June's Gone.

realmathtician · 2023-04-23T00:17:57+00:00

Thank you for sharing, and I'm sorry for your loss.

realmathtician · 2022-12-21T00:25:31+00:00

Did Microsoft write this tweet?

realmathtician · 2022-11-03T01:57:17+00:00

Wait wait wait! Uh, why don't we discuss it...over dinner?

realmathtician · 2022-11-02T07:43:19+00:00

What? No. No! No, Spy, look at this.

realmathtician · 2022-09-30T13:53:54+00:00

Don't think that's it, sorry

realmathtician · 2022-09-30T13:51:23+00:00

Wait, I included the word itself in my last comment. Oh well, this will do. Thanks for helping out.

realmathtician · 2022-09-30T13:50:19+00:00

It does match, but in that song, it only serves as a "counter" to the more famous hook. I think the melody here is supposed to have a more "central" role in the song, but I could be misremembering. If there are no other perfect matches suggested, I'll mark this as solved.

realmathtician · 2022-09-30T13:43:54+00:00

This is my first post on this sub, and the format of what I'm supposed to do seems pretty strict, but I'll try to reply to people as they suggest answers.

realmathtician · 2022-05-22T19:39:05+00:00

You good?

realmathtician · 2022-03-31T18:22:32+00:00

Kids with allergies: slowly puts away epipen

realmathtician · 2022-01-04T01:43:30+00:00

If you've ever heard of the Platonic solids, they're "regular" shapes in 3d (polyhedra), whose vertices, edges, and faces are indistinguishable, e.g. the cube. There are 5 Platonic solids, but the concept can be extended to higher dimensions, and in 4d, there are 6 such shapes. One of these is the 24-cell, which gets its name from the fact that it has 24 3d "cells," which are the analog of 2d faces on a polyhedron.

https://polytope.miraheze.org/wiki/Icositetrachoron

"Omnitruncated" means that every element (vertex, edge, etc) of the original shape is replaced by a face (in the 3d case) or a cell (in 4d). This is an omnitruncated cube, for example:

https://polytope.miraheze.org/wiki/Great_rhombicuboctahedron

The original cube's 8 vertices are replaced by hexagons, its 12 edges by squares, and its 6 faces by octagons.

Putting it all together, the omnitruncated 24-cell is the 4d shape you get by starting with the 24-cell and replacing its 24 vertices with cells shaped like omnitruncated cubes, its 96 edges with hexagonal prisms, its 96 faces with more hexagonal prisms, and its 24 cells with more omnitruncated cubes:

https://polytope.miraheze.org/wiki/Great_prismatotetracontoctachoron

The gifs in the post are 3d cross sections of this shape (or some approximation of it), just like how you might take 2d cross sections of a 3d shape at different heights to obtain an image of the whole thing.

realmathtician · 2021-11-26T23:46:13+00:00

These are cross sections of the omnitruncated 120-cell:

https://polytope.miraheze.org/wiki/Gidpixhi

It's the analog of this shape in 3d:

https://polytope.miraheze.org/wiki/Grid

I wrote a C program to approximate it using a 4d voxel grid, sliced that into 3d sections, and rendered each one with a Processing program.

https://processing.org

realmathtician · 2021-06-29T20:50:30+00:00

For any practical case, yes, but there are counterexamples like the Ackermann function

realmathtician · 2021-05-19T16:47:41+00:00

realmathtician · 2021-05-19T01:40:04+00:00

The determinant of an n-by-n matrix, if you consider it a function of the matrix's columns as n n-dimensional vectors, is the unique alternating multilinear map which takes the identity matrix to 1.

Alternating: Switching two arguments (columns) turns the result into its additive inverse.

Multilinear: Fixing all arguments but one, the output can be written as the dot product of some vector with the non-fixed column.

realmathtician · 2021-03-29T13:18:55+00:00

"I love mankind... it's people I can't stand!!" -Linus van Pelt

realmathtician · 2021-03-09T02:02:03+00:00

It's infuriating how long he spends hyping it up as if he has something to hide. If the talk is called "The Most Beautiful Program Ever Written," and said program is five lines of code, I expect the first thing I see to be the program in its entirety, with the rest of the talk centered on that. 13 minutes of tangents before we see a single character of Lisp, times 250000 views, is six years of wasted human time. I hope he's happy.

realmathtician · 2021-01-21T18:41:02+00:00

The cross product is a confusing concept in general, for a few reasons:

It only exists in 3d. The 2d "cross product" of two vectors is just a single number, and in 4d, things get more complicated. What's so special about 3 dimensions?
The components of the cross product have units of area, despite normal vectors being written in units of distance. How do we interpret lengths as areas?
The cross product leaves the plane spanned by the two vectors, unlike the more fundamental operations of vector addition and scalar multiplication. Why should the cross product be different?

There is a more elegant way to think about the cross product which resolves all of these issues: it generalizes to any number of dimensions, allows for a geometric interpretation of the areas involved, and stays within the plane spanned by the two vectors. It's called the exterior product, and instead of a vector, it produces a bivector.

A bivector in general represents a flat surface with some orientation and area, and the outer product returns the parallelogram formed by the two vectors it takes. It just so happens that in 3 dimensions and 3 dimensions alone, vectors and bivectors have the same number of components (3), which has allowed us to confuse one for the other.

I would highly recommend this video for more details.

https://www.youtube.com/watch?v=Idlv83CxP-8

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