# Talk:Jessen's icosahedron

I moved the sentence "This family was described by H. S. M. Coxeter in 1947." into the "related family" section.  Here is the relevant passage from Section 3.7 of  Regular Polytopes:

This [orientation of the edges of a regular octahedron] enables us to define, for any given ratio 
  
    
      
        a
        :
        b
      
    
    {\displaystyle a:b}
  
, twelve points dividing the respective edges in this ratio, so that the three points in each face form an equilateral triangle. In general, these twelve points will be the vertices of an irregular icosahedron, whose faces consist of eight such equilateral triangles and twelve isosceles triangles. Without loss of generality, we may suppose that 
  
    
      
        a
        >
        b
      
    
    {\displaystyle a>b}
  
. When 
  
    
      
        a
        
          /
        
        b
      
    
    {\displaystyle a/b}
  
 is large, the isosceles triangles have short bases; in the limit they disappear, as their equal sides coincide and lie along the twelve edges of the octahedron. But when 
  
    
      
        a
      
    
    {\displaystyle a}
  
 approaches equality with 
  
    
      
        b
      
    
    {\displaystyle b}
  
, the isosceles triangles tend to become right-angled; in the limit, pairs of them form halves of the six square faces of a cuboctahedron, as in Fig. 84a on page 152.* By considerations of continuity, we see that at some intermediate stage the isosceles triangles must become equilateral, and the icosahedron regular.  In fact, the squares of the respective sides are 
  
    
      
        
          a
          
            2
          
        
        −
        a
        b
        +
        
          b
          
            2
          
        
      
    
    {\displaystyle a^{2}-ab+b^{2}}
  
 and 
  
    
      
        2
        
          b
          
            2
          
        
      
    
    {\displaystyle 2b^{2}}
  
, which are equal if 
  
    
      
        
          a
          
            2
          
        
        −
        a
        b
        −
        
          b
          
            2
          
        
        =
        0
      
    
    {\displaystyle a^{2}-ab-b^{2}=0}
  
, so that 
  
    
      
        a
        
          /
        
        b
        =
        τ
      
    
    {\displaystyle a/b=\tau }
  
. (See 2.47.) Thus the twelve vertices of the icosa­hedron can be obtained by dividing the twelve edges of an octahedron according to the golden section.

I think it's clear that Coxeter is describing a one-parameter family of convex icosahedra, as illustrated at kinematics of the cuboctahedron, not Douady's six-beaked shaddocks.  Otherwise, the family would not include the regular icosahedron, and the equal sides of each isosceles face would not coincide in the octahedral limit.  As far as I can tell, Coxeter never considers polyhedra with reflex dihedral angles.
On a related note: The star in the quoted passsage is a citation of a 1940 paper by Coxeter [1], where he credits this construction to Alicia Boole Stott. — Jeﬀ Erickson (talk) 03:29, 5 April 2025 (UTC)

From your picture it looks like the vertices of the regular and Jessen's icosahedron coincide.  That's not true.  If you get a chance, please try to correct the picture.  Mhym (talk) 04:14, 23 November 2008 (UTC)

Sorry. From everything I can see and read, the vertices DO coincide. Can you explain how they don't? Tom Ruen (talk) 17:44, 23 November 2008 (UTC)
P.S. I have Cromwell's book, Polyhedra. I couldn't find a reference to this figure, not in the index or quick search anyway. Can you give a page number? Tom Ruen (talk) 17:46, 23 November 2008 (UTC)
I don't have Cromwell's book.  The website I checked in the article references says pages 239-246 in Cromwell.  I am using Pak's book, p. 174.  Prof. Pak started the Jessen's icosahedron article.  Mhym (talk) 17:53, 23 November 2008 (UTC)
The problem is that the dihedral angles are not right if you simply take the regular icosahedron.  You need to modify the icosahedron a bit, as explained in Jessen's and Goldberg's articles, and in Pak's book.  Otherwise the polyhedron is not shaky, i.e. otherwise it is both infinitesimally and continuously rigid (see Goldberg's article). Mhym (talk) 18:02, 23 November 2008 (UTC)
I took the model data from [2].
Oh, well...  Then [3] is incorrect either.  Check out the dihedral angles (simply rotate the polyhedron in the Java applet to see that they are not right in that model.  You might want to read the papers/books in my post above.  Mhym (talk) 19:48, 23 November 2008 (UTC)
I moved the text above from my talk page in case the article author can help. I'll try to learn more. Tom Ruen (talk) 00:28, 24 November 2008 (UTC)
I looked at data here [4], but also looks like exact icosahedral coordinates. Tom Ruen (talk) 00:42, 24 November 2008 (UTC)
That's correct.  Still, this is not Jessen's orthogonal icosahedron.  I am guessing this is a common mistake.  I am thinking this should be emphasized in the article.  Mhym (talk) 02:30, 24 November 2008 (UTC)
Here we go - I found Goldberg's article on the web.[5]  It clearly explains the importance of 90 and 270 degree angles.   Mhym (talk) 02:34, 24 November 2008 (UTC)
Finally, over 10 years later, I have uploaded a new picture. I'm sure someone else can do better but in the meantime we at least don't have the wrong geometry. —David Eppstein (talk) 01:40, 17 October 2019 (UTC)
@David Eppstein:
Your head section figure is fine. Suggestion #1: It would be even better if it was turned a few ° clockwise around the vertical axis, as the following 2 faces would appear a bit more: the back isosceles triangle on the right side, & the left isosceles triangle on the top. (I can't do it: i don't have a geometry program that can change an svg file...)
You've reverted my latest edit a bit hastily: with the cyclic permutations of (±2,±1,0) as vertex coordinates, your head section figure does use left/right x-axis & front/back y-axis, (if vertical z-axis). If it used back/front x-axis & left/right y-axis, (if vertical z-axis), then your orthogonal icosahedron would be turned 90° around the z-axis.
Suggestion #2: But perhaps i should replace my asterisks with adding, after "the cyclic permutations of (±2,±1,0)", a little sentence like: "These coordinates, on left/right x-axis, front/back y-axis, vertical z-axis, orthonormal axes, yield the head section figure, with this orientation."? (If necessary, please rectify my English...)
--JavBol (talk) 21:12, 14 September 2021 (UTC)
Perhaps you should START BASING YOUR WIKIPEDIA EDITS ON PUBLISHED SOURCES instead of trying to figure things out for yourself and then spewing the results into articles. For instance, what is your source for your weird coordinate notation with asterisks in front of the coordinates? And why should readers care how we label the axes in an illustration of this polyhedron? What useful information about this polyhedron does it give them? I hope we're not going to see the same pattern of endless revisions to your talk comments, with no hope of the results ever becoming encyclopedic filling up my watchlist as they already are for Talk:Dual polyhedron (hint: when you have nine edits in a row to the same talk page, over a spread of two weeks, it may mean that people have stopped listening to you).  —David Eppstein (talk) 21:25, 14 September 2021 (UTC)
My asterisk in front of the coordinates was not a coordinate notation, but just the 2nd asterisk that my 1st asterisk referred to. But indeed, it was a bit weird; hence my suggestion #2: adding, after "the cyclic permutations of (±2,±1,0)", a little sentence. I'm used to back/front x-axis & left/right y-axis (& vertical z-axis), so when i tried these coordinates on the head section figure, it didn't work.
(You're not going to see the same pattern of endless revisions to my comments on Talk:Dual polyhedron. This was a special situation: in my opinion, my 2nd construction shows that the DL construction part should not be moved to Uniform polyhedron; but as you guys didn't comment it, i thought it was not convincing without a "proof". (I wrote a "proof" of my 1st construction, as it requires simpler calculations; but it was more complicated than i had expected...) I'll never try to add any of these 2 constructions to the article itself. (I also did the corresponding calculations for my 2nd construction, but i won't fill Talk:Dual polyhedron with those...))
--JavBol (talk) 01:27, 15 September 2021 (UTC)
Coordinate ordering and orientation is an arbitrary choice:

—David Eppstein (talk) 01:50, 15 September 2021 (UTC)

Suggestion #3: after "the cyclic permutations of (±2,±1,0)", instead of a little sentence, perhaps one should add a 2nd figure of the Jessen's icosahedron, but with its three 2-fold symmetry / coordinate axes & their labels "x", "y", "z" sticking out of it: back/front x-axis, left/right y-axis, vertical z-axis, so that it would be turned 90° around the z-axis, & so would show a different aspect/side than the head section figure (since the orthogonal icosahedron has no 4-fold symmetry axis). --JavBol (talk) 16:46, 15 September 2021 (UTC)
Or you could turn your laptop sideways and see exactly the same thing without the clutter of the axes and labels and another redundant image in the article. —David Eppstein (talk) 17:37, 15 September 2021 (UTC)
@Steelpillow: Do you have any opinion on my suggestions #1, #2, #3, please? In advance, thank you for your answer! --JavBol (talk) 23:45, 6 October 2021 (UTC)
If anybody wishes to prepare a new image and upload it to the Commons, then we can form a consensus on whether to use it. I agree that coordinate axes are best left out of it, unless their purpose is to illustrate a significant discussion of them in the main text. However, when I turned my 24" desktop monitor sideways it knocked over my coffee and I had to get a new keyboard, so I am not convinced by that argument.  — Cheers, Steelpillow (Talk) 06:22, 7 October 2021 (UTC)

Opinions on which of these two images is better? The first is less busy, and has face shading that might be helpful in understanding the 3d shape, but the second shows more hidden detail because of the transparency, matches a more conventional coordinate system, and is in true perspective rather than orthogonal projection. —David Eppstein (talk) 07:32, 7 October 2021 (UTC)

I think the new perspective view gives a better feel for the overall 3D shape and symmetry. I am not sure that the transparency adds anything though, I have no strong opinion on that. — Cheers, Steelpillow (Talk) 08:12, 7 October 2021 (UTC)
Thank you both for your detailed answers. I prefer the translucent image. Suggestion #4: But, after "the cyclic permutations of (±2,±1,0)", instead of a little sentence, perhaps one should add this translucent image, & thus show 2 very different images of the Jessen's icosahedron? --JavBol (talk) 11:05, 7 October 2021 (UTC)
@Steelpillow: Do you have any opinion on my suggestions #2, #4 & #1, please? In advance, thank you for your answer! --JavBol (talk)
I don't see any need to elaborate; the images are there to clarify the text, not the other way round. I already answered #1. — Cheers, Steelpillow (Talk) 08:23, 13 October 2021 (UTC)
@Steelpillow: Thank you for your answer. Sorry: i don't understand where/how you answered my suggestion #1. :-P
You asked about a modified image. I replied, "If anybody wishes to prepare a new image and upload it to the Commons, then we can form a consensus on whether to use it." I cannot help with 3D software, though Inkscape is a free-to-use SVG editor and is widely used. — Cheers, Steelpillow (Talk) 08:48, 7 November 2021 (UTC)
@David Eppstein: Now, i understand that "give all the dihedrals right angles" is better than "give all the dihedral right angles". What is(are) the other grammatical disimprovement(s) in my latest edit on Jessen's icosahedron, please? If i don't understand it(them), i'm bound to make it(them) again on other Wikipedia pages & on my future edit summaries...
--JavBol (talk) 22:39, 6 November 2021 (UTC)
Why do you think repeating the word "faces" twice in the same sentence, using the vaguer "regular" in place of the previously-used "equilateral", and splitting a short three-sentence paragraph into a single-sentence paragraph and a two-sentence paragraph are improvements? More to the point, why do you think this adds so much value to the article to be worth arguing about and pinging other editors back to the argument rather than just moving on? —David Eppstein (talk) 22:02, 7 November 2021 (UTC)
@David Eppstein: Thank you for your answer. And what about my replacing:
"The shapes in this family range from cuboctahedron to regular octahedron (as limit cases), which can be inscribed in a regular octahedron." with:
"The shapes in this family range from cuboctahedron to regular octahedron (as limit cases), and can be inscribed in a regular octahedron.", please? I just want to finish what i started. --JavBol (talk) 20:14, 8 November 2021 (UTC)
I think the whole sentence needs to go. If you take the cuboctahedron and regular octahedron as limiting cases, and don't go beyond them, you don't get non-convex variants and in particular you don't get Jessen's icosahedron. Also, regardless of whether it uses "which" or "and", the grammar is awkward. —David Eppstein (talk) 23:09, 8 November 2021 (UTC)
Thank you for your answer. I leave the difficult question of removing or rephrasing this whole sentence to you; in the meantime, i'll just restore "and" in place of "which".
By the way: indeed, 《 using the vaguer "regular" in place of the previously-used "equilateral", and splitting a short three-sentence paragraph into a single-sentence paragraph and a two-sentence paragraph 》 are disimprovements; but you caused them, by reverting my latest edit on Jessen's icosahedron. --JavBol (talk) 16:02, 9 November 2021 (UTC)
The dimensions of Jessen's icosahedron are noteworthy for being the square roots of the integers 1 - 6. There is quite a bit more that is interesting about the shape of the Jessen's than that its dihedrals are 90° and its long edges are 4 when its short edges are √6. For another article, I have done this translucent illustration of the Jessen's showing the inscribed cube and its unique integer-square-root dimensions, with an explanatory caption. I wonder if you think my illustration and caption works, or if it is too busy or confusing, and if it would be an improvement to this article to replace the translucent illustration in the /* Construction and geometric properties */ section with a more detailed translucent diagram.Dc.samizdat (talk) 01:17, 24 April 2022 (UTC)

I think it's very busy, and would not be an improved replacement for the existing image. —David Eppstein (talk) 16:07, 25 April 2022 (UTC)
I've redone it without the hidden edges -- better? −Dc.samizdat (talk) 03:09, 28 April 2022 (UTC)
The text is still too tiny to read, and the shading appears a bit haphazard. Also, there's the issues raised above by User:JavBol: the current semitransparent image uses different coordinates, chosen to match the ones in the text. You appear to have rotated them back to match the other image. —David Eppstein (talk) 04:39, 28 April 2022 (UTC)
I've made the text font bigger. The shading isn't haphazard, it's just deliberately subtle (the concave faces are slightly less transparent than the equilateral faces); this was the way I found to make the image less cluttered and busy. I think it works now, for the purpose of illustrating the dimensions. That said, I don't think this diagram should replace any of the existing images in the article, each of which has its own purpose. I would just add this one as an additional image, with its caption; it is not redundant of any of the other images since it alone illustrates the inscribed cube and the dimensions; and neither is it the best image for any other purpose.
Since the purpose of this image appears to be to highlight the claim "The dimensions of Jessen's icosahedron are noteworthy for being the square roots of the integers 1 - 6", it would also be helpful to have a published source for this claim. (It is a simple calculation that the dimensions are as stated, but the claim that it is significant that they are square roots of consecutive integers goes beyond that calculation.) It is a little difficult to tell which dimensions go with which edges. Also, 
  
    
      
        
          
            1
          
        
      
    
    {\displaystyle {\sqrt {1}}}
  
, 
  
    
      
        
          
            4
          
        
      
    
    {\displaystyle {\sqrt {4}}}
  
, and 
  
    
      
        2
        
          
            4
          
        
      
    
    {\displaystyle 2{\sqrt {4}}}
  
? For numbers that are actually integers? That is pointless obscurantism. We should not be doing that. —David Eppstein (talk) 05:46, 3 May 2022 (UTC)
Agreed (about the integers) and fixed. Also agree that absent a citation we should not make any claim of significance; the present caption doesn't. --Dc.samizdat (talk) 17:44, 3 May 2022 (UTC)
I still don't understand why some points and distances are highlighted, and others not. Your highlighted points are the vertices of the polyhedron, its center of symmetry, the midpoints of its concave edges, and the centers of its equilateral-triangle faces; why those specific points, and not also (for instance) the midpoints of the convex edges? Your highlighted distances include edges, center to everything else, and the altitudes of the non-equilateral faces, and one pair of non-adjacent vertex-to-vertex, but not the altitudes or center-to-vertex distances in the equilateral triangle, and not the distances between any other kind of pair of non-adjacent vertex-to-vertex – why? —David Eppstein (talk) 18:03, 3 May 2022 (UTC)
I developed this diagram for use in the Kinematics of the cuboctahedron article. It turns out that these dimensions of the Jessen's are important in its role as the stable equilibrium point in the kinematic cuboctahedron transformations: they are dimensions that the Jessen's shares in some manner with one or more of the other three kinematically-related polyhedra (cuboctahedron, regular icosahedron, octahedron). Specifically, these dimensions are all related to radii of some kind of one of the four polyhedra in the rigid-edge transformation, from the long edge mid-edge radius √1 to the short edge length √6 which is of course also the cuboctahedron long radius. √3 is the long radius of the octahedron, and √5 is related to the radii of the regular icosahedron. (See the metrics table in the article). The only distance I label that isn't an edge or such a "linking radius" is √2, the altitude of the isosceles faces; it is labelled not because it is an altitude but because (as I footnoted in the article) it is crucially involved in the radii in that √2 is the product by which the long radius of the polyhedron expands (from limit-smallest-case octahedron to limit-largest-case cuboctahedron) in all the kinematic transformations (regardless of how parameterized). In other words, these dimensions of the Jessen's are all significant because they situate the Jessen's in relationship to these other most closely related polyhedra. That, rather than the "coincidence" that they are the square roots of the first six integers, is why they are significant enough that they must be labelled.--Dc.samizdat (talk) 01:40, 4 May 2022 (UTC)

The following sentence:
"If Jessen's icosahedron (or its variant with vertices in the position of a regular icosahedron) is nested inside another regular icosahedron, it is possible to replace pairs of an isosceles face of Jessen's icosahedron and a corresponding face of the outer icosahedron by tubes of six triangles, [...]" should be improved such as:
"If Jessen's icosahedron (or its variant with vertices in the position of a regular icosahedron) is nested inside an actual regular icosahedron, it is possible to replace the eight pairs of an equilateral face of the inner icosahedron and a corresponding face of the outer icosahedron by eight mouths of six triangles each, [...]"; shouldn't it?
Anyway, an image would help to understand this rather complicated construction. :-P --JavBol (talk) 21:29, 10 November 2021 (UTC)
@Steelpillow: Do you have any opinion on all these points, please? In advance, thank you for your answer! --JavBol (talk) 18:11, 11 November 2021 (UTC)
To be honest, I don't see this as encyclopedic material, it's just some random factoid of no mathematical significance; see WP:NOTEVERYTHING. I think the article is better off without it. — Cheers, Steelpillow (Talk) 20:51, 11 November 2021 (UTC)
Thank you for your answer. But i "Never can get enough" toroids. ;-) However: indeed, the following sentence:
"This surface can be viewed as a non-convex analogue of a Platonic solid." should be removed, because the (topological) surface in question is neither face-transitive, nor edge-transitive; is it? --JavBol (talk) 18:10, 12 November 2021 (UTC)
Nevertheless, the source directly states that it is an analogue of a Platonic solid, because it is (as a combinatorial structure, not geometrically) vertex-transitive and geometrically has the symmetries of a tetrahedron. I think that is the only reason for interest in it, and I am skeptical about how interesting it is to combine combinatorial symmetry for part of the definition and geometric symmetry for the other part. If you don't find that reason credible, that would be a justification for removing this part entirely. —David Eppstein (talk) 18:52, 12 November 2021 (UTC)
Thank you for your detailed answer (all the more so as my "not-so-smart"-phone couldn't open this source's file). I agree that if the surface in question is neither combinatorially face-transitive, nor combinatorially edge-transitive, then it's less notable. (Obsolete comment: Moreover: with homothetic original Jessen's (or pseudo Jessen's) icosahedra, (i think) each 6-triangle mouth has 3 pairs of coplanar adjacent faces (forming a 3-trapezium mouth).) --JavBol (talk) 23:08, 14 November 2021 (UTC)
The outer one is a standard regular icosahedron, so its connecting faces are twisted relative to the inner one. —David Eppstein (talk) 23:37, 14 November 2021 (UTC)
Thank you for your answer. By the way: Jessen's icosahedron should state what Jessen's icosahedron's symmetry group is, shouldn't it? (Wills's article states that the symmetry group of the toroid {3,9;7} is S2xA4 (not isomorphic to the full tetrahedral symmetry group S4), where A4 is the tetrahedral rotation group, & S2 is represented by an involution.) --JavBol (talk) 21:42, 19 November 2021 (UTC)
We would need a source that covers this specific shape, not just a vaguely-related source that states a symmetry group for a vaguely-related shape. —David Eppstein (talk) 22:22, 19 November 2021 (UTC)

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Jessen's icosahedron's infobox should state that Jessen's icosahedron is vertex-transitive, & that its isosceles-triangle faces are obtuse; shouldn't it? --JavBol (talk) 22:01, 27 November 2021 (UTC)

Added "isogonal" to the properties section of the infobox. "Obtuse" not added: we could go on and on in more detail about the exact shape of these faces, but that sort of detail is for article text, not infoboxes, and adding "obtuse" already is enough to make that line overflow. —David Eppstein (talk) 00:20, 28 November 2021 (UTC)
Thank you for your answer & for your edit. And what about adding "congruent": before "equilateral triangles", & before "isosceles triangles"? --JavBol (talk) 18:03, 28 November 2021 (UTC)
Again, infobox not for nuance. That's what article text is for. If you want to explain things carefully and in detail, the infobox is not for you. More specifically, I already said, adding even one more single word to the lines describing the faces causes them to be more than one line, and therefore not to be a line describing the faces. —David Eppstein (talk) 08:29, 29 November 2021 (UTC)

This review is transcluded from Talk:Jessen's icosahedron/GA1. The edit link for this section can be used to add comments to the review.
Reviewer: Kusma (talk · contribs) 20:55, 7 January 2022 (UTC)
I'll review this one over the next few days. —Kusma (talk) 20:55, 7 January 2022 (UTC)


### Overall progress and general comments
Nice article about an interesting geometric object. Great images, all correctly licensed. Happy with most things (not copyvio etc.), just some questions on what should be in the lead, a few clarifications and some minor points (see below). —Kusma (talk) 20:16, 8 January 2022 (UTC)


### Prose and content review
The six-beaked shaddock of Douady seems to have slightly more general coordinates, so it is a family of objects containing Jessen's icosahedron? (Or I misunderstand the paper). Douady isn't mentioned again outside the lead, but could be; see below.
You are correct; I found a source stating that Douady's family is more general but includes Jessen's icosahedron, and modified the lead to say so. —David Eppstein (talk) 00:11, 9 January 2022 (UTC)
right angles, even though they cannot all be made parallel to the coordinate planes "made" here refers to rigid motions?
Rewrote, "it has no orientation where they are all parallel". —David Eppstein (talk) 22:26, 8 January 2022 (UTC)
However, because its dihedral angles are rational multiples of pi, it has Dehn invariant equal to zero. Therefore Is this meant to be a free and modern reformulation of what Jessen says about this or are you using a different theorem about Dehn invariants here?
It is more or less what Jessen says in the third paragraph of his paper, except that I have substituted "rational multiples of pi" for "orthogonal", as a more general condition that would still lead to Dehn invariant zero. I wanted to avoid the false implication that only orthogonal angles would have this property. (Incidentally, if you have any suggestions for making Dehn invariant § Realizability less fearsome, I'd appreciate hearing them; the technicality of that section is a big part of why I haven't nominated that article for GA.) —David Eppstein (talk) 22:26, 8 January 2022 (UTC)
OK, makes sense. Basically what I was looking for is a clearer statement (e.g. in Dehn invariant) that just says "if the Dehn invariant is zero, the polyhedron can be reassembled into a cube". That is more or less in Jessen's article as you point out, so I won't argue this. —Kusma (talk) 09:56, 9 January 2022 (UTC)
provides a counterexample to a question of Michel Demazure here you could mention Douady again, who apparently introduced the shaddock as this counterexample (or even a family of counterexamples).
Done. —David Eppstein (talk) 00:11, 9 January 2022 (UTC)
constructed in 1949 by Kenneth Snelson is there a better source for this? The thesis just claims this with nothing to back it up. If it is correct, shouldn't it be in the lead?
Ugh. This turns out to be an enormous can of worms. The fact that Snelson introduced tensegrity to Buckminster Fuller in 1948, and then that Fuller took sole credit for the concept for some ten years until finally being cornered into giving credit to Snelson, is well documented but off-topic here. The bad blood remaining from those circumstances make all recollections of that time from either Snelson or Fuller suspect. The images of tensegrity structures in Snelson's 1958 patent [6] do not appear to include this specific shape. It is also not included in Fuller's 1961 tensegrity paper [7]. The earliest references I can find to this shape in connection with tensegrity are from the 1970s, although it's not impossible that something like this could already have been found in the 1920s by Karlis Johansons (e.g. see jstor:779210). I am not convinced the source I was using for the 1949 claim is credible for this claim (it is mostly not about history and as you say has no justification for the claim). I have edited the article to remove all historical claims about tensegrity, because it is not the place to go into the history of tensegrity in general and because we have no good sources that I know of for the history of the tensegrity application of this specific shape. —David Eppstein (talk) 01:45, 9 January 2022 (UTC)
I was about to suggest that you could mention Snelson with just a claim, but you are probably right that this is debate is better covered elsewhere. —Kusma (talk) 09:56, 9 January 2022 (UTC)
certain pairs of equilateral- by it would be easier to parse if you added "triangle". (Again a bit further on).
Copyedited, among other changes eliminating this awkward wording.
The vertices of Jessen's icosahedron are perturbed from these positions in order to give all the dihedrals right angles. does this say anything other than what we already know? (Jessen vertices are at different positions and have right dihedral angles).
Not really. I removed this sentence and instead added a little more about how it has the same combinatorial type and symmetry as Jessen's. —David Eppstein (talk) 00:18, 9 January 2022 (UTC)
one of a continuous family of icosahedra I'm not sure I understand the construction here. Is the continuous parameter the ratio in which I divide each edge?
Yes. I added a sentence saying so. —David Eppstein (talk) 22:26, 8 January 2022 (UTC)
I added a movable gif from Wikimedia which seems to be helpful in explaining what's going on. Igorpak (talk) 05:13, 9 January 2022 (UTC)
User:Igorpak: Thanks, but I think this is a bad image. This same image has been removed before because the claim that it makes in its title is false. It does not stop at Jessen's icosahedron. The position that it stops at is the one with the vertices at a regular icosahedron, not Jessen. More, it only loops back and forth over the range of parameter values from octahedron to cuboctahedron, within which the shapes are all convex (the convex hulls of what you see in the image), not the parameter values past that point where it becomes non-convex. —David Eppstein (talk) 07:13, 9 January 2022 (UTC)
An image that does what is promised would be nice, but if we don't have one, better not include one that doesn't show the Jessen icosahedron. —Kusma (talk) 09:56, 9 January 2022 (UTC)
I see.  Thanks for catching this issue.  Would indeed be nice to have a corrected image.  Igorpak (talk) 18:35, 9 January 2022 (UTC)
to an infinite family of rigid but not infinitesimally rigid polyhedra in which way infinite? More and more vertices, or infinitely many solutions with a fixed number of vertices?
Added "combinatorially distinct" to clarify that more vertices was the intended meaning. —David Eppstein (talk) 22:26, 8 January 2022 (UTC)
I was talking about this issue with respect to the Gor’kavy/Milka claim at the bottom of the article.
isogonal and weakly convex should probably use {{em}} per MOS:ITALIC.
Ok, done. —David Eppstein (talk) 22:26, 8 January 2022 (UTC)
I think that's all I have. —Kusma (talk) 20:15, 8 January 2022 (UTC)

@Kusma: All issues responded to; please take another look. —David Eppstein (talk) 01:45, 9 January 2022 (UTC)
@David Eppstein: I think we're done once you clarify Gor’kavy/Milka. Nice work. —Kusma (talk) 09:56, 9 January 2022 (UTC)
@Kusma: I added a sentence at the bottom.  I hope this suffices.  Igorpak (talk) 18:49, 9 January 2022 (UTC)
@Igorpak @David Eppstein it needs a bit of copyediting. distinct from each other or from Jessen's? larger symmetry groups larger than what? no longer simplicial no longer? what is the passage of time here? Also, the sentence needs to be cited to a source.
While we're here, the external link at the bottom is nice, but could be described a tiny little bit. —Kusma (talk) 18:57, 9 January 2022 (UTC)
I copyedited the new material and removed the external link, as it didn't say much beyond what was already in the article (WP:ELNO #1). It does include a nice 3d viewable image, but it turns out to be of the icosahedral fake version, not of the actual Jessen's icosahedron. —David Eppstein (talk) 20:24, 9 January 2022 (UTC)
@David Eppstein Thanks for doing this.  Igorpak (talk) 21:09, 9 January 2022 (UTC)
I was confused about the image in the link; viewing it from a better angle made clear that it is the correct orthogonal one. Restored the link, with description. —David Eppstein (talk) 21:15, 9 January 2022 (UTC)
Excellent. All done now, I'll go do the paperwork. —Kusma (talk) 22:44, 9 January 2022 (UTC)