4.8.8

Defined in src/semi_regular/tess_4_8_8.scad. The 4.8.8 tiling, also called the truncated square tiling, has one square and two octagons at every vertex.

Everything is sized by radius, the distance from an octagon’s centre to one of its corners. The octagon’s edge length follows from it, and the squares have the same edge length:

side_length = (radius * 2) / sqrt(4 + 2 * sqrt(2)); // about 0.765 * radius

Theory derives this.

A complete pattern

use <tessella/tess.scad>

radius = 10;
n = 4;
m = 3;

oct_centers = octagons_centers_rect(radius, n, m);
sq_centers = squares_8_8_centers_rect(radius, n, m);

side_length = (radius * 2) / sqrt(4 + 2 * sqrt(2));

generic_poly(
    vertices = octagons_vertices(radius, oct_centers),
    paths = [[0, 1, 2, 3, 4, 5, 6, 7, 0]],
    centers = oct_centers,
    color_scheme = "scheme2"
);

generic_poly(
    vertices = squares_vertices(side_length, sq_centers, angular_offset = 0),
    paths = [[0, 1, 2, 3, 0]],
    centers = sq_centers,
    color_scheme = "scheme1"
);

The squares are drawn with squares_vertices() from Squares, with angular_offset = 0. The gap between four octagons is a square standing on a corner, and the default offset of 45 would draw it axis-aligned.

octagons_centers_rect

octagons_centers_rect(radius, n, m)

Returns the centres of an n by m grid of octagons.

Parameter	Meaning
radius	Centre-to-corner distance
n	Number of octagons along X
m	Number of octagons along Y

The pitch in both directions is the octagon’s width across the flats, side_length * (1 + sqrt(2)), which is radius * 2 * cos(22.5). The first centre is at the origin. The result has n * m points.

use <tessella/tess.scad>

echo(octagons_centers_rect(10, 2, 2));
// ECHO: [[0, 0], [0, 18.4776], [18.4776, 0], [18.4776, 18.4776]]

squares_8_8_centers_rect

squares_8_8_centers_rect(radius, n, m)

Returns the centres of the squares that fill the gaps in an n by m grid of octagons.

Parameter	Meaning
radius	Centre-to-corner distance of the octagons
n	Number of octagons along X
m	Number of octagons along Y

Call it with the same arguments as octagons_centers_rect(). Each square sits where four octagons meet, so the result has (n - 1) * (m - 1) points, and is empty when n or m is 1.

use <tessella/tess.scad>

echo(squares_8_8_centers_rect(10, 3, 3));
// ECHO: [[9.2388, 9.2388], [9.2388, 27.7164], [27.7164, 9.2388],
//        [27.7164, 27.7164]]

octagons_vertices

octagons_vertices(radius, centers, angular_offset = 22.5)

Returns the eight corners of an octagon around each centre.

Parameter	Meaning
radius	Centre-to-corner distance of the octagons drawn
centers	List of centre points
angular_offset	Angle of the first corner from +X; 22.5 puts flat edges on the axes

Corners are listed anticlockwise. Keep the default angular_offset with centres from octagons_centers_rect().

Pass a radius smaller than the one the centres were laid out with to leave a gap. The gap between neighbouring octagons is 2 * cos(22.5), about 1.848, times the difference. To get the same gap between an octagon and a square, shrink the squares’ side by that same 1.848 times the amount you shrink the octagons’ radius.

Render with paths = [[0, 1, 2, 3, 4, 5, 6, 7, 0]].

Known issues

• There is no radial layout for this tiling.
• tess_4_8_8.scad calls squares_centers_rect() but does not import squares.scad. Imported on its own, both centres functions return undef with an Ignoring unknown function warning. Import tess.scad instead.
• The source comment for octagons_centers_rect() documents a rotate parameter. The function does not have one.