HPS 0410 | Einstein for Everyone |

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John D. Norton

Department of History and Philosophy of Science

University of Pittsburgh

- Unfamiliar Geometries Become Familiar
- The New Geometry of 5
^{NONE}is Spherical Geometry - Circles and Triangles
- Corrections to the Other Postulates
- Is the Geometry of 5
^{NONE}Consistent? - The Geometry of 5
^{MORE} - What you should know

Euclid's Postulates and Some Non-Euclidean Alternatives

In the last chapter, we explored the
geometry induced by the postulate 5^{NONE} 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'''.

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
5^{NONE} 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 5^{NONE} and the geometry of the other
postulate 5^{MORE} 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 geometry of 5^{NONE} 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
5^{NONE} 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.

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".

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.

In short, the new geometry of 5^{NONE} 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.

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.

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.

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.

Now that we have identified our geometry of 5^{NONE}
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:

Consider the geometry of 5^{NONE}; that is
the geometry that is deducible from the the fifth postulate
5^{NONE} 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. |
By contradiction, I mean "A and not A," for A some sentence. So if one's theory allows contradictions to be deduced, the theorist has a very serious problem. It may mean that someone working in dynamics can infer that a system both conserves energy ("A") and does not conserve energy ("not-A"). Which ought the theorist to believe?! |

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 5^{NONE} 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 5^{NONE} by constructing a sphere. Imagine that somehow we
could generate a contradiction within the geometry of 5^{NONE}. 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
5^{NONE}. 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 5^{NONE} and the clone geometer working in the Euclidean
space.

Geometer working with straight lines in geometry of 5 ^{NONE}. |
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 5^{NONE}, then the second must find a
contradiction in Euclidean geometry.

So, if the geometry of 5^{NONE} is inconsistent, then
Euclidean geometry must be inconsistent. Or turning it around, if Euclidean geometry is consistent, then so must the
geometry of 5^{NONE}. 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 5^{NONE}.

What of the geometry of 5^{MORE}? 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.

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 5^{MORE} fails; we can always add one
more. So the geometry of 5^{MORE} 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
5^{MORE} through step by step inference. Since we've seen it done once
for the geometry of 5^{NONE}, let us just skip to the final result. It
turns out that the geometry of 5^{MORE} is the geometry of a negatively curved surface of constant curvature like a
saddle or potato chip.

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.

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.

- How the geometries of 5
^{NONE}and 5^{MORE}are realized in surfaces of constant positive and negative curvature. - How each of the these geometries differs in its treatments of ordinary figures from Euclidean geometry.
- How the consistency of the non-Euclidean geometries is assured through a relative consistency proof.

Copyright John D. Norton. December 28, 2006, February 28, 2007; February 2, 9, 14, September 22, 2008; February 3, March 1, 2010; January 1, 2013.