Thank you for your time to deal with my problems, nad!

In fact the epitahedron is fromed from one pentagon, – a heptahedron which Hilbert apparently had in mind, when he was searching for a 3D version of the Moebius strip;- not exactly the known heptahedron in the list, homomorphous but its edges are in the golden ratio.

Many thanks you for your link, II did it not see as clearly before: The epitahedron(E±)seems to serve for a 3D version of Euklid’s 10th proposition:

1 E± corresponds to the pentagon, 2 E± form a hexagon and

5 E± form a decagon

And with your second question it converges to an interesting solution: No, the perpendicular planes are not important here; the axis of roational axes are corresponding to the planes of the pentagons in an icosahedron which are forming a great dodecahedron;- again in the 3D manner, which seems to prove the above assumption. q.e.d.

Alas, the above question how it all related to the icosahedral groups remains still open…

And thanx for the tip: we’ll work it out on a 3D viewer too.

If you like, you may see more details of the Epitahedron

I could discuss with Roger Penrose:

http://quantumcinema.uni-ak.ac.at/site/expert-talk/sir-roger-penrose/

So if I understood correctly your epitahedron is the last of the heptahedra in this Wikipedia list.

This is also hard to see in the video, are the 5 vertices, which touch the icosahedra of your heptahedron always the same? (It looks so, but a video is not so overly useful for studying your geometry. Why don’t you put your 3D model on your website via a 3D viewer?) It looks as if your 5 points are the 5 points you get if you take the five points which are on three perpendicular planes as shown here. I.e. it looks a bit as if you e.g. take the quadrilateral formed by two green and two red points and then one blue point for the tip. Is this correct?

I guess the answer needs somebody who is familiar with Group Theory AND Geometry like Plato, Galois, F. Klein or Bertram Kostant,- or somebody else strumbling along

THIS blog ?

And sometimes it seems that mathematicians nowadays rather have footballs in their minds instead of Plato’s triangles ;-)

I understand that you want John to answer your question. He is extremely busy. He has a lot of duties, like his students at his university for which he has to care for and so there is not so much time left for all that math orphans. This holds also true for other academic experts. See I have also a question for him ( or some other geometry wizzard), which he hasn’t answered yet.

That is I could of course try to find the answer myself, but this would be rather tedious and timeintensive, especially since I am not familiar with all the dialects spoken in this branch of mathematics and then in the end after a cumbersome and lonesome archeological and linguistical investigation someone would point out to you that this question was answered in a russian paper in the early seventies. (which might still be quite unreadable though).

]]>yes: 6 vertices, 7 faces, and 11 edges; it is one of the 34 topologically distinct convex heptahedra, but golden … we will publish it soon online too.

And sometimes it seems that mathematicians nowadays rather have footballs in their minds instead of Plato’s triangles ;-)

]]>John said: Yes, I’m all in favor of fighting boredom by being more boring than the people who bore me.

I like that! I wouldn’t have tried it at school though.

http://en.wikipedia.org/wiki/Football_%28ball%29 says:

“Rugby league is played with a prolate spheroid shaped football which is inflated with air.”

and

“The ball used in rugby union, usually referred to as a rugby ball, is a prolate spheroid essentially elliptical in profile.”

]]>Visualisation of the icosadedral group in a new 3D representation, In: Experience-centered Approach and Visuality in The Education of Mathematics and Physics, Ed.: Jablan Slavik et al., Kaposvar: Kaposvar University, pp.195-197. (2012)

Oh it looked, as if you wanted to publish it somewhere else. Sorry I have no library account, so I have only your video to look at. It looks as if your Epitahedron has 6 vertices. Is that correct?

I guess the answer needs somebody who is familiar with Group Theory AND Geometry like Plato, Galois, F. Klein or Bertram Kostant,- or somebody else strumbling along

THIS blog ?

Yes seems so.

]]>Thanks, for your reply :

The epitahedron E+ shares 5 vertices with the icosahedron.

Related findings are published already:

Visualisation of the icosadedral group in a new 3D representation, In: Experience-centered Approach and Visuality in The Education of Mathematics and Physics, Ed.: Jablan Slavik et al., Kaposvar: Kaposvar University, pp.195-197. (2012)

I guess the answer needs somebody who is familiar with

Group Theory AND Geometry like Plato, Galois, F. Klein or Bertram Kostant,- or somebody else strumbling along

THIS blog ?

As I understood you call this shape in the center an Epitahedron. Unfortunately it is a bit hard to see in the video, wether all its vertices always end up on vertices of the Icosahedron, so given this I unfortunately find it really hard to help you further than with this. sorry.

But other people here may eventually see more than me in this video, it looks nice.

Eventually the referees of the paper you are submitting to might also give you hints.

]]>here we found how an icosahedral group evolves towards the great dodecahedron:

http://quantumcinema.uni-ak.ac.at/site/research/topology-of-lie-algebra/

Is it possible to specify this subgroup on the what we call the

“rosetta path”?

We would be greatful for any hints !