The basic form of the Polyhextans is a hexagon composed of six right-angled, equilateral triangles. This particular shape has not yet been named in any established polyform classification. Since polyforms made from right-angled, equilateral triangles are generally called Polytans, the term Hextan was introduced for the underlying hexagon.

A special feature of this grid is that it does not allow a complete, gapless tiling of the plane. When arranging the Hextans, empty spaces inevitably occur – the resulting pattern forms a hole grid.

Connected Hextans form the so-called Polyhextans. The illustration shows all 20 Pentahextans – that is, all combinations of five connected Hextans. These shapes fit into a regularly structured grid, which is referred to as Bevelled Squares in the PolySolver program.

There are 1 Dihextan, 3 Trihextans, 7 Tetrahextans, 20 Pentahextans, 60 Hexahextans, and 204 Heptahextans.

Shapes and Symmetries

The underlying grid allows shapes with four lines of reflection and four-fold rotational symmetry. The two polyhextan puzzles presented here, which can be downloaded as 3D-printing templates from the download section, are shapes exhibiting maximal symmetry.

1. HexTan-Puzzle

Since no fully symmetric shape composed of all seven tetrahextanes exists, two trihextanes and one dihextan were added for the HexTan puzzle.

2. PentaHexTan-Puzzle

The PentaHexTan puzzle consists of all 20 pentahextans.

The 3D-print templates consist of the 10 Polyhextan puzzle pieces and a sliderbox.

The 3D-print templates consist of the 20 Pentahextan puzzle pieces and a sliderbox.