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July 13

halfway slices of n-cubes

are there names for the various cross sections of the N-cubes exactly between opposing vertices? For example, the slice of a 3-cube is a hexagon, the slice of a 4-cube is an Octahedron, I'm not sure if they have names (or wikipedia pages) beyond that.Naraht (talk) 02:17, 13 July 2026 (UTC)

@Naraht: I don't think the name exists in the mathematical literature, but per Polytope Wiki, they can be called the mesotruncated simplices. In other words, you ring the node(s) closest to the centre on a Coxeter-Dynkin diagram, ringing both if there is more than one. The series starts:

1. line segment;
2. hexagon;
3. octahedron;
4. bitruncated 5-cell (decachoron);
5. birectified 5-simplex (dodecateron);
6. tritruncated 6-simplex (tetradecapeton);
7. trirectified 7-simplex (hexadecaexon);

and so on. Double sharp (talk) 02:40, 13 July 2026 (UTC)

Is it me or is the definition in the lead of Rectification (geometry) not conducive to understanding? Does this phrase have a meaning, "cutting off its vertices at the midpoints of its edges"? The cited source, MathWorld, does not contain such languages, and assumes (unlike our article) that the polytope about to be rectified is a regular polyhedron.  ​‑‑Lambiam 06:54, 13 July 2026 (UTC)

MathWorld is wrong in that the cells of r{3,3,5} are 600 octahedra (not truncated tetrahedra as MathWorld claims) and 120 icosahedra. (The cells it lists are correct for t{3,3,5}.) Double sharp (talk) 12:20, 13 July 2026 (UTC)

Is there a reliable source that can be used for sourcing the definition?  ​‑‑Lambiam 12:38, 13 July 2026 (UTC)

In terms of coordinates, for even N you get all coordinate permutations of (1, 1, ... , 1, -1, -1, ... -1) where there are N/2 1's and N/2 -1's. For N odd you get all coordinate permutations of (1, 1, ... , 1, 0, -1, -1, ... -1) with (N-1)/2 1's and (N-1)/2 -1's. This means that the number of vertices is 2, 6, 6, 30, 20, 140, ... . OEIS: A056040 calls these numbers "swinging factorials" and the entry does mention the polytopes in question, but there isn't a name or a link given for them. This probably doesn't help answer the question but giving the vertices as a collection of coordinates will hopefully clarify what it is we're talking about. In might be worth noting that the Polytope Wiki and Wikipeda give different names for the N=5 case, which casts doubt on how standard any of the names are. Math does not have a naming authority like astronomy, and Wikipedia isn't supposed to decide such issues. --RDBury (talk) 16:37, 13 July 2026 (UTC)

@Naraht: when n is even this is one example of a hypersimplex. --JBL (talk) 17:06, 24 July 2026 (UTC)

July 16

Sum of all integers from 1-100

I remember hearing about how one famous mathematician, from the 1800s, I believe, was asked in class when he was about 13 to sum all the integers from 1-100, and within a few seconds gave the answer as 5050. Rather than try to manually sum 1+2+3+4…, he figured out that you can add up the numbers in pairs like 100+1, 99+2, 98+3, et cetera, all of which add up to 101, and of which there are 50, so the answer is 50 x 101. I don’t remember which mathematician it was, though I feel like it was Cantor or Gödel or someone from that general time period. Does anyone know who this was? Cheers, 𝔰𝔥𝔞𝔡𝔢𝔰𝔱𝔞𝔯 ^{(𝔱𝔞𝔩𝔨)} (any/all) In solidarity. 22:38, 16 July 2026 (UTC)+1

Carl Friedrich Gauss, allegedly, though it seems to be anecdotal. [1] AndyTheGrump (talk) 22:43, 16 July 2026 (UTC)

Thanks! (It was bugging me beyond belief not being able to remember.) Cheers, 𝔰𝔥𝔞𝔡𝔢𝔰𝔱𝔞𝔯 ^{(𝔱𝔞𝔩𝔨)} (any/all) In solidarity. 22:47, 16 July 2026 (UTC)

July 24

Geodesic n-dimension distance calculation

I was looking at distance calculations like all the p-value Minkowski distances, and I looked at geodesic distance. If radius is the same for every dimension (a circle in 2D, a sphere in 3D, a hypersphere in nD), the formula to calculate the shortest distance between two points (defined as vectors from origin) is easy. What if every dimension had it's own radius? I'm thinking about Earth. The first two dimensions have the same radius (creating the equator), but the third is shorter (the polar radius). If I wanted 4D, 5D, 6D? Is there a single formula where I can use the angle definitions of two vectors and a list of the radii and calculate the shortest distance along the surface of the curved shape? ~2026-35239-54 (talk) 17:36, 24 July 2026 (UTC)

Take the case p = 2 (Euclidean distance) in two-dimensional Cartesian space. What does it mean if the two dimensions "have" their own radii – say one "has" radius a and the other radius b? Do you mean they are rolled up, topologically forming a torus? But then, why mention a circle for the case that a = b?  ​‑‑Lambiam 21:33, 24 July 2026 (UTC)

Our article on ellipse describes a shape with a larger and smaller diameter. One is in one dimension (height) and the other is in another dimension (width). It mentions no confusion with a torus. ~2026-39968-68 (talk) 21:56, 24 July 2026 (UTC)

I do not think that there is any simple formula for the distance along a geodesic on an ellipsoid or spheroid. You will just have to integrate the metric numerically. JRSpriggs (talk) 22:48, 24 July 2026 (UTC)

There is not even a simple formula for the 2D case of the arc length along a segment of an ellipse.  ​‑‑Lambiam 10:08, 25 July 2026 (UTC)

I know you're referring to ellipsoids beyond just three dimensions, but geodesics on an ellipsoid might help for the 3D case. GalacticShoe (talk) 00:45, 25 July 2026 (UTC)

You could try using the calculus of variations and apply the constraint of staying on the ellipsoid with a Lagrange multiplier. This would give you a difficult set of differential equations to solve. JRSpriggs (talk) 11:42, 25 July 2026 (UTC)

The method from Geodesics on an ellipsoid § Geodesics on a triaxial ellipsoid should be generalizable but I'd expect it to get unpleasantly complicated. –jacobolus (t) 18:48, 25 July 2026 (UTC)