Einstein's General Theory of Relativity can be understood as saying that:

- Matter and energy distort space, and
- The distortions of space affect the motions of matter and energy.

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"?

**4.2 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:

**4.3 Ourland**

This bulging out into the 4th dimension that our universe seems to
do occurs on different scales: For example, micro-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
valleys, mountains and planes. On a smaller scale, the Earth's surface breaks into
individual rocks and clods of dirt.

It appears that much of our universe has a particular kind of curvature
that follows a *hyperbolic geometry*, yet this is not apparent on the scale
that we are accustomed to observing. The surface of the Earth's oceans curve around the globe, yet even as a few square miles of ocean appear flat,
a small chunk of hyperbolic geometry approximates Euclidean geometry. For example a triangle formed by three points of light in a classroom on Earth has an
angle sum so near 180 degrees that any difference is too small for us to notice (the sides curve inward by a very tiny amount).
Thus, it
usually appears we live in a Euclidean geometry; however,
triangles defined by a set of three distant stars can have angle sums measurably different from 180 degrees.

Even on a large scale, there are many commonalities between hyperbolic
geometry and Euclidean geometry, such as the base angles of isosceles triangles being congruent, but, on the large scale, there are also striking differences.

**4.4 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.