The Platonic planar tilings

The Platonic tilings of the plane have probably been known for much longer than the Platonic solids, since humans have been making paved floors from stone or ceramic patterns for many thousands of years. As with our usual usage of 'Platonic', the Platonic tilings are made from regular polygons that are all the same, that is, congruent. But unlike space-filling polyhedra, tilings only cover the plane and the vertex stars are completely coplanar. There are only three such tilings, made up of either 6 triangles, 4 squares, or 3 hexagons. Thus, they are the {3,6}, {4,4}, and {6,3} planar tilings, respectively. Some people wonder why there are only these three. Of course, it is not hard to see by simple trial and error that only these three work. But perhaps we are just being thick-headed and missing another one? In that case, here is a mathematical explanation.

However, this only assumes that the tiles in the tiling are solid colors. What happens if we apply "labels" to the dividing lines between the tiles (the edges of the vertex star) and apply the rules of isogonality in the way the vertex stars are laid out on the plane? When we do that we get many more ways to tile the plane:  in fact, we can now do it in 67 different ways! Below we list all of the different labeled tilings of the plane, along with VRML models of the three tilings and their vertex stars, as described by Grünbaum and Shephard in 1978  [The ninety-one types of isogonal tilings in the plane, Branko Grünbaum and G. C. Shephard, Transactions of the American Mathematical Society, Vol. 242 (1978), pp. 335-353.]  (That article covers both Platonic and Archimedean tilings, so it reports more than 67.)

On the other hand, planar tilings are coverings only of the plane. In other words, they can be perfectly depicted by drawing black lines on white paper. But we can also consider the two-dimensional vertex stars as merely coplanar vertex stars in three-dimensional space. In this case, even more possibilities can arise when we allow the vertex stars to be flipped over as a possible means of connecting them isogonally. Note that because a flip of a coplanar vertex star produces the same edge outlines as a reflection, no new patterns can be created that could not have been created originally using only reflections. But if we allow the two "sides" of the vertex star to be colored in different colors, then we introduce the possibility of colored isogonal patterns in addition to the edge patterns listed earlier. At this time we have not yet done this, though, because of ambiguities in what color to assign the polygons when different combinations of flipped and non-flipped vertex stars are arranged around the vertices of a polygon. But we hope to eventually work out a design rule. In the meantime, here are the two-dimensional tilings of the plane.

The 3 Platonic planar tilings and their isogonal labelings

• {3,6} planar tiling   (3.3.3.3.3.3)
1. [a a a a a a;  a]
2. [a+ a+ a+ a+ a+ a+;  a+]
3. [a+ a- a+ a- a+ a-;  a-]
4. [a b a b a b;  b a]
5. [a+ b+ a+ b+ a+ b+;  b+ a+]
6. [a b+ b- a b+ b-;  a b+]
7. [a+ b+ c+ a+ b+ c+;  a+ b+ c+]
8. [a+ b+ c+ a+ b+ c+;  a+ c- b-]
9. [a+ b+ c+ c- b- a-;  a+ b- c+]
10. [a+ b+ c+ c- b- a-;  a- c+ b+]
11. [a+ b+ c+ c- b- a-;  c- b- a-]
12. [a b+ c+ d c- b-;  d b+ c+ a]
13. [a b+ c+ d c- b-;  d c- b- a]
14. [a+ b+ c+ d+ e+ f+;  a+ e+ c+ d+ b+ f+]
15. [a+ b+ c+ d+ e+ f+;  a+ e- c+ f- b- d-]
16. [a+ b+ c+ d+ e+ f+;  a+ e+ d- c- b+ f+]
17. [a+ b+ c+ d+ e+ f+;  b+ a+ d+ c+ f+ e+]
18. [a+ b+ c+ d+ e+ f+;  b- a- f+ e- d- c+]
19. [a+ b+ c+ d+ e+ f+;  c- e+ a- f- b+ d-]
20. [a+ b+ c+ d+ e+ f+;  d+ e+ f+ a+ b+ c+]

• {4,4} planar tiling   (4.4.4.4)
1. [a a a a;  a]
2. [a+ a+ a+ a+;  a+]
3. [a+ a+ a+ a+;  a-]
4. [a+ a- a+ a-;  a+]
5. [a+ a- a+ a-;  a-]
6. [a b a b;  a b]
7. [a b a b;  b a]
8. [a+ b+ a+ b+;  a+ b+]
9. [a+ b+ a+ b+;  a- b+]
10. [a+ b+ a+ b+;  a- b-]
11. [a+ b+ a+ b+;  b+ a+]
12. [a+ b+ a+ b+;  b- a-]
13. [a+ b+ b- a-;  a+ b+]
14. [a+ b+ b- a-;  a- b-]
15. [a+ b+ b- a-;  b+ a+]
16. [a+ b+ b- a-;  b- a-]
17. [a b+ c b-;  a b+ c]
18. [a b+ c b-;  a b- c]
19. [a b+ c b-;  c b+ a]
20. [a b+ c b-;  c b- a]
21. [a+ b+ c+ d+;  a+ b+ c+ d+]
22. [a+ b+ c+ d+;  a- b+ c- d+]
23. [a+ b+ c+ d+;  a- b- c- d+]
24. [a+ b+ c+ d+;  a- b- c- d-]
25. [a+ b+ c+ d+;  b+ a+ c- d-]
26. [a+ b+ c+ d+;  b+ a+ d+ c+]
27. [a+ b+ c+ d+;  b- a- c+ d+]
28. [a+ b+ c+ d+;  b- a- d- c-]
29. [a+ b+ c+ d+;  c+ b+ a+ d+]
30. [a+ b+ c+ d+;  c+ b- a+ d+]
31. [a+ b+ c+ d+;  c+ b- a+ d-]
32. [a+ b+ c+ d+;  c- b+ a- d+]
33. [a+ b+ c+ d+;  c- b- a- d-]
34. [a+ b+ c+ d+;  c+ d+ a+ b+]
35. [a+ b+ c+ d+;  c- d+ a- b+]
36. [a+ b+ c+ d+;  c- d- a- b-]

• {6,3} planar tiling   (6.6.6)
1. [a a a;  a]
2. [a+ a+ a+;  a+]
3. [a+ a+ a+;  a-]
4. [a b+ b-;  a b+]
5. [a b+ b-;  a b-]
6. [a+ b+ c+;  a+ b+ c+]
7. [a+ b+ c+;  a- b+ c+]
8. [a+ b+ c+;  a- b- c-]
9. [a+ b+ c+;  b+ a+ c+]
10. [a+ b+ c+;  b- a- c+]
11. [a+ b+ c+;  b- a- c-]