# The Shape of Space

2.1  Curved Space
Einstein's General Theory of Relativity can be understood as saying that:
1. Matter and energy distort space, and
2. The distortions of space affect the motions of matter and energy.
Many present-day cosmologists feel that, we live in a three dimensional universe that is curved into the 4th dimension.   No one can point to the 4th dimension, yet it is all around us.  The 4th dimension is a direction different from all the normal directions.  It is neither up nor down, left nor right, in nor out.

Some people say that Time is the 4th dimension.  This is, in a sense, true.  However, time is not the "different direction" of which I was speaking.  If we want to count time as a dimension, then we should say that we live in a four dimensional space-time that is curved into the 5th dimension!  So where is this "different direction"?

2.2 Flatland
It is very difficult to visualize four dimensions of space.   In 1884, a Victorian schoolmaster, Edwin Abbott, published a book about an imaginary two-dimensional world called Flatland.  Flatland is inhabited by creatures that slide about.  We might think of them as being like coins on a tabletop.  Alternatively, we could think of them as colored patterns in a soap film.  The story focus on the adventures of A Square, Flatland's most famous citizen, and his path to an understanding of the third dimension.  By thinking about A Square's difficulties in understanding the third dimension, we become better able to deal with our own problems with the fourth dimension.   Early in the story, A Square and his wife are comfortably sealed up in the safety of their home, when suddenly a voice out of nowhere speaks to them.  And then, a moment later, a circle appears in the confines of their tightly locked house.  It is A Sphere, come to teach A Square about the third dimension:

"I am not a plane Figure, but a Solid. You call me a Circle; but in reality I am not a Circle, but an indefinite number of Circles, of size varying from a Point to a Circle of thirteen inches in diameter, one placed on the top of the other. When I cut through your plane as I am now doing, I make in your plane a section which you, very rightly, call a Circle. For even a Sphere - which is my proper name in my own country - if he manifest himself at all to an inhabitant of Flatland - must needs manifest himself as a Circle." [Abbott-1884]
Flatland does not have to be flat.  The pair of flatlanders shown above are in a two-dimensional universe that is curved into the 3rd dimension.  This universe would not appear curved to the Flatlanders since all objects in it (including their own bodies) curve with the two-dimensional surface.

2.3 Ourland
This bulging out into the 4th dimension that our universe seems to do occurs on three different scales:  Small-scale curvature results from each subatomic particle.  Medium-scale curvature is associated with gravitational attraction of stars, black holes, and galaxies.  Large-scale curvature is the overall shape of space, and is the cumulative result of the totality of all the mater and energy in space.  To make clear what is meant by different "scales of curvature", consider the following:  On the large-scale, we say that the surface of the Earth is curved into the shape of a sphere that bulges a bit at the equator.  On the medium, human-sized scale, we notice that the Earth's surface is covered with hills and valleys.  And on the small scale, the Earth's surface breaks into individual rocks and clods of dirt.

On the large-scale, much of space has a particular kind of curvature called Hyperbolic Space.   Hyperbolic Space has its own geometry, Hyperbolic Geometry,  which has many similarities to our usual Euclidean Geometry, but also has some striking differences.  In a sense, Euclidean Geometry can be thought of as a subset of Hyperbolic Geometry since a small chunk of Hyperbolic Space has small curvature, and  very closely matches Euclidean Space.  All geometric theorems of Euclidean Geometry, therefore, appear to be true in a small chunk of Hyperbolic Space.

2.4  Mercury's Orbit
The distance from the Earth to the Sun is much too small for our most sensitive instruments to measure any of the differences between Euclidean and Hyperbolic Geometry  caused by the large-scale curvature of space.  However,  our Sun causes some medium-scale curvature that - thanks to planet Mercury - we are able to measure.  Mercury is the closest planet to the Sun.   It is in a much higher gravitational field than is the Earth, and therefore, space is significantly more curved in its vicinity.   Mercury is close enough to us so that, with telescopes, we can make accurate measurements of its motion.  Mercury's orbit about the Sun is slightly more accurately predicted  when Hyperbolic Geometry is used in place of  Euclidean Geometry.

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