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

Oddity distribution
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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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...
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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...
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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...
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Collection of Desmos Graphs – Part 2

Just like last time, I'm going to share a few of my favorite recent graphs on Desmos. This is probably going to become a yearly thing - despite that the previous part was my first post ever, it remains the most popular. That's probably due to the fact that r/visualizedmath on Reddit won't let me post anymore to share my work. Anyway, on with the graphs... November 2019 MCC Puzzler Try it yourself: https://www.desmos.com/calculator/qsy5qk4sjv Last year, I took a couple math classes...
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Project Euler – Number Spirals

Another Project Euler problem has sparked my attention. This one isn't particularly difficult (much less than the last one I posted about - despite being in the same category of difficulty), but it's an interesting problem with a nice solution. Most people probably don't approach this in the way I did, but for once I think my solution is cleaner than the others that are posted on the forum. The problem statement can be found here: Problem 28 Statement. Like last time, I'm going to proceed un...
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Project Euler – Lattice Paths

Recently, I've been doing a bunch of Project Euler problems. If you're unfamiliar with Project Euler, it's a collection of challenging puzzles that require math and programming to solve. The puzzles are grouped by difficulty level. However, I've found that the difficulty can vary wildly within the same group. One challenge that stood out as much harder than the rest is Problem 15. The problem statement can be found here: Problem 15 Statement. Now, I'm going to continue under the assumption ...
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Contour plots and their surface counterparts

This is mostly a gallery post, showing the contour plots and 3D surfaces of several functions. All of these images were made with MathGraph3D. f(x,y) = (sin(x) ^ 2) * (sin(y) ^ 2) * exp((x + y) / 2) / 3 + sin(x + y) / 4 on the region [-4,4] x [-4,4]. f(x,y) = sin(x + cos(2y)) - cos(y + sin(2x)) on the region [-2,2] x [-2,2]. f(x,y) = 1 + (x^3 + y^3) / 32 - 0.25x^2 on the region [-4,4] x [-4,4]. f(x,y) = sin(x + cos(y))sin(y + c...
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