Polyhedra Viewer
by @tesseralis
For centuries, mathematicians and artists have been fascinated by the beauty in polyhedra. While most are familiar with only a few of them, such as the Platonic solids, prisms, or pyramids, there are many more polyhedra to discover, with interesting properties and relationships to each other.
This application visualizes the relationships between the convex, regular-faced polyhedra. The 120 solids presented here can be transformed into each other by a network of operations. Select a solid below to manipulate it and to explore its relationships with other polyhedra.
Uniform Polyhedra
The uniform polyhedra are the Platonic solids, the Archimedean solids, and the infinite set of prisms and antiprisms.
| tetrahedron | cube | octahedron | dodecahedron | icosahedron | |
|---|---|---|---|---|---|
| regular |  T |  C |  O |  D |  I |
| truncated |  tT |  tC |  tO |  tD |  tI |
| rectified | (O) |  aC |  aD | ||
| bevelled | (tO) |  bC |  bD | ||
| cantellated | (aC) |  eC |  eD | ||
| snub | (I) |  sC |  sD | ||
| prism | antiprism | |
|---|---|---|
| triangular |  P3 | (O) |
| square | (C) |  A4 |
| pentagonal |  P5 |  A5 |
| hexagonal |  P6 |  A6 |
| octagonal |  P8 |  A8 |
| decagonal |  P10 |  A10 |
Johnson Solids
The 92 Johnson solids, named after Norman Johnson, are the non-uniform convex regular-faced polyhedra—solids whose vertices aren't transitive.
Pyramids, Cupolæ, and Rotundæ
The majority of Johnson solids are created from combining pyramids, cupolæ, and rotundæ with prisms and antiprisms.
| ortho- | gyro- | ortho- | gyro- | |||||
|---|---|---|---|---|---|---|---|---|
| -- | elongated | gyroelongated | bi- | elongated bi- | gyroelongated bi- | |||
| triangular pyramid | (T) |  J7 | coplanar |  J12 |  J14 | coplanar | ||
| square pyramid |  J1 |  J8 |  J10 | (O) |  J15 |  J17 | ||
| pentagonal pyramid |  J2 |  J9 |  J11 |  J13 |  J16 | (I) | ||
| digonal cupola | (P3) | coplanar | concave | coplanar |  J26 | coplanar | coplanar | concave |
| triangular cupola |  J3 |  J18 |  J22 |  J27 | (aC) |  J35 |  J36 |  J44 |
| square cupola |  J4 |  J19 |  J23 |  J28 |  J29 | (eC) |  J37 |  J45 |
| pentagonal cupola |  J5 |  J20 |  J24 |  J30 |  J31 |  J38 |  J39 |  J46 |
| cupola-rotunda |  J32 |  J33 |  J40 |  J41 |  J47 | |||
| pentagonal rotunda |  J6 |  J21 |  J25 |  J34 | (aD) |  J42 |  J43 |  J48 |
Augmented, Diminished, and Gyrate Polyhedra
The next group of Johnson solids are defined by augmenting, diminishing, and gyrating uniform polyhedra.
| para- | meta- | |||
|---|---|---|---|---|
| augmented | biaugmented | triaugmented | ||
| triangular prism |  J49 |  J50 |  J51 | |
| pentagonal prism |  J52 |  J53 | ||
| hexagonal prism |  J54 |  J55 |  J56 |  J57 |
| dodecahedron |  J58 |  J59 |  J60 |  J61 |
| truncated tetrahedron |  J65 | |||
| truncated cube |  J66 |  J67 | ||
| truncated dodecahedron |  J68 |  J69 |  J70 |  J71 |
| para- | meta- | -- | augmented | ||
|---|---|---|---|---|---|
| diminished | bidiminished | tridiminished | |||
| icosahedron | (J11) | (A5) |  J62 |  J63 |  J64 |
| para- | meta- | para- | meta- | para- | meta- | ||
|---|---|---|---|---|---|---|---|
| -- | diminished | bidiminished | tridiminished | ||||
| -- | (eD) |  J76 |  J80 |  J81 |  J83 | ||
| gyrate |  J72 |  J77 |  J78 |  J82 | |||
| bigyrate |  J73 |  J74 |  J79 | ||||
| trigyrate |  J75 | ||||||
Elementary Johnson Solids
The remaining Johnson solids cannot be created by gluing together other polyhedra.
| digonal | triangular | square | |
|---|---|---|---|
| snub |  J84 | (I) |  J85 |
 J86 |  J87 |  J88 |  J89 |  J90 |  J91 |  J92 |
And Many More...
The polyhedra represented above are just a small subset of the wondrous world of geometric shapes and figures. For instance, the Kepler-Poinsot polyhedra are regular like the Platonic solids but non-convex, while the Catalan solids are the duals of the Archimedean solids and have non-regular faces. Beyond three dimensions, one can explore four dimensional shapes like the tesseract or the grand antiprism.
If you would like to learn more about polyhedra and other geometric figures, check out these links:
- Virtual Polyhedra by George W. Hart - an extensive encyclopedia of polyhedra and the major inspiration for this site
- Visual Polyhedra by David I. McCooey - More polyhedral models with extensive geometric data
- polyHédronisme by Anselm Levskaya - Build complex polyhedra using Conway operations
- Polyhedra by Stacy Speyer - Paper models of polyhedra and other artistic imaginings
- Johnson Solids by Allison Chen - Diagrams categorizing the Johnson solids based on their operations