The world of tilings has recently had a breakthrough discovery: for the first time a single shape has been found that can cover the plane only in aperiodic patterns. That means that, unlike many patterns you see on wallpapers, buildings, or floors, the patterns created by these new tiles do NOT repeat themselves. This is…
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Delighting mathematicians, artists, and children alike, the Klein Bottle is a whimsical and perplexing surface that was discovered by the German mathematician Felix Klein in late 19th century. While it is most frequently seen visiting our three dimensional world in the form below, its true home is 4 dimensional space; there it fits comfortably, without…
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Several years ago, I explored approximating Pi with polygons made of Geometiles. I described my findings in this blog. Fast forward to 2022, when Charles Fleischer showed me a polyhedral torus I’ve never seen before, and its boundary was a regular 30-sided polygon. The fancy name for a 30 sided polygon is triacontagon. Triaconta τριάκοντα…
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This story started with our desire to expand the capability of the Geometiles® to make transformable hinged constructions. In particular, we were interested in making flexagons using the equilateral and square Geometiles tiles. Flexagons are polygons typically made of paper, which can be folded to expose certain faces and hide others. They were invented by…
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