Mathematically analyzing the (n-1) dimensional surfaces of n-dimensional shapes is the basis for a branch of mathematics referred to as “topology”. More correctly, these “surfaces” are referred to as “topological manifolds”. This is unlike the book I mentioned in Post #8 (titled Geometry of Four Dimensions), for it did not deal with surfaces of the higher dimensions. It dealt with the shapes of the higher dimension.
We have no problem visualizing any one of us standing on the surface of a sphere, since the planet Earth is a spheroid. But let’s again think “outside the box” for a moment. Let’s consider all the large-matter in the Universe (stars, planets, comets, asteroids, etc.) and agree they too reside “on” the surface of this indescribable shape that we call the Universe. Its surface is covered with a transparent "fabric" we call space-time (thanks to Albert Einstein).
And if, in fact, the Universe is a 4-dimensional shape, we could describe this surface in terms of 3-dimensions. In other words, the shape of the Universe can be described in terms of its 3-dimensional topological manifold.
In order to imagine this surface of the Universe, we would have to “zoom outside” its entire shape and have a look. We can’t do this, but we can guess what it might look like and then test our guess using formal scientific method.
Let's stop a moment here and think about what choices this gives us further when selecting a geometry system for explaining the things around us. In the beginning, we had Euclid's plane geometry. That is probably what you learned in school, and certainly that is what I learned in high school.
For comparison reasons, let's consider the attempt of us to draw two parallel lines using the laws of Euclid. This would be easy to do on the the surface of a two-dimensional plane, and we could feel comfortable in visualizing these lines never intersecting or diverging from one another no matter how far we extended them in our defined space. The cartoon below makes a humorous use of the Euclidean postulate stating parallel lines never meet.
If the geometry of the Universe's surface were flat, this system of geometry and its associated mathematical calculations would work beatifully. We could enjoy the back-and-forth between the geometry and algebra working with a simple two-dimensional coordinate system and the rules of analytic geometry confined to that space. Life should be so simple!
But according to Einstein and many physical confirmations since his famous theories, we know life is not so simple as us being able to confine our thinking to the laws of Euclid. So now we look to some other form of geometry to help our descriptive process. We have Labatchevsky's hyperbolic/hyperspace geometry and we have Riemann's spherical/hyperspace geometry. Do either of these systems provide us with the math tools needed to describe the geometry of our Universe? Let's answer that just looking again at the concept of parallel lines.
Labatchevsky's hyperbolic/hyperspace geometry is based on shapes that translate down to the 3-dimensional level of a saddle-shape. In terms of the surface math, this is going to yield hyperbolic relationships. You may recall in math at some point in your education of plotting hyperbolas and being able to recognize hyperbolic equations.
The illustration shown to the right shows how "parallel" lines would look when superimposed on a hyperbolic surface. Notice that in such a system, the parallel lines are not the same distance apart forever, actually diverging at both ends. We say that such a surface has a negative curvature.
Einstein was practically going nuts between 1912 and 1915 trying to find some way to express his theories on gravitation mathematically. In a letter that Einstein wrote to his good friend and mathematician, Marcel Grossman, he expressed his frustrations.
Grossman researched the libraries for published math papers in an attempt to help his friend Einstein. He ran across a published lecture given by Riemann and passed it on to Einstein. The lecture had a unique way or relating the topology of surfaces in n-dimensional space. It made use of something called a tensor metric. Einstein went wild when he read the lecture papers. It was exactly what he needed in order to publish his General Theory of Relativity.
The work of Riemann is the manifold geometry commonly in use today for analyzing higher dimensional shapes and their surfaces. It was the geometry that Einstein used to explain the distortion of the time-space fabric in his explanation of gravity. Because it was based on a spherical model, as opposed to flat or hyperbolic models, it was perfect for explaining the distortional patterns of space caused by the massive spheroids consisting of planets and stars. However, as we shall see as we venture further, it was not the geometry of Riemann alone that completed the Einstein picture of what was happening in the geometry of the Universe. There were others that had extended the usefulness of the Riemann geometry.
So let's return to the title of this blog entry..."Believing in a World We Can't See". As I hope I have shown here, we are reaching out beyond the limits of our minds to visualize the shape of our Universe by examining only its surface. This precludes us from having to see the shape in our minds. Rather, we look at the geometric evidence of its surface geometry and test our physical theories accordingly. Isn't it wonderful how God has given us this ability? I leave the reader again with the quote from the Parker Manning's book I cited in Blog #8:
“Although it is doubtful whether we can ever picture to ourselves the figures of hyperspace in the sense that we can picture to ourselves the figures of ordinary space, we can nonetheless reason about them. And knowing that the validity of our geometry depends only on the logical accuracy of our reasoning, we can proceed to build upon any figure within our geometric system without waiting for a realization of it. Subsequently, we may in time acquire such facility in handling the geometrical proofs of the theorems. Thereby we are able to state precisely the forms and properties of the figures such that it is almost as if we could see them.”