Defining polyshapes

[Keiji]

Well, I just had a thought. Any polygon can be defined by a list of coordinates (x,y), based on their order.

But polyhedra can't be defined in this way - you would need a net* (2-d list) of coordinates (x,y,z) instead of just a list.

So, by analogy, to define polychora, you would need a 3-d list of coordinates (w,x,y,z).

Another way of defining polyhedra is to define them as a list of polygonal faces, and then define all the polygons required.

So, by analogy, the other way to define polychora is to define them as a list of polyhedral cells, and then define all the polyhedra required.

*A net can be defined by a list of data, one item for each item in the net, plus a list of connections, of which each item defines two items in the first list which are connected to each other.

... Hey, I've just got yet another idea for my office suite

[Aale de Winkel]

As a last contribution prior to a 1 1/2 week vacation I uploaded formulae for various hyperobjects onto my private page:
http://home.wanadoo.nl/aaledewinkel/Enc ... hapes.html
Note that this is currently priliminairy at best to be augmented next year.

The page will mention the various shapes by formulea, a future uploade might holf relevant links onto probably Polyhedron Dude's site, and will be the place where I'll try to augment the formulae listed in the first glossary posting.

not yet seen nor defined anywhere here a genaralisation of the n-Cone:
x[sub]k[/sub] x[sup]k[/sup] = x[sub]l[/sub] x[sup]l[/sup]; k < l <= n

a kind of sphere-balncing through the dimension

[Keiji]

Where did I say anything about formulas? idiot :x

[Aale de Winkel]

Where did I say anything about formulas? idiot :x

No You didn't I do, I'm not in the habit of starting threads or post single sentence postings merely for incrementing the posting count, the other forum is filled with thesde kind of posting and it is getting hard to find the more informative postings.
I merely found the thread you started a convenient relevant place to notify you all to the private page I added.

I just uploaded a further version of the page, mentioning all the regular polyaeder(with a private convenient notation), the planar stars I already defined in the context of magic stars.
The formulae section Iaugmented with the n-p-multispare of which the discussed tatraspace duocircle is the first example.
(A further generalisation of these formulae includes the cones, hwever I currently have not yet found a formaltion of this yet.

The added section of "operations" reflect the elswhere defined discussions of operations, the various listed samples are illustrative of these operations, but perhaps needs some rechecking as prior to the last upload I noticed that a lethed line forms the disc and not the circle.

the varous shapes operated on there I mentioned there by bracketed names, these names not solidly defined, but I reckon the names are used as they usualy reflects the shape.
As I don't know whether the "ring" is usually used for the subtraction of two circles or of two discs I used (open ring) for the first and (filled ring) for the latter.

[pat]

Well, I just had a thought. Any polygon can be defined by a list of coordinates (x,y), based on their order.

But polyhedra can't be defined in this way - you would need a net* (2-d list) of coordinates (x,y,z) instead of just a list.

The convex polychorons can be defined as the convex hull of a set of vertices. This places no requirement on ordering or specifying a net. A concave polychoron could be expressed as a union of convex polychorons.

I'm wondering if there are better ways though. I'm thinking that given a set of vertexes, there could be several concave polychorons around them. I was hoping that if one specified a set of vertexes, that the polychoron of minimal "surface area" with each of those vertexes actually on the "surface" would uniquely specify (up to rigid transformation) a polychoron. Aside from the fact that not all concave polychorons have minimal surface area, I'm certain that not all sets of vertexes would have a unique minimal surface.

Hmmm....