Talk:Midsphere

Where does the name "canonical polyhedron" come from?
This article cites Ziegler (1995), but Ziegler doesn't use this name. What Ziegler says explicitly is:

–jacobolus (t) 23:21, 5 February 2024 (UTC)


 * One source seems to be G. Hart (1997) "Calculating Canonical Polyhedra", Mathematica in Education and Research 6(3): 5–10. –jacobolus (t) 00:30, 6 February 2024 (UTC)
 * The earliest form of the MathWorld page on the same topic that I can find credits Hart. But Ziegler's phrasing is very close to this already. Schramm's 1992 "how to cage an egg", another early reference for this, does use the word "canonical" twice, but not for exactly this meaning. —David Eppstein (talk) 00:32, 6 February 2024 (UTC)
 * It seems like the name "Koebe polyhedron" is sometimes given to any polyhedron which has a "midsphere". Is this name widespread enough to be worth mentioning? –jacobolus (t) 00:38, 6 February 2024 (UTC)

The confusing canonical polyhedron's definition
The article says about the definition of canonical polyhedron in the following: "Any polyhedron with a midsphere, scaled so that the midsphere is the unit sphere, can be transformed in this way into a polyhedron for which the centroid of the points of tangency is at the center of the sphere. The result of this transformation is an equivalent form of the given polyhedron, called the canonical polyhedron, with the property that all combinatorially equivalent polyhedra will produce the same canonical polyhedra as each other, up to congruence"

The words seems a bit confusing and WP:TECHNICAL as I read, an example is "centroid of the points of tangency". Can you provide some explanation both in roughly speaking and fully meaning here, @David Eppstein? Also, MathWorld says that a polyhedra is canonical provided that the edges are touching the sphere as well, but I remember that MathWorld is not good source, so I leave it to you in this case. I was trying to understand the definition. Dedhert.Jr (talk) 12:46, 17 March 2024 (UTC)


 * MathWorld says "and the center of gravity of their contact points is the center of that sphere". This is the same as our "centroid of the points of tangency is at the center of the sphere". Scale and translate the polyhedron so that its midsphere is a unit sphere centered at the origin. Take all the points where the edges touch the sphere. Average their coordinates. The resulting average is itself a point, somewhere in space. If it's the origin, you're done: you have a canonical polyhedron. —David Eppstein (talk) 16:59, 17 March 2024 (UTC)
 * "Average their coordinates"? I don't get it in this case. Also, should you provide an example of canonical polyhedron in this case, an example is gyrate rhombicosidodecahedron? The infobox says it is canonical, but no source has ever mentioned this fact nowadays. Dedhert.Jr (talk) 01:36, 18 March 2024 (UTC)
 * You know. The average of (1,1,1), (-1,2,-7), and (3,-4,2) is ( (1,1,1) + (-1,2,-7) + (3,-4,2) )/3 = ($1-1+3⁄3$,$1+2-4⁄3$,$1-7+2⁄3$). Add up the vectors of Cartesian coordinates, coordinate by coordinate, and then divide by how many you added. Just like you compute any other kind of average.
 * In the case of the gyrate rhombicosidodecahedron, the rhombicosidodecahedron has enough symmetry to force the average of the points of tangency to be the center of the sphere, as long as you have a midsphere at all. And then spinning the cupolas doesn't change the existence of a midsphere or the position of the average of the tangency points. For the rhombicosidodecahedron itself, it's not completely obvious that it has a midsphere because it's not edge-transitive, but there's a simple argument that can be used to show it: because it is vertex-transitive, there is a circumsphere, through all vertices. And then the faces that are squares have their edge midpoints equally far from the circumcenter, by the symmetry of the square. But each edge of the rhombicosidodecahedron belongs to a square, so all edge midpoints are equally far from the circumcenter, and the circumcenter is also a midcenter.
 * I doubt you find these arguments in any published source, though, so it is probably original research. —David Eppstein (talk) 06:45, 18 March 2024 (UTC)
 * Ooooooohhhh... My bad. I was expecting that the word "average" has alternative meaning other than statistical definition. Dedhert.Jr (talk) 11:03, 18 March 2024 (UTC)