Talk:5-cell

message: Forgive my ignorance, but is this shape impossible in 3 dimensions? By the diagrams given, it appears so. If it is impossible, I think that would be useful to note somewhere in the article, particularly in the lead. -kotra (talk) 22:48, 18 October 2008 (UTC)
replies: text: Simple answer yes - I'll add to the intro. Going down a dimension you can ask if a tetrahedron can exist in a plane. The answer is YES, if you make it degenerate, like a pyramid with zeo height, but faces will overlap. Tom Ruen (talk) 00:42, 20 October 2008 (UTC)
replies: text: Thanks for your explanation. What you added to the lead is helpful. -kotra (talk) 18:16, 20 October 2008 (UTC)
replies: text: Is this how it was decided that the object was 4th dimensional?  See my post.  — Preceding unsigned comment added by MrJosiahT (talk • contribs) 22:45, 22 August 2011 (UTC)
thread_title:  Impossible? 

message: Is there a formula for the hypervolume of one of these things? Obviously it would be a function of the volume of a tetrahedral cell multiplied by the minimum distance from that cell to the opposing vertex, and then divided by God knows what (probably 4), but there must be something simpler.  —Preceding unsigned comment added by 99.48.55.129 (talk) 09:58, 4 August 2009 (UTC)
replies: text: According to Regular Polytopes the hypervolume of a regular pentachoron of unit edge is √5/90. —Tamfang (talk) 18:04, 5 August 2009 (UTC)
thread_title:  Hypervolume formula? 

message: Sorry if this comment is ignorant, since I haven't actually read the page, but I couldn't make it past the first sentence- how is the shape 4th dimensional when the 4th dimension is time!?
: Four geometric dimensions, see the link Four-dimensional space. SockPuppetForTomruen (talk) 23:12, 22 August 2011 (UTC)
thread_title:  4th dimension!? 

message: It would be easier to understand if you generate coordinates for each node progressively, like so:
For any set of points (A, B, C, etc) all equidistant (say 6ft) from each other, we generate coordinates by setting point A at origin, so Ax=0, as does Ay and so forth.
pointB has x=6 since it is 6 ft from A (By is left at zero).
pointC Cx=3 can be determined from law of cosines.  cosine for triangle ABC is 0.5*6=3. Cy = 5.196, which can be deduced from pythagorean theorem. squareroot of (6^2-3^2).
point Dx=3, point Dy =(((AD^2+AC^2-CD^2)/2)-(Cx*Dx))/Cy which is 1.73. Dz is deduced from pythagorean theorem sqrt(6^2-3^2-1.73^2) = sqrt24 = 4.89
so now we got
 	x	y    	z
A	0	0    	0
B	6	0    	0
C	3	5.196	0
D	3	1.73 	4.89
-	x	y   	z   	w
E	3	1.73	1.22	4.74
point Ex=3, Ey=1.73, Ez=(((AE^2+AD^2-DE^2)/2)-(Ex*Dx)-(Ey*Dy)/Dz =1.22 and Ew (4th dimension coordinate) can be deduced from pyth. theorem and will be 4.74
One can use this same algorithm for any combination of distances to generate valid coordinatesQdiderot (talk) 08:11, 12 July 2012 (UTC)
thread_title:  Do the cartesian coordinates need to be centered at origin? 

message: The 5-cell is also known as a pyrochoron, so maybe, someone should add that in as an alternate name. I don't want to do it myself because I don't have an account, and I've seen these things get inexplicably very contentious.--69.112.209.47 (talk) 03:58, 16 May 2018 (UTC)
replies: text: There are no references on that website, but I'll link it as an external reference. Tom Ruen (talk) 20:59, 23 May 2018 (UTC)
replies: text: Yeah, I guess it wouldn't be a good source for an encyclopedia—I'm pretty sure that the people on that website basically made up the names they use themselves even if their names are better than "n-cell"-style names. Thanks for adding it as an external link, though! 69.112.209.47 (talk) 05:03, 31 May 2018 (UTC)
thread_title:  Yet Another Alternate Name 

message: The "Related polytopes and honeycomb" section of the article states that "It [the 5-cell] is similar to three regular polychora: the tesseract {4,3,3}, 600-cell {3,3,5} of Euclidean 4-space, and the order-6 tetrahedral honeycomb {3,3,6} of hyperbolic [4] space. All of these have a tetrahedral cell." However, this is not referring to geometric similarity; instead, it's referring to "similarity" in a non-geometric way! Since this is arguably a geometry article, I think this is worth changing.  — Preceding unsigned comment added by 69.112.209.47 (talk) 04:07, 16 May 2018 (UTC)
replies: text: I reworded it, and it was wrong anyway, listing tesseract rather than 16-cell. Tom Ruen (talk) 20:41, 23 May 2018 (UTC)
thread_title:  Lexical Ambiguity Alert! 

message: An editor has asked for a discussion to address the redirect Hypertetrahedron. Please participate in the redirect discussion if you wish to do so.  signed, Rosguill talk 19:37, 26 May 2019 (UTC)
thread_title:  Hypertetrahedron listed at [[Wikipedia:Redirects for discussion|Redirects for discussion]] 

message: Listed as a name for a lower-symmetry variant of the 5-cell, added by User:Tomruen in 2014. Google gives no relevant results for "hyperdisphenoid" except copies of this very article. Where does it come from? The name i'm familiar with for this shape is "5-2 step prism", which seems to come from Wendy Krieger. --Goomba1729 (talk) 16:05, 26 April 2022 (UTC)
replies: text: Yes, it was an attempted descriptive dimensional analogue of the tetragonal disphenoid which has a cycle of 4 equal edges, and two equal cross-edges. The 5-cell form has a cycle of 5 equal edges and 5 equal cross-edges. I wish there was better term available to all of the n-simplices in this series. Their symmetries are isomorphic to the dihedral group of the regular polygons, which can be seen in these 2D projections, so the n-gonal description is an accurate generalization. Note: the 4 and 6 edge labels are not lengths, but imply edge lengths from positions for squares and hexagons in the honeycomb. Tom Ruen (talk) 16:48, 26 April 2022 (UTC)
thread_title:  "Pentagonal hyperdisphenoid" 

message: :
Not loving this language.  Any vertex inhabits an infinite number of hyperplanes.
:
—Tamfang (talk) 04:48, 20 July 2023 (UTC)
replies: text: I was about to object that this doesn't specify that every set of four must not be in one plane, and so on down; but then it hit me, if four are in a plane, then the five form a skew pyramid, which is in a 3-plane, so an explicit recursion is not necessary – but could it be mentioned somehow for clarity? —Tamfang (talk) 00:17, 8 November 2023 (UTC)
thread_title:  different hyperplanes 

message: Just a note that 5-cell, 16-cell, 120-cell, 600-cell, and 744 (number) all have the same problem. In some cases they have exactly the same notes, so if you have gone through one it might be easy for you to spot those on the other articles that duplicate material in other articles and can be dropped in favor of a link. -- Beland (talk) 18:25, 21 May 2024 (UTC)
thread_title:  Excessive explanatory footnotes