Polytopia 1: Tesselations and polyhedra
Polytopia 2: Honeycombs and polytopes
Virtual Image (Windows CD) +44 161 480 1915
Educational site licence £49.50+VAT, single user home licence £24.95 inc VAT each
ISBN 1 901579 03 4 and 04 1
Polytopia 1 uses text, tables, movies and interactive graphics to explore the ways polygons can spread out in sheets or curl up into balls. The sequel Polytopia 2 pursues these themes into higher dimensions.
The prettiest parts of Polytopia 1 are the movies: imaginatively conceived and nicely ray-traced, with objects rendered in a subtle colours and pleasingly textured materials including transparents. There is nice scenery, too: fractals, Leonardo sketches, the Pleiades, rippling water and reflective spheres in which you almost expect to catch the eye of Maurits Escher.
Polytopia 2 largely abandons the photorealistic world of bright, hard surfaces in favour of jaggy wire-frame graphics against a dark background. Polytopia 1 had its share of wire-frame graphics: they seemed crude and not always particularly useful. But they come into their own on the second disc as we meet objects whose structure is more than skin deep. What is lost in ray-traced eye appeal is gained in interactivity and potential for learning. If you cannot have a physical model of a polyhedron or honeycomb in your hand, it does help to be able to turn it around and view it from all angles on the computer. The 3D honeycombs and crystal structures leap into solidity when you spin them. But when higher-dimensional objects such as the awesome 600-cell are projected down to 2D as they must be, it is hard to gain any sense of their crystalline rigidity. Instead they morph and flow on the screen like soft, fleshy organisms. We get a sense of the objects' topology, but a full metrical intuition of any but the simplest 4D objects is probably beyond most human minds.
The sequel is intellectually more challenging, but visually the best thing on either disc is Polytopia 1's section on stellated polyhedra, including some gems from that rich vein, the stellated icosahedra. These are literally the stars of the show.
The mathematical content is leavened with history and philosophy. Kepler's musings on the polyhedral dimensions of the solar system, discussed on the first disc, seem less potty when we are reminded that regular polygons (triangles or hexagons) arise in the dynamics of the Trojan points, where rocks or asteroids can orbit in sync with a larger planet. Polytopia 2 reminds us that Immanuel Kant was the first to suggest that God might have created worlds with more than three dimensions of space. Kant also noted that human perception is attuned to three dimensions, and by implication will have serious trouble with any more.
Computer graphics like those of Polytopia 2 (or the pioneering work of Thomas Banchoff during the 1980s) are probably the closest we can get to mentally grasping the worlds of 4D, 5D and beyond. If you hope to prove Kant wrong, here is your chance. But keep the aspirin handy.