Let's back up and start the procedure over, with another kind of polygon. Continuing the step-by-step approach, we change from triangles to squares by increasing the number of sides by one. The method is reliable if plodding. What happens if two squares come together? Again, that's just a hinge (Fig. 4-3). So start with three. If there are three squares around one corner, the same must be true for all corners, and as before, the structure is self-determining. Continue to join three squares at available vertices until the system closes itself off. With six squares and eight corners, this is the most familiar shape in our developing family of regular polyhedra—the ubiquitous cube.

Fig. 4-3 Click on thumbnail for larger image.

      Next, we gather four squares at a vertex and immediately hit ground. Four times 90 degrees is 360 degrees, the whole plane again. And while such groups of four will generate graph paper indefinitely, they can never close off as a system. So that's it for squares.

Triangles, squares,..., now we come to pentagons. Three of them around each corner works. A system exists as soon as there are twelve pentagons and twenty three-valent corners. Called the pentagonal dodecahedron ("dodeca" is Greek for twelve), it adheres to the definition of regular polyhedra (Fig. 4-4).

Fig.4-4 Click on thumbnail for larger image

      Try four around a corner. Another new situation. The interior angle of a regular pentagon measures 108 degrees (see Appendix A); four of them together add up to 432 degrees, which is more than the planar 360. This indicates that four pentagons simply will not fit around one point. Not only have we reached a stopping point for regular polyhedra out of pentagons, but this example also shows that not all regular polygons can be made to fill a page (or tile a floor). Specific spatial constraints apply in two dimensions as well as in three. We shall investigate these patterns in more detail in Chapter 12.

Fig. 4-5 Click on thumbnail for larger image.

Having exhausted the pentagonal possibilities, we go on to hexagons. Three of their 120-degree angles total 360 degrees, and are therefore planar right away-creating the hexagonal pattern seen in honeycomb and frequently used for bathroom floor tiles (Fig. 4-5). Three heptagons with angles totalling 385.71 degrees just won't fit together. Neither will octagons, nor any polygons with more vertices.

A quick review reveals the surprisingly limited inventory of five regular polyhedra: tetrahedron, octahedron, icosahedron, cube, and pentagonal dodecahedron (Fig. 4-6). Like it or not, we have reached the end. A child in kindergarten, with the two rules carefully explained, will discover the same five shapes. Space takes over, imposing that upper limit. This idea runs counter to the bias of our mathematical background that space is passive emptiness and we impose desired configurations. Invisible, unyielding constraints sound more like mysticism than science.

Fig. 4-6. Five regular polyhedra. Click on thumbnail for larger image.

      If we dwell on this subject, that is because it is crucial to understanding the framework of Bucky's investigation. Space has specific characteristics, and we want not only to list and understand them, but also to begin to really feel their embracing qualities-a sense of structured space permeating all experience.

      The regular polyhedra provide a good starting point from which to branch out in all directions. An eighteenth-century mathematician, Leonhard Euler (1707-1783), greatly simplified our task with his realization that all patterns can be broken down into three elements: crossings, lines, and open areas. He thereby introduced the basic elements of structure (vertices, edges, and faces) which underlie all geometrical analysis. Bucky saw this contribution as a breakthrough of equal importance to the law for which Euler is known, for this precise identification of terms enabled Euler's other, more famous observation.