Tetrahedral gyrobicupola

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

Tetrahedral gyrobicupola

Postby quickfur » Thu Dec 02, 2010 12:14 am

Just as 3D cupolas can be thought of as the "expansion" of a polygon (start with the top face, expand its edges outward while moving downwards: it traces out the volume of the cupola), we can make 4D cupolas by starting with a cell and expanding it downwards, resulting in a polytope with the original polyhedron as the top cell, its cantellated counterpart as the bottom cell, and a bunch of prisms and other polyhedra in between.

For example, the tetrahedral cupola is formed by starting with the top tetrahedron, expanding it to a cuboctahedron (=runcinated tetrahedron), while filling in the gaps with triangular prisms and other tetrahedra. Specifically, the cupola consists of 5 tetrahedra, 10 triangular prisms, and a cuboctahedron.

Now a tetrahedral bicupola is formed by attaching two such cupola base-to-base. However, there are two ways of fitting the cupola together: either in such a way as to have 4 tetrahedra in one cupola share a face with 4 tetrahedra in the other cupola, which gives the orthobicupola (having bilateral symmetry), or rotated in complementary orientation so that no two tetrahedra share a face, which gives the gyrobicupola.

Now the latter, the tetrahedral gyrobicupola, has some rather interesting properties... which I'll refrain from stating for now, just to see if anyone else guesses it. :-) (Here's a hint: what's a triangular gyrobicupola?)
quickfur
Pentonian
 
Posts: 2935
Joined: Thu Sep 02, 2004 11:20 pm
Location: The Great White North

Re: Tetrahedral gyrobicupola

Postby Keiji » Sun Dec 05, 2010 12:31 pm

Is it a runcinated 5-cell?
User avatar
Keiji
Administrator
 
Posts: 1984
Joined: Mon Nov 10, 2003 6:33 pm
Location: Torquay, England

Re: Tetrahedral gyrobicupola

Postby quickfur » Sun Dec 05, 2010 3:49 pm

Yep!

And furthermore, just as the mid cross-section of the runcinated 5-cell is in the shape of a cuboctahedron (as a "runcinated tetrahedron"), so the mid cross-section of the runcinated hexateron (5-simplex) is in the shape of a runcinated 5-cell, and it, in turn, is the shape of the mid-cross section of the runcinated heptapeton (6-simplex), and so forth. The runcinated n-simplex in each dimension derives from the symmetry of the n-simplex, and since the n-simplex is self-dual, the n-simplex cupola will always have 2n facets in the shape of n-simplices, n of which corresponds with (n+1)-simplices, and the other n with n-simplex prisms. The symmetry guarantees that we can always fit the n facets belonging to the (n+1)-simplices in one cupola to the n facets belonging to the n-simplex prisms in a copy of the cupola. In other words, because the n-simplex is self-dual, the gyro part of the gyrobicupola is always well-defined (whereas for other cupola, there may not be a well-defined "gyro"bicupola because the two cupola won't fit together in dual orientation). Fitting such cupola together always produces a runcinated (n+1)-simplex.
quickfur
Pentonian
 
Posts: 2935
Joined: Thu Sep 02, 2004 11:20 pm
Location: The Great White North

Re: Tetrahedral gyrobicupola

Postby Keiji » Sun Dec 05, 2010 6:21 pm

By runcinated, I assume you mean xon-2x, so xx, xox, xoox, xooox etc. - this is cantellation in 3D of course, and probably something else in 5D!

The series could do with its own name, like mesotruncate for the oxo, oxxo, ooxoo series...
User avatar
Keiji
Administrator
 
Posts: 1984
Joined: Mon Nov 10, 2003 6:33 pm
Location: Torquay, England

Re: Tetrahedral gyrobicupola

Postby quickfur » Sun Dec 05, 2010 9:12 pm

Sigh I hate the current messy terminology. Maybe instead of using numbers or hard-to-remember arbitrary prefixes, we should just write xoxoxo exactly according to the Coxeter-Dynkin symbol.

What I mean by "runcinated" n-simplex, which is probably not exactly what others mean by it (which is why I'm fed up of this terminology... nothing about it tells you exactly what it is, and every dimension is different), is ... gah, I might as well start using the new terminology now ... the xoo...ox-aerotomes. Or using your notation, xo+x-aerotomes.

I hope this is clear(er).
quickfur
Pentonian
 
Posts: 2935
Joined: Thu Sep 02, 2004 11:20 pm
Location: The Great White North

Re: Tetrahedral gyrobicupola

Postby Keiji » Sun Dec 05, 2010 9:49 pm

Yes, I hate the current terminology too, thought I was the only one :sweatdrop:

I knew what you meant from your previous post, I just figured I'd point out that (as much as I'd like it not to) runcinate means xooxo+.
User avatar
Keiji
Administrator
 
Posts: 1984
Joined: Mon Nov 10, 2003 6:33 pm
Location: Torquay, England

Re: Tetrahedral gyrobicupola

Postby wendy » Tue Dec 07, 2010 10:15 am

Hi

I use words like 'runcinate', 'cantelate' etc, to describe specific processes, different to the sum of nodes of the dynkin symbol. Moreover, replacing the x with m creates the dual. Since writing this is essentially the same symbolic form of the conway-hart system, one can use the same for both.

Conway-Hart notation assumes the base polytope is taken as regular, and that all of the x-o operations happen in every flag (and transmitted by "flipping" the flag over).

runcinate refers to marking the first and last node only. It corresponds to the stott-style face-expansion.

n-rectate corresponds to the marking of a single node (counting the end at 0). It corresponds to a vertex at the centre of a surtope (n), eg n=1 gives edge.

truncate corresponds to the sum of adjacent rectates.

cantetruncate, refers to marking all of the nodes connected to a given node. It corresponds to the rectate of the figure with the marked node. Including the base node makes for the cantetruncate.
The dream you dream alone is only a dream
the dream we dream together is reality.

\ ( \(\LaTeX\ \) \ ) [no spaces] at https://greasyfork.org/en/users/188714-wendy-krieger
User avatar
wendy
Pentonian
 
Posts: 2014
Joined: Tue Jan 18, 2005 12:42 pm
Location: Brisbane, Australia

Re: Tetrahedral gyrobicupola

Postby Keiji » Tue Dec 07, 2010 1:22 pm

wendy wrote:runcinate refers to marking the first and last node only. It corresponds to the stott-style face-expansion.


Um, no it doesn't?

The "standard" "accepted" way of doing things is to use the same names for 3D abc and 4D abco. The tables on Wikipedia will tell you that.

If what you said was correct - and I wish it was - then either 3D's xox (cantellate) would be called runcinate, or runcinate would be called cantellate too.
User avatar
Keiji
Administrator
 
Posts: 1984
Joined: Mon Nov 10, 2003 6:33 pm
Location: Torquay, England

Re: Tetrahedral gyrobicupola

Postby wendy » Wed Dec 08, 2010 8:20 am

I copied the terms from Norman Johnson, and reallocated them. This is what the polygloss gives.

Actually, the desired alignment is this. Note that rr and rt are the symbols for xoox and xxox respectively.

rr = recto-rectate (cantellate) ie ..xox... (all others o)
tr = recto-truncate (cantetruncate) ..xxx... (all others o)

Still, I use constructions in the pseudoregular trace (eg Dt9 etc), because this leads to short names while still preserving the stott-sum.
The dream you dream alone is only a dream
the dream we dream together is reality.

\ ( \(\LaTeX\ \) \ ) [no spaces] at https://greasyfork.org/en/users/188714-wendy-krieger
User avatar
wendy
Pentonian
 
Posts: 2014
Joined: Tue Jan 18, 2005 12:42 pm
Location: Brisbane, Australia


Return to Other Polytopes

Who is online

Users browsing this forum: No registered users and 18 guests

cron