Talk:Johnson solid

Images
I have been modifying user:Cyp's povray macros to generate images of as many of the Johnson solids as I can. See User:AndrewKepert/poly.pov for what may be the latest version. Here is where I am tracking progress. Bold numbers have images.

Relocated to User:AndrewKepert/polyhedra

Images of the flat kind
Doesn't do 3d, and only knows 2 Johnson solids (so far), but here's makepolys.c.
 * [[image:pentagonal pyramid flat.svg|50px]][[image:square orthobicupola flat.svg|50px]]

&#922;&#963;&#965;&#960; Cyp  00:27, 5 Nov 2004 (UTC)

I'm making some "home-made" nets:
 * [[image:square_pyramid_net.png|50px]][[image:Triangular cupola net.PNG|50px]][[image:Square cupola net.PNG|50px]][[image:Pentagonal Cupola.PNG|50px]]

And the rest with Inkscape, now that I found out about it:
 * [[image:Pentagonal Rotunda Net.svg|50px]]

Now that there's enough nets for a whole section, anyone think we should incorporate them into the table?

Complete set of nets
I have Stella (software) which generates all the Johnson solids. Previously I didn't have the patience to try uploading all 92 nets, but figured easier for me than generating all from scratch. By default Stella colors faces by symmetry positions. I only had patience to upload them by indexed names. Here they all are! Feel free to "trace" or change arrangements in a complete set of SVG versions as your patience allows! I do think the symmetry coloring is worthy to use. Tom Ruen (talk) 23:46, 28 June 2008 (UTC)


 * I added the nets to stub articles J47-92. Patience exhausted for now. Tom Ruen (talk) 18:11, 29 June 2008 (UTC)


 * 1) [[Image:Johnson solid 2 net.png|80px]]
 * 2) [[Image:Johnson solid 3 net.png|80px]]
 * 3) [[Image:Johnson solid 4 net.png|80px]]
 * 4) [[Image:Johnson solid 5 net.png|80px]]
 * 5) [[Image:Johnson solid 6 net.png|80px]]
 * 6) [[Image:Johnson solid 7 net.png|80px]]
 * 7) [[Image:Johnson solid 8 net.png|80px]]
 * 8) [[Image:Johnson solid 9 net.png|80px]]
 * 9) [[Image:Johnson solid 10 net.png|80px]]
 * 10) [[Image:Johnson solid 11 net.png|80px]]
 * 11) [[Image:Johnson solid 12 net.png|80px]]
 * 12) [[Image:Johnson solid 13 net.png|80px]]
 * 13) [[Image:Johnson solid 14 net.png|80px]]
 * 14) [[Image:Johnson solid 15 net.png|80px]]
 * 15) [[Image:Johnson solid 16 net.png|80px]]
 * 16) [[Image:Johnson solid 17 net.png|80px]]
 * 17) [[Image:Johnson solid 18 net.png|80px]]
 * 18) [[Image:Johnson solid 19 net.png|80px]]
 * 19) [[Image:Johnson solid 20 net.png|80px]]
 * 20) [[Image:Johnson solid 21 net.png|80px]]
 * 21) [[Image:Johnson solid 22 net.png|80px]]
 * 22) [[Image:Johnson solid 23 net.png|80px]]
 * 23) [[Image:Johnson solid 24 net.png|80px]]
 * 24) [[Image:Johnson solid 25 net.png|80px]]
 * 25) [[Image:Johnson solid 26 net.png|80px]]
 * 26) [[Image:Johnson solid 27 net.png|80px]]
 * 27) [[Image:Johnson solid 28 net.png|80px]]
 * 28) [[Image:Johnson solid 29 net.png|80px]]
 * 29) [[Image:Johnson solid 30 net.png|80px]]
 * 30) [[Image:Johnson solid 31 net.png|80px]]
 * 31) [[Image:Johnson solid 32 net.png|80px]]
 * 32) [[Image:Johnson solid 33 net.png|80px]]
 * 33) [[Image:Johnson solid 34 net.png|80px]]
 * 34) [[Image:Johnson solid 35 net.png|80px]]
 * 35) [[Image:Johnson solid 36 net.png|80px]]
 * 36) [[Image:Johnson solid 37 net.png|80px]]
 * 37) [[Image:Johnson solid 38 net.png|80px]]
 * 38) [[Image:Johnson solid 39 net.png|80px]]
 * 39) [[Image:Johnson solid 40 net.png|80px]]
 * 40) [[Image:Johnson solid 41 net.png|80px]]
 * 41) [[Image:Johnson solid 42 net.png|80px]]
 * 42) [[Image:Johnson solid 43 net.png|80px]]
 * 43) [[Image:Johnson solid 44 net.png|80px]]
 * 44) [[Image:Johnson solid 45 net.png|80px]]
 * 45) [[Image:Johnson solid 46 net.png|80px]]
 * 46) [[Image:Johnson solid 47 net.png|80px]]
 * 47) [[Image:Johnson solid 48 net.png|80px]]
 * 48) [[Image:Johnson solid 49 net.png|80px]]
 * 49) [[Image:Johnson solid 50 net.png|80px]]
 * 50) [[Image:Johnson solid 51 net.png|80px]]
 * 51) [[Image:Johnson solid 52 net.png|80px]]
 * 52) [[Image:Johnson solid 53 net.png|80px]]
 * 53) [[Image:Johnson solid 54 net.png|80px]]
 * 54) [[Image:Johnson solid 55 net.png|80px]]
 * 55) [[Image:Johnson solid 56 net.png|80px]]
 * 56) [[Image:Johnson solid 57 net.png|80px]]
 * 57) [[Image:Johnson solid 58 net.png|80px]]
 * 58) [[Image:Johnson solid 59 net.png|80px]]
 * 59) [[Image:Johnson solid 60 net.png|80px]]
 * 60) [[Image:Johnson solid 61 net.png|80px]]
 * 61) [[Image:Johnson solid 62 net.png|80px]]
 * 62) [[Image:Johnson solid 63 net.png|80px]]
 * 63) [[Image:Johnson solid 64 net.png|80px]]
 * 64) [[Image:Johnson solid 65 net.png|80px]]
 * 65) [[Image:Johnson solid 66 net.png|80px]]
 * 66) [[Image:Johnson solid 67 net.png|80px]]
 * 67) [[Image:Johnson solid 68 net.png|80px]]
 * 68) [[Image:Johnson solid 69 net.png|80px]]
 * 69) [[Image:Johnson solid 70 net.png|80px]]
 * 70) [[Image:Johnson solid 71 net.png|80px]]
 * 71) [[Image:Johnson solid 72 net.png|80px]]
 * 72) [[Image:Johnson solid 73 net.png|80px]]
 * 73) [[Image:Johnson solid 74 net.png|80px]]
 * 74) [[Image:Johnson solid 75 net.png|80px]]
 * 75) [[Image:Johnson solid 76 net.png|80px]]
 * 76) [[Image:Johnson solid 77 net.png|80px]]
 * 77) [[Image:Johnson solid 78 net.png|80px]]
 * 78) [[Image:Johnson solid 79 net.png|80px]]
 * 79) [[Image:Johnson solid 80 net.png|80px]]
 * 80) [[Image:Johnson solid 81 net.png|80px]]
 * 81) [[Image:Johnson solid 82 net.png|80px]]
 * 82) [[Image:Johnson solid 83 net.png|80px]]
 * 83) [[Image:Johnson solid 84 net.png|80px]]
 * 84) [[Image:Johnson solid 85 net.png|80px]]
 * 85) [[Image:Johnson solid 86 net.png|80px]]
 * 86) [[Image:Johnson solid 87 net.png|80px]]
 * 87) [[Image:Johnson solid 88 net.png|80px]]
 * 88) [[Image:Johnson solid 89 net.png|80px]]
 * 89) [[Image:Johnson solid 90 net.png|80px]]
 * 90) [[Image:Johnson solid 91 net.png|80px]]
 * 91) [[Image:Johnson solid 92 net.png|80px]]

Elongated square gyrobicupola
The picture is wrong - that's obviously a rhombicuboctahedron. Compare:
 * No it is right. Look again. Andrew Kepert 03:47, 9 Nov 2004 (UTC)


 * It's definitely an image of the right polyhedron, but it's taken from an unflattering angle. Could someone POVRay up an image that is at first glance obviously not a rhombicuboctahedron? &mdash;ajo, 21 April 2005


 * I'm not sure that's possible. They don't call that the "pseudorhombicuboctahedron" for nothing.  RobertAustin 01:18, 8 November 2006 (UTC)


 * When you look at http://peda.com/posters/img/poly4.gif 10th row sixth picture from the left you can see a view of elongated square gyrobicupola which is very distinct of rhombicuboctahedron. 19:45, 17 April 2010 —Preceding unsigned comment added by 74.125.121.33 (talk)

The list
Usually it would be called good practice to make a list such as that in this article stand-alone. Not something to insist on, perhaps, in this case; but it is something to think about, in the way of writing the article so that it doesn't 'wrap' round having the list there in the current way. Charles Matthews 09:13, 17 Nov 2004 (UTC)
 * I don't understand this comment. Clarify?  dbenbenn | talk 05:54, 26 Jan 2005 (UTC)

Johnson numbers
Is the numbering of the Johnson solids arbitrary? If not, how are the Johnson numbers determined? I think this should be mentioned in the article. Factitious 19:25, Nov 21, 2004 (UTC)
 * Good point - the numbering was in Johnson's original paper. I have amended the article. Andrew Kepert 00:29, 22 Nov 2004 (UTC)

"simple" Johnson solids?
28 of the Johnson solids are "simple". Non-simple means you can cut the solid with a plane into two other regular-faced solids. But it isn't clear which ones. Anyone? dbenbenn | talk 05:52, 26 Jan 2005 (UTC)


 * Off the top of my head:
 * 1-6 (pyramids, cupolae & rotunda)
 * 63 (tridiminished icosahedron - can't chop any further)
 * 80 and 83 (parabidiminished & tridiminished rhombicosidodecahedra - ditto)
 * the "sporadics" 84-86 & 88-92, (87 is an augmented sporadic) They have no relation to platonics or archimedeans.
 * which makes 6+1+2+8 = 17. There are other components from the platonic, archimedean, prisms and antiprisms that could arguably considered as needed for a building any of the J solids, but these are not "of the J solids".  I think I have all or most of the list here, given your defn - well short of 28.
 * Where did you get 28? ... ah I see it in the mathworld article.  Google throws up no other ref to "simple johnson solid".  I suspect Mathworld is wrong, probably in the defn of "simple"  --Andrew Kepert 07:58, 27 Jan 2005 (UTC)


 * Okay, thanks. That's disturbing if MathWorld is totally wrong here.  dbenbenn | talk 22:15, 27 Jan 2005 (UTC)


 * Incidentally, the Wikipedia articles are using the term "elementary" instead of "simple," and upon incautious consideration I agree with Wikipedia's choice of terminology. &mdash;ajo, Apr 2005


 * I added a table of images at the end. Very useful.
 * Probably the list should be moved to "List of Johnson solids", and then this article can be shorter.
 * I'd like more statistics on these solids - Vertex, Edge, Face counts (and types of faces), Symmetry group. (I don't have this information) When this is available, making a data table would be more useful.
 * Tom Ruen 19:48, 15 October 2005 (UTC)


 * Actually, the Mathworld article was discussing all the simple convex regular-faced solids, including the simple Archimedean solids. There are 11 of these:
 * tetrahedron
 * dodecahedron
 * truncated tetrahedron
 * truncated cube
 * truncated octahedron
 * truncated cuboctahedron
 * truncated dodecahedron
 * truncated icosahedron
 * truncated icosidodecahedron
 * snub cube
 * snub dodecahedron
 * which when added to the 17 simple Johnson solids, make 28.
 * Mongo62aa (talk) 03:07, 15 August 2010 (UTC)
 * which when added to the 17 simple Johnson solids, make 28.
 * Mongo62aa (talk) 03:07, 15 August 2010 (UTC)


 * The cube is excluded because ...? —Tamfang (talk) 16:50, 16 August 2010 (UTC)


 * Because the prisms and antiprisms are excluded, otherwise the list would be infinite in length. Mongo62aa (talk) 14:14, 17 August 2010 (UTC)

NEW TABLE
I added a new table with columns: Name, image, Type, Vertices, Edges, Faces, (Face counts by type 3,4,5,6,8,10), and Symmetry.

I computed the VEF counts by the table from: http://mathworld.wolfram.com/JohnsonSolid.html
 * Total faces by: F=F3+F4+F5+F6+F8+F10
 * Computed total internal angle_sum=180*(F3+2*F4+3*F5+4*F6+6*F8+8*F10)
 * Used angle defect sum to compute vertices: V=chi+angle_sum/360 (chi=2 for topological spheres)
 * Computed edges by Euler: E=V+F-2

The results should be correct, but may not be correctly matched by names if the indices were inconsistent!

A Name for the #84 - #92 group? [Sporadics proposed 2009-02-21]
The series #84 - #92 are not derived from cut-and-paste of Platonics, Archimedians, and prisms. I put forth a trial name in the table: Johnson Special solids, after fiddling with a thesaurus for a while, thinking that they deserved better than "Miscellaneous". (One of them is actually an augmented Johnson special.) Other possibilities are Johnson Unique, Johnson Peculiar, Johnson Disctinctive, Johnson Elemental, etc.


 * Steven Dutch calls them "Complex Elementary Forms". &mdash;Tamfang 23:42, 8 July 2006 (UTC)


 * They're not really a set though, are they? As far as I can see, only the sphenocoronas form a set, and all the others are one-of-a-kind shapes. I think some sort of generic name like "Miscellaneous" or "Other" is the best way to describe them. "Special" indicates some sort of status they don't really have. Did Johnson himself give the group a name? In fact, did he group them at all? — sjorford++ 09:04, 10 July 2006 (UTC)

Very well, I will revert it back to Miscellaneous as I found it.

Any views on the name "Sporadics" for this part of the series? User:AndrewKepert used the term in passing, and I believe it fits the bill of not asserting commonality, whilst being less dismissive than "Miscellaneous". This collection is the most interesting to me because the faces generate new angles, and as I was modeling with Geomag, this gave new model possibilities.

Karl Horton (talk) 14:56, 22 February 2009 (UTC)


 * I like it well enough, but making up our own words is against the rules; we need to find a term already in use in the field. For whatever it's worth, this page calls them "Complex Elementary Forms".  —Tamfang (talk) 09:33, 23 February 2009 (UTC)

Table changes ongoing...
I removed the "type" column from the tables in favor of a list of types at the beginning of each section. It took too much screen width and redundant with polyhedron names.

I'd like to expand the table with a vertex configuration column, listing the counts and types of vertices for each form. I made an automated tally once somewhere and I'll see if I can merge it in sometime - NOW that there's some screen width to play with.

I have an old different tally on a test page - lists all reg/semireg/Johnson solids by vertex figure: User:Tomruen/Polyhedra_by_vertex_figures

Tom Ruen 07:56, 7 January 2007 (UTC)

Suspicious edits
All (it seems) of the individual Johnston solids pages were edited by 140.112.54.155 so that the table on each page listing the number of faces for the solid has entries like "3.5 triangles". They haven't responded for explanation that I've seen. Before I go fixing up 92 pages, is there any reason to believe this isn't vandalism? Thanks, Fractalchez (talk) 00:45, 6 December 2007 (UTC)


 * They look like honest edits, although notation could be confusing, 3.5 meaning 3×5=15 triangles, while could look like 3+1/2. It looks like an attempt to group the types of triangles - there's 3 sets of 5 triangles in equivalent positions of symmetry. I don't keep a watch on all the individual pages. Tom Ruen (talk) 01:06, 6 December 2007 (UTC)

Tetraeder
Why isn't the tetraeder 4 F3 on the list? Did I not understand the definitions enough? --Saippuakauppias ⇄ 11:14, 3 July 2008 (UTC)

It is on the list, under the name Gyrobifastigium. It's in the section of modified cupolas and rotundas, in that it can be viewed as a bicupola, but instead of the top being a polygon, it's a single edge, and the bottom is a square. You don't find a single one of these in normal cupolas/rotundas/pyramids though, because that would be simply a triangular prism. —Preceding unsigned comment added by Timeroot (talk • contribs) 19:15, 3 July 2008 (UTC)

Urgent!
I need to know the name of the Johnson solid with 42 faces, 80 edges and 40 vertices. Professor M. Fiendish, Esq. 11:10, 29 August 2009 (UTC)


 * You can search that for yourself - looks like at least two! Tom Ruen (talk) 22:48, 29 August 2009 (UTC)


 * In case the tables aren't clear enough for you: they are the elongated pentagonal birotundae. —Tamfang (talk) 23:16, 29 August 2009 (UTC)

Impossible Johnson solids
Proving the hexagonal pyramid with equilateral triangles is impossible uses the fact that 6 triangles add up to 360 degrees. But, here's a hard problem: prove the augmented heptagonal prism is not a valid Johnson solid. Georgia guy (talk) 22:06, 15 October 2010 (UTC)


 * hm, I guess I need to prove that &alpha;=atan(√2) &gt; 2π/7.
 * cos(&alpha;) = 1/√3, sin(&alpha;) = √(2/3)
 * exp(i &alpha;) = (1+i√2) / √3
 * exp(7 i &alpha;) = (43+13i√2) / 27√3, which is in the first quadrant, implying that either 2π/7&lt;&alpha;&lt;5π/28 or 0&lt;&alpha;&lt;π/14; the latter is ruled out because tan(&alpha;) &gt; tan(π/4).
 * What do I win? —Tamfang (talk) 04:07, 19 October 2010 (UTC)

I DISCOVERED A NEW JOHNSON SOLID
Really! here are pictures!

faces: 16 triangles, 3 squares, total 19 vertex figure: 1 (4,4,4), 3 (3,3,4,4), 3 (3,3,3,3,4), 5+5 (3,3,3,3,3) symmetry:C3v Discovered by me, David Park Jr.--David P.Jr. (talk) 09:44, 15 March 2011 (UTC)


 * Have you proven that the faces are flat and regular? Models can flex. —Tamfang (talk) 07:24, 16 March 2011 (UTC)

another near miss?
five squares eight triangles (eleven vertices) looks valid to me url= http://cs.sru.edu/~ddailey/tiling/hedra.html David.daileyatsrudotedu (talk) 02:30, 11 October 2018 (UTC) Jim McNeill [3] has demonstrated to my satisfaction that the referenced shape is indeed a near miss, having distortion mainly confined to the two isolated square faces. David.daileyatsrudotedu (talk) 12:02, 12 October 2018 (UTC)

failed
I installed Great Stella software and test it but some triangles are not quite regular. It has 3 squares, 6+9 isosceles triangles, and 1 regular triangle. T.T OTL How can prove or disprove no more Johnson solid? --David P.Jr. (talk) 12:48, 16 March 2011 (UTC)


 * A good attempt. I've never tried, but the proof was the intention of Johnson's paper! There's another open-ended category called near-miss Johnson solids, and some are listed here: . Tom Ruen (talk) 17:31, 16 March 2011 (UTC)

Previously discovered
This model is readily buildable with Polydrons. Jim McNeill keeps a catalog of near misses and lists this one.

This trisquare hexadecatrihedron has 16 triangular and 3 square faces, and looks somewhat like a cube embedded in an icosahedron (hence my informal name of 'cubicos'),. The squares are regular and the aggregate distortion in the lengths of the triangular edges is only about 0.1 in total (stress map). Distortion (E=0.10, P=0, A=18.3°). 

Karl Horton (talk) 11:32, 10 July 2013 (UTC)

duals of the Johnson solids
Is there a name for the set of 92 polyhedra that are duals of the Johnson solids? Other than "duals of the Johnson solids"? (By analogy with the way Catalan solids are duals of the Archimedean solids). --DavidCary (talk) 04:10, 6 April 2013 (UTC)


 * Not to my knowledge. I don't think they have even been enumerated in any reliable source. I'd probably call them "Johnson duals" for short. &mdash; Cheers, Steelpillow (Talk) 13:12, 6 April 2013 (UTC)


 * One problem is that for a geometric dual – rather than a mere topological dual – you need a center. How do you choose centers for the 56 that lack D symmetry? (Ch or Ci or S symmetry would also do, but there aren't any.)  —Tamfang (talk) 07:09, 27 February 2014 (UTC)


 * To define the centre usefully, it would need to remain static under duality - that is, the centre of the dual must be the same point as the centre of the original. This ensures that when you dualise the dual, you get back to the original form. It turns out that for some figures this is really hard, I seem to recall that even the "Stella" software author gave up on it and used a simpler algorithm. I think it would be fair to ignore centres and polar reciprocity but instead to require the dual condition, that all vertices be regular, i.e. having the same polygonal angle between adjacent edges. Not sure if that set of polyhedra would match the Johnson solids one-to-one, though: an interesting problem. &mdash; Cheers, Steelpillow (Talk) 10:37, 27 February 2014 (UTC)

Organizing the table
Can anyone edit this article so that there's one large table of all 92 figures rather than several small tables?? This way, the table can be re-sorted by the number of faces each polyhedron has or any other appropriate way. Georgia guy (talk) 21:38, 13 October 2010 (UTC)


 * I think the value of multiple tables is that it easier to edit, and there were distinct groupings by named categories from Johnson's numbering, but it looks easy to delete the sections and table headers to remerge into a single table if you want to try. Tom Ruen (talk) 21:48, 13 October 2010 (UTC)


 * Adding a "|-" before the headers seemed to do the trick! Tom Ruen (talk) 22:19, 13 October 2010 (UTC)


 * This was not a good change. The classes made it easier to figure out how the solids were made and where the regular variations started and stopped, so that if you needed to do something for a set of the solids you could work out your process from choices in each class and extend it to the rest regularly. So there's a Pareto principle in the information one needs to study these solids and learn their types. One can easily merge all the tables in a sandbox if one needs them ordered by a column of the table. ᛭ LokiClock (talk) 23:22, 31 July 2013 (UTC)


 * Perhaps both are useful, grouped solids here, and List of Johnson solids as a single sortable table? (I definitely use the sort feature, by face counts, edge counts, or symmetry) Perhaps the list here should be simpler, without element counts, symmetry, etc? Tom Ruen (talk) 00:08, 1 August 2013 (UTC)
 * I added List of Johnson solids as an experiment, copied from here, so this article could have a more compact summary by groupings? Tom Ruen (talk) 00:15, 1 August 2013 (UTC)
 * I started reworking the first ones into topological groups. I'm not sure if this helps LokiClock's purpose. Tom Ruen (talk) 02:40, 1 August 2013 (UTC)


 * Yes, that serves the original purposes I had used this article's classification for. If the solids are given in order in this table, then we don't need to group the solids in order here. Is there a reason for having the augmentation and diminishing subclasses as separate sections? Also, I found that if you use 2, 2 , the tables will still sort the numbers inside the tag properly, so perhaps the beginnings of the sections in the original numeration can be labelled inside the table. ᛭ LokiClock (talk) 06:06, 1 August 2013 (UTC)


 * I'm not sure I follow. I hope the groupings here are helpful. Myself, I'm interested in showing similar non-Johnson solids as well, whether regular, semiregular, or having coplanar faces, so I started adding some of these. I added the bottom rows of the table on "augmented from polyhedra" to help show their construction, since some of the views, even transparent, are confusing to see easily. Anyway, I'd do more when I have some time. Tom Ruen (talk) 06:12, 1 August 2013 (UTC)
 * p.s. I'm unsure if the nets are helpful here, so those rows might be removed. Tom Ruen (talk) 06:14, 1 August 2013 (UTC)


 * They are helpful. Looking again, they have different themes of construction, even if they're all the same type of modification, and some solids have more than one construction. I think the nets are helpful because they can give clues as to how the solids are similar to others and how to dissect them and put them together. It can be hard to figure out what the "others" and the rotunda are all-around using just the picture. Just now I used them to make sure the triangular hebesphenorotunda's squares all had 3 triangles attached, which suggested it had triangular symmetry (the triplet of pentagons and their center triangle has the same plane of rotation as the hexagon), which I then confirmed at its article. The information you just added it reinforced by the nets. Some time ago, when I was generalizing these solids to 4D I mainly interpreted the nets, and didn't have this information about how the icosidodecahedron was related to the rotunda and so forth. Around this same time I also noticed the wedging theme in constructing the "others" by looking at their nets, because when I saw the pictures of the solids my eye didn't group the faces by those wedges, but in the Bilunabirotunda (File:Bilunabirotunda.png) for example first separating it along one of the hexagons crossing the midpoint, then grouping the faces of each piece into the front faces and back faces. ᛭ LokiClock (talk) 07:43, 1 August 2013 (UTC)
 * p.s. on nets, the faces are colored by the symmetry, autogenerated by Stella (software), although manually made nets might pick different arrangements for seeing the figures better. Tom Ruen (talk) 02:13, 3 August 2013 (UTC)


 * Thanks, that's incredibly helpful to know about them! ᛭ LokiClock (talk) 10:23, 17 August 2013 (UTC)

Isomorphs
A new section on non-convex isomoprps has been added. I would suggest that these are not notable. Other classes of isomorph exist - convex and non-convex - but nobody has bothered to describe them, there is nothing notable about these ones either. A single fanboi web page does not constitute a reliable source. &mdash; Cheers, Steelpillow (Talk) 08:21, 19 April 2014 (UTC)
 * The crossed cupolae have probably been described more widely: Johnson has terminology for them, so he might mention them somewhere. But yeah, most of these are just trivial and don't really need to be here, and after all they are just cut-and-paste operations. So I removed it again. Double sharp (talk) 14:09, 22 April 2014 (UTC)
 * How can we know that "Johnson has terminology for them" unless we know whether or not he mentioned them somewhere? (Just teasing, thanks for the revert). &mdash; Cheers, Steelpillow (Talk) 17:04, 22 April 2014 (UTC).
 * The terms "semicupola" (cuploids) and "sesquicupola" (cupolaic blend?) have been attributed to Johnson on some websites, so it's quite possible that he mentions them in his (still) forthcoming book, or somewhere else. Double sharp (talk) 12:38, 24 April 2014 (UTC)

Convex regular-faced polyhedra with conditional edges
I found this interesting list Convex regular-faced polyhedra with conditional edges, Johnson solid failures due to adjacent coplanar edges, 78 forms, by Robert R Tupelo-Schneck. It says the listing was independently produced and proven complete in 2010 by A. V. Timofeenko. Tom Ruen (talk) 03:26, 14 April 2017 (UTC)

Strictly convex
The article says: A Johnson solid is a strictly convex polyhedron. As far as I know, a strictly convex polyhedron is a strictly convex set, and hence the edges can't contain straight lines. Madyno (talk) 17:22, 30 July 2017 (UTC)


 * How would you define it? The excluded cases have dihedral angles of zero, or having two faces in the same plane. Tom Ruen (talk) 15:12, 31 July 2017 (UTC)

Dual of Johnson solids
The dual of Archimedean solids are Catalan solids, and the dual of Platonic solids are also Platonic solids, but what are the dual of Johnson solids? 2402:7500:586:91EF:6911:7EBA:959B:3B90 (talk) 03:53, 7 September 2020 (UTC)


 * Less well defined, as discussed in the section duals of the Johnson solids above. —Tamfang (talk) 06:17, 30 October 2020 (UTC)

What is the set of the polyhedrons whose faces are all regular polygons? (not need to be convex or uniform, and there is no requirement that each face must be the same polygon)
Regular polyhedron does not need to be convex, the convex regular polyhedrons are the 5 Platonic solids, and there are 9 non-convex regular polyhedrons, including the 4 Kepler–Poinsot polyhedrons and the 5 regular compounds, and for the semiregular polyhedrons, there are 13 convex ones other than the convex prisms and the convex antiprisms, but what are the non-convex ones? And for the polyhedrons with each face regular polygons or regular star polygons, there are 92 convex ones other than the regular polyhedrons and the semiregular polyhedrons, but what are the non-convex ones? (These would include the 4 Kepler–Poinsot polyhedrons, the 5 regular compounds, the stellated octahedron, the 53 nonconvex uniform polyhedras, the uniform star prisms, the uniform star antiprisms, the augmented heptagonal prism, the pentagrammic prism, the deltahedras, the toroidal prisms, etc.)

(Polyhedrons whose faces are not all regular polygons, such as the Catalan solids, the hexagonal pyramid, the near-miss Johnson solids, the parallelepiped, the rhombic icosahedron, the Szilassi polyhedron, the Császár polyhedron; and the polyhedrons with 180° dihedral angles, such as ; and the non-connected polyhedrons, such as the crossed prisms; and the degenerate polyhedras, such as dihedron and hosohedron; and the infinity forms, such as triangular tiling, square tiling, hexagonal tiling, trihexagonal tiling, snub trihexagonal tiling, truncated trihexagonal tiling, apeirogonal prism, apeirogonal antiprism; are not in this set)

Reference: ——36.234.85.41 (talk) 10:03, 29 August 2021 (UTC)


 * The last row includes, for a start, a stack of n p-antiprisms joined at p-faces. (–hedra is plural, darn it.) —Tamfang (talk) 13:00, 5 August 2023 (UTC)
 * If I'm understanding the question correctly, keeping all faces regular or regular star polygons while allowing different types of vertex, self-intersection, dihedral angles >=180 degrees, I suspect there might be an uncountably infinite collection of such, or at least extremely hard to enumerate...
 * Among other things, you have:
 * ANy polyform with a platonic, archimedean, keplar-poinsot, uniform star polyhedral, or Johnson solid monoform. This includes the aformentioned antiprism stacks, polycubes, polytetrahedra, polytruncated octahedra, polyiamond prisms, polyhex prisms... and those are just the poly forms already mention or which have a tiling of the plane or pace as a limiting case.
 * Non-convex augmentations, including more than one type of face or augmenting adjacent faces that result in the biaugmented edges being non-convex. Just with cubes augment and para biaugmented are already coveredby the elongated square pyramid and elongated square bipyramid, but there's the meta biaugmented cube, two formas of triaugmented cube, two tetraagumented cubes, and the pentaaugmented and hexaaugmented cube. With 92 faces to pick from, the snub dodecahedron could potentially have hundreds or thousands of non-convex augmentations.
 * mix stacks of prismatic forms. For every regular n-gon, there's a prismatic stack for every bit sequence where 0 and 1 represent prisms and anti-prisms... and then there's cupolae and pyramids to add to the mix... for example, you could take an elongated pentagonal copula and put a elongated pentagonal pyramid on its pentagonal face.
 * Augmenting with prismatic stacks.
 * Any connected subset of a honeycomb where all faces are regular.
 * Biform star polyhedra with all regular faces. E.g. the cousins of the uniform star polyhedra with exactly two types of vertex. Then the triform, tetraform, etc. At least, my intuition is that you need to group these by number of unique vertex types to have any chance of listing them since I don't think the term convex is well defined for self-intersecting forms... though I could be wrong and there's a finite set of regular faced, self-interesecting forms with all convex dihedral angles.
 * And I'm sure there are forms that are regular faced but don't fit any of the above categories.
 * Regular faced forms beyond the Johnson Solids and the Uniform star polyhedra strike me as being pretty deep waters that are far from well explored, or if they have, than much of the information is locked up in obscure places... and keep in mind, it took over two thousand years to go from the Archimedean solids to the Johnson Solids, the Archimedean solids where lost for much of that time, Johnson had to invent terminology to describe most of the Johnson Solids, and it's been less than 60 years since Johnson enumerated the Johnson Solids. 2603:6080:7001:8205:0:0:0:115C (talk) 20:08, 1 June 2024 (UTC)

Completely irrelevant and distracting
The definition of a Johnson solid absolutely should not confuse readers with all the things that it is not. Or any things that it is not. Like the sentence in the introduction:

"There is no requirement that each face must be the same polygon, or that the same polygons join around each vertex."

This just confuses people. The stated definition prior to this ridiculous sentence is crystal clear, and we should leave it at that.

I hope someone knowledgeable about this subject will remove this idiotic sentence. 2601:200:C082:2EA0:2494:C097:5957:E04C (talk) 02:32, 8 July 2023 (UTC)

Improve the description under the image with the 3 examples
Referring to this text:


 * "The following are three examples of solids. The first solid, elongated square gyrobicupola, is Johnson solid because it has the convexity property. The second solid, stella octangula is not Johnson solid because it is not convex, meaning whenever two points are interior, the connecting line may not. The last solid is not a Johnson solid because it is not convex, meaning every face is planar or the dihedral angles of two adjacent faces have 180°."

it inconsistently uses "Johnson solid" as an adjective and then and a noun, i.e. sometimes prefixed with an article, sometimes not. It also omits articles for the named polyhedra. Overall it reads a little verbose and clunky. here's my proposed alternative:

Introscopia (talk) 15:29, 16 July 2024 (UTC)
 * Among these three polyhedra, only the first, the elongated square gyrobicupola, is a Johnson solid. The second, the stella octangula, is not convex, as some of its diagonals lie outside the shape. The third presents coplanar faces.


 * I like the brevity. We have an enthusiastic new editor who makes occasional lapses in English, likely including these missing articles; let's be patient and correct them as needed. —Tamfang (talk) 23:06, 16 July 2024 (UTC)