#geometricmodels

Stereographic projection of regular polychora (4D polytopes) by emileokada.

“Having made a demonstration that successfully manages to project regular 3D polytopes I decided it would be worth a shot trying to generalize to higher dimensions. Doing some minor computations shows that generalizing to higher dimensions is in fact trivial! ”

Some major points to take note of are:

(1) You want to “explode” the polytope onto the n-sphere. Take a point in n-space  given by the position vector , 

then it’s image on the n-sphere is given by 

(2) You want to project a point on the n-sphere into a space tangent to the n-sphere i.e. a subspace of n-1 dimensions. Thus the stereographic projection of a point in n-space to a point in n-1 space is given by

 where 

is a point on the n-sphere. Taking these facts into account it was easy to modify the code to produce the following animations of the simplex, hypercube, 16-cell and 24-cell:

GIF 1: simplex.gif

GIF 2: tesseract.gif

GIF 3: hexadecachoron.gif

GIF 4: Icositetrachoron.gif

 The 120-cell and the 600-cell as they took way to long to render. 

Curved folds:

Frangipani Box, Variation 5

Swirling Salt Cellar

92 Gears - Caged - Render

(Digital Art / Animation © TaffGoch on DeviantArt)

Added external cage, to support gear axis spindles, which also adds reflectivity & shadow interest to the overall render.

• Red gears: 20 teeth, 12 count.

• Blue gears: 24 teeth, 20 count.

• Yellow gears: 13 teeth, 60 count.

Red gears aligned with icosahedron vertices • Blue gears aligned with dodecahedron vertices. Polyhedral basis for red & blue gears: buckyball. Regarding all 92 gears, the central rotation axes are aligned with the vertices of a frequency 3v, class-I geodesic tessellation of an icosahedron.

Here is Gear axes are centered on vertices of a rhombic-enneacontahedron (as shown in animation): 32 large gears - 30 teeth each 60 small gears - 16 teeth each

See more: 92 Gear Spheres with Zero Phase Error by Paul Nylander at [Bugman123.com/Gears/92GearSphere]