Non-Euclid's Elements

One video that I watched on my favorite math channel, 3Blue1Brown, was about one of the most influential math textbooks of all time, Euclid’s Elements. Euclid’s Elements is described as the first large-scale deductive treatment of mathematics, consisting of a large number of geometric theorems; which Euclid called propositions. However, Euclid did a lot more than just prove a ton of theorems, he established an entire universe of mathematics. This is now referred to as Euclidean Geometry, which is built entirely on Euclid’s five postulates, statements which are assumed to be true without proof.

 

After watching this video, I was inspired to create a project in Desmos, which would be an interactive version of Euclid’s Elements. While I was in the middle of this project, one Youtube video, called “The Beautiful Story of Non-Euclidean Geometry”, caught my eye. It explained the significance of the fifth postulate in Euclid’s Elements, which states that if a straight line intersects two other straight lines, and the sum of the interior angles on one side of the transversal is less than two right angles (180°), then the two lines being intersected will eventually intersect themselves. This proposition was significantly more complicated than others, and mathematicians wondered whether or not it deserved to be called a postulate. To test this, they created a geometry where the first four postulates held but the fifth did not. This is called hyperbolic geometry, and upon learning about it, I felt as if I had discovered a whole new universe.

 

After learning about hyperbolic geometry and its connection to the Desmos project I was working on, I had a brilliant idea: I was going to make my interactive version of Euclid’s Elements in hyperbolic geometry. After all, the first few propositions in Euclid’s Elements never even mention the fifth postulate, so mathematically, this would be possible. The way I planned to go about graphing hyperbolic geometry was to use the Poincare Disk Model of hyperbolic geometry. A much more intuitive cousin of hyperbolic geometry is spherical geometry, which is geometry on the surface of a sphere–let’s say the Earth. On the surface of the Earth, it’s possible to walk in one direction, make a right turn, make another right turn, and end up back where you started. What you’ve done is that you’ve constructed a triangle with two right angles, which demonstrates the property of spherical geometry that when looked at from a Euclidean perspective, triangles seem to bulge outward. On the other hand, hyperbolic geometry can be interpreted as geometry on the surface of a curve that constantly curves away from itself, like a saddle shape. When looked at from a Euclidean perspective, hyperbolic triangles seem to bulge inwards. One way that mathematicians visualize hyperbolic geometry is by projecting it into two dimensions, specifically the entire infinitely expansive space would get projected into the unit disk.​ The result is called the Poincare Disk Model.