The term "stage" in the fine dining industry originates from the French word "stagiaire," meaning trainee or intern, and refers to a cook or chef working briefly for free in another kitchen to learn new techniques and cuisines. Ultimately the etymolgy traces to Latin, drawing on the sense of

* * *

In high-school pre-calc (we called it EMA for "Elementary Mathematical Analysis" and my teacher was Mrs Burden), I was briefly obsessed with a framework for identifying coordinates on a sphere for any arbitrary number of maximally-spaced points.

For 2 points the solution looks like points at the North and South poles on a globe

For three points it looks like an equilateral triangle on the equator

For four points, it looks like a four-sided die (d4; i.e. a regular tetrahedron)

Mrs Burden did not have a solution for 5, 6, 7 points. Or if she did, she thought I should figure it out for myself, but I never did.

I did at least quickly intuit that the solutions for 4, 6, 8, 12 and 20 points are classically known regular polyhedra ("dice shapes"). In mathematics these are known as the Platonic Solids. I also realized that [the vertices of the 6-point solution] map to [centers of the polyhedral faces of the 8-point solution], and vice versa. We might call them transformations of each other. Likewise, the vertices and face-centers of 12- and 20-point solutions map to each other, another transform pair.

Turns out this question I posed to my EMA teacher is a classic problem in mathematics, known as the Tammes Problem after Dutch botanist Peter Merkus Lambertus Tammes, from his 1930 paper about optimal distribution of pores on the surface of spherical pollen grains. For smallish numbers (up to 12) there are known configurations. For larger N, there's a formula mapping the Fibonacci spiral onto a sphere (the "Fibonacci Sphere"), and it's a handy approximation but not perfect for all N. Beyond "approximate", solutions have to be tested on case-by-case basis. The problem can be rephrased as "maximizing minimum angular distance between n points on a unit sphere" or "packing non-overlapping circular caps on a unit sphere".

There's another type of geometrically elegant solid figures know as Archimedean Solids. Platonic Solids are defined as having regular polygon faces all of one type: triangle, square, pentagon. Archimedean Solids have regular polygonal faces of more than one type. There's a NYT article currently trending about the geometry of the soccer ball (because FIFA fever). A soccer ball is a rounded Archimedean Solid composed of 20 hexagonal and 12 pentagonal faces. It is not a requirement that the polygon faces be of equal area, and in the classic soccer ball "truncated icosahedron" construction, they're not.

It sounds like this "truncated icosahedron" soccer-ball surface would be a good basis for a hypothetical 32-sided die. BUT you'd want all the die faces to be of equal area; the more disproportionate the face areas, the more likely the die will land on larger faces. It defies intuition somewhat but it turns out there is no 32-sided geometric solid that simultaneously satisfies both constraints of "regular polygonal faces" and "equal face areas" for the socer-ball/truncated-icosahedron configuration. Either you have to warp some polygons into irregular shapes to satisfy a constraint of equal face areas, or you're forced to use polygons with differing area measures to satify the regularity constraint. In soccer it doesn't matter; nothing changes about the game just because a kicked ball is more likely to come to rest on a hexagonal face. But in a dice game, the relative odds of results are fundamental to the game and it changes everything if there are different odds for each possible result of a die roll.

In fact, Archimedean Solids by definition have more than one type of face polygon, therefore no Archimedean Solid can be used as a "fair" die (equal chances for every side), because no pair of regular polygons with different numbers of sides but the same length of sides will have the same area formula.

We do see 30-sided and 100-sided dice in the tabletop gaming industry. 30-siders belong to a category of geometric solids called Catalan Solids. They're mathematically fair, but their faces are not regular polygons; they're diamond-shaped rhomboids. 100-siders ("Zocchihedron") are not precisely mathematically fair; the original 1985 Lou Zocchi product was demonstrated by extensive sampling to significantly favor non-extreme results (between 9 and 92). In percent-based games like Runequest where extremes indicate critical successes or fumbles, this would be disappointing.

While exotic dice designs informed my naive exploration of this Tammes Problem, that wasn't why I was interested in it as a teen. I had some ideas about world-building, where I wanted to populate my homemade D+D setting with X number of imprisoned demon-lord bosses, or scatter legendary wondrous items throughout a setting "somewhat evenly". As long as I postulated 4, 6, 8, 12 or 20 ... things ... to be placed equidistant around a game-setting world, I could math out relative locations. But I had no solution for, say, "lost palaces of the FIVE legendary beast princes".

Thanks to internets, now I'm assured the 5-points solution looks like a point at each pole, plus 3 more points defining an equilateral triangle around the equator. I had intuited that, but had no resources to prove or verify it.