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Joseph Mellor
Joseph Mellor

3.5K Followers

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Published in

Cantor’s Paradise

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An Intro to Topology

Topology is the study of how you can squish and stretch things without gluing or cutting them, so why is it so complex? — I took a lot of Math and Science classes in college, but I had three glaring omissions: Complex Analysis, Atomic Physics, and Topology. I had heard about Topology from talks of doughnuts and coffee cups and a few 3blue1brown videos, but I didn’t pursue it because I didn’t think it…

Mathematics

22 min read

An Intro to Topology
An Intro to Topology
Mathematics

22 min read


Published in

Cantor’s Paradise

·Pinned

The Road to Quantum Mechanics, Part 1: Intro, Coordinates, and Kinematics

It’s a long road to understanding Quantum Mechanics, and there are no shortcuts. In this series, I’ll show you Quantum Mechanics from the ground up. The word “quantum” roughly means “amount.” In the modern world, it functions as a shorthand for “Quantum Mechanics,” especially when used as an adjective. When…

Physics

20 min read

The Road to Quantum Mechanics, Part 1: Intro, Coordinates, and Kinematics
The Road to Quantum Mechanics, Part 1: Intro, Coordinates, and Kinematics
Physics

20 min read


Published in

Cantor’s Paradise

·Pinned

How to Discover Finite Fields While Bored in Class, Pt. 1: Cellular Automata

I thought I was just drawing pretty lines, but my high school doodles turned out to be Linear Algebra over a finite field. When I was in high school and college, I would always try to read ahead in my classes so I’d be prepared when the topics came up…

Mathematics

11 min read

How to Discover Finite Fields While Bored in Class, Pt. 1: Cellular Automata
How to Discover Finite Fields While Bored in Class, Pt. 1: Cellular Automata
Mathematics

11 min read


Nov 25

Representations of Finite Fields

There’s little to no easy-to-use, freely available software for working with general finite fields. Let’s fix that. Over the past few years, I’ve spent a lot of effort trying to figure out the best way to explain concepts to a wide audience, which requires that I first figure out the…

Math

19 min read

Representations of Finite Fields
Representations of Finite Fields
Math

19 min read


Sep 4

Eigenvalues and Eigenvectors

Eigenvalues and eigenvectors are the most powerful tools in Linear Algebra. We’ve talked about matrices and linear transformations in the previous article in the context of solving systems of equations, but there are often many cases where we have a linear transformation of a space to itself. …

Math

16 min read

Eigenvalues and Eigenvectors
Eigenvalues and Eigenvectors
Math

16 min read


Sep 3

Update: I’m Still Alive and Making Content

I haven’t published an article in a few months because I’ve been incredibly busy, but I’m starting to get more time to make content. I apologize for the radio silence. It’s been about two months since I’ve published an article or video. …

Science

3 min read

Update: I’m Still Alive and Making Content
Update: I’m Still Alive and Making Content
Science

3 min read


Jun 26

Differential Forms and Tensors

The Levi-Civita symbol allows us to take determinants and rewrite Differential Forms in terms of tensors. We’ve spent roughly five articles (or six if you count Vectors and Covectors) talking about Differential Forms, but concepts in Physics are expressed mostly in terms of tensors. If we want our progress in…

Math

15 min read

Differential Forms and Tensors
Differential Forms and Tensors
Math

15 min read


Jun 4

A Mathematical Approach to Tensors

A matrix is a linear map of a vector to another vector. A tensor is a multilinear map of a tensor to another tensor. There are a lot of bad explanations in science and math. “Spin is like a ball that’s spinning, except it’s not a ball and it’s not…

Science

17 min read

A Mathematical Approach to Tensors
A Mathematical Approach to Tensors
Science

17 min read


Jun 3

Update: I Can Now Make Vector Fields in Blender

The article on tensors will come out soon, but I might also want to work on making stream lines. Blender is quickly becoming one of my favorite tools to use. It’s incredibly powerful, runs at breakneck speeds, and it has a great workflow once you get used to it. The…

Blender

11 min read

Update: I Can Now Make Vector Fields in Blender
Update: I Can Now Make Vector Fields in Blender
Blender

11 min read


May 18

Update: How Should I Depict Tensors?

I only need to make a featured image before I can publish my next article. Unfortunately, I can’t find any depictions of tensors. Before I get into my main topic, I want to announce that I’ve finally finished the Differential Forms arc (previously named the Mathematical Objects arc) by publishing…

Mathematics

6 min read

Update: How Should I Depict Tensors?
Update: How Should I Depict Tensors?
Mathematics

6 min read

Joseph Mellor

Joseph Mellor

3.5K Followers

BS in Physics, Math, and CS with a minor in High-Performance Computing. You can find all my articles at https://josephmellor.xyz/articles/.

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