Planes tiling

The quotient of a map can be a map with loops and multiple edges. Consider, for example, the 4-valent plane tiling 4,4 (see Section 1.5) formed by 4-gons and the group Z2 acting by translations on it. There is one orbit of vertices, two orbits of edges, and one orbit of faces under Z2 so the quotient 4,4)/ 2 is a torus represented by a single vertex and two loops. 

Only r-gons and non-Platonic plane tilings r, q are isotoxal their respective automorphism groups are Crv and T 2, r, q). The group Aut([r, q] — f) is Crv in five Platonic cases none is isotoxal, isogonal, or isohedral polycycle, except of isohedral 3,3 — / = (3,3)-star. 

A normal tiling T is a tiling by a set of plane tiles T = 7i, 72,... with the following properties ... 

Note that in the next section we do the enumeration of planes for every case, without assumption of balancedness and normality. It turns out that, for every case, except possibly Case 29, all obtained planes are balanced and normal, but this was not guaranteed a priori. If we did not restrict ourselves to normal and balanced planes, we would have obtained hyperbolic plane tilings, which, as is well known, cannot be classified easily. 

Case 30, i.e. ( 3,6, 4)-plane 3R0 and 6/ o- The condition 3Ro, 6/ o implies that every vertex has corona 3636. It is easy to see that there is a unique plane tiling, called the Kagome tiling, shown on Figure 9.12. 

The practice of employing reusable thermal protection systems for reentry is becoming more common. These are essentially ablative materials exposed to environments where veryHtde ablation actually occurs. Examples iuclude the space shuttle tiles and leading edges, exhaust no22le flaps for advanced engines, and the proposed stmctural surface skin for the National Aerospace plane. 

Figure 7 Example of a periodic model of CO on a slab representing a Rh( 100) surface. A unit cell containing several metal atoms and the adsorbate is tiled in the x, y, and z direction. This produces a metal slab with on one side molecules adsorbed. The slab extends along the xy plane, and is separated by some empty space from its image in the z direction...

On the other hand, given a torus with non-trivial translation group, there exists a unique minimal torus with the same universal cover and trivial translation subgroup. Those minimal tori correspond, in a one-to-one way, to periodic tilings of the plane. 

Classify the tilings by regular r-gons. This is done for the sphere by Johnson (see [Joh66, Zal69]) and for the plane in [Cha89]. 

We will give in Table 9.3 all 33 parameter set for face-regular ( a, b], fc)-plane 21 continua and 12 sporadic cases (when the tiling is unique). The use of continua to describe discrete structures is not new ... 

The crucial hypothesis is the fourth one. It excludes hyperbolic tilings of the plane, like those which we can see in some of Escher art (planar ones but we cannot draw them with given metric constraints). 

Let Dip, P) represent a circular disk in the plane with center P and radius p. Let Aip, P) denote the patch of tiles that is the set of all those tiles of T, whose intersection with D p, P) is non-empty, together with the minimum number of additional tiles needed to make the union of the tiles of Aifi, P) a topological disk (i.e. simply connected). Write pip, P), eifi, P) and u(p, P) for the number of tiles, edges, and vertices in Afp, P). It is proved in [GrSh87a, 3.2] that, for any fixed number a, we... 

Our basic example is the graphite lattice sheet, i.e. the 3-valent tiling 6, 3 of the plane by 6-gons. At every vertex of this tiling, we can substitute a 0-elementary (5, 3)-polycycles, either Ei or C3. If we substitute only Ei, we obtain a ( 5, 12, 3)-plane that is 12R0. In order to obtain a ( 5, 13, 3)-plane, we need to substitute a part of the E, by some C3, such that every 6-gon is incident... 

GrSh83] B. Griinbaum and G. C. Shephard, The 2-homeotoxal tilings of the plane and the 2-sphere, Journal of Combinatorial Theory, Series B 34 (1983) 113-150. 

Mor97] J. F. Moran, The growth rate and balance of homogeneous tilings in the hyperbolic plane, Discrete Mathematics 173 (1997) 151-186. 

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