Regular hexagon and square tiling

The excellent ‘Tyler’ app by Melinda Green and Don hatch, lets you explore planar tilings using regular polygons (from 3 to 12 sides). The zoom facility is fantastic. Get it here… https://superliminal.com/geometry/

There are some really good examples, as shown in the gallery… https://superliminal.com/geometry/tyler/gallery/ but in this instance, I am using only hexagons and squares.

To make it more interesting, I put a constraint on the placement of both hexagons and squares, i.e., no hexagon may touch another (except at vertices) and likewise for squares. As well as this, the gaps must be consistent; in this case, thin rhombi.

Below left shows the only possible dodecahedron configuration of hexagons and squares excluding rotations. Also, the ‘propeller’ to the right of it, will always appear in any pattern.

If we consider only squares when continuing a pattern from a dodecahedron at its centre, there arises many combinations.

In the example below, the ‘disk’, far right, is not permissible, as a hexagon within the dodecahedron will touch another on the outer rim.

When extending each of the twenty five arrangements a little further, ‘bites’ sometimes occur. These can be filled in three different ways.

Below – six examples of central cores other than the dodecagon arrangement.

I finish with an attractive aperiodic tessellation.

There is also another article here on these type of formations but without the constraints that I set. http://followinglearning.blogspot.com/2015/07/tiling-again.html

Joshua Socolar discovered a substitution tiling rule for this set of shapes… https://tilings.math.uni-bielefeld.de/substitution/socolar/

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