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4D Dual-morphing

 
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David



Joined: 05 Jun 2011
Posts: 1

PostPosted: Sun Jun 12, 2011 7:15 am    Post subject: 4D Dual-morphing Reply with quote

Hi! I'm new here. I like polytopes, and stella program! Smile

I have idea for 4D dual-morphing.

4D morph by truncation method is...

parent -> truncated -> rectified -> bitruncated -> rectified dual -> truncated dual -> dual

And, 4D morph by expension (morph with rectangles) method is...
parent -> expanded (in 4D, it also be runcination) -> dual

If you can't understand, see here.
[url]http://en.wikipedia.org/wiki/Uniform_polychoron#Geometric derivations for 46 nonprismatic Wythoffian uniform polychora[/url]
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daniela



Joined: 03 Oct 2010
Posts: 6

PostPosted: Sun Jun 19, 2011 1:48 pm    Post subject: Reply with quote

I second that, in fact, I logged in because I was looking for this feature Smile
Daniela
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robertw
Site Admin


Joined: 10 Jan 2008
Posts: 394
Location: Melbourne, Australia

PostPosted: Fri Jun 24, 2011 2:33 pm    Post subject: Reply with quote

I thought about 4D dual morphing before, but haven't got around to it yet. I guess all you'd be able to see during the morph is the 3D projection, ie not cross-sections or nets, though you could convert a morph to be the new base model to obtain those.

For morph by rectangles (maybe I should rename this morph by expansion?), we have the following:

  • each vertex becomes a new small dual cell that grows
  • each cell shrinks to nothing, becoming a dual vertex
  • each edge becomes a long thin cell that grows wider and shorter till it becomes a dual face
  • each face expands into a short prism that grows longer and thinner till it becomes a dual edge

That's probably the easiest one. Morph by tilting quads may not be too hard either if implemented as the dual of a morph by rectangles.

Morph by truncation is trickier to do in a general way (ie not limited to uniform polytopes), as there is no rectified form in general, so I don't rely on that in 3D.

I'll probably look into it sometime.

Thanks,
Rob.
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