Circumradius of Rapsady

[Mecejide]

I think the circumradius of the retroantiprismatosnub dishecatonicosachoron (rapsady) is equal to the golden ratio. This is because when augment the vertex figure of rapsady with that of the small swirlprism (sisp) at the vertex figure of a pentagonal antiprism (pap), the adjacent pap vertex figure is blended out, meaning the two have the same dichoral angle and therefore circumradius.

[Klitzing]

That is interesting!
Did you just observe that within stella visually, or could you derive that algebraically too?
--- rk

[Mecejide]

I only observed it. I don’t know how to derive circumradii.

[username5243]

I only observed it. I don’t know how to derive circumradii.

Klitzing's website has a spreadsheet that derives the circumradius, but it only works for Wythoffians (and even then I'm not sure how exactly it works). Rapsady is not Wythoffian, nor does it even have any snub description that I'm aware of. I therefore have no clue how one would go about calculating the circumradius. Maybe if someone had coordinates for the vertices of the shape? Not sure how one would generate that.

[wendy]

I wrote the core of Richard's spreadsheet. It's basically a dot-normal of stott vectors, something I wrote back in the seventies. The x3o3o3x style notation was my invention to automate the process. Richard did the fancy stuff about multiple-size arrays etc.

Now what on earth is rapsady? retroantiprismatosnub dishecatonicosachoron The second bit looks like {5,3,3} twelftychoron. Any takers?

[username5243]

It's a particular non-convex uniform polychoron that's notable mainly for including certain 3D snub polyhedra as cells. Definitely isn't Wythoffian.

[Klitzing]

Hi Wendy,
thanx for pointing to the spreadsheet, you once came up with. - As to your Question: here is a link to Jonathan's corresponding webpage. Or to mine.
--- rk

[Mecejide]

On Hedrondude's regiment list, rapsady's symbol is listed as (s,[s*'s),s]. I'm not sure what the brackets indicate.

[Klitzing]

I too find these brackets quite hard too read. But at least he means the following Stott-Dynkin diagram

Code: Select all

    s---3---s
   / \     /
 3/2 5/2  3 
 /     \ /   
s--3/2--s   


I myself have not yet verified it. But at least I recognize the lower left circuit as sirsid and the upper right one as seside.

--- rk

[wendy]

I did a dynkin and stott matrix for this beast. It's got a negative determinate, so i don't know what to do next.

Dynkin
2 -1 -0.618034 1
-1 2 -1 0
-0.618034 -1 2 1
1 0 1 2

-6.4721361006

Stott
4 5.236068 6.472136 -5.236068
5.236068 1.9999999497 5.236068 -5.236068
6.472136 5.236068 4 -5.236068
-5.236068 -5.236068 -5.236068 1.9999999497

It looks hyperbolic to me, prolly from 5.3.4 group.

[Klitzing]

Well, as I already said, I don't have too much belief into that diagram. Infact, I could not find this diagrammal structure as some member of the 120-cell symmetric (= hyic) Goursat tetrahedron addition list. Even so, both mentioned subgraphs are the ones of seside and sirsid. So I suppose, Jonathan simply pasted these together, supposing that this diagram might work… - Or, I simply read his bracket notation wrongly.
--- rk

[wendy]

It's OK. As i was dropping off to sleep, i figured out how this dynkin symbol was indeed a subgroup of {5,3,4}, of order 4.

The Stott matrix should have a positive determinate.

I even played around with a notation for the Gossart tetrahedra, similar to the Schwarz-triangles. It's 3 ordered pairs, where one face appears as the first of each pair, and the remaining faces by one top, two bottom. The odd series are vertices.

[URL]

Stella gives a circumradius of 1.61083. Hopefully when I make my verf → coordinates code, I'll be able to verify that empirically.