Saturday, June 7, 2014

Penrose Tile Quilt: Christmas Tree Skirt - Design Phase

I made this Christmas Tree Skirt as a wedding present for my niece and nephew-in-law:

It's based on a Penrose tiling, which is an aperiodic tiling derived from tiling pentagons. (I'll let wikipedia explain it).

I used a combination of machine piecing and  English paper piecing to construct it using two types of diamonds:

and I lay them out in this base tiling:

Then I do two tricky things that exploit some of the niftier properties of Penrose tilings:  matching rules and inflation/deflation.  These need to be explained in more detail...

Constructing Penrose Tiling using Matching Rules

These rhombi (plural of rhombus, a fancy word for diamond) can be tiled in a lot of ways that are not actually Penrose tilings.  They can be tiled in ways that are periodic, or completely non-periodic:

and you can certainly make some lovely quilts doing this:

This was made by Domesticat

This was made by Dorothea
However, these are not strictly Penrose tilings.  To make a quilt that has that quirky stars-with-ripples effect that the aperiodic tiling creates, you need to follow the matching rules:

My first plan was to design a quilt that copied the curved design as shown in the above diagram (but just the pink lines):

NOTE:  Penrose tiling is beyond the powers of Electric Quilt, so I designed these using Geometer's Sketchpad instead.

I made this pot holder as a test to see if it was feasible to make the entire full sized quilt this way:

Although this was strictly feasible, I found that all the curved lines and sharp points made this too tricky for my skill level.

So I decided to tweak the matching rules into straight lines:

This way I can machine piece these straight lines first and paper piece the rhombi.

Constructing Penrose Tiling using Inflation/Deflation Properties

One of the mind-bending properties of a Penrose tiling is that if all the rhombi in a given tiling are broken up like so:

The resulting tiling is also a Penrose tiling of smaller tiles.  In this example I deflate this simple tiling one level:

Now I can deflate this tiling down another level (I shaded a couple of rhombi from the previous level to illustrate this):

You can continue this indefinitely.  Here's 3 and 4 level deflation:

The flip side of this is inflation; any tiling can be combined into larger tiles by reversing this process. So any tiling, no matter how large, is only a subset of a single rhombus (mind = blown).

This inspired me to design a quilt that exploits this property.  I first thought it would be interesting to have a quilt that uses different sized rhombi:

Then I thought it would be cool if I designed a quilt where all the rhombi were the same size, but by using different colors and/or values (light and dark), an inflated level of Penrose tiling would emerge:

The combination of this design with the matching rule design above became the final design for the Christmas tree skirt: