A catalog of convex deciphis.
Version of Wednesday 5 December 2012.
Dave Barber's other pages.

This report looks at a class of polygons meeting two criteria:

Such a figure we call a deciphi, from deci- (36° being one-tenth of a circle) and -phi (Greek letter φ denoting the golden ratio). A deciphi can be categorized as:

The obtuse deciphis form a subset of the convex, and the convex form a subset of the proper. Propriety is introduced to limit the figures to "plain old" polygons.

The side-length ratio φ and the 36° angle go hand in hand, as illustrated by the following two well-known triangles:

The area of the larger triangle (0.769421) is φ times the area of the smaller (0.475528).


While the non-convex deciphis are infinite in number, there are precisely 98 convex deciphis, listed in the table below. Each of these figures is assigned an arbitrary three-digit model number for convenience of reference, and all are drawn to the same scale.

Below each polygon is given its area in terms of a = 0.5 cos 18° and b = 0.5 cos 54°. If the number of sides is even, so is the coëfficient of a; similarly for odd. This a & b notation also aids in determining how a convex deciphi might be partitioned into smaller convex deciphis.

3-gons
one axis of reflective symmetry:
aa + b
4-gons
180° rotative symmetry,
two axes of reflective symmetry:
2b2a + 2b2a4a + 2b
one axis of reflective symmetry:
2a + 2b2a + b
180° rotative symmetry:
2a2a + 2b
5-gons
72° rotative symmetry,
five axes of reflective symmetry:
3a + b7a + 4b
one axis of reflective symmetry:
3a + 3b
no symmetry:
3a + 2b5a + 2b
3a + 3b5a + 3b
6-gons
180° rotative symmetry,
two axes of reflective symmetry:
4a + 2b4a + 6b6a + 6b10a + 6b
2a + 4b6a8a + 2b8a + 6b
one axis of reflective symmetry:
6a + 2b6a + 6b8a + 4b4a + 4b6a + 5b
180° rotative symmetry:
6a + 2b8a + 4b
4a + 4b6a + 4b
no symmetry:
4a + 3b6a + 3b
7-gons
one axis of reflective symmetry:
5a + 6b7a + 5b9a + 4b11a + 6b13a + 7b
no symmetry:
7a + 5b9a + 7b
9a + 7b9a + 4b
8-gons
180° rotative symmetry,
two axes of reflective symmetry:
6a + 6b12a + 6b12a + 8b18a + 12b
one axis of reflective symmetry:
8a + 8b12a + 8b8a + 5b
14a + 7b12a + 8b16a + 8b
180° rotative symmetry:
8a + 8b10a + 4b
10a + 10b14a + 4b
16a + 8b14a + 10b
no symmetry:
10a + 7b
9-gons
one axis of reflective symmetry:
11a + 9b17a + 13b
no symmetry:
15a + 9b
10-gons
To save space, the prefix for the model number is 1- and not 10-.
36° rotative symmetry,
ten axes of reflective symmetry:
10a + 10b30a + 20b
72° rotative symmetry,
five axes of reflective symmetry:
22a + 9b
180° rotative symmetry,
two axes of reflective symmetry:
14a + 10b20a + 8b16a + 14b
26a + 16b24a + 10b20a + 16b

Here is a sample tessellation:

Related are the penrose tilings, which were developed in a search for aperiodic tessellations.