Editing Talk:Johnson solid (section)

== 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 polyhedron|convex]], the convex regular polyhedrons are the 5 [[Platonic solid]]s, and there are 9 non-convex regular polyhedrons, including the 4 [[Kepler–Poinsot polyhedron]]s and the 5 [[Polytope compound#Regular compounds|regular compound]]s, and for the [[semiregular polyhedron]]s, there are 13 convex ones other than the convex [[prism]]s and the convex [[antiprism]]s, but what are the non-convex ones? And for the polyhedrons with each face [[regular polygon]]s or regular [[star polygon]]s, 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 polyhedron]]s, the 5 [[Polytope compound#Regular compounds|regular compound]]s, the [[stellated octahedron]], the 53 [[nonconvex uniform polyhedra]]s, the [[Uniform_polyhedron#.28p_2_2.29_Prismatic_.5Bp.2C2.5D.2C_I2.28p.29_family_.28Dph_dihedral_symmetry.29|uniform star prism]]s, the [[Uniform_polyhedron#.28p_2_2.29_Prismatic_.5Bp.2C2.5D.2C_I2.28p.29_family_.28Dph_dihedral_symmetry.29|uniform star antiprism]]s, the [[augmented heptagonal prism]], the [[pentagrammic prism]], the [[deltahedra]]s, the [[toroidal prism]]s, etc.)

{|class="wikitable"
|+ Polyhedrons whose faces are all regular polygons (or regular star polygons)
|Convex?
|Uniform? (i.e. Identical vertices?)
|Each face are the same polygon? (i.e. Identical faces?)
|Class
|-
|True
|True
|True
|5 [[Platonic solid]]s
|-
|True
|True
|False
|infinite convex uniform [[prism]]s, infinite convex uniform [[antiprism]]s, 13 [[Archimedean solid]]s
|-
|True
|False
|True
|8 convex [[deltahedra]]s
|-
|True
|False
|False
|92 [[Johnson solid]]s
|-
|False
|True
|True
|4 [[Kepler–Poinsot polyhedron]]s, 5 [[Polytope compound#Regular compounds|regular compound]]s
|-
|False
|True
|False
|infinite [[Uniform_polyhedron#.28p_2_2.29_Prismatic_.5Bp.2C2.5D.2C_I2.28p.29_family_.28Dph_dihedral_symmetry.29|uniform star prisms and uniform antiprisms]], 53 [[nonconvex uniform polyhedra]]s
|-
|False
|False
|True
|infinite [[Deltahedron#Non-convex forms|non-convex deltahedra]]s
|-
|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 solid]]s, the [[hexagonal pyramid]], the [[near-miss Johnson solid]]s, the [[parallelepiped]], the [[rhombic icosahedron]], the [[Szilassi polyhedron]], the [[Császár polyhedron]]; and the polyhedrons with 180° dihedral angles, such as [https://en.wikipedia.org/wiki/File:Partial_cubic_honeycomb.png this one]; and the non-connected polyhedrons, such as the [[crossed prism]]s; 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: [https://archive.ph/tthvD]——[[Special:Contributions/36.234.85.41|36.234.85.41]] ([[User talk: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.) —[[User:Tamfang|Tamfang]] ([[User talk: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. [[Special:Contributions/2603:6080:7001:8205:0:0:0:115C|2603:6080:7001:8205:0:0:0:115C]] ([[User talk:2603:6080:7001:8205:0:0:0:115C|talk]]) 20:08, 1 June 2024 (UTC)