Talk:Simplex

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==Volume formulas== this is not true

The formula given for the oriented volume of an n-simplex in n+1-dimensional space with vertices (v0, ..., vn),

{1\over n!}\det \begin{pmatrix} v_0-v_1 & v_1-v_2& \dots & v_{n-1}-v_{n} & v_n-v_0 \end{pmatrix}

appears to be spectacularly false, since,

Vol_{n-simplex} = {1\over n!}\det \begin{pmatrix} v_0-v_1 & v_1-v_2& \dots & v_{n-1}-v_{n} & v_n-v_0 \end{pmatrix}= {1\over n!}\det \begin{pmatrix} v_0-v_1 & v_1-v_2& \dots & v_{n-1}-v_{0} & v_n-v_0 \end{pmatrix},

where equality follows from alternating multilinearity of the determinant, i.e., adding a multiple of one column (resp. row) to another column (resp. row) does not change the value of the determinant. Continuing this, telescoping all the way to the farthest left row, we obtain:

Vol_{n-simplex}=\dots={1\over n!}\det \begin{pmatrix} v_0-v_1 & v_1-v_0& \dots & v_{n-1}-v_{0} & v_n-v_0 \end{pmatrix}={1\over n!}\det \begin{pmatrix} 0 & v_1-v_0& \dots & v_{n-1}-v_{0} & v_n-v_0 \end{pmatrix}=0.

I am unwilling to be persuaded that the oriented volume of every n-simplex in n+1-dimensional space is 0.

For the skeptical, one may quickly check this for the 1-simplex in 2-space, v0 = (0,0),v1 = (1,1). The length is \sqrt 2\neq 0.

Kneedan (talk) 18:53, 2 February 2008 (UTC)

I propose the following edit: The oriented volume of an n-simplex in n-dimensional space with vertices (v0, ..., vn) is

\pm{1\over n!}\det \begin{pmatrix} v_0 & v_1& \dots & v_{n-1} & v_n\\ 1 &1&\dots&1&1 \end{pmatrix}

where each column of the n+1 × n+1 determinant is one of the vi written as a column vector,v_i=(v_{i1} \dots v_{in})^t with a 1 appended at the bottom of the column.

Kneedan (talk) 22:56, 2 February 2008 (UTC)

I thought the formula was:
{1\over n!}\det \begin{pmatrix} v_1-v_0 & v_2-v_0& \dots & v_{n-1}-v_{0} & v_n-v_0 \end{pmatrix}

Tom Ruen (talk) 23:56, 2 February 2008 (UTC)

I should have posted here after making the change. Indeed, Kneeden was (and is) quite right in pointing out the error in the formula. I changed the formula to give at least one correct formula (for the volume of an n simplex in Rn.) However this was not what the text purported to give originally, which was the n-volume of an n-simplex in Rn+1 (i.e. a codimension 1 simplex). Kneedan's new formula is also correct (up to a sign which needs to be checked). However, neither of these gives the n-simplex in Rn+1. For that, I think we need a more complicated construction like
\frac{1}{n!}\sqrt{\det A^TA}
A=\begin{bmatrix}v_1-v_0&v_2-v_0&\dots& v_n-v_0\end{bmatrix}
This formula, suitably understood, also works for simplices of arbitrary codimension. Silly rabbit (talk) 00:24, 3 February 2008 (UTC)

Contents

[edit] Linear independence, general position

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)

Insertformulahere

[edit] References

Having a look at the first referenced book "Principles of mathematical analysis", chapter 10 is headed "Integration of Differential Forms", so I don't see the relationship to topology and simplexes. Maybe it should be chapter two, "Basic topology"!? 87.78.67.128 (talk) 11:15, 6 March 2008 (UTC)

No, Chapter 10 is correct. Chapter 2 deals with point set topology, not the topology of simplexes. Chapter 10 deals with simplexes, since these are of fundamental importance to dealing with the integration of differential forms. Silly rabbit (talk) 12:50, 6 March 2008 (UTC)