Talk:Simplex

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Hello, AxelBoldt! I see that you replaced "linearly independent points" with "points in general position in some Euclidean space". I'm not sure what you mean by "in general position" - is this a technical expression? Is it more accurate than saying the points have to be linearly independent? Thanks for any clarification you can give in this matter! -- Oliver Pereira 23:11 Nov 23, 2002 (UTC)

"Linearly independent" is technically incorrect: for example, the points (1,1), (1,0), (0,1) in R^2 are linearly dependent, but they span a 2-simplex. If you require linear independence, there won't be any 2-simplices in R^2.

There is probably a technical definition of "in general position", but I don't know it. Typically, the term is used to describe points that don't satisfy "more equations than necessary"; for instance if you have four points that all lie on a circle, or three points that all lie on a line, then they wouldn't be in general position.

It's probably not the best term to use here. Maybe we should go with the formally correct "affinely independent", which precisely means what we want: any m-plane contains at most m+1 of the points. AxelBoldt 19:46 Nov 24, 2002 (UTC)

Oh, of course! Silly me. I clearly wasn't thinking straight about the linear independence thing. It was late, after all. :) Thanks for clearing up my confusion. -- Oliver Pereira 21:01 Nov 24, 2002 (UTC)

I don't get this. Shouldn't it say that

∑	ti
i

is less than or equal, not equal to 1 in the geometric definition?

No, the n-simplex is given as a subset of Rⁿ⁺¹ not Rⁿ. It must therefore lie in an n-dimensional (affine) hyperplane. -- Fropuff 20:13, 2005 May 23 (UTC)

I'd just like to give a big THANK YOU to whoever described a k-chain as a set instead of a formal linear combination. Now that I (finally) know what is meant by a "formal linear combination," I'm kind of disgusted that such an obfusticated term exists for such a simple thing!

[1] This link might fit well on this page? TallAlex 16:28, 25 March 2006 (UTC)

Could the term "unit simplex" used under the section of random sampling please be clarified. Is this the same as "standard simplex" referred to elsewhere in this article? -- 2 August 2006.

"A regular n-simplex may be constructed from a regular (n − 1)-simplex by connecting a new vertex to all original vertices by the common edge length."

That only gives you the (n-1) simplex plus n line segments extending from the new vertex to the original vertices. You need to join the new vertex to every point in the the (n-1) simplex to create a new simplex.--129.15.228.164 00:06, 28 August 2006 (UTC)

[edit] Names for higher dimensional simplices

Who was the first to use words like hexateron? Do they appear in any scholarly publications? —Keenan Pepper 04:11, 4 September 2006 (UTC)

Hexa- is a standard prefix for a 6-faceted polytope.

The term polyteron is a proposed term for 5-polytopes comes from the same group authors as polychoron for 4-polytope, the active researchers into classifying higher dimensional polytopes.

Jonathon Bowers: [2], George Olshevsky: [3], Guy Inchbald: [4], Wendy Kreiger: [5]

I have yet to see a printed resource that offers specific dimenstional names for 4-polytopes or higher.

1. Branko Grünbaum's book Convex polytopes uses dimensional terms: d-polytope, d-simplex, d-cube, d-crosspolytope, d-prism, d-pyramid, d-bipyramid.
2. T. Gosset: On the Regular and Semi-Regular Figures in Space of n Dimensions, Messenger of Mathematics, Macmillan, 1900

   He calls an n-simplex a 'n-ic pyramid'.

So that's where I'm at for sources. Tom Ruen 04:50, 4 September 2006 (UTC)

Tom Ruen 04:50, 4 September 2006 (UTC)

Well, I'm against using these words, but I won't make a fuss about it. —Keenan Pepper 05:26, 4 September 2006 (UTC)

[edit] Graphs

The graph for the Tetrahedron seems to be wrong. No projection of a Tetrahedron results into a square.

No, a tetrahedron can project onto a square. A tetrahedron has four vertices. If you project each onto a different corner of a square, you get a square. —Ben FrantzDale 13:29, 20 December 2006 (UTC)

The correct graph should show an isosceles triangle with three line segments running from its vertices to a point at the centre of the triangle. -- Ross Fraser 06:19, 13 January 2007 (UTC)

There's many different ways to show simplex graphs. The graphs shown are not projections, but simply complete graphs of n+1 points on a circle. However at least for the tetrahedron, a square with two diagonals is an actual orthographic projective view of a tetrahedron as viewed along the center of two opposite edges. Tom Ruen 10:38, 13 January 2007 (UTC)