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:
 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 onetenth 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 sidelength ratio φ and the 36° angle go hand in hand, as illustrated by the following two wellknown triangles:
The area of the larger triangle (0.769421) is φ times the area of the smaller (0.475528).
While the nonconvex deciphis are infinite in number, there are precisely 98 convex deciphis, listed in the table below. Each of these figures is assigned an arbitrary threedigit 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 mirrorimage pairs which are drawn in the same compartment of a table.
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.
3gons 
one axis of reflective symmetry: 

 a  a + b




4gons 
180° rotative symmetry, two axes of reflective symmetry: 
 one axis of reflective symmetry: 

 2a + 2b  2a + b

 180° rotative symmetry: 



5gons 
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




6gons 
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: 



7gons 
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




8gons 
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 + 8b  10a + 4b
 
 10a + 10b  14a + 4b
 
 16a + 8b  14a + 10b

 no symmetry: 
 10a + 7b




9gons 
one axis of reflective symmetry: 

 11a + 9b  17a + 13b

 no symmetry: 
 15a + 9b




10gons  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




Here is a sample tessellation:
Related are the penrose tilings, which were developed in a search for aperiodic tessellations.