Classification of Heptiamond Shapes with 12-fold Symmetry
By Number of Holes
There are 226 559 shapes of 168 triangles which have dihedral symmetry of order 6. Only 182 of them can be tiled by the whole set of heptiamonds. If we consider the number of holes in the tilable shapes, we obtain the following results:
| Number of holes | Number of shapes | Tilability rate |
|---|---|---|
| 0 | 8 | 2.31% |
| 1 | 35 | 0.031% |
| 6 | 27 | 2.51% |
| 7 | 73 | 0.12% |
| 12 | 16 | 1.8% |
| 13 | 23 | 0.064% |
| Any | 182 | 0.08% |
As can be seen, there are fewer tilable shapes with an even number of holes, but their tilibality rate is much higher.
By Enclosing Shell
Draw a straight line from the center of the shape, parallel to the y axis, until it crosses the border of the shape. There are three possible angles:
- 90°: family of the Hexagonalia;
- 30°: family of the Stellaria;
- 150°: family of the Floralia.
The following abbreviations are occasionally used in the tables: H. = Hexagonus, Hh. = Heptahexus, S. = Simplex, St. = Stella, L. = Latus/Lata/Latum.
Details of naming conventions and an English dictionary of all names.
Hexagonalia
Because of the quantity of shells that fall in the category of Hexagonalia, they are presented in two distinct tables. The second contains the shells resulting from the Heptahexus S. base.
Stellaria
| Stellaria | |||||
|---|---|---|---|---|---|
 Stella Simplex | |||||
 St. Terribilis |
 Stella Duplex |
 Stella Mammata |
 Stella Corrupta | ||
 Stella Arcis |
 Stella Triplex |
 Stella Tridentata |
 Floccus Nivalis | ||
 St. Chrysanthemi |
 Stella Quadruplex |
 Stella Spinata | |||
Floralia
| Floralia | ||||
|---|---|---|---|---|
 Flos Simplex | ||||
 Bifidus S. |
 Chrysanthemum Simplex |
 Flos Arcis |
 Flos Spinatus | |
 Bifidus Duplex |
||||
 Bifidus Triplex |
 Bifidus Explosus | |||
Name formation
The basic principle applied in this classification is that of complexity by use of transformations. The name of a shell is intended to give an idea of the most basic structure of the shell and of the transformations that have been applied on it.
All shells are composed of six repeating segments, which have axial symmetry and are arranged around the center of the shell.
The word Simplex is used as a mere placeholder, to make all names fit in the binominal scheme.
The word Duplex indicated the addition of a single "bump" between the branches of a shell. In a similar fashion, Triplex means two such bumps and Quadruplex − three of them. In the case of Arx shells, Duplex is used differently.
Explosus (Explosa in the feminine, Explosum in the neuter) indicated a furher expansion of a bump into something which can be seen as a double bump.
The adjective Latus (Lata in the feminine, Latum in the neuter) indicated a flat segment has been lengthened. This is meaningful only for Hexagonalia shells, since the others have spikes or inner creases. Latior (Latius in he neuter) is used for a futher lengthening.
A tooth (Dens) is a triangle, connected to the shell by a single edge. When a shell has 12 of them (two for each segment), the adjective Dentatus, a, um is used. If there are 24 teeth, Terribilis, e is used instead. In the case of 18 teeth, 3 per segment, Tridentatus, a, um is used.
A spike (Spina) is a diamong (group of two triangles) which is connected to the shell by a single edge. Whenever these appear, the adjective Spinatus, a, um is used.
Corruptus, a, um is used for shells with indentations at the base of the symmetric segments.
The genitive of some names is used to indicate a mixed type, as is the case with Stella Arcis, Stella Chrysanthemi and Flos Arcis.
Developed by Todor Tchervenkov: tchervenkov@gmail.com