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Creating convex polychora from vertices

 
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Jabe



Joined: 12 Jan 2008
Posts: 46
Location: Somewhere between Texas and the Fourth Dimension

PostPosted: Sat Jan 12, 2008 8:32 am    Post subject: Reply with quote

Stella4D is a great program Cool - One feature I would like to see added is the ability to generate a convex polychoron, by using only the vertices - sort of a convex hull feature.
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robertw
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Joined: 10 Jan 2008
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Location: Melbourne, Australia

PostPosted: Sun Jan 13, 2008 5:23 am    Post subject: Reply with quote

Jabe wrote:
One feature I would like to see added is the ability to generate a convex polychoron, by using only the vertices - sort of a convex hull feature.


Ah, this is already possible! Just select "Poly->Create Convex Hull" from the menu with a 4D polytope loaded.

Rob.
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Jabe



Joined: 12 Jan 2008
Posts: 46
Location: Somewhere between Texas and the Fourth Dimension

PostPosted: Mon Jan 14, 2008 7:28 am    Post subject: Reply with quote

robertw wrote:
Jabe wrote:
One feature I would like to see added is the ability to generate a convex polychoron, by using only the vertices - sort of a convex hull feature.


Ah, this is already possible! Just select "Poly->Create Convex Hull" from the menu with a 4D polytope loaded.

Rob.


What I'm refering to, is taking a list of vertices (but you dont have a polychoron loaded) and finding the convex hull - as opposed to writing a huge 4D OFF file describing edges, faces, and cells.
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robertw
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PostPosted: Tue Jan 15, 2008 5:04 am    Post subject: Reply with quote

Jabe wrote:
What I'm refering to, is taking a list of vertices (but you dont have a polychoron loaded) and finding the convex hull - as opposed to writing a huge 4D OFF file describing edges, faces, and cells.


Oh, OK. I can probably add that sometime. If a 4D OFF is loaded with only vertices, it should ask if you want to create the hull instead.

Rob.
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hedron



Joined: 28 Jan 2008
Posts: 2
Location: Wales and Thailand

PostPosted: Mon Jan 28, 2008 2:06 pm    Post subject: Reply with quote

As far as 4D OFF is concerned, they can be read in without cells or edges being defined, as this is what HEDRON produces. It does fail without faces though.

A 3D OFF with vertices only does work as that was the method I used to generate a number of the sub-zonohedrifications of the dodecahedron at http://www.orchidpalms.com/polyhedra/rhombic/rh90/rh90.htm

Jim
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robertw
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PostPosted: Mon Jan 28, 2008 2:55 pm    Post subject: Reply with quote

Actually there is no way to define edges in an OFF file, but yes you often don't need to define cells (although sometimes you do).

I'll make it possible to load a vertex-only 4D OFF file at some point. If you just want a convex hull of the verts, you're not going to want to figure out how to link them all up with faces first.

Speaking of zonohedra, I've been working on improving Stella's support for zones Wink

Rob.
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oxenholme



Joined: 16 Jan 2008
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Location: North West England

PostPosted: Tue Jan 29, 2008 5:48 am    Post subject: Reply with quote

hedron wrote:
As far as 4D OFF is concerned, they can be read in without cells or edges being defined, as this is what HEDRON produces. It does fail without faces though.

A 3D OFF with vertices only does work as that was the method I used to generate a number of the sub-zonohedrifications of the dodecahedron at http://www.orchidpalms.com/polyhedra/rhombic/rh90/rh90.htm

Jim


I love that first zonohedron and would like to make one from card. What are the angles or diagonals of the two types of rhombi?
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robertw
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PostPosted: Tue Jan 29, 2008 6:08 am    Post subject: Reply with quote

That's the zonohedrification of the dodecahedron. Just load a dodecahedron in Stella and hit "z" (or "Poly->Zonohedrify").

Stella can then print nets for you, and show angles if you need them. Here's one I made out of paper:
http://www.software3d.com/Enneaconta.php

Rob.
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oxenholme



Joined: 16 Jan 2008
Posts: 83
Location: North West England

PostPosted: Tue Jan 29, 2008 12:45 pm    Post subject: Reply with quote

robertw wrote:
That's the zonohedrification of the dodecahedron. Just load a dodecahedron in Stella and hit "z" (or "Poly->Zonohedrify").

Stella can then print nets for you, and show angles if you need them. Here's one I made out of paper:
http://www.software3d.com/Enneaconta.php

Rob.


It looks good! I had in mind using the standard five colours for each of the twelve groups of rhombi and a sixth colour for the narrow rhombi whose long diagonals coincide with the dodecahedral edges.

If I use double tabbing throughout and 160gsm card am I likely to get the problem with rigidity? I had in mind about 1.5 to 2.0cm edge length...
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robertw
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PostPosted: Wed Jan 30, 2008 3:18 am    Post subject: Reply with quote

I used (and almost always use) 120gsm paper. I found 160 was just a bit bulky, but weight doesn't always correlate to thickness. Actually I think the more recent paper I got was 140gsm which was fine.

My model has edge length 2cm, giving the model a diameter of about 10.5cm. Yes, it could have used some internal support for extra rigidity! When picking the model up, a vertex will often push in if I'm not very careful. Then it can be tricky to get out again.

I would suggest a single strut across each 3-fold symmetry vertex would make a big difference, as shown below. You can create these using faceting mode. Once the facet is created it will appear as an extra net.



Rob.
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Nordehylop



Joined: 27 Feb 2008
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Location: Illinois, USA

PostPosted: Thu Feb 28, 2008 1:01 am    Post subject: Reply with quote

I recently made a model of this polyhedron with 6 colors in the way you described, oxenholme. It turned out fairly well with 65 lb (about 245 gsm) card. Mine had an edge length of about 4.318 cm (the length of the short diagonal of the slim rhombus was 1 inch).

It is, as Rob said, a very delicate model. My little sister dropped it and a section caved in, but I was able to pop it back out. I like the extra strut idea. You could possibly even just extend the tabs on some of the narrow rhombi out farther and have them grip each other.

This is a model where you want to be as precise as possible. I ended up with a hole about 1/8" larger than I needed for the last part. Possibly you could prepare an "endcap" of 5 wide and five narrow rhombi to put on as the last part.
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dysphras



Joined: 10 Jun 2008
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PostPosted: Wed Jun 11, 2008 11:08 am    Post subject: polychora from OFF vertex coordinates and cell description Reply with quote

Quote:

Stella4D is a great program - One feature I would like to see added is the ability to generate a convex polychoron, by using only the vertices - sort of a convex hull feature.
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I was just dreaming about the same matter. I would love to see that ability implemented, too.
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Jabe



Joined: 12 Jan 2008
Posts: 46
Location: Somewhere between Texas and the Fourth Dimension

PostPosted: Tue Dec 09, 2008 11:31 pm    Post subject: Reply with quote

The potential "generate polychoron/polyhedron from vertices" feature could also use a set of selected vertices from a model as well as an OFF file - lets say I wanted to generate the heptachoron - a step tegum, I could load up the 7-7 duoprism, select the correct 7 vertices - then activate the convex hull using vertices feature, then look at the dual - viola!

This is a must have feature Wink .
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robertw
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PostPosted: Wed Dec 10, 2008 9:55 am    Post subject: Reply with quote

Or maybe rather than selecting 7 vertices, there could be a feature to generate the convex hull minus the currently selected vertex. Then you would be able to delete one at a time and see the effect.

For now, you could create a faceting using just the vertices you want, then create the convex hull.

Rob.
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Jabe



Joined: 12 Jan 2008
Posts: 46
Location: Somewhere between Texas and the Fourth Dimension

PostPosted: Thu Dec 11, 2008 7:37 am    Post subject: Reply with quote

Here's a good way to impliment the remove vertices feature:

left click on vertex - removes vertex, finds convex hull of remaining verts

right click on vertex - highlights vertex to allow many to be removed at same time

left click on vertex after several are highlighted - removes vertex and all highlighted ones - convex hulls the rest

shift-left click on vertex after several are highlighted - removes all but the highlighted vertices and the one selected - convex hulls the survivors.

The dual process might be worth doing also:

left click on face/cell - removes face/cell, finds convex core of remaining

right click on face/cell - highlights it

left click face/cell after some are highlighted - removes it and the highlighted one - find convex core of remaining

shift-left click face/cell after several are highlighted - removes all but the selected one and highlighted ones - convex core the ones left.

The reason why we may need multiple vertex removal would be because after removing one vertex at a time, the original shape would be so chopped up we may get disoriented as to which vertex needs to be removed next - but with multiple removal we could still see the original structure as we highlight the vertices to be removed.

Make sure to make it possible for 4-D as well as 3-D, in 3-D we have the faceting feature to fall back on, but not so in 4-D yet - so the heptachoron (and many other exotic 4-D dice polytopes) appears to be "unStellable" at the current time - but that would change with the vertex removal feature.
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