It may seem that hyperbolic and elliptic geometry have no relevant meaning in the physical world, because everything here is purely, and simply, Euclidean. However, the surface of the Universe itself has a basis in the other types of geometry. In fact, there isn't any perfect Euclidean geometry in the physical world at all!
The geometry of the surface of space is not actually the shape of the Universe as a whole. Even if the geometry of the surface of the Universe was hyperbolic, the Universe could still be finite. This is because the surface of space isn't really a physical thing. In essence, it is what all the objects in the Universe "rest on". The gravitationally heavier objects, such as stars, make a dent in this surface and cause nearby objects to roll towards it. For example, the Sun creates a small depression in the center of the solar system, and all the planets roll toward it. However, the planets don't go hurtling into the Sun because each of them has a forward velocity. The closer you get to the Sun, the faster the forward velocity is needed to not roll into the Sun, because the depression is "steeper" in this area. Some objects are theorized to be so "heavy" on this surface, that they actually create a "hole" in space, with a steep well surrounding it. These objects are black holes. The hole in space could actually open a path to a parallel plane of space, an alternate universe!
With all these dents and holes, the Universe clearly isn't homogeneous or isotropic. Therefore, the crucial part is finding what the overall curvature of the Universe's surface is, i.e. what it would be without any of the inhomogeneities. As odd as it seems, the density of the Universe determines its geometry. It is known that there is a specified value in the Universe called the critical density of it. The critical density is the density of the Universe at which the curvature is zero and the surface of the Universe is Euclidean. If the the actual density is smaller than the critical density, the surface of the Universe is hyperbolic. If the density is higher than the critical density, the surface of space is elliptic.
In this image, the omega symbol represents the quotient of the actual density and the critical density of the Universe. For example, if the critical density equals the actual density, the omega would equal 1.
To calculate the density of the Universe, one must take into account not only density from the matter. One must also consider the mass contributed by energy. There is energy in empty space, called dark energy. The famous equation E=mc^2 shows that energy, E is equivalent to a certain amount of mass, m. Therefore, dark energy also adds density to the Universe. Regular matter and dark matter are attracted to each other gravitationally, but, oddly enough, dark energy seems to exert antigravity, or a force that pushes matter apart. The result of this is the creation of more space in the Universe, which in turn creates more dark energy, which accelerates the expansion even further. Despite the fact that the expansion of the Universe lowers the density of matter, the production of dark energy keeps the overall density constant throughout the lifetime of the Universe.
Using the Cosmic Background Radiation, one can easily infer how much matter existed at the beginning of the Universe. Using the speed at which different galaxies are receding from us, one can infer how much dark energy there is. Therefore, the densities of both dark energy and matter have been calculated. They do indeed add up to the critical density. To be exact, matter and dark matter makes up 27% of the actual density and dark energy makes up 73%. At the beginning of the Universe, the density of matter was minutely close to 100% of the critical density, and there was almost no dark energy. Since the amount of matter and dark matter hasn't changed for billions of years (since matter stopped spontaneously annihilating and turning into energy 380,000 years after the Big Bang) its density has steadily decreased. To varies with this change, the density of dark energy has increased. Therefore, the surface of the Universe is indeed flat. What does this tell us? It tells us that the Universe will never stop expanding. A value for the actual density that is the critical density or less dooms us to a fate of an infinitely expanding Universe. Only if the actual density is greater than the critical density will matter dominate. If this was the case, gravitational forces between matter would eventually cause the Universe to contract. Therefore, the Universe will continue to expand, at an ever increasing rate, until the Universe is infinitely large and infinitely old.
Overall, the fallacy of Euclid's parallel postulate has lead to to not only discover other geometries and theorems in mathematics, but has also determined the fate of our Universe.
This post and the others in this series were inspired by the following books: The Road to Reality by Roger Penrose Chapter 2, and Origins by Neil De Grasse Tyson and Donald Goldsmith Chapter 5.
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