3-Uniform (2 Vertex Types)6-4-3,4-6,3/m30/m(2c)6-3-3,3/m60/r30(2e)3-3,3-6/m90/r35(2e)3-4-3,3/m30/m(3e)4-6,4-0,3,3,4/m90/r30(2e)4-6,4-0,3,3/m90/r15(2e)6-3,0,3,3,3,3/r90(1e)/r300(1e)/r240(1e)6-3,3-0,3/r/r(1e)/r45(3e)4-4-4-3/m90/r5(4e)4-4-3/r(1e)/m90(1e)/r10(3e)4-4-3-3-4/m90/r90(1e)/r10(4c)4-3,4-3,3-4/m90/r15(4e)4-4,3,4-3,3,3-3,4-3-4/r/r100(2e)/r15(5e)4-3,3-0,4,3/r/r10(2e)/r70(3e)4-3,0,3-3-3/r25(4e)/r165(2e)/m90(1e)4-3,0,3-3/r25(3e)/r165(2e)/m90(1e)4-3-3-3-3-3/m90/r5(6e)4-3-3-3-3/m90/r0(3e)/m90(1e)
3-Uniform (3 Vertex Types)12-6,4-3,3,4/m30/r30(2c)12-3,4,6-3/m60/m90(1c)6-4-3,3-12-0,0,0,3/m30/r60(2c)12-4,3-6,3-0,0,4/m30/m30(2c)12-3,4-3-3-3/m30/m30(2e)12-3,4-3,3/m30/r60(2e)6-3-3-4-3,3/m30/m45(5e)6-4-3,4-3,3/m30/m30(2c)3-4-3,6-4/m30/r60(2c)3-4-3-3/m30/r30(2e)12-4-3,3/m90/r20(2e)6-4,3-3,0,4-6/m90/r45(2e)6-4,3-3,3,4-0,0,6,3/m90/m(2e)/r60(2c)4-3-3-0,4/r90/r20(3e)6-4,3,3-4,4-4/m30(2c)/r/r120(1e)6-4,3,3-4/m30(2e)/r/r120(1e)4-6-3,0,3,3-0,0,4/r20(2e)/m904-6,4-0,3,3/r0(2e)/m90/m90(2c)4-6,4-0,3,3-0,3,3/r/r(1c)/r90(2e)6-6-3,3,3/r60/m(2c)6-6,6,3-3,3/m/r35(3e)/r145(2e)6-3-3/m/r135(1e)/r20(2e)3-6/r60/m30(1c)6-3-3,3-3,3-0,3/r90(1e)/r225(1e)/r30(3e)3-3,6-3/m/r35(2e)/r135(1c)6-3-3/m90/r15(3e)3-3,3-3,6,3/m90/r90(2e)/r(2e)3-3-6-0,3/r60/m60(2c)3-3-6/r60/r30(2e)4-4-3-3/m90/r(3e)/r90(1e)4-4-3-3-3/m90/r5(5e)/r90(1e)4-4-3-3-3/r0(1e)/r0(3e)/m90(1e)4-4-3-3-3/r0(1e)/r7(5e)/m90(1e)
4-Uniform (3 Vertex Types)6-3-3-3/m90/r15(4e)
(3⁶)²tells us there are 2 vertices (denoted by the superscript 2), each with 6, 3 sided polygons (equilateral triangles). With a final vertex (
3⁴.6) of 4 more 3 sided polygons and a single 6 sided polygon (hexagon).
(3⁶)²; 3⁴.6notation as an example. If a single vertex was placed, surrounded by 4, 3 sided polygons and a 6 sided polygon, there would be 3 other vertices with 2, 3 sided polygons. From here either the vertex type of
3⁴.6is possible, and the notation gives no indication to which is correct.
/) separated blocks. When split up, the very first block is the "Shape placement" stage, this takes care of placing the regular polygons on the plane. The blocks after this are the transformation functions, of which there can be two or more of.
-) separated phases. Similar to the Cundy & Rollett's notation, each number represents the number of sides on the polygon. The very first phase will always contain a single number of either 3, 4, 6, 8 or 12* (see 'Angles' at the end to understand why these are the only possible shapes). This defines the 'seed' shape which is the first shape to be placed at the center of the area to be covered.
0to skip a side of a polygon.
(3⁶)²; 3⁴.6configuration as an example. With this new notation as shown above, the shape placement stages consist of:
m' (mirror) applies a reflection transformation and a '
r' applies a rotation transformation. When specifying the origin of the transformation, it also slightly changes the behaviour of the ensuing transformation result.
(3⁶)²; 3⁴.6configuration as an example. The first transformation operation is
m60(reflect at the center of the plane by 60°). So all of the transformational operations performed would be:
r30(2e). Imagine a line drawn starting from the center of the plane with an endpoint at 30°. The second edge intersecting that line would become the transform's new point of origin. In which the shapes are then rotated by 180°. However this 180° is now relative to the intersecting line, and the center point of the plane becomes 0°.
(3⁶)²; 3⁴.6configuration as an example. The second transformation function is
r30(2e)(rotate 180° at the edge of the second intersecting shape at 30°).
r30(2c), except this origin type is represented by a `
c`. Instead of specifying the number of intersecting edges, it's simply the number of intersecting shapes (excluding the seed shape). This also allows for some flexibility in the angle of the intersection line as it does not have to target the center point of the shape directly. The shape simply needs to intersect on that line.
[3³.4²; 3².4.3.4]¹configuration as an example. The second transformation function is
m60(2c)(reflect over a line 180° relative to the intersection angle, at the center of the second intersecting shape at 60°).
(3⁶)²; 3⁴.6configuration as an example. Both transformation functions
r30(2e)are continuously repeated, each time taken the shapes that are currently on the plane.
* Angles - For a valid tessellation that produces no gaps, the sum of the interior angles of every shape around a vertex must equal 360°. For example, the interior angle of an equilateral triangle is
180°/3=60°, therefore you can have 6 triangles around a vertex as
Taking the pentagon (unusable for regular tessellations) as an example the interior angle is
540°/5=108°. While you could place this around a vertex with a regular square and a icosagon (108°+90°+162°=360°), the next available vertices would not allow for this same configuration.
In total, there are seventeen combinations of regular polygons whose internal angles add up to 360°. However only eleven of these can occur in regular polygon tilings.
Gomez-Jauregui, Valentin & Otero, César & Arias, Ruben & Manchado, Cristina. (2012). Generation and Nomenclature of Tessellations and Double-Layer Grids. Journal of Structural Engineering. 138. 843–852. 10.1061/(ASCE)ST.1943-541X.0000532.