Four-armed spiral tiling of scalene triangles

Tis Veugen

tis.veugen@gmail.com

1 April 2021

Introduction

Robert Fathauer presented in [1] a tessellation based on a four-armed spiral. The triangles in that tiling have a right angle. In this paper a generalization of the tessellation is addressed using scalene triangles, that can be acute and obtuse. We show that for any given set of triangle angles there is only a single space filling configuration.

Analysis

Figure 1 sketches the addressed spiral configuration based on an arbitrary triangle. We will show that the two drawn iterations can be extended by smaller counter-clockwise iterations and bigger clockwise iterations, filling the whole plane.

Figure 1: Spiral configuration

The 4 seed triangles at the border of the parallelogram are ABC (red), ABC (purple), BAD (green), and BAD (blue). These 4 triangles have angles called : , , . Note that for triangles ABC and ABC the order of these 3 angles is counter-clockwise, whereas for triangles BAD and BAD the order is clockwise: the latter 2 triangles are flipped compared to the former 2 triangles. Triangles ABC and ABC have the same size with sides equal to a, b and 1. Due to the normalization of side AB to 1 the values of a and b can be computed from the angles. The size of triangles BAD and BAD differs a factor t with triangle ABC, so that their sides equal a*t, b*t and t. The middle of the parallelogram, considered as origin, is a point of symmetry. Triangles ABC and ABC are rotation symmetric with respect to the origin, and the same holds for triangles BAD and BAD.

Consider parallelogram ADAD, that has the same angles as parallelogram CBCB at their corresponding corners. This can be easily verified using the property: . Furthermore, we demand that the 2 parallelograms are similar. The scale factor between both parallelograms equals s. So side AD is s times BC, and side AD is s times CB. The latter gives the equation:

(1)

Also side AB imposes a restriction to s and t :

(2)

Combining equations (1) and (2) gives a quadratic equation in s (or t) :

(3)

The discriminant of (3) is always positive, so that there are 2 real, different roots. The product of the roots is 1. So, one root is bigger than 1 and the other, desired one is smaller than 1. This proves that for any given set of triangle angles there is a unique pair of parameters s and t.

For example, the configuration in [1] :

(4)

yields

(5)

Parallelogram ADAD is obtained from parallelogram CBCB by scaling with a factor s and rotating by angle . This process can be repeated unlimited. Parallelogram CBCB can in the same way repeatedly be enlarged by scaling with a factor 1/s and rotating by angle , filling the whole plane.

References

[1] Robert W. Fathauer, “New tessellation based on a four-armed spiral tiling of right triangles.”, https://twitter.com/RobFathauerArt/status/1358449514244280320