Vincent Pantaloni
Consider the following operation on polyhedra where the midpoint of each edge becomes a vertex of a new polyhedron.
I noticed that this applied to the icosahedron or to the dodecahedron gave the icosidodecahedron (see previous toots). So I wanted to see what this would do to the cube. The 3rd iteration (blue) surprised me ! #MathGIF
#GeoGebra https://www.geogebra.org/m/kg68u34z
New polyhedron by joining midpoints of edges of a cube

New polyhedron by joining midpoints of edges of a cube

GeoGebra
Nov 21, 2022 at 5:26pmWeb
Show threadVincent PantaloniNov 21, 2022

I was very happy to discover this transformation but of course I wasn't the first to think of this. I read it's what Conway called the "ambo" operation with prefix a.
https://en.wikipedia.org/wiki/Conway_polyhedron_notation

It's also called rectification.
https://en.wikipedia.org/wiki/Rectification_(geometry)

Conway polyhedron notation - Wikipedia

Show threadJon AwbreyNov 21, 2022

@panlepan

Vaguely reminds me of #LineGraphs (I think, it's been a while) in #GraphTheory ...

Show threadBen ReinigerNov 21, 2022
@Inquiry @panlepan
The wiki article on rectification mentions&links the medial graph (https://en.wikipedia.org/wiki/Medial_graph). The difference from the line graph is that two edges that don't share a face won't get connected; that doesn't happen with the cube, but in the next step it does. The medial graph preserves planarity, but the line graph doesn't (generally).
Medial graph - Wikipedia

Show threadBen ReinigerNov 21, 2022
@panlepan
This operation is "rectification"? https://en.wikipedia.org/wiki/Rectification_(geometry)
Rectification (geometry) - Wikipedia

Show threadVincent PantaloniNov 21, 2022
@bmreiniger Yep, thanks. I first found the Conway operations then "ambo" led me to rectification. I have seen it mentioned before but never had really looked up what it was. @tomruen must have mentioned it several times though... shame on me.
Show threadMatt McIrvinNov 21, 2022

@panlepan If you do rectification/ambo to a cube you get the cuboctahedron. If you do it to an octahedron (the dual), you also get the cuboctahedron.

There's an operation where you expand edges into faces, and that gives you the rhombic dodecahedron from both the cube and the octahedron. That is the dual of the cuboctahedron.

Show threadMatt McIrvinNov 21, 2022
@panlepan ...And if I recall correctly, these operations both have counterparts in 4 dimensions, and if you do either of those to a hypercube or its dual, you get the 24-cell, which is actually regular and self-dual.
So the 4D counterpart of the cuboctahedron and the rhombic dodecahedron are, in a sense, the same thing--a 6th "Platonic" polytope.
Show threadJohn Carlos BaezNov 21, 2022

@mattmcirvin @panlepan - besides "rectification" it's good to read about "cantellation":

https://en.wikipedia.org/wiki/Cantellation_(geometry)

In 3d, at least, to cantellate a polyhedron is to rectify it twice!

Cantellation (geometry) - Wikipedia

Show threadVincent PantaloniNov 21, 2022

@mattmcirvin Thanks !
I see that, and many things make more sense now. :)

The other operation you're talking about is chamfer(ing) ?
(chanfreiner in French :) )
https://en.wikipedia.org/wiki/Chamfer_(geometry)

Chamfer (geometry) - Wikipedia

Show threadMatt McIrvinNov 21, 2022
@panlepan I think it is not quite chamfering because that leaves a face in place for every original face. It's the one that Conway calls "join". Not sure if it has other standard names.
Show threadRahul NarainNov 21, 2022
@panlepan Ooh, so two iterations of rectification give you Doo-Sabin subdivision! https://en.wikipedia.org/wiki/Doo%E2%80%93Sabin_subdivision_surface
(Topologically, at least, though not geometrically.)
Doo–Sabin subdivision surface - Wikipedia

Show threadVincent PantaloniNov 21, 2022
@narain Ohh that's interesting (and smooth). I'd say a² is cantellation. Some nice animations on this wiki page :
https://en.wikipedia.org/wiki/Cantellation_(geometry)
But there seems to be different levels of cantellation and what I get with the cube seems to be a 1/4 cantellated cube or "beveled cube". Calling @tomruen for help :)
Cantellation (geometry) - Wikipedia

Show threadsofia ☮️🏴Nov 21, 2022
@panlepan that looks like it would converge to a sphere 🤔. does it?
Show threadVincent PantaloniNov 21, 2022
@sofia That's also what my gut tells me.
At least it seems to be rounder and rounder :)
Show threadrobinhoustonNov 21, 2022
@panlepan There's a nice web app called polyHédronisme where you can type in an expression in Conway notation and it will show a 3D model of it: http://levskaya.github.io/polyhedronisme/?recipe=aaaC
polyHédronisme

Show threadVincent PantaloniNov 21, 2022
@robinhouston OMG this is awesome !
I could test (visually) the natural conjecture that a^(n)C would converge (whatever that means) to the sphere.
It doesn't seem true actually.
https://levskaya.github.io/polyhedronisme/?recipe=aaaaaaaaaaaaC
Cc @sc_griffith @sofia
polyHédronisme

Show thread🏳️‍⚧️⛧ Fawn>Fun ⛧🏳️‍⚧️Nov 21, 2022
@panlepan @robinhouston @sc_griffith @sofia this looks like some nerd shit, but y’all seem stoked about it so I support this, uhhh, this thing is thingin’
Show threadrobinhoustonNov 22, 2022
@panlepan @sc_griffith @sofia There's an interesting comment on the MO page, suggesting that the limit can never be a sphere, for any starting polyhedron
Show threadVincent PantaloniNov 22, 2022
@robinhouston @sc_griffith @sofia Interesting... I can believe the conclusion but I don't follow every step of this proof. Why would inscribed in a sphere => edges have same length ?
Show threadrobinhoustonNov 22, 2022
@sofia @sc_griffith @panlepan I don't fully follow it either. The small part I think I maybe do understand: there's a point in the middle of each face that remains in every iteration, so if the limit is a sphere then it's an inscribed sphere of the original polyhedron. I don't know about the circumscribed sphere