there are 48 regular polyhedra
Why Hacker News recommends it
HN commenters point to it as a favorite math rabbit-hole video and as a corrective to the claim that the five Platonic solids are the only regular polyhedra.
Recommended by 2 people across 2 different months, 2021 to 2026.
Why Mathematics Stack Exchange recommends it
Mathematics Stack Exchange users link it when questions turn on what counts as a regular polyhedron, since it lays out why the answer can be 48 rather than 5 or 9.
Recommended by 2 people across 2 different months, 2020 to 2023.
What it is
jan Misali walks through a comprehensive list of all 48 regular polyhedra in 3D Euclidean space, working from the Platonic solids through Kepler-Poinsot, tilings, Petrie-Coxeter, Petrials, and blended and pure Grünbaum-Dress apeirohedra. It is a narrated, visual explainer based on a published paper and a companion visualization tool.
Cited in 1 Wikipedia article, including Regular polyhedron.
What people said
Strong rec macromaniac · 2023-07 · Hacker News · on “Super Mario 64 – Watch for Rolling Rocks – 0.5x a Presses [video]”
Can anyone recommend other rabbit hole videos like the parents? It can be in any topic. I love these types of videos that show just how deep things are. [this video] is my current favorite.
Recommends Dr. Richard Klitzing · 2021-09 · Mathematics Stack Exchange · on “Why are there nine regular polyhedra?”
In fact there are much more than just 9 regular polyhedra, in fact there are 48! Just cf. this video [this video] addressing to this well-proven fact.
Recommends No Name · 2023-07 · Mathematics Stack Exchange · on “Do non-convex platonic solids exist?”
What makes a polyhedron regular is not just that each vertex has the same number of edges meeting at it, but that the vertices are congruent , just like the faces. Your example, the icosahedron with a vertex popped in, fails because the five vertices around the popped vertex are not congruent to the other seven. In technical terms, a regular polyhedron is "transitive on all its flags" [LINK] , where a flag is, in three dimensions, a face, edge and vertex, all adjacent, considered as a unit, and transitive means that you need only rotate or translate to get one flag into the place of another, and there will be no outward difference unless you labeled. A slightly more stringent condition than
Recommends ffhh · 2026-10 · Hacker News · on “Great Dirhombicosidodecahedron ("Miller's Monster")”
You are assuming that the faces don't intersect, and that the faces have edges that don't intersect. The easiest example of a polyhedron with self-intersecting behavior to wrap your head around is IMO the small stellated dodecahedron, made from twelve pentagrams. If you want to dive in a bit into strange polyhedra, I can really recommend this YouTube video: [this video]
Cites user813374 · 2020-08 · Mathematics Stack Exchange · on “I heard there are 48 regular polyhedrons. With what Jan Misali calls regular polyhedrons, are there any more?”
I heard there are 48 regular polyhedrons. With what Jan Misali calls regular polyhedrons, are there any more? Assumptions: A polyhedron must lie in 3D Euclidean space. It must be a single connected shape. It's invalid for two vertices edges or faces to have the exact same location while remaining distinct. If there are only 48 polyhedrons, what about 4D polytopes? Watch this video if this info isn't full enough here: [this video]
Cites David Spector · 2020-11 · Mathematics Stack Exchange · on “Can Penrose tilings generate regular polyhedra?”
Physicist Roger Penrose discovered several pairs of polygons that can each tile the infinite plane aperiodically (see [LINK] ). Is there way to fold such a tiling, or a finite strip of such tiling, or a finite regular pentagonal subset of such a tiling, so as to form a regular but aperiodic polyhedron? There would clearly be many follow-on questions if this were possible, but they are beyond my ability to imagine. This question was stimulated by the very interesting video there are 48 regular polyhedra [this video] .
Cites trevortheblack · 2021-10 · Hacker News · on “Conterintuitive facts in mathematics, CS, and physics”
> 21. In two dimensions, there are infinitely many regular polygons. In three dimensions, there are five Platonic solids. In four dimensions, there are six platonic polychora. In all higher dimensions than four, there are only ever three regular polytopes. (Maths 1001, Richard Elwes) This implies that Platonic Solids are the 3D analogue of the 2D regular polygon. This is not the case. The Platonic Solids are merely all of the convex solid regular polyhedra. When the caveats are removed there are (at least) 48 regular polyhedra: [this video]
Cites Dr. Richard Klitzing · 2022-05 · Mathematics Stack Exchange · on “Why are there nine regular polyhedra?”
In fact it indeed depends on context, cf. the following video on the nowadays proven 48 regular polyhedra [this video]