2 More than π(radius)2
Surface area of a sphere Less than 4π(radius)2 4π(radius)2 More than 4π(radius)2
Volume of a sphere Less than 4π/3(radius)3 4π/3(radius)3 More than 4π/3(radius)3

In very small regions of space, the three geometries are indistinguishable. For small triangles, the sum of the angles is very close to 2 right angles in both spherical and hyperbolic geometries.

Dropping the Embedding Space

What made visualizing these non-Euclidean geometries easy was that we embedded the non-Euclidean space in a higher dimensioned Euclidean space. That took an unfamiliar and even disquieting geometry and made it familiar. However in the end, we must dispense with these higher dimensional embedding spaces and simply take the new geometries as worthy geometries in their own right. There are three problems if we do not dispense with the embedding space.

One is technical. Sometimes the embedding cannot be implemented fully. The two dimensional negatively curved saddle shape can only be embedded into a three dimensional space in pieces; the full surface cannot be embedded.

Another is practical. The real gain is to our imagination. Imagine a three dimensional curved space that is curving into the fourth dimension of a four dimensional Euclidean space. Well--that's the problem. You cannot imagine it. So the practical gain to visualization is lost in this case. It is replace by a new problem: how are we to visualize the curving of the three dimensional space into a four dimensional space?

The final problem is the most serious. If our geometry turns out to be factually one of the curved geometries, then the supposition of a higher dimensioned Euclidean space is a falsehood and a potentially very misleading one. For, if we take it seriously, we end up believing that space is really Euclidean after all, but only in some higher dimension to which we have no access. If all we know is the three dimensions of space in which we measure, then we have no license to conjure up an otherwise inaccessible higher dimensioned Euclidean space for it to curve into. What makes us think such a higher dimensioned space exists?

outside the cosmos

What you should know

Copyright John D. Norton. December 28, 2006, February 28, 2007; February 2, 9, 14, September 22, 2008; February 3, March 1, 2010.