Colton Bukowsky 

Footnote 18 

Euclid's geometry of the universe is the geometry most people are familiar with. In terms of the x,y, and z axis, there are three spatial dimensions that exists in our universe which objects can exist in. In Euclid's geometry, dimensions can be thought of in the following way: one dimension extends along a strait line into infinity and each dimension is at a right angle to each other dimension. Our universe in this geometry can be modeled as expanding from the point of the Big Bang, resembling a spherical structure. The formula for voume of these three dimensions would be . Up until right after the beginning of the 20th century, this has been the trend of thought concerning the nature of the universe; that the space contained within it followed the laws of Euclid's geometry.  

When Einstein's Theory of General Relativity was released, the implications were astounding. His theory says that space can be curved by objects of mass, and that the effect of this curvature is gravity. This ability to bend means that space itself may have a different structure than the popular Euclidean geometry. Overall, there are three  possibile shapes of the universe, all of which are dictated by the amount of gravity, or curvature, of the universe. The most difficult part of the curvature of space to grasp is that our familiar three dimensions must curve into a fourth spatial dimension. Just like a sheet of paper can only be curved into the a third dimension, three curved dimensions must have a curvature into a fourth dimension, but this does not mean objects in three dimensional space must have a fourth dimension. The different universal structures are best examined by considering an analogous case of each in 2, 3, and then 4 dimensions. 

The first is that of Euclid, which is a universe which has the exact amount of mass and gravity that there is no overall curvature of the univsere. A universe of this type would be expanding, but slowing down into infinity, such that there is no limit to the acutal volume of the universe. The rate of the expansioin of space would always approch, but never reach zero. Because there is no overal curvature in this universe, the three dimensions we experience are the familiar Euclidean dimensions. This would be analogous to a flat plane as below, or simply an ever expanding sphere within which our universe is contained. 

The next possible universe is a closed universe. This simply means that the universe does not expand into infinity becuase there is sufficient matter and gravity to reverse the expansion of the universe. There is so much curvature in this universe that the dimensions actually curve back into themselves, much like a circle or sphere. This is a universe with positive curvature, and the dimensions of this universe can be thought of in the following ways: 

Imagine a 1 dimensional universe that curves into two dimensions with positive curvature. This would be a familiar circle. This makes sense because in terms of units, we measure the circumference of a circle with meters- only one spatial dimension. The maximum space in this dimension is the circumference, which can be expressed as a function of its radius R. This can be found using calculus to calculate the arc length of the function of a circle. 

The formula for arc length is the integral of the square of 1 plus the first derivative squared from a to b 

For a circle centered at the origin, the integral should be evaluated from -R to R for the top half of the circle. To simplify the integral further, the arc length is evaluated from zero to R and multiplied by two, assuming symetry about x=0. 

To account for the bottom half of the arc, we multiply by two 

This shows that for a curved universe of one dimension, the total space would be the circumfrence of the circle.  

A two dimensional space with positive curvature would curve into a third Euclidean spatial dimension. The space of this universe would be equivalent to the surface area of a sphere. 

The formula for the surface area of a surface is the integral of the square root of 1 plus the partial derivatives of the function squared. 

The formula for the top half of a sphere is  

Polar coordinates were used to simplify the integrals. 

To account for the bottom half of the sphere, we again must multiply by two. 

If a two dimensional universe existed with positive curvature, the space in those two dimensions would be limited to the surface area of a sphere, which curves into the third dimensions of Euclidean definition. In order to help visualize what a closed two dimensioinal universe looks like, imagine a two dimensional creature, like a circle, living on the sphere below. It could travel what it would think is a strait line into infinity, yet end up at the same point. 

It can be seen that there is a space which comprises two dimensions, but not as defined according to Euclid. Angles for a shape in this space would be less than what they would be in Euclid's space; a triangle would have less than 180 degrees in its angles. 

Further, we can extend the analogy of the space on a circle and the space on a sphere to the space on a hypersphere. This would be the equivalent of a three dimensional universe which curves into a fourth Euclidean dimension.We can extend the formula for surface space again for four dimensions by adding the derivative with respect to z squared as part of a triple integral. Spherical coordinates are used to make the integrals simpler. 

The equation for the top half of a hypersphere is: 

A table of Integration was used to integrate with respect to rho 

Again, the result is multiplied by two to account for the bottom half of the hypersphere, and we get the formula for the surface space, or volume of a hypersphere. 

We can look at a level curves of the original function to get an idea of what this hypersphere might look like. 

The level surfaces for the hypersphere indicate that further from the center of the hypercube, the volume and radius of the three dimensional spheres which form its surface decrease to zero. It is difficult to understand the structure of a hypersphere, yet mathematically it is easy to work with. 

The last possibility is a universe with less gravity and curvature than required to stop the universe's expansion. This is analogous to a two dimensional universe upon a saddle point plane, which extends into forever. The volume of this kind of universe would be infinite, so there is no formula for it. In two dimensions this negative curvature follows the shape below 

Having a hyperbolic structure of space means that our three dimensions still curve into a forth Euclidean dimension, but without enough curvature to eventually curve back into themselves, as in the hyperspherical geometry. Again, we uselevel surfaces to help visualize what three dimensional space embedded in a fourth euclidean dimension might look. 

The level surfaces maintain their saddle point shape, but move down the z-axis. The hyperbolic geometry is even more difficult to grasp than the hyperspherical! The purpose of this study was to demonstrate the implications of Einstein's discovery that space has the ability to curve itself. I each case, there is a different fate of the univserse. Perhaps the most interesting is the closed universe with hyperspherical geometry, because just like the 2-d creature on the surface of a sphere, theoretically, we could travel as far as possible in a straight line, and come right back to the same spot and the distance travelled would be equal to 2Rπ in any direction! This is because we would be travelling around a circle on the hypersphere, but without knowing it.  

Sources: 

Thanks Ms. Brewer for all your help! 

"The Geometry of the Universe", http://csep10.phys.utk.edu/astr162/lect/cosmology/geometry.html  

Gammow and Stannard, "The New World of Mr. Tompkins",Cambridge UK, Cabridge University Press, 2001.