This isn't a novel idea, but I was looking for an iterative method for easing between two values. In this post, I describe and analyze the method of iterating a weighted arithmetic mean. I'm going to examine the effect of changing the weight factor and the rate at which the limiting value is approached as iterations increase. The end goal of this is to provide an alternative easing method to the standard parametric methods that use easing functions such as smoothstep or logistics. This allows t...

Read More
# Math

# Collection of Desmos Graphs – Part 4

In a Desmos Global Math Art Contest-induced fit of madness, I almost forgot to do the annual Desmos graph showcase. But it's not 2023 yet (hours away!) so nobody can say I didn't do it. From harmonographs and obscure coordinate projections to artistic animations, this has probably been my most active year with Desmos, which is really saying something, since it has been my primary hobby for quite some time now. As such, attempting to comb through and select only a few graphs that represent the y...

Read More
# Computational Geometry and Topographical Maps

Everyone has seen a topographical map at some point during their life. In addition to the information that regular maps have, topographical maps typically show land features and elevation. How are these maps generated? Given that it's not possible to sample the elevation at every single point in the area of interest, we must use some sort of interpolation. To tackle this mathematically, we should make a few simplifications.
As you might agree, the Earth is round. However, if we consider the ...

Read More
# Bring Math Class to the Modern Age

The feelings of hatred, apprehension, and outright fear associated with doing math are so prevalent and extreme that they have been given a proper term: “math anxiety”. Math anxiety is largely a result of poor experiences in introductory and secondary mathematics classes. As a society, it is our responsibility to resolve this issue because math teaches critical problem-solving skills that are helpful in life, even if the specific concepts taught in class are not directly applied. A major proble...

Read More
# Parametric Flowers

"Mathematical beauty" usually refers to the elegance of a proof, that is, how cleanly some mathematical result is proven with a convincing argument. That certainly sounds appealing to those who are already immersed in math, but it's not clear how such a result is "beautiful" in the general sense of the word. The goal of this post is to give an example of how math can be beautiful in a more accessible and universal way: the creation of art. We'll create flower designs with a large degree of arti...

Read More
# Differential Equation Model for Language Learning

Due to the COVID pandemic, the Susceptible-Infectious-Removed (SIR) model for disease spread has grown wildly in popularity. SIR is a system of differential equations that models the evolution of a disease over time. Knowledge of the SIR model is not necessary to understand this post, but there are many great videos about it online if you want to learn more. My favorites are: Simulating an epidemic by Grant Sanderson, Oxford Mathematician explains SIR Disease Model for COVID-19 (Coronavirus) by...

Read More
# Randomblings – Oddity and the Collatz conjecture part 1

It means "random ramblings". No, it's not an actual word, but I've never understood why people keep words separate that are meant to be squished together. If you had a crocodile made out of chocolate, is it a chocolate crocodile? Obviously not - it's a chocodile. Why does everybody always talk about stray dogs? They're straynines. What if your sister likes to read the Communist Manifesto? Is she your Marxist sister? I think you get the idea. I suppose the ultimate opportunity for word squishifi...

Read More
# Creation of MathGraph3D (Part 3 – Coloring and Lighting)

This series focuses on the creation of the original version of my 3D plotting software MathGraph3D. The first part was about the overall structure of the software. The second part was concerned with all algorithms for smoothing and optimizing surfaces. This third part discusses coloring, lighting, and styling the objects that MathGraph3D plots. I'll try to keep this part shorter than the other ones... posts in this series have a tendency to spiral out of control.
In MathGraph3D, every plotta...

Read More
# Creation of MathGraph3D (Part 2 – Surface Algorithms)

Finally, after a few months of promising this post, here it is. This series focuses on the creation of the original version of my 3D plotting software MathGraph3D. The first part was about the overall structure of the software. This second part is concerned with all algorithms for smoothing and optimizing surfaces.
The basic idea of writing a program to plot 3D surfaces isn't too difficult once you get all the 3D space handling techniques covered. A naive script would map a mesh of input poi...

Read More
# The infamous Goat Problem; my fruitless efforts…

The Goat Problem is a centuries-old geometry problem with no closed form solution. If you tie a goat to the boundary of a circular fence that bounds 1 acre of area, how long does the rope need to be to allow the goat to roam exactly half of this area? Back in December 2020, Quanta Magazine posted a story announcing that Ingo Ullisch, a German mathematician, had reached the first exact solution. The trouble with this solution, however, is that it can only be evaluated iteratively. This is due to...

Read More