Tetracairos belong to the Polycairo family. Each Tetracairo consists of four Cairo pentagons, each with two right interior angles. These pentagons can tile the plane.

There is one Monocairo, two Dicairos, five Tricairos, 17 Tetracairos, 55 Pentacairos, 206 Hexacairos, 781 Heptacairos, and 3099 Octacairos.

On the grid, every Tetracairo piece can be rotated by 90°. The points of its pentagons then face either horizontally or vertically.

Regardless of rotation, 12 of the 17 Tetracairos have half of their pentagons pointing horizontally and the other half vertically. The remaining five Tetracairos have an asymmetric distribution: three pentagons point in one direction and only one in the other (either 3:1 or 1:3).

When a shape is assembled from all 17 Tetracairos, the numbers of horizontally and vertically oriented pentagons therefore differ by 2, 6, or 10.

A mirror-symmetric shape would require an even distribution—that is, a difference of 0, 4, 8, and so on. Since this condition is not met, no mirror-symmetric shape made from all 17 Tetracairos can exist.

The 3D-printing files include the 17 Tetracairos and a box.

The online Tetracairo solver runs directly in your browser—no installation or download required. You can arrange the pieces with a mouse click and then solve the resulting shape.

The solver can also find solutions for shapes made from 2 to 16 pieces, as well as shapes using all 17 Tetracairo pieces.

Solutions are not grouped by symmetry: the solver counts each solution separately, including solutions that can be transformed into one another by rotation or reflection. For some shapes, the solver may take a few minutes to find its first solution.

The interface is in English and easy to use.