### The Archimedean planar tilings

The Archimedean tilings of the plane have probably been known for much longer than the Archimedean solids, since humans have been making paved floors from stone or ceramic patterns for many thousands of years. As with the Archimedean solids, the Archimedean tilings are made from more than one type of regular polygon, which distinguishes them from the Platonic tilings. Nevertheless, like the Platonic tilings, the Archimedean tilings only cover the plane and the vertex stars are completely coplanar. But while there are only 3 Platonic tilings, there are 8 Archimedean ones.

On the other hand, also like the Platonic tilings, this only assumes that the tiles in the tiling are solid colors. If we again apply "labels" to the dividing lines between the tiles (the edges of the vertex star) and apply the rules of isogonality to the way the vertex stars are laid out on the plane, we get many more ways to tile the plane. In fact, the number of labeled tilings increases from 8 to 26. Because the Archimedean tiling vertex stars must contain at least two kinds of polygons, the numbers of symmetries of the vertex star are greatly reduced. In fact, one of them, the (4.6.12) vertex star, is completely asymmetric. This greatly reduces the number of labelings that can be applied to the tilings, and this explains why the Archimedean tilings only change from 8 unlabeled to 26 labeled tilings, while the Platonic tilings change from 3 unlabeled to 67 labeled tilings.

Below we show all of the different labeled Archimedean tilings of the plane, along with VRML models of their tilings and vertex stars, as listed 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.]  Since this article covers both Platonic and Archimedean tilings, it contains the 67 Platonic ones plus the 26 Archimedean ones for a full total of 93. The reason the article title only claims ninety-one is because only 91 of the 93 labeled tilings can be realized as pure tilings using unlabeled, but curved, shapes. The remaining two tilings require markings on the tiles. Here we use markings on straight-edged polygons, so we can claim the full 93. One additional thing to note is that because there are so many more possible choices of which edges to apply which labels to than with the completely symmetric Platonic vertex stars, the different ways to label the edges means that there are many different incidence symbols that all describe the same polyhedron. We mostly use here the choices made by Grünbaum and Shephard in 1978. (This problem also occurs in the Platonic tilings, but far less often.)

But, again, planar tilings are coverings only of the plane. And while they can be perfectly depicted by drawing black lines on white paper, 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 planar tilings. In addition to the (4.6.12) vertex star being completely asymmetric and producing only an asymmetric tiling, note that the (3.3.3.3.6) vertex star, even though it has a reflective symmetry, also produces only an asymmetric tiling.

#### The 8 Archimedean planar tilings and their isogonal labelings

1. (3.3.3.3.6) planar tiling   (3.3.3.3.6)
1. [a+ b+ c+ d+ e+;  e+ c+ b+ d+ a+]

2. (3.3.3.4.4) planar tiling   (3.3.3.4.4)
1. [a b+ c+ c- b-;  a b- c+]
2. [a+ b+ c+ d+ e+;  a+ e+ c+ d+ b+]
3. [a+ b+ c+ d+ e+;  a- e+ c+ d+ b+]
4. [a+ b+ c+ d+ e+;  a+ e+ d- c- b+]
5. [a+ b+ c+ d+ e+;  a- e+ d- c- b+]

3. (3.3.4.3.4) planar tiling   (3.3.4.3.4)
1. [a b+ c+ c- b-;  a c+ b+]
2. [a+ b+ c+ d+ e+;  a+ c+ b+ e+ d+]
3. [a+ b+ c+ d+ e+;  a+ d- e- b- c-]

4. (3.4.6.4) planar tiling   (3.4.6.4)
1. [a+ b+ b- a-;  a- b-]
2. [a+ b+ c+ d+;  a- b- d+ c+]
3. [a+ b+ c+ d+;  b+ a+ d+ c+]

5. (3.6.3.6) planar tiling   (3.6.3.6)
1. [a+ a- a+ a-;  a-]
2. [a+ b+ a+ b+;  b+ a+]
3. [a+ b+ b- a-;  a- b-]
4. [a+ a- b+ b-;  b- b+ a- a+]
5. [a+ b+ c+ d+;  d+ c+ b+ a+]

6. (3.12.12) planar tiling   (3.12.12)
1. [a b+ b-;  a b-]
2. [a+ b+ c+;  a+ c+ b+]
3. [a+ b+ c+;  a- c+ b+]

7. (4.6.12) planar tiling   (4.6.12)
1. [a+ b+ c+;  a- b- c-]

8. (4.8.8) planar tiling   (4.8.8)
1. [a b+ b-;  a b-]
2. [a+ b+ c+;  a+ b- c-]
3. [a+ b+ c+;  a- b- c-]
4. [a+ b+ c+;  a+ c+ b+]
5. [a+ b+ c+;  a- c+ b+]

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Last updated: April 18, 2019