## Tuesday, September 20, 2016

### octagonal star and monster tilings

Regular octagons can be placed edge to edge so that their centers lie on the vertices of a square.

These form a nice tiling of octagons and squares (a semi regular tiling):

Regular octagons can also be placed edge to edge on the vertices of a larger octagon by skipping an edge - this leaves a star in the center.

These rings of octagons also fit together nicely, leaving those square gaps, so we see both the 4-rings (whose centers are squares) and the 8-rings (whose centers are eight-pointed stars).

In a couple of posts a while back (here and here), there is a rule that describes how copies of the same regular polygon can fit into rings like these. I have been playing around with these, noticing that there are rules for which rings can be imposed on regular tilings, and that sometimes you can make rings of regular polygon rings.

Another nice way that polygons can fit together is around a fused polygon, also sometimes called monsters (thanks to Kepler - see Craig's comment to this post). Just as regular pentagons may be placed around the vertices of a monstrous fusion of two decagons (as in the Kepler pentagonal tiling), octagons can be placed around the vertices of a monstrous fusion of two octagons.

These double rings of octagons also fit together reasonably nicely, in a way that also includes the 4-rings, 8-rings, and another ring-around a monstrous fusion of four octagons.

Here are the stars and monsters in this pattern:

Of course, you might consider making a ring of rings around monsters:

You get a similar central star in this pattern from an earlier post on rings of rings of polygons.