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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
Doesn't do 3d, and only knows 2 Johnson solids (so far), but here's makepolys.c.
Κσυπ Cyp 00:27, 5 Nov 2004 (UTC)
I'm making some "home-made" nets:
And the rest with Inkscape, now that I found out about it:
Now that there's enough nets for a whole section, anyone think we should incorporate them into the table?
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)
The picture is wrong - that's obviously a rhombicuboctahedron. Compare: [1]
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)
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)
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)
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
The results should be correct, but may not be correctly matched by names if the indices were inconsistent!
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.
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 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)
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)
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)
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)
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)
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)
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)
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)
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)
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)
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)
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)
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)
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)
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)
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.)
Convex? | Uniform? (i.e. Identical vertices?) | Each face are the same polygon? (i.e. Identical faces?) | Class |
True | True | True | 5 Platonic solids |
True | True | False | infinite convex uniform prisms, infinite convex uniform antiprisms, 13 Archimedean solids |
True | False | True | 8 convex deltahedras |
True | False | False | 92 Johnson solids |
False | True | True | 4 Kepler–Poinsot polyhedrons, 5 regular compounds |
False | True | False | infinite uniform star prisms and uniform antiprisms, 53 nonconvex uniform polyhedras |
False | False | True | infinite non-convex deltahedras |
False | False | False | ? (this is my question in this talk, what is the set of such polyhedrons, I know that this set include the augmented heptagonal prism) |
(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 this one; 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: [5]—— 36.234.85.41 ( talk) 10:03, 29 August 2021 (UTC)
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)
Referring to this text:
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)
![]() | This article is rated B-class on Wikipedia's
content assessment scale. It is of interest to the following WikiProjects: | ||||||||||||||||||||
|
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
Doesn't do 3d, and only knows 2 Johnson solids (so far), but here's makepolys.c.
Κσυπ Cyp 00:27, 5 Nov 2004 (UTC)
I'm making some "home-made" nets:
And the rest with Inkscape, now that I found out about it:
Now that there's enough nets for a whole section, anyone think we should incorporate them into the table?
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)
The picture is wrong - that's obviously a rhombicuboctahedron. Compare: [1]
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)
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)
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)
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
The results should be correct, but may not be correctly matched by names if the indices were inconsistent!
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.
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 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)
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)
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)
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)
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)
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)
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)
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)
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)
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)
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)
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)
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)
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)
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)
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.)
Convex? | Uniform? (i.e. Identical vertices?) | Each face are the same polygon? (i.e. Identical faces?) | Class |
True | True | True | 5 Platonic solids |
True | True | False | infinite convex uniform prisms, infinite convex uniform antiprisms, 13 Archimedean solids |
True | False | True | 8 convex deltahedras |
True | False | False | 92 Johnson solids |
False | True | True | 4 Kepler–Poinsot polyhedrons, 5 regular compounds |
False | True | False | infinite uniform star prisms and uniform antiprisms, 53 nonconvex uniform polyhedras |
False | False | True | infinite non-convex deltahedras |
False | False | False | ? (this is my question in this talk, what is the set of such polyhedrons, I know that this set include the augmented heptagonal prism) |
(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 this one; 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: [5]—— 36.234.85.41 ( talk) 10:03, 29 August 2021 (UTC)
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)
Referring to this text:
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)