Tuesday, March 31, 2015

The Adams "World in a Square" projection and knotology weaving

The square module of Adams's "World in a Square II" conformal projection of the globe is similar to the square module of knotology weaving.

A Belyi function maps an orientable surface to the sphere with at most 3 singular points (critical values.) The pre-images of these points on the orientable surface are called critical points. In the general case, the critical points form an Eulerian triangulation of the orientable surface (a triangulation with an even number of triangles meeting at each vertex.) The smallest such triangulation has just two triangles: it is, so to speak, a triangular envelope.

From a number theorist's point of view, the three critical values on the sphere should be located at 0, 1, and infinity in complex (Riemann sphere) coordinates. These points on the Riemann sphere share a great circle (the real axis) at spacings of 90°-90°-180°. The Adams "World in a Square II" projection also has critical values along a great circle (the extended prime meridian that includes 180° longitude,) and they are also spaced at 90°-90°-180° (the South Pole, the Pacific Point, i.e., the antipodes of the longitude/latitude origin, and the North Pole,) so the correspondence is pretty exact.

The several appearances of the extended prime meridian in Adams's projection (the four sides of the square plus a diagonal of the square) show that Adams's projection is really two triangles joined along a shared edge. The four corners of the square fall into two classes: those that are composed of a single triangle angle (the two appearances of the Pacific Point) and those that are composed of two triangle angles (the North and South Poles.) Each triangle angle represents 180° on the earth's surface (the angle between two segments of a straight line is always 180°) so the North and South Poles each represent a pair of triangle angles, 2 x 180° = 360°, or a full turn. Each of the two appearances of the Pacific Point represent a single triangle angle, 1 x 180° = 180°, or a half-turn.

A "World in a Square" knotology weaver folded the natural (prime-median) way.

The way nature intends us to fold Adams's square is along the prime meridian: that turns the square into two triangles with every triangle corner = 180°. The unnatural way is to fold the square along the equator, then the right-angled corners at the poles represent 360° on the earth (each being actually two triangle corners), while the 45° corners represent 90° on the earth (each being 180° on the earth split down the middle by the fold.)

Nature's way requires an Eulerian triangulation (just as we would expect in a Belyi surface;) ensuring that every vertex of the triangulation gets an integral multiple of 360°. The unnatural way allows any number of triangles at the "rangles," the places where right angles (each representing 360° of earth surface) meet, but the price is that we need doubly-Eulerian vertices (multiples of four triangles) at the "nooses," the places where the 45°-angles (each representing 90° of earth surface) meet. That sounds strange, but it is actually convenient to the way knotology weaving is frequently done. Often, we are weaving a deltahedral surface that is, so to speak, omnicapped by cube corners, so we need an odd number of triangles (3) at the rangles. Getting doubly-Eulerian vertices for the nooses may sound difficult, but, since every each omnicap contributes a pair of triangles to that vertex, the underlying deltahedral triangulation only needs to be Eulerian.

A "World in a Square" knotology weaver woven the unnatural (equatorial) way.

In weaving the enveloping surfaces of vox-solids the placement of the oblique knotology creases are irrelevant as they are not folded, but the number of squares around a vertex can be 3 (the head of a corner,) 4 (flat ground,) 5 (the corner of a building rising from flat ground,) and 6 (a square well touching corners with a building rising from flat ground.) Those odd numbers—3 and 5—cause problems for the Pacific Point since it supplies only half-a-turn at each corner of the square. For example, we cannot "world-weave" the surface of a 1-voxel cube because the Pacific Point would be forced to make an appearance at certain "heads of corners;" but, we can world-weave the surface of an 8-voxel cube if we place Pole Points at the corners, thus keeping the Pacific Point safely in the middle.

A portion of the enveloping surface of a vox-solid.
It is interesting to note that an 8-voxel knotology cube can be viewed as a deltahedron omnicapped with cube corners, the underlying deltahedron is an octahedron—and the surface of an octahedron is indeed an Eulerian triangulation. Any vox-solid we make out of these larger "octo-voxels" can indeed be world-woven.