Layouts

A layout is the set of tile centres. The regular tilings each have two: rectangular, sized by a column count n and a row count m, and radial, sized by a number of levels out from a central tile.

Rectangular

Function	Tiles	Column pitch (X)	Row pitch (Y)	Stagger
`squares_centers_rect(side, n, m)`	n × m	`side`	`side`	none
`hexagons_centers_rect(radius, n, m)`	n × m	`radius * sqrt(3)`	`radius * 1.5`	odd rows shifted half a column
`triangles_centers_rect(side, n, m)`	n × m	`side * sqrt(3)/2`	`side / 2`	odd rows shifted by `side * sqrt(3)/6`
`octagons_centers_rect(radius, n, m)`	n × m	`radius * 2 * cos(22.5)`	same as X	none

Rows are numbered from zero, so the first row is even.

A 9 by 6 grid of squares with alternate columns partly removed A rectangular hexagon layout with tiles removed from two corners

In the triangle layout the triangles point along X, alternately left and right, so a “row” is a horizontal band half a side apart. n = 9, m = 15 with side = 20 covers about 160 × 160 units.

A rectangular triangle layout with a zigzag band removed

Radial

Function	Tiles for `levels = L`	L = 1, 2, 3, 4, 5
`hexagons_centers_radial(radius, levels)`	3L² − 3L + 1	1, 7, 19, 37, 61
`squares_centers_radial(side, levels)`	3L² − 3L + 1	1, 7, 19, 37, 61
`triangles_centers_radial(side, levels)`	1 + 3L(L − 1)/2	1, 4, 10, 19, 31

levels = 1 is a single tile at the origin. The hexagon count is derived in Theory.

The hexagon layout is a hexagon of hexagons: each level adds a complete ring.

Five levels of hexagons with a wedge removed

The square layout has the same row structure as the hexagon one. The middle row has 2 * levels - 1 squares, each row above and below has one fewer, and every row is centred. Adjacent rows are therefore offset by half a square, like courses of brick. It is not the edge-to-edge 4.4.4.4 tiling; use squares_centers_rect() for that.

Five levels of squares in staggered rows, with tiles removed near the top

The triangle layout grows as a hexagon with alternate sides shorter.

Sixteen levels of triangles with the four central triangles removed

Radial triangle centres need rect = false when they are turned into vertices. See triangle_vertices().

Moving a layout

The centres are plain points, so you can transform them yourself. For most tilings it does not matter whether you move the centres or the finished geometry.

Triangles are the exception. triangle_vertices() decides which way each triangle points from the absolute coordinates of its centre, so centres that have been shifted come out with the wrong orientation. Compute the vertices from the centres as the library returned them, then move the result with translate():

use <tessella/tess.scad>

centers = triangles_centers_rect(side = 10, n = 6, m = 8);
vertices = triangle_vertices(side = 10, centers = centers);

translate([30, 15, 0])
    generic_poly(vertices = vertices, paths = [[0, 1, 2, 0]]);

4.8.8

The 4.8.8 tiling has two kinds of tile, so it has two centres functions that are called with the same arguments:

use <tessella/tess.scad>

radius = 10;
// 4 × 3 octagons
oct_centers = octagons_centers_rect(radius, 4, 3);

// the 3 × 2 squares between them
sq_centers = squares_8_8_centers_rect(radius, 4, 3);

n and m count octagons. The squares fill the gaps between four octagons, so there are (n - 1) × (m - 1) of them. 4.8.8 shows the full pattern.