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.

Wednesday, September 14, 2016

islamic geometric patterns of Eric Broug

based on Kairouan Mosque pattern (p. 30)

I first heard about Eric Broug's work from a post by Alex Bellos, mentioned here. Alex provided a set of instructions by Eric that showed how to construct a design like the one below.

similar to Mosque of al-Nasir Muhammad,
but on a square grid instead of hexagonal (p. 82)

I was reminded of Eric's work after playing around with sliceformstudio, and ordered a copy of his book Islamic Geometric Patterns. It provided exactly the sort of instructions I was looking for, and has been a lot of fun to play with - the page numbers in the image captions of this post refer to that book. 

These are pure straight-edge and compass constructions and building up the elaborate designs using just those tools is key to appreciating their structure.  As a preamble, the book provides a quick overview of some standard ruler and compass constructions that may or may not be familiar to you, depending on when and where you learned school geometry. As pleasurable as it is to produce the patterns the old fashioned way (although not quite the authentic way: the original artists may have only had rope, rather than a standard geometry set), the desire to make larger versions of the patterns may prompt you to reach for more contemporary tools. Using geometry software, you can more easily explore and apply the techniques that Eric describes for drawing the thick interwoven lines that are characteristic of the art form, but that are difficult to do with pencil and paper.

applying some line drawing techniques (p. 22)

Rotating copies of the tile on the top right (medium sized lines), constructed in GSP using instructions from the book, I was able to make my own coloring page of the Kairouan mosque tile. (See also this earlier attempt at celtic knot tiles, that weave in a similar way.)

another view of Kairouan Mosque pattern (p. 30)

Rooted in an ancient tradition and found at culturally important sites, these patterns will, for some, have an aura of significance that stands apart from their geometric beauty. Although Eric's book pays tribute to those traditions, making these patterns understandable and reproducible may, in some sense diminish that aura; but if something is lost, something more is gained. In explaining, popularizing and providing the means of participating in the creation of these patterns, a tradition is both remembered and transcended. Rather than sacred and mysterious, I prefer my geometry to be rational and accessible - just as it is in Eric's instructions.