6-4-3

After the regular tiling’s, we can consider those that use more than 1 regular shape to tessellate the plane. This chapter will illustrate how to construct the 6-4-3 tiling which is made up of hexagons, squares and triangles.

Following up on the PIC method, once a grid is created, a second set of lines on top of this grid creates the final pattern. By varying the angle that the lines are drawn, different patterns can be revealed.

In order to tessellate the plane, we aim to produce a rectangular section that looks like the picture below. This is made up of the 6-4-3 pieces, but as a rectangle, it can easily be repeated to fill a region.

The construction begins with a circle centered on a horizontal line, and the 2 other circles drawn on either side. These additional circles have the same radius and are centered on the original circle.

The brown rectangle, cornered at the intersection of these circles, will be the rectangular unit to tessellate.

Prepared by circleofsteve.com The midpoint of the vertical side, J, is found and a circle is drawn around it that is tangent to the two vertical sides. Where this circle intersects the central vertical line identifies the radius of the inner circle.

Placing a similarly sized circle above and below that middle circle is helpful. The central circle now holds the hexagon for the final pattern. The hexagon can be drawn by marking off the radius of the circle along the circumference beginning at the top point, K.

In the second picture, a similarly sized circle is drawn at the rectangle’s corner. Points U, W and Q are found at the intersections of the drawn circles and form a square.

Prepared by circleofsteve.com { With these lines drawn in so far, we can now see why the inner circle constructed the way it was will fit this pattern. The top half of the rectangle is shown below in more detail with some of the lengths identified.

In this example, the large radius OH = 2.0. The smaller radius which I call r, is the length of the sides of the polygonal pieces, example OK , KW and WH. Now OH is also made up of 3 segments, O-KQ , KQ- WU , and WU-H. The middle segment KQ-WU is just r; but the other two segments are altitudes of an equilateral triangle of side r, each of which is rcos(30°).

So we get that 2 = rcos(30°) + r + rcos(30°) = r(1+√3). From which r = 2/(1+√3) = √3 – 1. (approx = 0.73).

Now this value can be found exactly because OJ = √3 and JK = 1. The inner circle is thus found by “subtracting” JK from OJ. }

Continuing to the other corners, a circle drawn identifies the remaining squares in the pattern. Triangles above and below the central hexagon can now be completed as well. A pair of horizontal and vertical lines now identify the rest of the pattern.

Prepared by circleofsteve.com The final pattern with guidelines removed appears below, which can be tessellated. The PIC method proceeds by finding the midpoint of each side and choosing an angle from which to introduce the upper set of lines.

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The example below shows a line drawn at a 55% angle to the original grid. The following picture draws a similarly angled line at each midpoint. These individual segments terminate when they intersect a segment coming off the midpoint of an adjacent side. The third picture illustrates the original grid, in line, and then the overlay pattern appears as tiles. The final picture removes the underlying grid completely.

A wide range of angle choices produce different patterns.

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Prepared by circleofsteve.com As an alternative, once the polygonal grid is established, the overlay design need not be based on the midpoints of the sides. In this example, line segments originate from the vertices and continue until they intersect each other.

Prepared by circleofsteve.com A classic example is based on a 30° angle which converts the triangles into perfect hexagons.

This pattern can also be tessellated, with some overlap, to fill larger regions.

Prepared by circleofsteve.com Choosing different angles produce different patterns, as illustrated with 45° and 60° below.

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