# Collection of Desmos Graphs – Part 5

So I broke the streak that I had since 2019 of posting these before the end of the year. I never thought the day would come, but due to the struggles of employment and graduate level study of two difficult disciplines, my Desmos use dramatically decreased in 2023. Naturally, there are still plenty of graphs to choose from. I’m going to exclude graphs from the new Geometry tool, because I made most of them while working as a contractor for Desmos. They’re listed here. 2023 saw the release of Beta 3D, Desmos’ long-awaited 3D graphing calculator.

## Origami Crane

This was my submission to the 2022 Desmos Global Math Art Contest (yeah, the deadline for the 2022 contest was in 2023). It’s likely the last time I’ll put enough effort into the contest to be competitive, because I was one of the winners in my age category. The crane itself is a just a series of rotations around a bunch of different axes applied to a set of polygons. It starts with the net (unfolded outline of the crane) and folds as the animation slider continues. Here’s a video in case the graph is a bit laggy.

The background uses a parallax that shifts as the graph rotates, and the design on the ground plane is based on hyperbolic coordinates (related to research I was doing on tilings in the PoincarÃ© disk).

## Convergence of Newton’s Method

Newton’s method is a well-known iterative technique for finding the zeros of a function. The associated fractal, created by assigning a color to each point in $\mathbb{C}$ based on which zero it eventually converges to (if it converges). This graph is related to that, but not quite the same. Here, each point is colored by how many iterations of Newton’s method it takes to get within a certain distance of a zero (shown as white points). Interestingly, the overall pattern is visually similar to Newton’s fractal itself.

I made this after watching 3Blue1Brown’s videos on Holomorphic Dynamics, which are well worth a watch.

## Indra’s Pearls Loxodromic Circle Pairing

During my research, I read an amazing book called Indra’s Pearls, an informal visual text about MÃ¶bius transformations in complex space. Loxodromic transformations are complex maps that trace out a certain type of spiral when fractionally applied to a fixed number $z$. These spirals trace “loxodromes” when lifted to a sphere via inverse stereographic projection, thus the name. A loxodrome is a curve on a sphere that intersects all circles formed by intersecting parallel planes with the sphere at the same angle. For example, a loxodrome on the Earth would intersect all longitudes at the same angle.

We can use a certain type of loxodromic map to “pair circles”. We say a map pairs two circles if it maps the exterior of one circle to the interior of the other (or vice versa). We can iterate this process to generate nice diagrams of circles of infinitely decreasing size. Such maps form a special group of orientation-preserving isometries of hyperbolic space called Schottky groups.

## Schwarz-Christoffel Map and Inverse for Regular Polygons

I will not attempt to explain the mathematical content of this graph in this post. This map was at the center of my research on hyperbolic space. But here’s the gist:

A conformal map $\mathbb{C}\to\mathbb{C}$ preserves angles. That is, if two differentiable curves in the complex plane intersect at some angle, their images under the map intersect at the same angle. The Riemann Mapping Theorem guarantees a conformal bijection between any two open subsets of $\mathbb{C}$. The Schwarz-Christoffel map is a construction of this conformal bijection for the special case of polygons and the open unit disk. The linked Wikipedia article incorrectly states that the map is valid only for simple polygons. Really, it works for open polygons as well, or divisions of the plane into two regions using a finite number of segments, two of which are infinite rays.

This graph explicitly calculates the Schwarz-Christoffel map and its inverse for regular polygons, creating a conformal map from one regular polygon to another.