Talk:Johnson solid

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I have been modifying user:Cyp's image:Poly.pov 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[edit]

Doesn't do 3d, and only knows 2 Johnson solids (so far), but here's makepolys.c.

Pentagonal pyramid flat.svgSquare orthobicupola flat.svg

Κσυπ Cyp   00:27, 5 Nov 2004 (UTC)

I'm making some "home-made" nets:

Square pyramid net.pngTriangular cupola net.PNGSquare cupola net.PNGPentagonal Cupola.PNG

And the rest with Inkscape, now that I found out about it:

Pentagonal Rotunda Net.svg

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

Complete set of nets[edit]

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)

Elongated square gyrobicupola[edit]

The picture is wrong - that's obviously a rhombicuboctahedron. Compare: [1]

  • 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? —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 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 (talk)

The list[edit]

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[edit]

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?[edit]

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. —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:
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)
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)


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:

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][edit]

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". —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...[edit]

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[edit]

All (it seems) of the individual Johnston solids pages were edited by 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)


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 (talkcontribs) 19:15, 3 July 2008 (UTC)


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[edit]

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


Really! here are pictures!
New johnson polyhedron 1.jpgNew Johnson polyhedron 2.jpgNew Johnson polyhedron net.jpg

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)
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)


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: [2]. Tom Ruen (talk) 17:31, 16 March 2011 (UTC)

Previously discovered[edit]

This model is readily buildable with Polydrons. Jim McNeill [3] 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°). [4]

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

duals of the Johnson solids[edit]

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. — 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. — Cheers, Steelpillow (Talk) 10:37, 27 February 2014 (UTC)

Organizing the table[edit]

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 <abbr title="heynow">2</abbr>, 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)


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. — 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? Face-wink.svg (Just teasing, thanks for the revert). — 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)

Théby cant stop the star of david — Preceding unsigned comment added by (talk) 16:14, 18 August 2015 (UTC)

Convex regular-faced polyhedra with conditional edges[edit]

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)