Some New Tilings of the Sphere with Congruent Triangles Robert J. MacG. Dawson Dept. of Mathematics and Computing Science Saint Mary's University, Halifax, Nova Scotia
Not every tiling of the sphere arises from a polyhedron; it is possible that the vertices of a spherical tile may not be coplanar. However, in the special case of triangular tiles, meeting edge to edge, there is a bijection between monohedral tilings of the sphere and inscribed polyhedra with congruent triangular faces. This is one of the various reasons why it was reasonable for Sommerville [5] (in 1923) and Davies [1] (in 1967) to restrict their attention to tiles that met edge-to-edge. Ueno and Agaoka [6], expanding on Davies’ work, did the same. Even under this assumption there are some unexpected and beautiful tilings. See, for instance, Figure 1. The tiling on the left can be seen as related to the union of an Note that neither of these tilings is isohedral; in one case the tiles form two orbits under the symmetry group, in the other there are many orbits. For more information on isohedral tilings, see [7].
This correspondence breaks down if the tiles do not meet edge-to-edge – or at any rate, it gets more complicated. An arc, that is the union of edges in one way on one side and in another way on the other, acts like a degenerate tile with area 0 and angles of 0° and 180° . In the corresponding polyhedron, this becomes a face in the form of a crossed polygon, lying in a plane through the center of the sphere. Such a polyhedron certainly won’t have a single face type.
Combinations of angles that do fit at a point correspond to solutions in natural numbers (
Except for a few exceptional cases, we can show that
"With a few important exceptions, spherical triangles with rational angles don’t have rational relations between their edge lengths."
This is left vague because there doesn’t seem to be a hope of proving it without a major advance in transcendence theory. In the same way, we cannot prove that p is irrational (though most mathematicians would offer excellent odds that it is!) However, as our purpose is to show that certain sums of edge lengths cannot be obtained in more than one way, we just need to know that the edges ( For any tiling with
We can also make more subtle uses of this principle. If there is only one configuration in which a certain angle appears in at least the target proportion, then that configuration must appear in any tiling. Moreover, if it can be shown – as is often the case – that such a configuration is always accompanied by nearby split vertices with a strong surplus of a different angle, it may still be impossible to create a tiling with vertices in the ratios in which the triangle provides them. In practice, if you try to create such a tiling, you end up with more and more of the hard-to-use species of angle on the boundary, until eventually you get stuck.
If a tile will tile a lune with polar angle f
, and also some convex polygon P with angles (a
The three special cases are shown in Figure 8. They are the (80° , 60° , 60° ), (100° , 60° , 60° ), and (150° , 60° , 60° ) triangles, which tile with 36, 18, and 8 copies respectively. The first of these tiles in exactly three distinct ways [2], the second and third uniquely. Note that the (100° , 60° , 60° ) tiling is two-colorable.
Every isosceles triangle that tiles the sphere, of course, gives rise by bisection to a right triangle tile. The (90° , 60° , 40° ) tile obtained by bisecting the (80° , 60° , 60° ) triangle tiles the sphere in many ways, none of them edge-to-edge, and might make a nice spherical puzzle!
There are also a number of other right triangles that tile the sphere. These include the (90°, 60°, 54°) triangle, three of which make up one of the (90° , 108° , 54° ) tiles mentioned above. (The smaller tile was discovered first; my wife, Bridget Thomas, was looking at my sketch and noticed that they fitted together into larger tiles.) It also tiles in several other ways. Another right-angled tile is the (90° , 72° , 30° ) triangle, 60 copies of which tile the sphere. While both (72° , 72° , 60° ) and (144° , 30° , 30° ) triangles appear in this tiling, neither of them tiles the sphere on its own. The (90° , 105° , 45° ) triangle (on the right of Figure 10), like the tile in Figure 3, is "right-obtuse". The twelve tiles have a delightfully twisty configuration. The (90° , 75° , 45° ) triangle tiles a 120° lune, three of which tile the sphere (Fig. 11, left). Moreover, by picking the orientations of these lunes properly, two of the three joins may be made edge-to-edge. The only non-edge-to-edge boundaries in this tiling are along the single meridian seen through the transparent front section of this sphere. While in one sense this tile comes extremely close to tiling the entire sphere edge-to-edge, it was easily ruled out by Davies because the (0,3,3) vertex, needed at the poles, could not be realized.
The final right-angled triangle tile, with angles of (90°, 78.75°, 33.75°), was discovered by Mr Doyle, in the last week of our search. It has only one tiling, a structure of great complexity (Fig. 11, right). In particular, the tiling is notable for having a very small symmetry group (order 4) for a tiling with so many (32) tiles. It has 8 different orbits under its symmetry group, a record for the "least symmetric most symmetric" tiling.
The tiling in Figure 13 uses 48 (120° , 45° , 30° ). triangles. It has polar stars like the tilings in Figure 12, but it has a double belt of quadrilaterals around its equator. The complex but symmetric pattern is remniscent of a Fabergé egg. There are not any other closely related tilings with more or less than 12 triangles meeting at the poles. To see this, note that with four different vertex figures, the set of linear equations determining the angles is overdetermined. Thus, in particular, the (3, 0, 0) vertex and the (1, 0, 2) and (0, 4, 0) splits force the (0, 0, 12) polar configuration. This tiling is also unusual in that all of its triangles are paired into kites. Thus, we also have a non-edge-to-edge tiling of the sphere with 24 congruent (90° , 120° , 60° , 120° ) kites.
References [1] Davies, HL. , Packings of spherical triangles and tetrahedra, [2] Dawson, R., An isosceles triangle that tiles the sphere in exactly three ways, [3] Dawson, R., Tilings of the sphere with isosceles triangles, [4] Dawson, R., Single-split tilings of the sphere with right triangles; in
[5] Sommerville, D.M.Y, Division of space by congruent triangles and tetrahedra, [6] Ueno, Y, and Agaoka, Y., Classification of the Tilings of the 2-Dimensional Sphere by Congruent Triangles, Technical Report 85, Division of Mathematics and Information Sciences, Hiroshima University, 2001 [7] Grünbaum, B., and Shepherd, G.S., "Spherical Tilings with Transitivity Properties", |