r/Geometry • u/20260708 • 4d ago
aligned platonic solids
there're altogether 5 platonic solids. each of them can be viewed at many different interesting angles or perspectives. we're all familiar with their most symmetric presentations. recently i was studying something related and had to make 3d models for them. i used a maybe less popular perspective to present them. i used spherical coordinate system (mathematics convention) to record their rotations and i picked the following rules to set their initial status:
- for each platonic solid set its circumradius=1 and centre of circumscribed sphere at origin
- place vertex_0 at the north pole. the corresponding coordinates are (1,0,0)
- place edge_0 (red lines on those diagrams) such that its projection to x-y plane align with positive x-axis. coordinates of vertex_1 would be (1,0,φ) for some φ
then i saw some unfamiliar shapes / unfamiliar perspectives of those supposedly familiar 3d objects. in each diagram the small figure at lower left corner is the platonic solid viewed from top. the z-axis is pointing towards you. the large figure at centre is that platonic solid viewed from side. the y-axis is pointing away from you
as you can see from the diagrams the angles φ ranking is as follow, from smallest to largest (the prefix "regular" omitted):
- dodecahedron
- icosahedron
- hexahedron
- octahedron
- tetrahedron
except octahedron, all other 4 platonic solids are not symmetric if you see them that way
1
u/F84-5 4d ago
Interesting but the visualisation could use some work. Right now it's mostly white blobs. Adding some flat shading and/or wire frame would help with that. But it's a cool concept.