**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.

**A****nalysis**

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.

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