In geometry, the **gyrobifastigium** is the 26th Johnson solid (*J*_{26}). It can be constructed by joining two face-regular triangular prisms along corresponding square faces, giving a quarter-turn to one prism. It is the only Johnson solid that can tile three-dimensional space.

It is also the vertex figure of the nonuniform *p*-*q* duoantiprism (if *p* and *q* are greater than 2). Despite the fact that *p*, *q* = 3 would yield a geometrically identical equivalent to the Johnson solid, it lacks a circumscribed sphere that touches all vertices, except for the case *p* = 5, *q* = 5/3, which represents a uniform great duoantiprism.

Its dual, the elongated tetragonal disphenoid, can be found as cells of the duals of the *p*-*q* duoantiprisms.

In mathematics, **Alcuin's sequence**, named after Alcuin of York, is the sequence of coefficients of the power-series expansion of:

- ${\frac {x^{3}}{(1-x^{2})(1-x^{3})(1-x^{4})}}=x^{3}+x^{5}+x^{6}+2x^{7}+x^{8}+3x^{9}+\cdots .$

The sequence begins with these integers:

- 0, 0, 0, 1, 0, 1, 1, 2, 1, 3, 2, 4, 3, 5, 4, 7, 5, 8, 7, 10, 8, 12, 10, 14, 12, 16, 14, 19, 16, 21 (sequence A005044 in the OEIS)

The *n*th term is the number of triangles with integer sides and perimeter *n*. It is also the number of triangles with *distinct* integer sides and perimeter *n* + 6, i.e. number of triples (*a*, *b*, *c*) such that 1 ≤ *a* < *b* < *c* < *a* + *b*, *a* + *b* + *c* = *n* + 6.

If one deletes the three leading zeros, then it is the number of ways in which *n* empty casks, *n* casks half-full of wine and *n* full casks can be distributed to three persons in such a way that each one gets the same number of casks and the same amount of wine. This is the generalization of problem 12 appearing in Propositiones ad Acuendos Juvenes ("Problems to Sharpen the Young") usually attributed to Alcuin. That problem is given as,

- Problem 12: A certain father died and left as an inheritance to his three sons 30 glass flasks, of which 10 were full of oil, another 10 were half full, while another 10 were empty. Divide the oil and flasks so that an equal share of the commodities should equally come down to the three sons, both of oil and glass.

The term "Alcuin's sequence" may be traced back to D. Olivastro's 1993 book on mathematical games, *Ancient Puzzle: Classical Brainteasers and Other Timeless Mathematical Games of the Last 10 Centuries* (Bantam, New York).

The sequence with the three leading zeros deleted is obtained as the sequence of coefficients of the power-series expansion of

- ${\frac {1}{(1-x^{2})(1-x^{3})(1-x^{4})}}=1+x^{2}+x^{3}+2x^{4}+x^{5}+3x^{6}+\cdots .$

This sequence has also been called Alcuin's sequence by some authors.

What the heck is a Chu space? And whatever it is, does it really belong with all the rich mathematical structures we know and love?

Say you have some stuff. What can you do with it?

Maybe it’s made of little pieces, and you can do a different thing with each little piece.

Sometimes you discover a nice-looking path through the forest, and you follow it thinking it’ll lead you to some new, exciting place. But as you walk and walk amongst the trees, you come to realize it’s actually leading you somewhere you already knew. This article is a story about (re)discovery, and how new things can make us understand old things better.

Before we start, though, I want to make clear I’m

in no wayan “expert mathematician”. Every single piece of mathematical knowledge I’ve used in this article, I was taught in high school. And I have forgotten so much of it, in fact, that for the integrals in this article I had to resort to Wolfram Alpha every single time. ¯_(ツ)_/¯This also means this article

is likely to have innacuracies, and a fair share of things an actual mathematician would consider silly. For any such things (or anything else you want), you can reach me by email atjonat this domain, or on Twitter at jon_valdes.

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In my recent post on mapping Perlin noise to angles, I was put on to the subject of Curl noise, which I thought I understood, but did not. I figured out what Curl noise

reallywas in a subsequent post and then posted my earlier incorrect (but still interesting and perhaps useful) concept of Curl noise in yet another post. Although I kind of understood what Curl noise was at that point, I wanted to give myself a more complete understanding, which I usually do by digging into the code, making sure I understand every line 100% and seeing what else I can do with it, trying to make multiple visualizations with it to test my understanding, etc.

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Mathematician: That depends on what you mean by “universe.” Here’s a framing:A circle of radius R centered at a point P is the set of all points in the plane with distance R from P. The diameter D of this circle is twice the radius, but can also be thought of as the longest possible straight-line path from a point on the circle to another point on the circle. The circumference C of the circle is its arc length. By definition, Pi = C/D.

The Collatz conjecture is an allegedly-unsolved problem in mathematics.

I say allegedly there because, reading those words, I feel like mathematicians are going to jump out from my closet yelling, “You got punk’d!” for being duped into believing it’s unsolved.

We’ve all heard of integers, rationals, reals, even complex numbers, but what on earth are surreal numbers? They are a beautiful way of defining a class of numbers which includes all reals, but also ordinal numbers; i.e. all the different infinities and even infinitesimal numbers. Not only this but we get a full system of arithmetic for all these numbers. Ever wondered what (∞−1) is, or √∞ ? Before we get stuck into that, let’s learn some history.

It all started a long, long time ago in a galaxy far, far away (Cambridge in the 1970s). A man by the name of John H. Conway was playing Go, an ancient Chinese game that’s very elegant in itself. After much thought, he realised that the later stages of the game could be thought of as the sum of many smaller games. Conway then applied his ideas to other games like Checkers and Dominoes. It seemed that these games were behaving as if they were numbers.

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