#penrose tile

[reblogs-the-art]

"Why Penrose Tiles Never Repeat"

[paulfc]

Penrose tile Origami

Nerd Alert.  There are rules associated with the aperiodic (non repeating) tiling of the plane with the Penrose kites and darts, certain juxtapositions of  tiles break these rules and are disallowed.  Certain parts of the pattern do however show local five fold symmetry, while still adhering to the rules, and these areas often have what is referred to as a sun, five kites points inward, at their centres.

So I started with a sun and then two rotations of tiles around this and printed them out on A4 and A3 and tried to fold them, the first thing you find is you cannot do this because Origami has its own rules.  There is a huge amount of maths that has been done about Origami and there are Theorems would you believe, one of which is about folds that result in a vertex, where lines come to a point, such as the centre of a sun shape, you have to have a valley for every mountain for it to be foldable that is an even number of lines and obviously five fold symmetry is an odd number, so the kites have to be divided into two triangles as do the darts, and this gets an even number.

Having mangled a few complete sheets I cut one up into the separate symmetry areas and they fold beautifully, they also are flat foldable which has its own theorem.  It’s when you have five of them in a circle it gets interesting and there are choices do you start from the centre and work out or the edge and work in?  Do you start with this fold as a mountain or a valley, this results in structures being either concave or convex.  

I cannot be the first person to try this so somewhere there is a lovely Origami Penrose tile structure but it is not in my studio.  I have not finished with this just yet but if you like challenges I can recommend this as an introduction to Penrose tiling and Origami.

[nakedgeometry]

the penrose tile is a powerful aperiodic tiling.  i use it in some of my art.