A catalog of convex deciphis.
Version of Saturday 6 January 2018.
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 = ½ cos 18° and b = ½ 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. Also, each diagonal of a polygon is shown if the angle it forms with an edge is a multiple of 36°.

3-gons
one axis of reflective symmetry:
a a + b
4-gons
180° rotative symmetry,
two axes of reflective symmetry:
2b 2a + 2b 2a 4a + 2b
one axis of reflective symmetry:
2a + 2b 2a + b
180° rotative symmetry:
2a 2a + 2b
5-gons
72° rotative symmetry,
five axes of reflective symmetry:
3a + b 7a + 4b
one axis of reflective symmetry:
3a + 3b
no symmetry:
3a + 2b 5a + 2b
3a + 3b 5a + 3b
6-gons
180° rotative symmetry,
two axes of reflective symmetry:
4a + 2b 4a + 6b 6a + 6b 10a + 6b
2a + 4b 6a 8a + 2b 8a + 6b
one axis of reflective symmetry:
6a + 2b 6a + 6b 8a + 4b 4a + 4b 6a + 5b
180° rotative symmetry:
6a + 2b 8a + 4b
4a + 4b 6a + 4b
no symmetry:
4a + 3b 6a + 3b
7-gons
one axis of reflective symmetry:
5a + 6b 7a + 5b 9a + 4b 11a + 6b 13a + 7b
no symmetry:
7a + 5b 9a + 7b
9a + 7b 9a + 4b
8-gons
180° rotative symmetry,
two axes of reflective symmetry:
6a + 6b 12a + 6b 12a + 8b 18a + 12b
one axis of reflective symmetry:
8a + 8b 12a + 8b 8a + 5b
14a + 7b 12a + 8b 16a + 8b
180° rotative symmetry:
8a + 8b10a + 4b
10a + 10b 14a + 4b
16a + 8b 14a + 10b
no symmetry:
10a + 7b
9-gons
one axis of reflective symmetry:
11a + 9b 17a + 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 + 10b 30a + 20b
72° rotative symmetry,
five axes of reflective symmetry:
22a + 9b
180° rotative symmetry,
two axes of reflective symmetry:
14a + 10b 20a + 8b 16a + 14b
26a + 16b 24a + 10b 20a + 16b

The polygons can be sorted by area:

approxexactpolygons
0.47553 a 300
0.58779 2b400
0.76942 a + b301
0.951062a 402, 406, 407
1.244952a + b405
1.538842a + 2b401, 404, 408, 409
1.720483a + b500
2.014373a + 2b503, 504
2.126632a + 4b604
2.308263a + 3b502, 507, 508
2.489904a + 2b403, 600
2.783794a + 3b621, 622
2.853176a 605
2.965435a + 2b505, 506
3.077684a + 4b611, 617, 618
3.259325a + 3b509, 510
3.440956a + 2b608, 613, 614
3.665474a + 6b601
3.734856a + 3b623, 624
 
approxexactpolygons
4.028746a + 4b619, 620
4.141005a + 6b700
4.322636a + 5b612
4.392018a + 2b606
4.504277a + 4b501
4.616536a + 6b602, 609, 800
4.798167a + 5b701, 705, 706
4.979808a + 4b610, 615, 616
5.273698a + 5b806
5.455329a + 4b702, 711, 712
5.567588a + 6b607
5.9308510a + 4b812, 813
6.155378a + 8b804, 810, 811
6.337009a + 7b707, 708, 709, 710
6.5186410a + 6b603
6.8125310a + 7b822, 823
6.9941711a + 6b703
 
approxexactpolygons
7.4696912a + 6b801
7.6942110a + 10b814, 815, 100
7.8329714a + 4b816, 817
7.8758411a + 9b900
8.0574812a + 8b802, 805, 808
8.2391213a + 7b704
8.7146414a + 7b807
9.5963214a + 10b820, 821, 103
9.7779615a + 9b902, 903
9.9595916a + 8b809, 818, 819
11.7229516a + 14b105
11.8617120a + 8b104
11.9045817a + 13b901
12.0862218a + 12b803
13.1066622a + 9b102
14.2128520a + 16b108
14.3516024a + 10b107
17.0660226a + 16b106
20.1437030a + 20b101

Here is a sample tessellation:

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