Climbing the Pinwheel

Pinwheels
Pinwheels, LEA26

In February 2014, Mac Kanashimi unveiled Dragon Curves at LEA26. A stunning 3D piece of fractal art which quite captivated those who visited – including myself.

Now, as a part of the Artist In Residence (AIR) round 7 series, Mac is back at LEA26, this time with Pinwheels, which he invited me over to see on Saturday August 2nd, not long after he’d set it up in the region. Pinwheels is another remarkable mathematical structure with something of a fractal bent, and which uses Charles Radin’s pinwheel tilings, themselves based on the Conway triangles, to tremendous visual and artistic effect.

Pinwheels, LEA26
Pinwheels, LEA26

To explain this requires delving into a little bit of maths and geometry, but bear with me. A Conway triangle is a right-angle triangle with sides of 1, 2 and \sqrt{5}. which can be divided into five isometric copies of itself by the dilation factor of 1/\sqrt{5}  (see the image below), and when suitably rescaled and translated / rotated, can produce an infinite growing pattern of isometric copies of the original.

A Conway triangle divided into 5 isometric copies of itself
A Conway triangle divided into 5 isometric copies of itself (via wikipedia)

A pinwheel tiling is essentially a pattern of these isometric triangles where one tile may only intersect another either on a whole side or on half the 2 side (which actually makes the Conway triangle itself a pinwheel tiling – again, look at the image on the right and see how the five smaller triangles are positioned relative to one another). There’s actually more to the math than this, but I’ll let wikipedia explain the rest.

Like Dragon Curves, Mac’s Pinwheels is a huge piece, measuring 256 x 256m, but this time is confined vertically to a height of 256 metres as well, so to get the full measure of the piece – and to appreciate the overall complexity and beauty of the piece, you’ll need to ramp-up your draw distance to at least 600 metres, and cam out.

When you do so, the patterns of pinwheels and triangles and triangles within triangles becomes apparent. Each Conway triangle forms an individual segment made up of five prim isometric triangles of a similar shade (blues, greens, reds, etc, sometimes mixed with whites), which helps the eye to define individual groupings. These segments in turn are arranged to form pinwheels among themselves – although you’ll need to cam overhead to see them clearly.

Pinwheels
Pinwheels, LEA 26

Nor is it static; sections of the design rise and fall, creating an ever-changing landscape of colour and form, with only the arrival point, which is itself quite fascinating to watch. However, this motion isn’t in any way random; the triangles making up a particular pinwheel pattern all move together, and in doing so, they communicate their height and position to one another and to the surrounding segments.

The result of all this is that as the landscape changes and triangles and patterns rise and fall, paths can be found running through the entire construct, allowing the visitor to walk through it starting at the landing point (itself a static platform of 5 Conway triangles), with the individual prim triangles within each larger Conway triangle suitably adjusted so that they form steps for you to follow.

Pinwheels, LEA26
Pinwheels, LEA26

Just how artful this is requires you to walk through the piece. In this way you get to experience how the motion of segments works – no matter how the triangles on which you stand rise or fall relative to one another or to the surrounding patterns, no matter how high the plateau on which you find yourself lifted, or how far down into the depths of the piece you are carried, a footpath can always be found before you and behind you, leading you through the piece without ever necessarily reaching an end.

Pinwheels is another mathematical masterpiece from Mac, and will remain open through until the end of December. If you enjoyed Dragon Curves or if you’re into maths-based art, it’s a recommended visit.

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