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Jim's Gasket - 2014

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H: 29cm W:30cm Detail from Pi & Phi III. [not for sale] H: 42cm W:42cm

This is a perfect space filling fractal. There are seven generations of pentagons in the blue version above. If the process were carried on infinitely, the blue areas would become infinitesimally small – much like in Sierpinski’s gasket.

Work in Progress?

After I first designed and constructed this fractal, I was struck by how Islamic the design looked. I have an idea that it might look well as a large marble paving in a mosque perhaps. It would be fairly easy to cut, polish and lay all the variously sized pentagons to as many generations as is practicable. Special tiles could be made to fill the spaces in between, or they could be filled with resin and the whole surface polished. Is there an architect out there who would like to collaborate with me on the project?

If the largest marble pentagon had a side length of 60cm, the pentagonal paved area would have a side length of 60 x Φ = 97cm. If the same sized largest pentagon were used with a fractal-fractal version of the gasket, the paved area would have a side length of 60 x Φ2 = 157cm.

Jim's gasket 10 generations Jim's gasket pentagon fractal fractal

Perhaps the design would make an interesting stained glass window?

Maths Challenge

Constructing an image of the fractal in a computer is challenging.

Another challenge is to find a formula for the sum of the pentagonal fractional areas whose limit is the area of the fractal itself.

Jim's gasket 10 generations - main

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