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 Bet...

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# The Last Time I’ll Ever Think About Factoring

Factoring quadratics is a completely solved problem. Any polynomial of degree $n$ breaks into $n$ linear factors in the splitting field of its ring of coefficients. That's cool, but in practice, it's nice to have a direct way to break a quadratic in $\mathbb{Z}[x]/(x^{3})$ into its two linear factors without exactly solving for the roots.*
Say we have a quadratic $Ax^2+Bx+C$ with two solutions $x=-p/q$ and $x=-r/s$. Then we have the following:
$$ \left(x+\frac{p}{q}\right)\left(x+\frac{r...

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# Iterated Weighted Arithmetic Means

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...

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# 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...

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# 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 ...

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# Collection of Desmos Graphs – Part 3

As is tradition, I'll share a few of my favorite recent Desmos graphs. Since last year, Desmos has done some significant updates, including performance improvements and new feature implementations. The largest new features are polygon lists, actions, and list comprehensions, which essentially trivialize many things that were previously difficult on Desmos. While this is obviously great for Desmos (and the calculator is much better as a result), it takes away a lot of the challenge. Creating an ...

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# 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...

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# 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...

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# 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...

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# 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...

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