Talk:Simplex

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Volume formulas

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

{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, $v_{0}=(0,0),v_{1}=(1,1)$ . The length is ${\sqrt {2}}\neq 0$ .

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

I propose the following edit: The oriented volume of an n-simplex in n-dimensional space with vertices (v₀, ..., v_n) 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 $v_{i}$ 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)[reply]

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)[reply]

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^{T}A}}

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)[reply]

I want to criticize the description of the origin of the 1/n! factor in terms of sorting. The vectors v_1-v_0 , ..., v_n-v_0 are not necessarily the closest vertices of the parallelepiped to the origin. A counterexample exists in as few as 2 dimensions: if v_1-v_0 and v_2-v_0 are equal in length and the angle between them is > 2pi/3 (120 degrees), then v_1+v_2-2v_0, the remaining vertex of the parallelogram, is a shorter vector than either of v_1-v_0 or v_2-v_0. DavidLHarden (talk) —Preceding undated comment was added at 18:43, 25 July 2008 (UTC)[reply]

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 $\sum _{i}t_{i}$ 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)[reply]

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)[reply]

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)[reply]

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.

Branko Grünbaum's book Convex polytopes uses dimensional terms: d-polytope, d-simplex, d-cube, d-crosspolytope, d-prism, d-pyramid, d-bipyramid.
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)[reply]

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

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

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)[reply]

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)[reply]

$Insertformulahere$

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)[reply]

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)[reply]

-1 simplex

I extended the table to include the -1-simplex, because it is just as well defined as n-simplices for higher n, because without it, pascal's triangle is incomplete, and because it is essential for an elegant definition of reduced singular homology. sephia karta 23:47, 8 October 2008 (UTC)[reply]

I have a few objections to the addition. First and foremost among my concerns is that it makes a table which is already too long even longer. Moreover, since the −1-simplex is not a particularly standard thing, putting it in the very first row and column is quite confusing. In fact, I think it makes it better to do away with the table altogether. The primary purpose of the table is, it seems to me, to provide useful information, along with links to other Wikipedia articles on various simplices. We don't have an article on the −1-simplex.

My second objection is more firmly rooted in the Wikipedia verifiability policy. The table was, if I recall, adapted from the Coxeter's description of the n-simplex (including keeping consistency with his notation). Coxeter does not treat the −1-simplex.

That said, I would not object to a line or two (with references) stating that the −1-simplex can be defined, and indicating briefly why one might want to do it. However, I don't see it as deserving such a prominent place in the article. By the way, most treatments of singular homology do not use −1-simplices. (The boundary operator is simply taken to terminate on 0-simplices. The emptyset is not regarded as −1 dimensional, etc.) siℓℓy rabbit (talk) 01:31, 9 October 2008 (UTC)[reply]

I have to agree, doesn't help the article to add this column to the table. One wiki reference I can find for -1 is in Abstract_polytope#Example. So, if there was some application of abstract simplicies perhaps a section explaining this usage and value might be interesting? Tom Ruen (talk) 19:58, 9 October 2008 (UTC)[reply]

I agree that it the -1-simplex should not get a prominent place in the article, but that is not what this is about. I think we disagree what the purpose of the table is. To me it seems that its purpose is to show how the numbers of the faces of n-simplices follow Pascal's triangle. And Pascal's triangle is simply incomplete without the column of the -1-simplex. If the table is really too big (I don't think it is), removing the 10-simplex and the 9-simplex would be a much better idea, because it is clear how Pascal's triangle goes on once one has seen how it starts.

Concerning the source of the table, perhaps this is different for the 3rd edition of Coxeter's book, but at least in the first edition this table does not feature in this form, so it is an adaptation anyway. sephia karta 12:01, 10 October 2008 (UTC)[reply]

The verifiability issue isn't whether the table appears in Coxeter, but whether Coxeter recognizes the −1-simplex, which he does not. As for including the −1-simplex in order to complete Pascal's triangle, I find that a very weak reason for inclusion. First, I don't think it is necessary to have a complete Pascal triangle at any cost here, and secondly, it is borderline OR to insist on using a non-standard definition in order to achieve this. siℓℓy rabbit (talk) 12:19, 10 October 2008 (UTC)[reply]

It is true that Coxeter does not recognise the -1-simplex, but since the table is not his, we don't need to follow him. You are right that the -1-simplex is not considered a face of a polytope by all authors, but for one that does, see for example [6], pages 8 and 9. It is also stated in the article on Convex polytopes.

I think that the text that precedes the table should anyway be updated to state that some authors consider the empty set to be a face of an n-simplex and some don't. And if my inclusion of the -1-simplex in the table is kept, there should be a qualifying statement to this extent also. But while some authors may not include the empty set as a face of polytopes, I disagree that seeing the empty set as the -1-simplex requires a non-standard definition. It follows directly from the definition of the n-simplex for n=-1.sephia karta 10:59, 11 October 2008 (UTC)[reply]

There are many, many reasons for regarding the empty set as a (-1)-polytope and the minimal face of any polytope. For example, the Hasse diagram (face lattice) of any polytope and its dual are inversions of each other, so accepting the whole polytope as the maximal face clearly requires seeing the the null polytope as the minimal one. Then, as the article points out, the face lattice of an n-simplex is the graph of an (n+1)-hypercube, but not without the (-1)-face. (In fact, all n-polytopes have lattices that are (n+1)-polytopic graphs). Euler's V+E=F+2 generalises better also: the number of faces of even and odd dimension are equal for spherical polytopes (genus 0). More formally, ∑(-1)ⁿ.Z_i = 0 (for -1 ≤ i ≤ n), where Z_i is the number of i-faces.

The Indians invented zero - where would we be without that? Or the empty set in set theory? We all revere Coxeter, but he would be the last person to wish his subject remain static for all eternity. Where would modern abstract geometry be if we clung to Euclid's parallel postulate?

Well yes, a null polytope has no point at all - zero vertices! But zero has much more than zero value! SLWoolf (talk) 04:00, 28 October 2008 (UTC)[reply]

Alright, I'll bite. Frankly, I don't understand what's the big deal with including the (-1)-simplex. All it does on this page is to clutter the table with a redundant column of 1's which provides absolutely no useful information whatsoever (every polytope contains one instance of nothingness! woohoo! stating the obvious since 1983!). Can we not just mention once that the null polytope may be considered as a minimal face in every simplex, and be done with it? I don't understand what's this obsession over empty sets. The empty set is the empty set is the empty set, and there's all there is to it. We only need to mention it once. I see no point in repeating something this obvious on every row of the table. Also, this article is not about Pascal's triangle, but about simplices. We can always just link to Pascal's triangle if that is really necessary. Why must we replicate Pascal's triangle here? That's what hyperlinking is for.—Tetracube (talk) 19:53, 28 October 2008 (UTC)[reply]

Seconded (thirded?) None of the ardent supporters of the (-1)-simplex above have given any compelling reason why it needs to be included umpteen different times in the table. A single mention of the (-1)-simplex in the text should be more than adequate. As far as I can tell, the only reason that has been advanced for having this in the table is to complete Pascal's triangle which, as you point out, is a very weak reason indeed in an article about simplices. siℓℓy rabbit (talk) 20:22, 28 October 2008 (UTC)[reply]

Hi guys, let's not add to global warming by getting overheated about NOTHING.... Well I don't think you have given a compelling reason for excluding the (-1)-faces. If they are obvious, then so are the numbers of 0-faces. If reducing the table size is paramount, why not remove the name column? The best name for an n-simplex is just that and not some impossible-to-remember Greek unpronouncable. Or the symbol column. Both of these take up far more screen width. As for the first three columns - these are the most useless of all! But more constructively, the numerous number of element columns could be narrowed considerably by creating another table heading row with text such as "Number of Faces of Rank:" under which the numbers -1, 0, 1, ... appear in the narrower columns. I don't know how to do that, so can one of you ardent tables-mustn't-get-too-big-ists? Then I think we'll all be happy and can move on to the next galactic crisis. SLWoolf (talk) 14:05, 29 October 2008 (UTC)[reply]

Ok. Given that no good reason has yet been given for including the new column in the table, I am going to remove it. siℓℓy rabbit (talk) 15:36, 29 October 2008 (UTC)[reply]

I find it a bit smug to state that no good reason has been provided, given that you didn't even go to the trouble of responding to my last arguments. So I will repeat these here. The table purports to list all the faces of the n-simplices. It may be obvious that the -1-simplex is a face of every other simplex, but that does not make it any less true. The table is simply incomplete without it. Furthermore I put forward that the table is not at all too large or too cluttered. And I state this looking at a small screen. But if you really want to reduce it's size, remove some columns from the right side, those really are not very interesting. sephia karta 22:39, 29 October 2008 (UTC)[reply]

Can we please just add a sentence stating that the (-1)-simplex is sometimes considered to be the minimal face of all simplices, and leave it at that? The issue here is neither that the table would be too large nor that it would be too cluttered with the extra column. The issue is that the additional column does not provide any useful information, save the trivial fact that every simplex contains a null polytope. This piece of information can be stated in one short sentence and we can be done with it. Why do we absolutely have to express this in the most verbose way possible? If you really want to, add a hyperlink to Pascal's triangle somewhere, where the reader can see the "complete" table if he so wishes.—Tetracube (talk) 23:37, 29 October 2008 (UTC)[reply]

I entirely agree with Sephia's points - even if Silly Rabbit has read and understood the various points, it is poor form to act unilaterally without responding and trying to reach a civil consensus. If abstract mathematicians can't resolve this, what hope is there for the human race in a hi-tech world? As Sephia points out, the table provides a list of elements and it is INCOMPLETE if you omit the (-1) dimension, and mathematics must be precise. The leftmost 3-columns are far more useless and obvious than the fact that all polytopes have (-1) faces, which is far from obvious. So remove those first 3 boring columns. And it takes more words and space to state this fact in words, and less elegantly. It is not promoting the wonders of the Pascal triangle, though this connection is interesting and relevant. It is about, if you are going to make a list of something, then don't omit parts of it which you personally consider obvious. Or is it that you really do not accept the (-1)-face concept?SLWoolf (talk) 09:38, 30 October 2008 (UTC)[reply]

I have added a mention of the (-1)-simplex to the text. I have also included a reference to the OEIS as a source for the table (which in turn is sourced to Grunbaum's Convex Polytopes). If the table is incomplete, as you contend, then so is the table in the OEIS and Grunbaum's book, and the definition given by Coxeter is wrong as well. The fact is, there is far from a universal agreement that the (-1)-simplex should even be considered. Per WP:NPOV, the encyclopedia article should not present a minority viewpoint as though it were the majority one. Giving it such a prominent place in the table clearly gives it WP:UNDUE weight. I have added a mention of it in the text, which is pretty much all it deserves. Can we please move on now? siℓℓy rabbit (talk) 11:45, 30 October 2008 (UTC)[reply]

Looks good to me, although I moved the pargraph up a section, below the table where it makes sense. Maybe this was where it was intended to be? Tom Ruen (talk) 18:46, 30 October 2008 (UTC)[reply]

It certainly fits better immediately after the table.—Tetracube (talk) 19:11, 30 October 2008 (UTC)[reply]

Well, you were honest enough to admit you don't really believe in (-1) polytopes. But I have to say here that I think majority (I wouldn't dream of saying mob) rule has triumphed over good mathematics. I would love to see the big boys' (Grunbaum, Mullen, Shulte) comments here! Let us always have civil words, but "I'll be back", I know your address there! SteveWoolf (talk) 07:58, 31 October 2008 (UTC)[reply]

Maybe a good time for Wikipedia to add an entry nullitope if it's important? There's also a statement here: Polygon#Generalizations_of_polygons (P.S. I initiated the simplex table here with Pascal's triangle) Tom Ruen (talk) 16:46, 31 October 2008 (UTC)[reply]

I would prefer null polytope instead.—Tetracube (talk) 18:44, 31 October 2008 (UTC)[reply]

Well, that is nice, because the OEIS mentions Wikipedia as a source for the table. Grunbaum doesn't provide one, so this is a case of circular referencing. The truth is probably that the table was constructed by some Wikipedia-editor to begin with. Grünbaum doesn't take an explicit position on the -1-simplex. But note that he explicitely says on page 17 that the empty set is a face of every convex polytope, and that his definition of a simplicial complex as one that contains only simplices only works if one accepts the empty set as the -1-simplex.

I refuse to subscribe to your claim that the concept of the -1-simplex is a minority viewpoint. Do a Google Scholar search on "empty simplex". Furthermore, I contend that in the majority of cases where the -1-simplex is not mentioned, it is ignored because it is so trivial that it makes no difference for the relevant discussion. In this way this is no different from the 10-simplex, for which a search reveals even less results. Surely we cannot deduce from the absence of results that other authors question the concept of the 10-simplex.

But in an overview of the faces of the n-simplex that purports to be complete and that makes mention of Pascal's triangle, it is highly relevant to include the -1-simplex. Again I restate what I said earlier: I'm not advocating that the -1-simplex should get a prominent place within this article, and your claim that I am is misleading. My comments are solely directed towards the contents of this table. Either do it properly and include the -1-simplex or don't do it and remove the table completely.sephia karta 11:23, 31 October 2008 (UTC)[reply]