@HigherMathNotes

Motivated, accessible math. YouTuber | Open-source developer | Aspiring independent researcher. Prev: @Wikimedia #Eventmath grantee. Views are my own.

 Denver, CO
youtube.com/c/HigherMathNotes
Joined June 2016
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I just released WebGL2 & GLSL: A zero-to-hero guide. It’s a full curriculum for internalizing the graphics pipeline—not just copy-pasting code. It uses spaced repetition to guide you from a blank canvas to building a minimal 3D engine from scratch. 🧵👇 github.com/GregStanton/webgl…
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Replying to @CoryMSimon
Nice! I love how Strang emphasizes the fundamentals. My favorite approach is based on a simple visualization of the dot product: xitter.cf/HigherMathNotes/status… This picture makes its geometric properties intuitive, including the (Cauchy–Bunyakovsky–)Schwarz inequality.
Two vectors v and w are drawn in R^2, in standard position. A line is drawn through one of them (w) and turned into a number line in the direction of w. Both vectors are projected orthogonally onto this line (v projects onto the number -4, and w projects onto its magnitude, which is shown as the number 3). The dot product is the product of these projections: -3*4 = -12.

ALT Two vectors v and w are drawn in R^2, in standard position. A line is drawn through one of them (w) and turned into a number line in the direction of w. Both vectors are projected orthogonally onto this line (v projects onto the number -4, and w projects onto its magnitude, which is shown as the number 3). The dot product is the product of these projections: -3*4 = -12.

Greg Stanton@HigherMathNotes3 Apr 2020
#MTBoS How do you motivate the dot product for math students? I think my latest approach is my best yet! Extend ℝ to ℝ^n. We can multiply numbers in ℝ; how might we multiply vectors in ℝ^n? Hint: We can multiply numbers on a number line. Reduce to that. 1/
Two vectors v and w are drawn in R^2, in standard position. A line is drawn through one of them (w) and turned into a number line in the direction of w. Both vectors are projected orthogonally onto this line (v projects onto the number -4, and w projects onto its magnitude, which is shown as the number 3). The dot product is the product of these projections: -3*4 = -12 ("comp" is used to denote the projections).

ALT Two vectors v and w are drawn in R^2, in standard position. A line is drawn through one of them (w) and turned into a number line in the direction of w. Both vectors are projected orthogonally onto this line (v projects onto the number -4, and w projects onto its magnitude, which is shown as the number 3). The dot product is the product of these projections: -3*4 = -12 ("comp" is used to denote the projections).

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Are you a Google Docs user who often needs to convert drafts to or from Markdown, e.g. for GitHub? Productivity tip: * select a range of text * right click * select "Copy as Markdown" It'll format links, lists... and yes, TABLES! Can also "Paste from Markdown."
Context menu with the following items: * Cut * Copy * Copy as Markdown * Paste * Paste without formatting * Paste from Markdown * Delete

ALT Context menu with the following items: * Cut * Copy * Copy as Markdown * Paste * Paste without formatting * Paste from Markdown * Delete

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Nice application of linear programming to improve harvesting of drinking water! Even uses a reduced 2D linear program, perfect for teaching. One axis for mass of each candidate material. Solution is best combination, subject to min water delivery constraints. #math #iteachmath
A linear program in two dimensions, with one axis for each of the two candidate materials (metal-organic frameworks), measured in kg. Constraints are based on drinking-water delivery and mass. The corners of the feasible region are marked, with the optimal composition being located at a corner in the interior of the first quadrant, and the optimal single-material designs located on the axes.

ALT A linear program in two dimensions, with one axis for each of the two candidate materials (metal-organic frameworks), measured in kg. Constraints are based on drinking-water delivery and mass. The corners of the feasible region are marked, with the optimal composition being located at a corner in the interior of the first quadrant, and the optimal single-material designs located on the axes.

Cory
@CoryMSimon15 Apr 2025
🥤"Optimizing mixtures of metal–organic frameworks for robust and bespoke passive atmospheric water harvesting" by C. Harriman, Q. Ke, T. Vlugt, A. Howarth, C. Simon. feedback welcome on our new preprint! 💡the key idea is to optimize the composition of *mixtures* of metal-organic frameworks (MOFs) to comprise the adsorbent bed of a passive atmospheric water harvester. with a mixture, each MOF can specialize in delivering water under different conditions. we frame a linear program for the optimal design of the composition of the MOF adsorbent bed that is: (1) bespoke, i.e., tailored to a declared geographic region and time frame and (2) robust, i.e. satisfies minimum daily water delivery requirements despite variance in weather. both weather data and MOF water adsorption data are inputs to our optimization problem.
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Replying to @CoryMSimon
Looks like a cool and valuable application of linear programming! I appreciate the inclusion of an LP in a reduced, 2D design space, and the clear visualization. Also the CC-BY 4.0 license :) Potentially makes it a good application for teaching. © 2025 Simon et al.
A linear program in two dimensions, with one axis for each of the two candidate materials (metal-organic frameworks), measured in kg. Constraints are based on drinking-water delivery and mass. The corners of the feasible region are marked, with the optimal composition being located at a corner in the interior of the first quadrant, and the optimal single-material designs located on the axes.

ALT A linear program in two dimensions, with one axis for each of the two candidate materials (metal-organic frameworks), measured in kg. Constraints are based on drinking-water delivery and mass. The corners of the feasible region are marked, with the optimal composition being located at a corner in the interior of the first quadrant, and the optimal single-material designs located on the axes.

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Very cool🧵here about a cutting-edge application of topology to drug discovery! Great example of fundamental mathematical modeling principles, including the use of simplifying assumptions: "If you forget the chemical information for a second," we're sort of left with... this?
Vidit Nanda@viditnanda7 Mar 2025
Among the most common questions that I get from data scientists is "Are there any compelling applications of Topological Data Analysis?" The answer is yes, with high multiplicity. Would you like to see a brand new one? 1/
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In case the world needs one more Tower of Hanoi animation, here you go😅 This one is made with love and @p5xjs. It's customizable and open source. Read on for some interesting facts, and guidance on how to customize the animation for your own needs.
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Replying to @matthen2
Thanks for sharing this! It inspired me to finally fix a small issue with a logo animation I coded six years ago. It uses a cycloid with a similar design concept! khanacademy.org/computer-pro…
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Replying to @hankgreen
American white pelican in Denver, Colorado. P.S. I'm a Hodgkin's lymphoma survivor. If you ever want to reach out, please do. I'd be happy to talk anytime. P.P.S. Kim's Convenience and The Neighbor are lighthearted Netflix comedies, if you need a distraction!
Four American white pelicans on a pond, seen through tree branches.

ALT Four American white pelicans on a pond, seen through tree branches.

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Wouldn't it be great if beginners could easily program #math animations directly in the browser, share them, and *interact* with them? Well... youtu.be/Br40z-lpC3o

Mathemagical.js: Code math visuals on the web (teaser)

Hi everyone!It's been a while since I've posted a video! A long w...

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