Triangles

Defined in src/regular/triangles.scad. Triangles are equilateral and sized by side, the edge length. A centre is the triangle’s centroid.

Neighbouring triangles in the 3.3.3.3.3.3 tiling point in opposite directions, so the vertices function has to work out an orientation for each centre. That makes triangles the one tiling where the centres function and the vertices function must be told about each other, through the rect parameter.

triangles_centers_rect

triangles_centers_rect(side, n, m)

Returns the centres of m rows of n triangles.

Parameter	Meaning
side	Edge length
n	Number of triangles in each row
m	Number of rows

The triangles point along X. Triangles in a row are side * sqrt(3) / 2 apart, rows are side / 2 apart, and odd rows are shifted by side * sqrt(3) / 6 toward +X and point the other way. The first centre is at the origin. The result has n * m points.

use <tessella/tess.scad>

echo(triangles_centers_rect(10, 3, 2));
// ECHO: [[0, 0], [8.66025, 0], [17.3205, 0], [2.88675, 5], [11.547, 5],
//        [20.2073, 5]]

triangles_centers_radial

triangles_centers_radial(side, levels)

Returns the centres of a pattern that grows outward from a single triangle at the origin.

Parameter	Meaning
side	Edge length
levels	Size of the pattern; 1 is one triangle

Here the triangles point along Y. The result has 1 + 3 * levels * (levels - 1) / 2 points: 1, 4, 10, 19, 31 for levels 1 to 5. That formula matches the output at every level checked, 1 to 8 and 16.

use <tessella/tess.scad>

echo(triangles_centers_radial(10, 2));
// ECHO: [[0, 0], [0, -5.7735], [-5, 2.88675], [5, 2.88675]]

triangle_vertices

triangle_vertices(side, centers, angular_offset = 0, rect = true)

Returns the three corners of a triangle around each centre, flipping alternate rows so the triangles interlock.

Parameter	Meaning
side	Edge length of the triangles drawn
centers	List of centre points
angular_offset	Rotation applied to every triangle
rect	true for centres from triangles_centers_rect, false for radial ones

The two layouts need different arguments:

Centres from	angular_offset	rect
triangles_centers_rect	0	true
triangles_centers_radial	30	false

use <tessella/tess.scad>

side = 10;

rect_centers = triangles_centers_rect(side, 6, 8);
rect_vertices = triangle_vertices(side, rect_centers);
generic_poly(vertices = rect_vertices, paths = [[0, 1, 2, 0]]);

radial_centers = triangles_centers_radial(side, 5);
radial_vertices = triangle_vertices(
    side,
    radial_centers,
    angular_offset = 30,
    rect = false
);
translate([90, 30, 0])
    generic_poly(vertices = radial_vertices, paths = [[0, 1, 2, 0]]);

Pass a side smaller than the one the centres were laid out with to leave a gap. The gap between neighbouring triangles is the difference divided by sqrt(3).

Render with paths = [[0, 1, 2, 0]].

Known issues

• Orientation depends on absolute position. The function decides which way a triangle points from its centre’s X coordinate (rect = true) or Y coordinate (rect = false), measured from the origin. Centres that have been moved since the centres function returned them come out pointing the wrong way. Compute the vertices first and move the geometry with translate(), as the example above does.
• The source comment documents a parameter named is_radial. The parameter is rect, with the opposite sense.