There is a fiction abroad in the world that geometry is a branch of mathematics. In fact, it is one of the three foundations of physics (the other two being time and matter-energy). It is said that the Egyptians discovered geometry in constructing their buildings and pyramids, that through trial and error they learned how to set corners at right angles and to incline slopes to their prescrip-tions. In the process they developed empirical rules for the relationships among spatial measures, rules that the Greeks later formally transformed into what we now call geometry (literally, the measure of the earth).
That mathematicians managed to appropriate geometry for themselves says something not only about mathematicians but also about the nature of the universe. For the mathematicians reduced geometry to a series of axioms and postulates, from which all else is derived. Since in fact geome-try does describe relationships among the measured properties of space, that such properties may be placed in axiomatic and postulate form means that at its foundations, the universe is governed by number and reason.
This article is not a comprehensive treatise on geometry. It is intended rather to be a kind of
introduction to those aspects of geometry which the physicist finds basic to describing the struc-ture of the universe.
The Geometry of Space.
Point and Distance.
Like time and matter-energy, it is not possible to define space in terms of simpler physical entities. Space simply exists. It can be defined only in terms of its properties. Those properties are what we call geometry. Two of these properties are the concept of point and the shortest distance between two points.
In order to specify points and distances, it is necessary to use material objects. Nevertheless it is taken for granted that space and its geometry exist independently of the presence of matter. We suppose that a completely empty space is possible even though its emptiness precludes the detection of any property of the space. Furthermore the properties of space may be described in the absence of the passage of time, although it is conceivable that these properties change in time.
Points are identified by building a rigid grid throughout space, a mental construct which allows us the freedom to imagine the grid to be composed of infinitely thin wires woven into an infinitesi-mally small mesh. Distances are measured using rigid rods. Exactly what these concepts imply is a matter for philosophy. We accept them as defined from experience rather than from a logical deduction from assumptions a priori.
Given two points in space, we define a geodesic as the line of shortest length spanning the space
between the two points. Experimentally, we mean that we lay tiny measuring rods end-to-end
between the points and choose that line along which the fewest number of rods lies.
The Dimension of Space.
The dimensionality of the space is the number n, where n+1 is the maximum number of points which may be mutually equidistant. For example, in the trivial case of a one-dimensional space, only two points can be mutually equidistant. For a two-dimensional space, no more than three points can be mutually equidistant. (These points lie at the vertices of an equilateral triangle. A fourth point cannot be located in the two-dimensional space so that all four points are separated by the same distance.) For a three-dimensional space, no more than four points can be mutually equidistant. (These points lie at the vertices of a regular tetrahedron.)
Physical space is three-dimensional. Why space is so limited we do not know. This property does
not seem to derive from any other character of space or from other laws of physics.
The geometry of space is embodied in the set of postulates which govern the relationships among geometric figures constructed from points and lines and curves lying the space. From these postulates, the properties of space may be derived under logical and mathematically rigorous methods. That nature yields measures in agreement with these predictions is both the faith and foundation of all science.
For all practical purposes, it appears that the geometry of physical space is Euclidean in the neighborhood of the earth, though the general theory of relativity predicts that there are immeasurably small departures from the Euclidean due to the massiveness of the earth.
There are many ways in which we may define what is meant by Euclidean geometry. We choose here to define it in terms of the properties of a circle. First, a circle is defined to be that curve such that every point on the curve is equidistant from a single point, called the center of the circle. The radius of the circle is the length of the geodesic drawn from the center to any point on the circle.
A necessary and sufficient condition that a space be Euclidean is that the ratio of the circumference of any circle to its radius be the same value everywhere in the space. It then follows that this value must be 2(pi) = 6.2832... It is not possible to construct a geometry in which this ratio is some constant other than 2(pi).
Figure 1. Circle drawn on Spherical Surface.
|If a space is found in which the ratio for a
given circle is some number other than
2(pi), say N, then other circles drawn in
the space can always be found for which
the ratio is not N.
As an example from the world of two dimensions, we consider the surface of a sphere, as shown in Fig. 1. This surface is a curved space embedded in a three-dimensional Euclidean space. Consider a point C on the surface. From this point we construct a curve every point of which is the same distance r from C. Here r is measured out from C, along the surface of the sphere. That is, a line drawn on the surface of the sphere, from C to the curve is a line whose length is r.
This curve is a circle which lies both on the sphere and in an imaginary Euclidean plane passing through the surface. We call the circumference of the circle s. In a sense, the circle has two centers: the point C, the true center lying on the spherical surface, and a point C', an imaginary mathematical center lying in the Euclidean plane. (We say "imaginary" because, since reality is limited to the two-dimensional surface of the sphere, the Euclidean space inside and outside that surface does not exist except in our imaginations.) The radius of the circle measured from C' is r'. Clearly r' is less than r because, from our three-dimensional view, r' is "straight", while r, lying on the spherical surface, is "curved."
The ratio s/r' is 2(pi) because r' is the Euclidean radius. But the ratio s/r is less than 2(pi) because r is greater than r'. Furthermore, the larger the true radius r, the less the ratio s/r. If the radius r is large enough that the circle divides the sphere into two equal halves (that is, if the circle is an equator of the sphere), then the ratio s/r equals 4. If the true radius r of the circle increases further, the value of s decreases until it vanishes at a point on the sphere opposite C.
There are other properties of this spherical surface which are not the properties of a flat surface. Consider, for example, constructing a large triangle on the surface. A triangle is built first by picking three points, A, B, and C, which are the corners or vertices of the triangle. Then shortest distances, or geodesics, between these three points are constructed. If the surface were a plane instead of a spherical surface, we would call these geodesics "straight" lines. But on the surface of the sphere, as we view it from outside the sphere in our three-dimensional world, these geodesics are "curved." An example is shown in Fig. 2. The angles at vertices B and C are
Figure 2. A spherical triangle
|right angles, and the angle at vertex A is less
than a right angle. Thus we have constructed
a triangle on the surface of the sphere such
that it contains two right angles and the sum of
its angles is greater than 180 by the amount
of the angle at vertex A. It may be remember-ed from high school geometry that a plane
triangle can contain only one right angle and
that the sum of the angles is 180.
There is one other significant difference be-tween the natures of spherical and plane surfaces. The area of a plane surface is infinite, unless one imposes a boundary, like a wall, to confine the region being considered. On the other hand, the area of a spherical surface is finite even though there is no boundary to make it so.
A Spherical Two-Dimensional Universe.
While a two-dimensional universe does not exist, we can imagine what it would be like, at least in terms of the measurement of space, if the space had the geometry of a spherical surface. As creatures on this surface, we would be like shadows, with only a two-dimensional existence.
First, if we make measures over a very small regions of our universe, we find the geometry to be the same as Euclidean geometry. The ratio of the circumference of a circle to its radius is 2(pi), the sum of the angles of a triangle is 180, and right triangles can have only one right angle.
But if we now stretch our measurements to cover a large region of our space, we would find that for very large circles, the ratio of the circumference to radius of a circle is less than 2(pi) and that the larger the circle the smaller the ratio. Stretching (two-dimensional) strings between fixed points in the universe would define shortest distances between those points. Connecting three points with shortest distances would produce triangles who angular sum is greater than 180.
If we mark off small approximately rectangular sectors of equal area in our universe, we would find we can fit only a finite number of them in the universe, even though we encounter no wall or boundary to limit our movement. We would discover as we proceed laying down sectors that they cannot be made exactly rectangular without leaving gaps between the sectors. Furthermore the shapes of the sectors cannot all be the same.
Next suppose a kind of two-dimensional light beam can exist in our universe and that this light is used to see objects at a distance. We require this light to travel along geodesics and that its speed be the same as in our real universe, namely about 186,000 miles per second. Light spreading out
Figure 3. The Divergence and Convergence of Light from a Point Source.
|from a point source would travel through
our universe and meet again at a point on
the opposite side of our universe and con-tinue through this point back to the original
source. Figure 3 is an illustration of the
divergence of light from a source A and its
convergence onto a second point B on the
"opposite" side of the universe.
A shadow eye near the point A and looking at this source would see a nearby point of light, since light here diverges from A onto the eye. But if this eye is placed far from A and near B, light from A converges on the eye, which would see only a line of very dim light spread out across the flat horizon of its universe.
If the eye near B is next turned around to look at B, it would see light diverging from
B, this light having originated at A and having traveled toward B on the side of the universe facing away from the reader. That is, the eye would see a point of light at B.
Finally, imagine that the Euclidean radius of this spherical two-dimensional surface universe
begins to increase in time. (It should be remembered, however, that we shadow creatures living in
this universe cannot be aware of anything off the surface -- that is, outside our universe. For us,
the radius of our spherical universe exists only mathematically.) The universe would then be
expanding in the geometric sense that its surface area increases and its geometry becomes locally
more Euclidean. If we had filled the surface with small, approximately rectangular sectors, empty
spaces would appear and grow between the sectors, and the distance between points A and B
would increase. It is conceivable that the size of our universe increases so rapidly that the
distance between "opposite" points (like A and B) grows so rapidly that light starting out from A
never reaches B, in which case we would not see light circumnavigating the universe, and B
would not appear to be a source of light.
A Spherical Three-Dimensional Universe.
We live in a spatially three-dimensional universe. The geometry of this universe appears to be Euclidean (that is, flat). However, our measurements are limited to the earth where, as far as we can tell, the sums of angles of triangles (after accounting for the earth's roundness) is 180 and the ratio of the circumference of a circle to its radius is 2(pi). What would we expect the universe to be like if its overall geometry were that of the surface of a sphere? For even though Euclidean geometry is the simplest of geometries, it is not necessarily the geometry that describes the entire universe.
First, it is important to understand that our universe has only three spatial dimensions. To say that this universe is "curved" does not mean that there is a fourth spatial dimension into which our three dimensions can be bent in the same way we bend the two-dimensional surface of a sphere into the three dimensions of our real space. It means instead that the geometry of our space is not Euclidean. This point is worth elaboration.
Imagine this experiment. We nail three pegs into fixed positions on rigid structures. Perhaps one sits on Mt. Rainier, one sits in Death Valley, and the third is at the Denver airport. We clear the space between these three pegs of all earth so that we can stretch weightless strings freely, with-out obstruction, between the pegs. (The strings have to be weightless so that they don't sag under their own weight.) We next make a careful measure of the angles between these strings at the pegs. If the sum of the angles is 180 (and 180 for all other such triangles me might con-struct), our space is flat. If the sum is not 180, our space is curved.
If instead of strings (which actually would sag in spite of our best efforts to stretch them tight), we use laser beams in evacuated tubes, we could determine the same result. (Light is also affected by gravity and the motion of the earth, and its paths are not exactly along the shortest distances between the pegs. But the effect is probably too minuscule to be detected by spatial measures on the earth.) Either the sum is 180 (flat space) or it is not (curved space).
All our experience assures us that the geometry of space in the neighborhood of the earth, indeed
in the neighborhood of our galaxy and nearby galaxies is flat. We come to this conclusion from
the evidence of light received from distant sources. We cannot of course make measurements of
the angles of triangles because at least one of the angles would have to be off the earth. But we
can infer geometric properties from the light from distant objects.
A Bit of Optics.
Figure 4a. Light Diverges from a Point Source and Enters the Eye.
Figure 4b. The Angle of Divergence of Enter-ing Light is Small for Distant Sources
|Figure 4a shows light emanating from a point source. This light diverges -- that is, the rays of light spread out. Some of this light enters the eye, whose lenses converge the light onto the retina where (if vision is perfect) the rays meet at a point. The brain then "sees" the point source of light. Figure 4b shows the same situation from a point source that is farther from the eye than the source in Fig. 4a. The only difference in the two situations, as far as the eye is concerned, is that the angle between the incoming rays from the nearer source is greater -- that is more divergent -- than light from the farther source. The brain uses this information unconsciously to give an indication of whether the source is near or far. If the source of light is infinitely remote from the eye, the rays of light enter the eye along parallel lines. That is, the angle of|
divergence is zero.
If, as in Fig 4c, we place a wavy piece of glass between the source and eye, the image of the point is distorted and unfocused because the rays of light, as they pass through the glass, get bent every which way, and they no longer enter the eye along the straight lines connecting the source with
Figure 4c. A Wavy Piece of Glass Distorts the Image of the Point Source.
|the eye. (It is through this phenomenon
that corrective lenses compensate for
whatever imperfections lie in the optical
structure of the eye.)
If we ignore the effects of gravity, we can say that in a Euclidean space, rays of light travel along the straight lines lying between the source and the observer. However, in a curved space, light travels along the shortest paths connecting the source with the observer.
Let us next return to Fig. 3, showing how light from a point source on a spherical surface and following geodesics on that surface first diverges from the source (at A) and then converges onto a point on the opposite side of the sphere (at B). When the eye is close to the source, it sees light diverging from the source just as it would in a flat space. But if the eye is placed halfway between points A and B, light coming from A strikes the eye in parallel lines, just as it would were the source infinitely remote from the eye. If the eye is in the lower position in the figure, far from A, light entering the eye is converging (as though the light had passed through a magnifying glass) and the image is blurred.
When we view the universe through our most powerful telescopes, we see images of distant galaxies undistorted. There is no evidence that light from these sources has been "bent" by a "curvature" of space. The message this light sends us is that the spatial geometry of the universe out to the distances we can detect is Euclidean.
Two notes of caution:
(1) The geometry of space is warped by the presence of massive objects. Light passing near the surface of the sun, for example, is bent in toward the sun, a phenomenon that has been observed during total eclipses of the sun.
(2) Light falls in a gravitational field just as do material objects. However, light travels so rapidly, is passes by the attracting source before it can fall very far, and so its fall is not easily observed.
Thus light passing by the surface of the sun is bent toward the sun for two reasons -- because of the warping of space and because of its inward. These phenomena are local and do not affect the overall geometry of the universe.