Circumradius of Rapsady

Discussion of tapertopes, uniform polytopes, and other shapes with flat hypercells.

Circumradius of Rapsady

Postby Mecejide » Sun Jan 05, 2020 6:57 pm

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.
Mecejide
Trionian
 
Posts: 64
Joined: Sun Mar 10, 2019 1:58 am
Location: Minnesota

Re: Circumradius of Rapsady

Postby Klitzing » Mon Jan 06, 2020 12:34 pm

That is interesting!
Did you just observe that within stella visually, or could you derive that algebraically too?
--- rk
Klitzing
Pentonian
 
Posts: 1536
Joined: Sun Aug 19, 2012 11:16 am
Location: Heidenheim, Germany

Re: Circumradius of Rapsady

Postby Mecejide » Mon Jan 06, 2020 8:11 pm

I only observed it. I don’t know how to derive circumradii.
Mecejide
Trionian
 
Posts: 64
Joined: Sun Mar 10, 2019 1:58 am
Location: Minnesota

Re: Circumradius of Rapsady

Postby username5243 » Mon Jan 06, 2020 9:20 pm

Mecejide wrote: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.
username5243
Trionian
 
Posts: 110
Joined: Sat Mar 18, 2017 1:42 pm

Re: Circumradius of Rapsady

Postby wendy » Tue Jan 07, 2020 9:45 am

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?
The dream you dream alone is only a dream
the dream we dream together is reality.

\(\LaTeX\ \) at https://greasyfork.org/en/users/188714-wendy-krieger
User avatar
wendy
Pentonian
 
Posts: 1925
Joined: Tue Jan 18, 2005 12:42 pm
Location: Brisbane, Australia

Re: Circumradius of Rapsady

Postby username5243 » Tue Jan 07, 2020 11:00 am

It's a particular non-convex uniform polychoron that's notable mainly for including certain 3D snub polyhedra as cells. Definitely isn't Wythoffian.
username5243
Trionian
 
Posts: 110
Joined: Sat Mar 18, 2017 1:42 pm

Re: Circumradius of Rapsady

Postby Klitzing » Tue Jan 07, 2020 5:51 pm

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
Klitzing
Pentonian
 
Posts: 1536
Joined: Sun Aug 19, 2012 11:16 am
Location: Heidenheim, Germany

Re: Circumradius of Rapsady

Postby Mecejide » Tue Jan 07, 2020 8:25 pm

On Hedrondude's regiment list, rapsady's symbol is listed as (s,[s*'s),s]. I'm not sure what the brackets indicate.
Mecejide
Trionian
 
Posts: 64
Joined: Sun Mar 10, 2019 1:58 am
Location: Minnesota

Re: Circumradius of Rapsady

Postby Klitzing » Tue Jan 07, 2020 10:34 pm

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
Klitzing
Pentonian
 
Posts: 1536
Joined: Sun Aug 19, 2012 11:16 am
Location: Heidenheim, Germany

Re: Circumradius of Rapsady

Postby wendy » Wed Jan 08, 2020 9:40 am

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.
The dream you dream alone is only a dream
the dream we dream together is reality.

\(\LaTeX\ \) at https://greasyfork.org/en/users/188714-wendy-krieger
User avatar
wendy
Pentonian
 
Posts: 1925
Joined: Tue Jan 18, 2005 12:42 pm
Location: Brisbane, Australia

Re: Circumradius of Rapsady

Postby Klitzing » Wed Jan 08, 2020 5:31 pm

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
Klitzing
Pentonian
 
Posts: 1536
Joined: Sun Aug 19, 2012 11:16 am
Location: Heidenheim, Germany

Re: Circumradius of Rapsady

Postby wendy » Thu Jan 09, 2020 10:39 am

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.
The dream you dream alone is only a dream
the dream we dream together is reality.

\(\LaTeX\ \) at https://greasyfork.org/en/users/188714-wendy-krieger
User avatar
wendy
Pentonian
 
Posts: 1925
Joined: Tue Jan 18, 2005 12:42 pm
Location: Brisbane, Australia


Return to Other Polytopes

Who is online

Users browsing this forum: Google [Bot] and 5 guests

cron