Here's the familiar tessellation of the regular hexagon:

Any hexagon with opposite sides of equal length and opposite angles equal can form a periodic tessellation:

K Reinhardt in his 1918 doctoral thesis Über die Zerlegung der Ebene in Polygone found that there are just three distinct cases of convex hexagons (that is, hexagons with all interior angles less than 180 degrees) that tessellate.

The three cases are:

I am currently trying to translate the thesis, as I suspect that Rheinhardt identified the 3 cases of hexagon, but not necessarily different ways to tessellate them.

An example of the first case is illustrated here:

An example of the second case is illustrated here. Note that half the hexagons have been turned over:

The following illustrates an example of the third case:

Here is a radial hexagon tessellation which I found a while ago. The hexagon has all side lengths equal and two opposite angles of 90 degrees:

There is a whole family of such radial tessellations (angle 72 degrees and 5 hexagons round the centre point, angle 60 degrees and 6 hexagons round the centre point, and so on..). There are also tessellations with a square or octagon at the centre.

For some reason I can't find any reference to these radial tessellations anywhere on the web. If you find one, do let me know. If it's new (very unlikely indeed) let me know too!


If we consider hexagons which are not convex, a whole range of possible tessellations arise. One interesting shape I've looked at is a chevron, made of 4 equilateral triangles. 6 chevrons can form a hexagon. Also 4 chevrons can form a hexagon. Both of these can themselves tile the plane of course:

You could also surround either of these with a new ring(s) of chevrons, to enlarge the hexagon.

Here's a spiral tessellation which is especially groovy:

There are loads of other possibilities, both for the chevron and other concave hexagons.


page date: 23Oct05.      I enjoy correspondence stimulated by this site. You can contact me here.