A catalog of convex deciphis.
Version of Friday 27 August 2021.
Dave Barber's other pages.

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

• The polygon does not intersect itself.
• The angle between any two sides is a multiple of 36°.
• All sides are either:
• One unit long, or
• φ units long, where φ is the golden ratio, approximately 1.618034.

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:

• proper if the angle between any two adjacent sides is 36°, 72°, 108°, 144°, 216°, 252°, 288° or 324°
• convex if each angle is 36°, 72°, 108° or 144°
• obtuse if each angle is 108° or 144°

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.

If a polygon has axes of reflective symmetry, one of them is drawn vertically to make the symmetry more obvious. Polygons lacking an axis of symmetry form mirror-image pairs which are drawn in the same compartment of a table. Their model numbers can be notated with a double-ended arrow, for instance 406 ↔ 407.

Below each polygon is given its area in terms of a = ½ cos 18° and b = ½ cos 54°. This a & b notation also aids in determining how a larger convex deciphi might (or might not) be partitionable into smaller convex deciphis. Below each area is then given the polygon's perimeter. There are important even-odd relationships among these:

areaperimeternumber
of sides
coefficient
of a
coefficient
of b
integercoefficient
of φ
eveneveneveneveneven
evenoddoddoddeven
oddoddoddevenodd
oddevenevenoddodd

Equilateral polygons are shown on a yellow background; there are seven in each of the two side lengths.

A diagonal of a polygon is shown if the angle it forms with an edge is a multiple of 36°. The length of such a diagonal equals an integer added to an integral power of φ. The lengths of other diagonals are not easy to characterize.

Two elementary properties of n-gons when n is even:

• If any polygon has 180° rotative symmetry, opposite sides must be parallel.
• If any polygon has 180° rotative symmetry, opposite sides must be the same length.
Neither converse is always true, however, as demonstrated by 10-gon #102.

Tiles can be manufactured in these shapes as an educational toy. Few sets would include all 98 polygons; the youngest children might benefit from a set limited to equilateral polygons.

3-gons
one axis of reflective symmetry:
 a2 + φ a + b1 + 2φ
 same sequence of angles: none same sequence of sides: none
4-gons
180° rotative symmetry,
two axes of reflective symmetry:
 2b4 2a + 2b4φ 2a4 4a + 2b4φ
one axis of reflective symmetry:
 2a + 2b2 + 2φ 2a + b3 + φ
180° rotative symmetry:
 2a2 + 2φ 2a + 2b2 + 2φ
 same sequence of angles: 400, 401, 406, 407 402, 403, 408, 409 same sequence of sides: 400, 402 401, 403 406, 407, 408, 409 same both: 406, 407 408, 409
5-gons
72° rotative symmetry,
five axes of reflective symmetry:
 3a + b5 7a + 4b5φ
one axis of reflective symmetry:
 3a + 3b3 + 2φ
no symmetry:
 3a + 2b4 + 1φ 5a + 2b2 + 3φ 3a + 3b3 + 2φ 5a + 3b1 + 4φ
 same sequence of angles: 500, 501 503, 505 504, 506 507, 509 508, 510 same sequence of sides: 503, 504 505, 506 507, 508 509, 510
6-gons
180° rotative symmetry,
two axes of reflective symmetry:
 4a + 2b6 4a + 6b4 + 2φ 6a + 6b2 + 4φ 10a + 6b6φ 2a + 4b6 6a4 + 2φ 8a + 2b2 + 4φ 8a + 6b6φ
one axis of reflective symmetry:
 6a + 2b4 + 2φ 6a + 6b2 + 4φ 8a + 4b2 + 4φ 4a + 4b4 + 2φ 6a + 5b3 + 3φ
180° rotative symmetry:
 6a + 2b4 + 2φ 8a + 4b2 + 4φ 4a + 4b4 + 2φ 6a + 4b2 + 4φ
no symmetry:
 4a + 3b5 + 1φ 6a + 3b3 + 3φ
 same sequence of angles: 600, 601, 602, 603, 613, 614, 615, 616 604, 605, 606, 607, 617, 618, 619, 620 608, 609 621, 623 622, 624 same sequence of sides: 600, 604 601, 605, 613, 614, 617, 618 602, 606, 615, 616, 619, 620 603, 607 608, 611 609, 610 621, 622 same both: 601, 613, 614 602, 615, 616 605, 617, 618 606, 619, 620
7-gons
one axis of reflective symmetry:
 5a + 6b6 + 1φ 7a + 5b5 + 2φ 9a + 4b4 + 3φ 11a + 6b2 + 5φ 13a + 7b1 + 6φ
no symmetry:
 7a + 5b5 + 2φ 9a + 7b3 + 4φ 9a + 7b3 + 4φ 9a + 4b4 + 3φ
 same sequence of angles: 700, 703, 711, 712 701, 702, 704, 709, 710 705, 707 706, 708 same sequence of sides: 705, 706 707, 708 709, 710 same both: 709, 710
8-gons
180° rotative symmetry,
two axes of reflective symmetry:
 6a + 6b8 12a + 6b4 + 4φ 12a + 8b4 + 4φ 18a + 12b8φ
one axis of reflective symmetry:
 8a + 8b6 + 2φ 12a + 8b4 + 4φ 8a + 5b7 + φ 14a + 7b3 + 5φ 12a + 8b4 + 4φ 16a + 8b2 + 6φ
180° rotative symmetry:
 8a + 8b6 + 2φ 10a + 4b6 + 2φ 10a + 10b4 + 4φ 14a + 4b4 + 4φ 16a + 8b2 + 6φ 14a + 10b2 + 6φ
no symmetry:
 10a + 7b5 + 3φ
 same sequence of angles: 800, 801, 802, 803, 810, 811, 812, 813, 814, 815, 816, 817, 818, 819, 820, 821 804, 805 806, 807, 808, 809, 822, 823 same sequence of sides: 801, 802, 814, 815 810, 811, 812, 813 816, 817 818, 819, 820, 821 same both: 801, 802, 814, 815 810, 811, 812, 813 816, 817 818, 819, 820, 821
9-gons
one axis of reflective symmetry:
 11a + 9b7 + 2φ 17a + 13b3 + 6φ
no symmetry:
 15a + 9b5 + 4φ
 same sequence of angles: all same sequence of sides: none
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 + 10b10 30a + 20b10φ
72° rotative symmetry,
five axes of reflective symmetry:
 22a + 9b5 + 5φ
180° rotative symmetry,
two axes of reflective symmetry:
 14a + 10b8 + 2φ 20a + 8b6 + 4φ 16a + 14b6 + 4φ 26a + 16b2 + 8φ 24a + 10b4 + 6φ 20a + 16b4 + 6φ
 same sequence of angles: all same sequence of sides: none

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.