Talk:Simplex

message: I think that a reference to simplices in functional analysis is missing here. Terminologies such as "Choquet simplex" or "Bauer simplex" are often met. As far as I understand, they are infinite-dimensional extensions of the notion of simplex discussed in that article.  — Preceding unsigned comment added by Gamesou (talk • contribs) 14:52, 6 December 2019 (UTC)
replies: text: Which article?  I don't see any reference to Choquet simplex or Bauer simplex in the functional analysis article.—Anita5192 (talk) 17:22, 6 December 2019 (UTC)
thread_title:  Simplices in functional analysis 

message: It appears that the normal vectors in the dihedral angles sections are wrong: https://en.wikipedia.org/wiki/Simplex#Dihedral_angles_of_the_regular_n-simplex
For example, in 2d you would get (0, 1) and (1, 0), which can not be the correct normal vectors of an equilateral triangle. Further, calculating the angle between two such vectors leads to arccos(1/(n-1)), which is not correct.
Both of those issues appear if the normals are instead given by (-n, 1, ..., 1) and permutations, I think that's the correct answer.  — Preceding unsigned comment added by 2001:16B8:A0C4:1800:7500:BF95:344F:B5AC (talk) 09:01, 13 October 2020 (UTC)
replies: text: I did the calculations and got the same result as you, so I changed it.—Anita5192 (talk) 21:19, 13 October 2020 (UTC)
thread_title:  Normal vectors and dihedral angles 

message: The section Geometric properties contains this incoherent sentence:
"If P is a general parallelotope, the same assertions hold except that it is no longer true, in dimension > 2, that the simplexes need to be pairwise congruent; yet their volumes remain equal, because the n-parallelotope is the image of the unit n-hypercube by the linear isomorphism that sends the canonical basis of \mathbf R^n to e_1,\ldots, e_n."
This is way too confusing. 
Much better: State exactly what does hold true, instead of trying to use a previous paragraph with modifications.01:51, 10 November 2020 (UTC)
thread_title:  Incoherent sentence 

message: I removed this bullet that reads like original research, but I got reverted twice by someone who disagrees. The part about methane sounds particularly problematic.
*In chemistry, the hydrides of most elements in the p-block can resemble a simplex if one is to connect each atom. Neon does not react with hydrogen and as such is a point, fluorine bonds with one hydrogen atom and forms a line segment, oxygen bonds with two hydrogen atoms in a bent fashion resembling a triangle, nitrogen reacts to form a tetrahedron, and carbon forms a structure resembling a Schlegel diagram of the 5-cell. This trend continues for the heavier analogues of each element, as well as if the hydrogen atom is replaced by a halogen atom.
–LaundryPizza03 (dc̄) 06:16, 5 January 2021 (UTC)
replies: text: This is definitely not original research.  The well-known molecular geometry in this example can be found in most any first-year college chemistry textbook and also at Molecular geometry.  See Molecular geometry#VSEPR table.  In particular, the geometry of methane, CH4, is shown in that table, and can be compared to the Schlegel diagram of the 5-simplex in :File:5-simplex verf.png.—Anita5192 (talk) 06:59, 5 January 2021 (UTC)
thread_title:  Questionable "Applications" entry 

message: et al.:  Is it reasonable to add the following volume formula?
The expression
\mathrm{Volume} =
\frac{1}{n!}
\sqrt{
\det \left[
\begin{pmatrix}
v_0^T & 1 \\
\vdots & \vdots \\
v_n^T & 1
\end{pmatrix}
\begin{pmatrix}
v_0 & \ldots & v_n \\
1 & \ldots & 1
\end{pmatrix}
\right] -
\det \left[
\begin{pmatrix}
v_0^T \\
\vdots \\
v_n^T
\end{pmatrix}
\begin{pmatrix}
v_0 & \ldots & v_n
\end{pmatrix} \right]}\,,
is symmetric in the vertices and works even when the n-simplex's vertices are in a Euclidean space with more than n dimensions.
Alternatively,
Let
:M = \begin{pmatrix}
v_0^T \\
\vdots \\
v_n^T
\end{pmatrix}
\begin{pmatrix}
v_0 & \ldots & v_n
\end{pmatrix}
=
\begin{pmatrix}
v_0 \cdot v_0 & v_0 \cdot v_1 & \cdots & v_0 \cdot v_n \\
v_1 \cdot v_0 & v_1 \cdot v_1 & \cdots & v_1 \cdot v_n \\
\vdots & \vdots & \ddots & \vdots \\
v_n \cdot v_0 & v_n \cdot v_1 & \cdots & v_n \cdot v_n
\end{pmatrix}, \qquad U = \begin{pmatrix}
1 & 1 & \cdots & 1 \\
1 & 1 & \cdots & 1 \\
\vdots & \vdots & \ddots & \vdots \\
1 & 1 & \cdots & 1
\end{pmatrix}\,.
Then
\mathrm{Volume} = \frac{1}{n!} \sqrt{\det(M+U) - \det(M)}
is symmetric in the vertices and works even when the n-simplex's vertices are in a Euclidean space with more than n dimensions.
Or mix and match somehow.  Thank you —Quantling (talk | contribs) 14:59, 21 February 2022 (UTC)
replies: text: That’s cute but probably WP:OR. Another cute formula: \mathrm{Volume} = \tfrac1{n!} \sqrt{\sum_{i,j}(\operatorname{adj} M)_{ij}}. —Anders Kaseorg (talk) 20:13, 21 February 2022 (UTC)
replies: text: I believe that it is the square root of the sum of the (un-squared) elements of  rather than the Frobenius norm that goes in that formula.  If by original research you mean that no one has ever discovered and published this before now, ... I very much doubt that.  But if you mean that my library isn't big enough that I could find the citation myself ... then I agree!  I will let it sit here for a while and hope that someone has a better familiarity with the literature than I.  Thank you —Quantling (talk | contribs) 22:22, 21 February 2022 (UTC)
replies: text: Right about that formula, corrected above. —Anders Kaseorg (talk) 00:38, 22 February 2022 (UTC)
replies: text: If you know of a citation that supports the notability of \mathrm{Volume} = \tfrac1{n!} \sqrt{\sum_{i,j}(\operatorname{adj} M)_{ij}}, I'd want to add that formula to the article.  —Quantling (talk | contribs) 02:51, 23 February 2022 (UTC)

text: Only a little off topic ... do you know of an algorithm that computes \sum_{i,j} (\mathrm{adj~} M)_{ij}, or any of these other formulations for volume, asymptotically more quickly than the running time of a matrix inversion? —Quantling (talk | contribs) 15:46, 9 May 2024 (UTC)
thread_title:  Another volume formula 

message: : \mp should indeed be changed to \pm. Just look at the case where n=1, for example. 2601:547:500:E930:944C:F88D:FBB3:E92F (talk) 02:30, 23 March 2022 (UTC)
replies: text: I agree with you now.  Thank you for being persistent.  —Quantling (talk | contribs) 01:22, 24 March 2022 (UTC)
thread_title:  Signs 

message: I am moving a discussion with  to this talk page.  The Simplex article is a little inconsistent about the dimensionality of the space containing a simplex.  Specifically, if a -simplex is non-degenerate, we'll need at least  dimensions for placing its  vertices; such as the 3 vertices of a triangle (2-simplex) fit well in a 2-dimensional plane.  But one of our main examples is placing a -simplex into  dimensions; such as a triangle with vertices at , , and .  So, how should we modify the article to clarify this seeming inconsistency?  As far as I am concerned, anyone with ideas should go ahead and make edits.  If we find that there is contention over those edits we can then continue the discussion here.  Thanks! —Quantling (talk | contribs) 16:46, 6 February 2023 (UTC)
replies: text: Ha I edited before I saw this comment, so I'm glad you endorsed that :).  In the lead section, I don't see any reason to specify the ambient space in general -- it is equally true that three affinely independent points span a triangle in \mathbb{R}^{17} as in \mathbb{R}^{2} and \mathbb{R}^{3}.  Does this inconsistency extend to the body, as well?  --JBL (talk) 20:16, 6 February 2023 (UTC)
replies: text: I think this makes sense, the definitions of linear independence and affine independence do not specify dimensions. Dukeleimao (talk) 16:35, 7 February 2023 (UTC)
thread_title:  Dimensionality of space containing the simplex 

message: The volume is shown as |det...| but shouldn't that be either det[...] or |...| for the short form of the determinant?  I didn't know the formula, so I don't want to change it myself, but it looks like a notational error.  It is confusing (to me), however.  — Preceding unsigned comment added by 2A02:58:157:9B00:70B:DBE4:BC88:CA19 (talk) 10:41, 28 October 2023 (UTC)
replies: text: In the first formula, the determinant could come out negative.  It is the absolute value of that possibly negative value that is the actual volume, which is what the  is indicating.  —Quantling (talk | contribs) 14:44, 28 October 2023 (UTC)
thread_title:  Notation error in volume formula?