# Talk:Simplex

WikiProject Mathematics (Rated B-class, High-importance)
This article is within the scope of WikiProject Mathematics, a collaborative effort to improve the coverage of Mathematics on Wikipedia. If you would like to participate, please visit the project page, where you can join the discussion and see a list of open tasks.
Mathematics rating:
 B Class
 High Importance
Field:  Geometry
WikiProject Uniform Polytopes
This article is within the scope of WikiProject Uniform Polytopes, a collaborative effort to improve the coverage of uniform polytopes on Wikipedia. If you would like to participate, please visit the project page, where you can join the discussion and see a list of open tasks.

## Volume formulas

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

${\displaystyle {1 \over n!}\det {\begin{pmatrix}v_{0}-v_{1}&v_{1}-v_{2}&\dots &v_{n-1}-v_{n}&v_{n}-v_{0}\end{pmatrix}}}$

appears to be spectacularly false, since,

${\displaystyle Vol_{n-simplex}={1 \over n!}\det {\begin{pmatrix}v_{0}-v_{1}&v_{1}-v_{2}&\dots &v_{n-1}-v_{n}&v_{n}-v_{0}\end{pmatrix}}={1 \over n!}\det {\begin{pmatrix}v_{0}-v_{1}&v_{1}-v_{2}&\dots &v_{n-1}-v_{0}&v_{n}-v_{0}\end{pmatrix}},}$

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:

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

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

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

${\displaystyle \pm {1 \over n!}\det {\begin{pmatrix}v_{0}&v_{1}&\dots &v_{n-1}&v_{n}\\1&1&\dots &1&1\end{pmatrix}}}$

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

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

I thought the formula was:
${\displaystyle {1 \over n!}\det {\begin{pmatrix}v_{1}-v_{0}&v_{2}-v_{0}&\dots &v_{n-1}-v_{0}&v_{n}-v_{0}\end{pmatrix}}}$

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
${\displaystyle {\frac {1}{n!}}{\sqrt {\det A^{T}A}}}$
${\displaystyle A={\begin{bmatrix}v_{1}-v_{0}&v_{2}-v_{0}&\dots &v_{n}-v_{0}\end{bmatrix}}}$
This formula, suitably understood, also works for simplices of arbitrary codimension. Silly rabbit (talk) 00:24, 3 February 2008 (UTC)

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)

It may be helpful to undergraduate students to have a sketch of the proof of the volume formula:

${\displaystyle {\frac {\alpha ^{n-1}{\sqrt {n}}}{(n-1)!}}=\int _{x_{1}+\cdots +x_{n}=\alpha }dx}$

Using techniques accessible to students that have had exposure to multivariable calculus, specifically the n-dimensional generalization of the surface integral. Beginning by noting that ${\displaystyle x_{n}=\alpha -(x_{1}+\cdots +x_{n-1})}$ and thus:

${\displaystyle {\begin{array}{rcl}\int _{x_{1}+\cdots +x_{n}=\alpha }dx&=&\int _{x_{1}+\cdots +x_{n-1}<\alpha }{\sqrt {1+\left({\frac {\partial x_{n}}{\partial x_{1}}}\right)^{2}+\cdots +\left({\frac {\partial x_{n}}{\partial x_{(n-1)}}}\right)^{2}}}dx_{n-1}\cdots dx_{1}\\&=&{\sqrt {n}}\int _{x_{1}+\cdots +x_{n-1}<\alpha }dx_{n-1}\cdots dx_{1}\\&=&{\frac {\alpha ^{n-1}{\sqrt {n}}}{(n-1)!}}\end{array}}}$

## 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 ${\displaystyle \sum _{i}t_{i}}$ 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!

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)

## 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)

## 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)

${\displaystyle Insertformulahere}$

For consistency with other dimensions and the hypercube article, shouldn't the graph for a one-dimensional simplex include its single edge (Complete_graph_K2.svg)? I will add a similar question on the cross-polytope article. 24.6.65.170 (talk) 16:23, 13 July 2009 (UTC)

In my system (Windows XP Pro) the miniature pictures in Elements look different from the full size. This applies at least to the 3, 5, 6, 7, 8, 9 simplices. Is this error caused when creating the miniature png, or is caused by Windows rendering the png poorly? -- Mikael —Preceding unsigned comment added by 213.204.183.82 (talk) 10:33, 28 January 2010 (UTC)

I agree (and I'm on a Mac - it's not caused by Windows). The 3-simplex, 5-simplex, 7-simplex and 9-simplex all seem to be missing horizontal edges, the 6-simplex, 8-simplex and 10-simplex are missing a vertical edges, at the size in the Elements section. This is not a problem with the larger versions just below.
I've made them 30% bigger to fix it (10% and 20% didn't resolve the issue). The other way would be to modify the source art to thicken up the lines a bit but I don't feel confident to do this.--JohnBlackburnewordsdeeds 10:50, 28 January 2010 (UTC)
I removed the graphs from the table, more readable compacted, since there's larger images 1-20 in the next section. Tom Ruen (talk) 21:56, 28 January 2010 (UTC)

## 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)

## -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)

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)
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)
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)
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)
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)
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.Zi = 0 (for -1 ≤ i ≤ n), where Zi 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)

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)
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)

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)

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)
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)
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)

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)

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)
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)
It certainly fits better immediately after the table.—Tetracube (talk) 19:11, 30 October 2008 (UTC)
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)
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)
I would prefer null polytope instead.—Tetracube (talk) 18:44, 31 October 2008 (UTC)
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)

Coming late to the fray, here are a few observations. Coxeter sometimes made small slips in his writings. If somebody pointed one out to him, he would be very pleased and amused. Taking an uncorroborated reference from him is therefore always a dangerous thing to do. Recall too that he came very much from the early 20th century geometrical tradition, and postwar set theory was a latecomer to his understanding. Grünbaum's Convex polytopes is also rooted in geometry - convexity is irrelevant to combinatorics - yet it provides much food for purely combinatorial thought. The book frequently omits the term "convex", leaving it understood - with the result that extracts often get quoted out of context, even to some people believing vehemently that a polytope is BY DEFINITION convex. It is possible that he frequently omitted the −1-polytope for the same reason. I can ask him if you like. Geometrically, one can argue that since −1-space does not exist, the −1-polytope does exist either and so can not be part of anything. I do not have Convex polytopes to hand, so cannot comment on the p.17 remark mentioned above other than that the "empty set" remarked on is not a geometrical construct but an algebraic one. Conceptually one can argue that the empty set is not per se a polytope, and does not need to be for a combinatorial approach to be consistent - whether we interpret the elements of some set as "polytopes" is entirely optional, and there is no need for all-or-nothing: some elements might be deemed polytopes, some not. If we are considering combinatorially the quadrilateral a, b, c, d then we need only be interested in the combinations a, b, c, d, ab, bd, cd, ca, abcd, and we need not care about ac, bd, abc, etc. - or about φ. Simplices do not have most of these "false" combinations, so it is easy to forget that φ still raises this issue. We need to keep distinct the ideas that apply in the contexts of geometry, combinatorics and set theory. Unfortunately, few authors attempt this with any rigor (the distinction between earlier combinatorial approaches and modern abstract set theory being particularly new), and without being able to do original research we Wikipedians are sometimes forced to leave our articles equally unclear - without explaining why! Overall, my own view is that the −1-polytope certainly needs explaining and setting in context, for example to mention its presence in every abstract simplex, but to include it throughout the main material would not be supported by the evidence so far discussed. — Cheers, Steelpillow (Talk) 09:59, 19 June 2010 (UTC)

I think such explaining remarks can be contained in footnotes without being considered WP:OR. As long as they stick obviously correct explaining/commenting the given situation in literature rather than primarily focusing on/pushing personal views on the subject, it should be fine.--Kmhkmh (talk) 07:12, 20 June 2010 (UTC)
As a Ph.D. topologist, I completely agree with Sephia that the (-1)-simplex deserves inclusion. Omitting it from a list of n-simplexes is like omitting 0 from the ordinal numbers.Daqu (talk) 00:19, 7 July 2010 (UTC)

## in the plane or under it?

The article now says:

The standard n-simplex is the subset of Rn+1 given by
${\displaystyle \Delta ^{n}=\left\{(t_{0},\cdots ,t_{n})\in \mathbb {R} ^{n+1}\mid \Sigma _{i}{t_{i}}\leq 1{\mbox{ and }}t_{i}\geq 0{\mbox{ for all }}i\right\}}$

Shouldn't ${\displaystyle \Sigma _{i}{t_{i}}\leq 1}$ be ${\displaystyle \Sigma _{i}{t_{i}}=1}$? —Tamfang (talk) 17:59, 28 June 2009 (UTC)

No, I think it's correct as written. This definition uses ${\displaystyle n}$ variables to define a set in ${\displaystyle \mathbb {R} ^{n+1}}$, and that's why the condition is ${\displaystyle \Sigma _{i=1}^{n}{t_{i}}\leq 1}$. If you wanted, you could define an extra variable ${\displaystyle t_{n+1}=1-\Sigma _{i=1}^{n}{t_{i}}\,}$, and then the condition would be ${\displaystyle \Sigma _{i=1}^{n+1}{t_{i}}=1}$. I don't think this is necessary, though, since the simplex is only defined by ${\displaystyle n}$ independent variables. --Apocralyptic (talk) 19:13, 15 July 2009 (UTC)
Are we looking at the same thing? As I read it, the definition uses ${\displaystyle n+1}$ variables (${\displaystyle t_{0},\cdots ,t_{n}}$), not n variables, to define a set in ${\displaystyle \mathbb {R} ^{n+1}}$. If the intention is to define something in an n-dimensional subspace, it ought to be more like a surface than a volume, thus somewhere there ought to be a strict equality, not only a series of inequalities. Take n=2: the definition given is equivalent to { (x,y,z) | x+y+z ≤ 1 and x,y,z ≥ 0 }, which defines a (right) tetrahedron, not a triangle. —Tamfang (talk) 05:16, 16 July 2009 (UTC)
I agree with you now... I think I was confused by the variables starting with t0 instead of t1. What I said before made no sense. The correct definition should be
${\displaystyle \Delta ^{n}=\left\{(t_{1},\cdots ,t_{n+1})\in \mathbb {R} ^{n+1}\mid \Sigma _{i=1}^{n+1}{t_{i}}=1{\mbox{ and }}t_{i}\geq 0{\mbox{ for all }}i\right\}}$. --Apocralyptic (talk) 14:40, 16 July 2009 (UTC)

Now, if in the above definition you lower all subscripts by 1, you'll be onto something.Daqu (talk) 01:01, 7 July 2010 (UTC)

## Cartesian Coordinates section has major problems

Property 2 is not defined directly through the dot product, but by the trigonometric interpretation of a dot product, however the entire section refers to it in the context of a dot product, not of trigonometry. Using the trigonometric definition just obfuscates the text by requiring the additional step of recognizing the function as that of the dot product and translating the statement. The logical derivation of the property is not given, which I assume would be in terms of the dot product in the first place.

Example in 3-dimensions (N=3): While the last step reads, "The above step is repeated for vi (i=3,...,N). In particular, we choose vi such that only the first i components are nonzero." However, the algorithm is not defined in a way that makes it iterable. The constants are not defined in terms of the previous vertices' coordinates. The train of thought is not detailed at all. For example, statements are made such as, "In particular, we choose vi such that only the first i components are nonzero," without explaining how the author expects you to do this. The pythagorean theorem's application for length calculation, as j22=1-j1 (where j is what's referred to in the article as v2), is used without explaining why the described process lends to the application formula. It's an unnecessary leap of logic.

Matlab code for arbitrary dimension N: All programmers can make use of pseudocode, while only MATLAB coders can use MATLAB code directly. The code may even be entirely opaque to those not familiar with MATLAB syntax and conventions. Pseudocode does not rely on such specifics, beyond a few common and widely readable conventions. Defining an implementation of an algorithm instead of a blueprint biases the information for the benefit of one cross-section of people using a proprietary language but the great detriment of the general programmer. LokiClock (talk) 08:31, 2 February 2010 (UTC)

I could follow the reasoning, so if I needed I could do it myself, but it wasn't easy as it's not very clearly presented. It could do with a rewrite to make the process clearer, perhaps with a better worked example.
I agree on Matlab. I program for a living, so know a few programming languages and a lot of maths, but I can't make sense of that. Pseudocode would be better, though if the paragraphs before were better written it could be done without resorting to programming at all. It's something I've noticed elsewhere so I've raised it here, to get some feedback on it as I think it's of wider concern.--JohnBlackburnewordsdeeds 10:36, 2 February 2010 (UTC)
I've done the bold thing and rewritten it from scratch to make it much clearer what's going on. I've removed the Matlab code as I don't think it helped. There's no step-by-step process for doing it in n-dimensions, e.g. no pseudocode, but hopefully now the algorithm is clearer so it's not needed: psudocode's not much use if you want to work through it using pen and paper for example.--JohnBlackburnewordsdeeds 14:23, 2 February 2010 (UTC)
For reference here's the Matlab code I removed. I discovered how to syntax highlight it after, so was interested to see what this looked like properly presented: much more readable though still pretty impenetrable to a non-Matlab programmer like me.--JohnBlackburnewordsdeeds 17:35, 2 February 2010 (UTC)
function v = NdimSimplexGen(N)

v = zeros(N+1,N);
v(1,1) = 1; v(2:(N+1),1) = -1/N*ones(N,1); % Case N=1 is already done

if(N>1) % The following assumes N>1
for i = 2:N
% Determine the new element v(i,i) by normalization
c = sqrt(1-sum(v(i,1:(i-1)).^2));
v(i,i) = c;

% Determine all elements below v(i,i) by the acos(-1/n) property
w = v(i,1:(i-1));
a = -1/c*(1/N+sum(w.^2));
v((i+1):(N+1),i) = a*ones(N-i+1,1);
end
end

Cool. If anyone is interested i created a piece of Java-code solving the same problem. Someone might convert it into proper pseudocode. RasF (talk) 13:02, 11 August 2012 (UTC)
double d = 3;
double angle = -1.0/d;
Point[] ret = new Point[d+1];
for(int s=0;s<d+1;s++) ret[s] = new Point(new double[d]);

double tmp = 0;
for(int i=0;i<d;i++){
double nCoord = Math.sqrt(1-tmp);
ret[i].set(i, nCoord);
double rCoord = (angle - tmp)/nCoord;

for(int s=i+1;s<d+1;s++) ret[s].set(i, rCoord);

tmp += rCoord*rCoord;
}

I forgot to watch this page. You are wonderful, thank you! I edited in some of the things I suggested. LokiClock (talk) 15:28, 4 February 2010 (UTC)
I had to take out the first thing you added as it did not make sense - the vectors are not given at that point so there's no dot product. After that I went through and copyedited a lot more, fixing some other things, "i.e." is bad style, and I don't like linking equations: better to use the name as a guide, which is also appropriate here and I think reads better.--JohnBlackburnewordsdeeds 17:37, 4 February 2010 (UTC)
Maybe out of fashion, but I've never known "i.e." or "that is" to be considered "bad," even by the arbitrations of linguistic prescriptionists. The reason I used the formula instead of the name was because it's possible the reader might not know the association between the Pythagorean theorem and line lengths, and so might not understand the expected application. Gaps in knowledge come in where you least expect them to, so I try when convenient to keep prior knowledge from being assumed. I suppose, though, if they can grasp the geometric action of the dot product then it's considerably more unlikely. LokiClock (talk) 19:36, 4 February 2010 (UTC)
My point was that if you find yourself writing "i.e." then "that is" is better. It's here under "Do not use unwarranted abbreviations", and Im sure I've read it elsewhere too. I've had someone copyedit them out of something I wrote, and since then I try a and avoid them myself, and will change them if I see them as bad style - unencyclopaedic/lazy/too conversational. After that I went on to copy edit it further, as I find I do after making a few changes.--JohnBlackburnewordsdeeds 19:51, 4 February 2010 (UTC)
The style guide makes an exception for these abbreviations. I don't mind replacing it with "that is." LokiClock (talk) 13:21, 5 February 2010 (UTC)
Well I'd consider the difference between "i.e." and "that is" simply a personal choice of the author. One might argue that the latter is a bit easier to read (for people not being aware of the expression i.e./id est), but i don't think you can claim it is too "difficult" to be used in WP. After all this is the English WP not the simple English WP. Improving readability is an important and worthwhile goal, but it should concentrate on improving language/grammar/accessability in cases where it is really (and often desperately) needed) rather than quarreling over marginal differences in style and taste. Also keep in mind that the style guide is not the "bible" and that the Englisch WP is written and read by a large amount of non native speakers as well (they may in fact even constitute the majority). As for the matlab code I completely agree, if at all any algorithm or procedure should be given in pseudo code.--Kmhkmh (talk) 14:50, 5 February 2010 (UTC)

───────────────── The "i.e." issue's not a big one: I won't edit something just to fix it but I will change it if copyediting as it's better style. Even common abbreviations are bad style, and ones like "i.e" and "e.g" are worse as they're not contractions of English words like e.g. TV, US, so need to be learned separately from their meaning, i.e. they may be harder for non-native or less experienced readers. See also the second paragraph of WP:ABBR. As for Matlab there's a largish thread about it here.--JohnBlackburnewordsdeeds 16:33, 5 February 2010 (UTC)

## Please try to be more exact in your wording, and avoid totally illogical sentences!

The introductory paragraph includes this sentence:

"Specifically, an n-simplex is an n-dimensional polytope with n + 1 vertices, of which the simplex is the convex hull."

Huh? If X = "n-simplex", and Y = "n-dimensional polytope with n + 1 vertices", then this sentence is of the form:

   "An X is a Y, of which the X is the convex hull."


Any idea how illogical that is? May I please suggest that you read and think about how your writing sounds before clicking on "Save page".Daqu (talk) 23:52, 6 July 2010 (UTC)

Uh, the "which" in the subordinate clause is referring to "n+1 vertices", not to "n-dimensional polytope". Granted, though, it could use a clearer phrasing.—Tetracube (talk) 00:07, 7 July 2010 (UTC)
Uh, let me repeat it then: May I please suggest that you read and think about how your writing sounds before clicking on "Save page".Daqu (talk) 14:40, 8 July 2010 (UTC)
Chill out, man! I'm not the one who wrote that sentence! Please take a deep breath before you hyperventilate.—Tetracube (talk) 20:33, 8 July 2010 (UTC)

## Wikipedia is not an outlet for the massive egotism of certain individuals.

The use of the individually coined words "pentachoron", "hexateron", and "heptapeton" may inflate the ego of whoever wishes to promote the use of these words.

But Wikipedia is specifically designated as not a place to include one's own inventions in an article. It is a shame that some people will gladly violate the rules of Wikipedia in order to use it as a vehicle for inflating their own egos by strategically placing nonexistent words in many articles.

I can't make you stop, but I can say this: You know who you are, and so do we. This behavior is a pathetic way to make yourself feel better by imposing your nonexistent words on millions of unsuspecting readers of this encyclopedia.Daqu (talk) 14:37, 8 July 2010 (UTC)

Well, you know, every term of art was coined by someone. Are you sure that n-choron was coined by the same one (or group) who coined n-teron and n-peton? (I don't know who they are, so who's "we"? I personally find the last two unbeautiful and their successors worse, but that's irrelevant.) —Tamfang (talk) 01:37, 11 July 2010 (UTC)
I highly doubt those who coined the words are the same people who added those words into these articles.—Tetracube (talk) 17:44, 13 July 2010 (UTC)
Yes, Tamfang, we all know that terms had to begin somewhere. Wikipedia is a good place to define and explain existing (i.e., generally accepted) terms relevant to the subject of a given article. But it's *not* the place to introduce new coinages, as certain people have been doing. When this happens systematically, it's a strong indication that someone is trying to hijack Wikipedia to propagate their coinages and hence inflate their ego.
Some of these "terms" came from the Uniform Polychora Project (and please note that polychora is not an accepted term as of this writing); some of them are in the PolyGloss glossary (replete with many terms unaccepted as of this writing) compiled by Wendy Krieger.
Simply trying to come up with appropriate terms or compile them is not what I'm complaining about. It is rather using Wikipedia as a means of propagating terms that have not gained acceptance. As determined by their not appearing even once in the MathSciNet database of all words used in all math articles published in peer-reviewed journals.Daqu (talk) 09:15, 26 July 2010 (UTC)
I agree. As far as uncommon terms are concerned, whoever wants such a term in the article needs in doubt produce a reputable source using that term, if he can't or there is none, the term stays out. The WP guidelines are rather clear on that. WP reflects existing (somewhat) established knowledge and terms, but it does not introduce new ones.--Kmhkmh (talk) 10:29, 26 July 2010 (UTC)
I would like to believe that, as you say, "the term stays out" -- except that your statement is contrary to fact. In particular, where the word "polychoron" is used, apparently whenever possible, in Wikipedia. And certainly in the article that is unfortunately so named. At some point quite a while ago I mentioned that in the entire MathSciNet database of peer-reviewed journal articles and books, there was not even one instance of the word. But still the term does not "stay out". (Nor do its equally nonexistent relatives, tetrachoron, pentachoron, et al.)Daqu (talk) 05:48, 26 September 2010 (UTC)
I assume you know how WP works. What the guidelines or maybe even common understanding suggest and what individual editors might do are two different things. In other words you are probably fine on removing those terms. If there are objections raise issue the at associated portal (in this case Portal:Mathematics might be the best choice)--Kmhkmh (talk) 14:00, 28 September 2010 (UTC)

## Applications section

This is very weak. A simple way to improve it is to look at the book by Saul Gass that I cite on Talk:Linear programming#Article not accessible redux / providing mainstream coverage Michael P. Barnett (talk) 02:18, 10 May 2011 (UTC)

## Random Walk Section

A previous version of this article had a very useful, if incomplete, discussion of Simplex Point Picking and random walks. It was deleted and, as far as I can tell, vanished into thin air. The only current discussion related to probability is a link to a very scant section of categorical distributions. Even though a simplex is a geometrical object, it does play a role in probability theory. So, even though the "geometrical purity" of this article might be somewhat sullied by the re-inclusion of material about random sampling and random walks, I would as that this be done as it is very helpful. I found several articles on the internet that actually linked this article because of the (former) section on random sampling and walks. Right now, they are bad links. —Preceding unsigned comment added by Skinnerd (talkcontribs) 16:48, 13 May 2011 (UTC)

It looks like it was deleted in November 2010 due to a lack of reliable sources, BUT the historical versions can be accessed (and linked as old copies), like [7]. Tom Ruen (talk) 20:14, 13 May 2011 (UTC)

I've created a stub for this topic at Sampling in order, with references. Feel free to expand it! Melchoir (talk) 00:09, 6 January 2014 (UTC)
The deleted material being discussed is perhaps heavily referenced, in part because of https://stackoverflow.com/questions/3010837/sample-uniformly-at-random-from-an-n-dimensional-unit-simplex - 75.173.139.216 (talk) 10:28, 28 October 2017 (UTC)

## Euclidean simplex

As I can understand, the article describes the same thing as the standard simplex. In any case, I do not see any merit in existence of a separate article. Incnis Mrsi (talk) 17:38, 27 April 2013 (UTC)

I agree. I believe that the two articles are about the same thing, and should therefore be merged. Maproom (talk) 11:41, 8 October 2013 (UTC)
Yes, merge. It looks like the ONLY article that references it is Pseudomanifold. Tom Ruen (talk) 05:30, 9 October 2013 (UTC)

DoneDavid Eppstein (talk) 21:35, 21 April 2015 (UTC)

Hi, I understand this is probably a complicated subject, but the intro is impenetrable to anyone not versed in this subject. Could someone simplify it perhaps? I came to get a general idea of the subject and I'm none the wiser. Malick78 (talk) 15:09, 1 May 2014 (UTC)

I agree that this is very technical. Would the following be helpful (and accurate)? A k-simplex is the simplest shape that requires k dimensions. Consider a line segment [A,B] as a "shape" in a 1-dimensional space (the 1-dimensional space is the line that the segment lies in). We can place one new point C somewhere off the line. The new shape, triangle ABC, requires 2 dimensions; it cannot fit in the original 1-dimensional space. So the triangle is the 2-simplex, the simplest shape that requires 2 dimensions. Consider a triangle ABC, a shape in a 2-dimensional space (the plane that the triangle lies in). We can place one new point D somewhere off the plane. The new shape, tetrahedron ABCD, requires 3 dimensions; it cannot fit in the original 2-dimensional space. So the tetrahedron is the 3-simplex, the simplest shape that requires 3 dimensions. Consider tetrahedron ABCD, a shape in a 3-dimensional space (the 3-space that the tetrahedron lies in). We can place one new point E somewhere out of the 3-space. The new shape ABCDE, called a 5-cell, requires 4 dimensions and is called the 4-simplex; it cannot fit in the original 3-dimensional space. (It also cannot be easily visualized.) The idea can be iterated, i.e., adding a single new point off the currently occupied space, which requires going to the next dimension up to hold the new shape. The idea can also be worked backward: the line segment that we started with is the simplest shape that requires a 1-dimensional space to hold it; so the line segment is the 1-simplex. The line segment itself was formed by starting with a single point in 0-dimensional space (this initial point was the 0-simplex) and adding a second point, which required the increase to 1-dimensional space. Michaelaoash (talk) 12:46, 5 April 2018 (UTC)

## Oriented volume

The formula for the oriented volume of a simplex is expressed by an absolute value. Shouldn't it be algebraic? maimonid (talk) 19:16, 3 May 2016 (UTC)

Hello fellow Wikipedians,

I have just modified one external link on Simplex. Please take a moment to review my edit. If you have any questions, or need the bot to ignore the links, or the page altogether, please visit this simple FaQ for additional information. I made the following changes:

• Added archive https://web.archive.org/web/20090505034911/http://cg.scs.carleton.ca/~luc/rnbookindex.html to http://cg.scs.carleton.ca/~luc/rnbookindex.html

When you have finished reviewing my changes, you may follow the instructions on the template below to fix any issues with the URLs.

As of February 2018, "External links modified" talk page sections are no longer generated or monitored by InternetArchiveBot. No special action is required regarding these talk page notices, other than regular verification using the archive tool instructions below. Editors have permission to delete the "External links modified" sections if they want, but see the RfC before doing mass systematic removals. This message is updated dynamically through the template {{sourcecheck}} (last update: 15 July 2018).

• If you have discovered URLs which were erroneously considered dead by the bot, you can report them with this tool.
• If you found an error with any archives or the URLs themselves, you can fix them with this tool.

Cheers.—InternetArchiveBot 00:58, 3 December 2017 (UTC)