"An Historical Chart of Polyhedra"

A beautiful illustration by Ugo Adriano Graziotti.

Great grand stellated 120-cell

A 3d projection of a four-dimensional figure (one of several "star polytopes"), which looks every bit as impressive as its name sounds.

Image attribution: Robert Webb, created with Stella software.

Polychora

These are examples of four-dimensional geometric figures-- in three-dimensional sections-- rendered by Jonathan Bowers. (Their odd names are abbreviations of complex mathematical designations.)

Polyhedral planets

This interesting article describes several hypothetical cubical worlds-- a hypothesized Planet X from 1884; a world of chimeric creatures called Aocicinori (illustrated in tracts handed out at Rice University, Texas, by a "mysterious, well-dressed gentleman" and created by a mental patient); and a cubical version of our own Earth:

Contemporary cosmologist Karen L. Masters also finds the topic of cube worlds fascinating -- especially the atmospheric possibilities. As she explains in Cornell's Ask a Physicist feature, all six faces of the [planet] would boast temperate weather, centralized bodies of water and none of them would feature polar or equatorial weather. What's more, the pointy edges of the cube would actually poke through the planet's atmosphere like titanic mountains.

 I am reminded of the far-future tetrahedral Earth described in a magazine article from 1918:

The world is now the shape of a globe, the shape which gives the biggest possible bulk for its surface, but the inside of the earth is still cooling and condensing, and the internal changes are slowly changing its shape. The surface, already condensed to its utmost, will not change with the core; it cannot reduce its area, but it adapts itself to the shrinking interior by taking a shape which occupies less bulk. So the earth is to become a tetrahedron, a sort of pyramid, the shape which gives the smallest bulk for its surface. Let us think about it all.

Temari

These traditional Japanese craft items (many more images here) sometimes display symmetries resembling the Platonic polyhedra; compare this figure for instance.

Toroid

If this polyhedron were made of rubber which could be stretched without tearing, it could be morphed into a donut or a coffee cup.

Stars in hyperspace

This clip shows four-dimensional geometric forms as they would look passing through our three-dimensional world, like the sphere passing through the plane in Flatland.

Elements of geometry

Illustrations by Johannes Kepler depicting the traditional correspondences between Platonic solids and classical elements. Source.

Subdivided Platonic solids

We aim to show that a single deterministic process can generate a heterogeneous set of forms with an astounding degree of complexity.

Some of these forms resemble Ernst Haeckel's drawings of the marine protozoa called radiolaria:

They also look like less-chaotic versions of another beautiful 3D fractal, the Mandelbulb.

2D images of 3D representations of 4D geometric figures

Cantellated 120-cell

Cantellated 600-Cell

Small icosicosidodecahedron

From V. Bulatov's Polyhedra Collection, which has many more of these beautiful computer-generated images.