Below are some math ideas I find fascinating.

Geometric constructions
During elementary school, I was fascinated by symmetric objects. I tried to construct platonic solids, such as dodecahedron and icosahedron, by folding paper. These required drawing precise constructions on paper. After reading books on this, I learned how to draw constructions, such as angle bisectors, and shapes, such as pentagons, using a straightedge and compass. Recently, I re-explored geometric constructions, specifically which lengths/angles are not constructible, after the guest speaker at our Math club, Dr. Rubinstein-Salzedo from the Euler Circle, discussed geometric constructions. I learned that we can use field theory to prove “unconstructability” of certain lengths, such as the cube root of 2, from the book “What is Mathematics?” by Courant and Robbins.

Dandelin’s proof (link)
I used to think that when a cone was cut with a plane, the resulting cross section would look like the outline of an egg, with one side more distorted than the other. I eventually learned that this cross-section is actually an ellipse, a theorem known since 200 BC. But I still couldn’t wrap my head around the symmetry of this cross-section. I was amazed to find a simple and elegant proof of this fact by G. P. Dandelin in the book “What is Mathematics?” by Courant and Robbins. Let’s say that we have a cone that is cut with a slightly slanted plane, dividing it into two parts and forming an elliptical cross section. The proof only requires two constructions: a sphere inscribed in the upper division of the cone tangent to the plane, and another bigger sphere inscribed in the lower division of the cone tangent to the plane. The proof cleverly uses these two spheres to equate the sum of the distances between an arbitrary point on the ellipse and its foci to another length that is known to be constant, and the proof shows that the foci of the ellipse are, in fact, the points of tangency of the spheres with the plane.
I appreciate the simplicity of this proof. I didn’t have to sift through much notation to understand it; I just had to read a picture and a few statements. In addition, since I comprehend information through pictures more easily, I could grasp the proof with little difficulty. I couldn’t help but wonder: how could a mathematician conceive of adding two spheres within a cone? After the fact, it seems easy, but it requires ingenuity to create a simple geometric proof, and that is what I admire in this proof.

Infinite Hotel Paradox
A few years ago, when I attended the Stanford Math Circle, I learned about a fascinating topic: infinite sets. I discovered that infinite sets have a way of breaking axioms and giving non-intuitive results, and one example of that is the problem of Hilbert’s Infinite Hotel Paradox. The Hilbert’s Infinite Hotel problem involves a hotel with an infinite number of rooms and which is “fully booked” at the moment. The problem asks how to rearrange the hotel occupants to accommodate (a) 1 incoming guest, (b) an infinite queue of incoming guests, (c) an infinite number of infinite queues of guests, and (d) an infinite number of an infinite number of infinite queues of guests. Surprisingly, all of these questions have solutions, and those solutions show that it is definitely possible to “fit” an n-dimensional countably infinite array of guests into a 1-dimensional countably infinite set of rooms without having to kick out any guests. This seems to go against intuition. The key insight is that a countably infinite set can be partitioned into infinitely many countably infinite sets.

Theory of Impartial games and Boolean operations
In the Euler Circle this past summer (2021), I learned about Combinatorial Game Theory (CGT). Combinatorial games are 2-player games where each player has complete control over his/her decisions (as opposed to rolling dice) and each player can know all possible future states of the game (including the moves that the other player can make). A branch of CGT that I found interesting was the theory of impartial games. Impartial games are games where both players have the same moves available to them (eg. Nim, where the two players can work with the same pieces). In an impartial game, it can be precisely determined whether the first player wins or the second player wins, assuming that both players play optimally.
The first interesting connection that I learned about was the connection between Nim and the Boolean XOR operation. A multiple-stack Nim game can be shown to be equivalent to a single-stack Nim game using the XOR operation. I also learned that the analysis of any impartial game is derived from the analysis of Nim. This neat connection is encapsulated in the Sprague-Grundy Theorem.
Another connection I learned was that Hamming codes – error-correcting codes with binary strings where, if a single bit is corrupted, it is possible to repair it – can be proven to be correct using impartial game theory and Nim.