Crystalline geometry; a projection from five dimensions into three.

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This sculptural, low-relief “slice” of crystalline geometry is a mathematical projection from five-dimensions into three.

When you move from one dimension to another circles become ellipses and angles change. My sculpture mimics a plane cast obliquely through a grid of hypercubes showing the conjoined surfaces. Fragments of the square faces of hypercubes have become skewed into diamond shapes. The sculpture is made from just one identical repeating shape, a rhombus with diagonals in the Golden Ratio.

The painted glass tiles are assembled according to certain mathematical matching rules that I’ve coded into a Baroque design, carved into rubber and printed on the glass. Each tile is printed and kiln-fired with traditional stained glass enamels, then painted several more times before being wrapped in adhesive copper tape. The tiles are then soldered together to build the sculpture. Below, copper-foiled glass tiles stacked in boxes and a small section tack-soldered together.

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The tiles fit together easily to form convex or concave rosettes with five ‘petals’. They also form groups of three, either shallow dishes or steeper ‘ears’. When the tiles are correctly conjoined my painted pattern flows through from tile to tile, creating a cohesive overall design. If each tile is also located correctly in three-dimensions the sculpture can grow infinitely without repeating itself, by simply adding more tiles.

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When light is cast through the sculpture onto a parallel surface the stained glass projects a two-dimensional image of another geometric pattern. Discovered in the late 1970’s and calledĀ  Penrose tiling, this pattern is made up from two different tiles. This also is beautiful and fascinating, and I have been obsessed with it since I was a post-grad student at the Royal College of Art in 1984.

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I had the lovely opportunity to meet Sir Roger last year for lunch at Yale and show him my model for this sculpture. I also took a stained glass panel I’d made some years before with his famous tiling pattern in it. This was the last of my Menfolk series and the beginning of my full-time focus on stained glass geometry.

I’ve received a lot of encouragement along the way, especially from computer scientist Duane Bailey who has been researching Penrose tiling for decades. When I met Duane almost four years ago we were both working in two-dimensions. This past July we started working together in three.

We have built three sculptural models so far, each with over 500 tiles the size and shape of business cards. We have designed four more sculptures. Each uses colour to explore some particular aspect of the geometry.

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The assembly was fiddly and time consuming, but well-worth the effort.

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The reverse of the sculpture is beautiful too. Each tile is made from a plain folded business card, and the triangular folded-back corners cluster together like the petals of mountain laurel.

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I learned such a lot about the this geometry. That it grows from seeds that changeĀ  shape depending upon where you begin. That it has kingdoms, worms and skeps, and more rules called inflation hierarchies to be taken into account. The surface can fold up to nest over itself in curious ways.

Remarkably, this pattern also models the structure of quasicrystals, materials that exhibit long-range order at a molecular level but lack any of the classic symmetries that characterize conventional crystals. Quasicrystals are neither amorphous solids (like glass or plastic) or regular crystalline materials (like salt or diamonds). They exhibit a host of unusual physical properties and can be made to self-assemble from nanoparticles to make invisible materials.

Making life-sized models of molecular structures in glass is pretty neat, and building something by hand (rather than modeling on a computer) provides wonderful insights. It may be difficult to visualize a skewed five-dimensional cube being ‘projected’ into another dimension) but the relationship between sculpture and cast image is really quite simple, and elegant to witness.

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