The polycairos are based on the cairo tiling, dual of the 3,3,4,3,4 semiregular tiling. It consists of identical, irregular pentagons (
). The name comes from the city of Cairo, Egypt, where many of the streets are paved in this design.
Enumeration[]
The number of polycairos of size N
| Type | Start | OEIS |
|---|---|---|
| Free | 1, 2, 5, 17, 55, ... | Sloane A159866 |
| Single Sided | 1, 3, 8, 31, 103 ... | Sloane A151534 |
| Fixed | 4, 10, 32, 112, 412 ... | Sloane A196991 |
| Free Path / Strip | 1, 2, 4, 10, 25, ... | Sloane A151536 |
| Single Sided Path / Strip | 1, 3, 7, 19, 47 ... | Sloane A151535 |
Shape[]
The base shape is an irregular pentagon with two non consecutive 90o angles, and the four edges adjacent to them unit length. The final edge can vary from almost 0 to almost , which determines the other angles.
Two that look relatively balanced are the one with three 120o angles, and the one with all five edges being unit length.
List[]
Monocairo[]
There is one free monocairo
Dicairos[]
There are two free dicairos
Coded as J and I
Tricairos[]
There are five free tricairos
They are coded as D, U, Y, J, and I
Tetracairos[]
There are 17 free tetracairos
They are encoded as
S, U, L, I, J, N
C, B, G, D, Y, R
Q, T, P, X, O
Pentacairos[]
There are 55 free pentacairos
Hexacairos[]
There are 206 free hexacairos
Heptacairos[]
There are 781 free heptacairos