The result generalizes to any n-dimensional simplex in the obvious way. If the set of vertices of a simplex is v0,...,vn, then considering the vertices as vectors, the centroid is C = \frac{1}{n+1}\sum_{i=0}^n v_i.
--- In Polytopia@yahoogroups.com, Adrian Rossiter <adrian_r@...> wrote:
> Alan is right about this number being related to dimension. In > an n-dimensional pyramid the centroid is at height 1/(n+1)
> This means that the dual of a regular symplex making vertex > to cell contact will have 1 - 1/(n+1) of its height equal > to 1/(n+1) of the larger simplex's height, giving a large > to small ratio of > scale ratio = (1 - 1/(n+1)) / (1/(n+1)) > = ((n+1) - 1) > = n
... I think that he's talking about the "concave rhombic dodecahedron" that's formed by the cube's diagonals. ... The box bounds the half of the cube that...
The result generalizes to any n-dimensional simplex <http://en.wikipedia.org/wiki/Simplex> in the obvious way. If the set of vertices of a simplex is...
Hi Alan ... It may be a bit difficult to pick out the individual polyhedra in my model as the triangle faced polyhedra all have the same colour struts. What I...
I should have read through a bit further. Here is the relationship stated, hopefully, a bit more clearly... Take each face of an icosahedron and divide its...
the [3-fold] apexes of the tetrahedra that are in the same positions as … the [3-fold] vertices of the cube (hexahedron) that are in the same positions as...
... Here is an image that illustrates this better http://www.freewebtown.com/adrian/tmp/ch_cfg02.jpg The parts of the tetrahedron which are external to the ...
... and also face-to-face (in case it adds confusion if I don't mention it). Adrian. -- Adrian Rossiter adrian@... Home: http://antiprism.com/adrian...
... The 3-valent VERTICES of the stella octangula make contact with the 3-valent triangle faces of the octahedron or icosahedron. The 3-valent FACES of the...
Could you tell me if you know of any Visual Basic related code snippets available online for creating models of geometric shapes, solids and rotating them? ......
... In my prefabricated conceptual cosmic heirarchy yes the Pen-Dod. would be inside the Ico. Cube inside of Oct. Does the geometry of equal unitary edged...
Hi Adrian Thanks for commenting on the Great Dodecahedron file. All was gererated in Turbocad v9 Pro. The Great Dodecahedron was formed by sectioning a...
Hi Adrian In reference to your observations, I should explain a little further that the Great Dodecahedron is contained in a glass icosahedron with metal...
Hi Adrian That is an interesting link, never saw those at that sire before. This is a truely amazing field to delve into - polyhedron construction, morphing,...
Hi Craig ... Do you mean a formula to turn a smooth surface into a mesh, like this sort of thing? http://gts.sourceforge.net/gallery.html Or something like the...
Hi Adrian There is sure a lot of activity around here to day I would like to know if it is possible to create a polyhedron from a mathematical surface from a...
Hi Craig ... The surface of a polyhedron is made from polygons, which lie on sections of planes. The polygons may cover parts of their section more than once...
Hi Craig ... I meant to say, the book I mentioned the other day, The Geometrical Foundation of Natural Structure by R Williams, has a lot of ideas for this...
http://mathworld.wolfram.com/GreatDodecahedron.html http://mathworld.wolfram.com/Cumulation.html http://mathworld.wolfram.com/Invaginatum.html Adrian Rossiter...
Thanks Allan for that terminology. This is a learning forum. Invaginatum...I love that term. Is it true that for every Elevatum there is an equal and opposite ...
... Now put the cursor on the image in the MIDDLE, and you will see the octahedron "grow" to a rhombic dodecahedron "grow" into a stella octangula! If you...
The great dodecahedron <http://groups.yahoo.com/group/Polytopia/files/Craig%27s%20Polytopes/Gre\ at%20Dodecahedrons03.png> appears to rest on a pattern with a...
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