Tessellation

A tessellation or tiling of the plane is a pattern of plane figures that fills the plane with no overlaps and no gaps. One may also speak of tessellations of parts of the plane or of other surfaces. Generalizations to higher dimensions are also possible. Tessellations frequently appeared in the art of M. C. Escher.

- Types of hexagon tessellation -

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:

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:

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.

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:

- How to make your own Hexagon Tessellation -

Draw a hexagon to use as the basis of your tessellation.

It needn't be a regular hexagon, but make sure that you can draw a line between opposite corners of your hexagon that passes through the centre, and that opposite corners are the same distance from the centre (shown by the solid green line here). This makes sure it will tessellate.

Mark the corners of the hexagon and remove the sides.

Now draw a curve through four of the corners, replacing three sides of your previous hexagon.

Take a copy of one of the sides and paste it exactly onto the opposite side. The corners should line up.

Do the same with the next side.

And finally the last side.

If you found that there was an overlap when you pasted, you'll have to go back and redraw the first curve.

Now decorate your shape

It tessellates like this

If you start with a concave hexagon, and go wild with the curves, you might end up with something like this:

which tessellates, believe it or not