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Spidroses - Penrose Tiling ... What "Aperiodic Tiling" means?
“Spidrose” Tiles
The next interesting result is that we were able to transform Penrose tiles into plane figures delimited by special spidron edge sequences. This eliminates the need for the markers enforcing the matching rules described by Conway, this way, the rules are encoded in the shapes.
With a clever idea, Marc Pelletier replaced the edges of the darts and kites of the Penrose tilings with special Spidron edge sequences. This change makes the matching markers unnecessary.
Nicely shaped aperiodic tessellation corresponding to Penrose tiling
Definitions and remarks
Aperiodic Tile: A set of tiles that can tile the plane non-periodically,
but cannot tile the plane periodically.
Periodic Tiling: A tiling is periodic, if it can be translated in any way, which leaves the tiling invariant.
Non-periodic Tiling: A tiling which doesn't have the property of the Periodic Tiling, but the tiles, makes possible to make a periodic tiling from them.
Listen: Only the tiling can be periodic or non-periodic and only a set of tiles can be aperiodic!
Nobody have found one unique tile which is aperiodic itself. The smallest group of aperiodic tiles has two units. (For example Penrose tiles, Goodmann-Strauss Tiles)
Matching Rules: To make a set of certain tiles aperiodic we have to add matching rules. To show and describe the matching rules, there are different ways. We can make it with coloring, with markers, but we also can do it with help of only the shape of the tiles itself.
The Spidroses above are one of the examples where we could represent the matching rules only with the shape.This tiling was presented to Mr Roger Penrose last year, who appreciated it. In 2008 in the proceedings of the the BRIDGES Conference (Leeuwarden, NE) we published a paper with Walt van Ballegooijen, where we presented Marc's idea, the spidronized Penorse tiling.
Dániel Erdély and Walt van Ballegooijen
The next interesting result is that we were able to transform Penrose tiles into plane figures delimited by special spidron edge sequences. This eliminates the need for the markers enforcing the matching rules described by Conway, this way, the rules are encoded in the shapes.
With a clever idea, Marc Pelletier replaced the edges of the darts and kites of the Penrose tilings with special Spidron edge sequences. This change makes the matching markers unnecessary.
Nicely shaped aperiodic tessellation corresponding to Penrose tiling
Definitions and remarks
Aperiodic Tile: A set of tiles that can tile the plane non-periodically,
but cannot tile the plane periodically.
Periodic Tiling: A tiling is periodic, if it can be translated in any way, which leaves the tiling invariant.
Non-periodic Tiling: A tiling which doesn't have the property of the Periodic Tiling, but the tiles, makes possible to make a periodic tiling from them.
Listen: Only the tiling can be periodic or non-periodic and only a set of tiles can be aperiodic!
Nobody have found one unique tile which is aperiodic itself. The smallest group of aperiodic tiles has two units. (For example Penrose tiles, Goodmann-Strauss Tiles)
Matching Rules: To make a set of certain tiles aperiodic we have to add matching rules. To show and describe the matching rules, there are different ways. We can make it with coloring, with markers, but we also can do it with help of only the shape of the tiles itself.
The Spidroses above are one of the examples where we could represent the matching rules only with the shape.This tiling was presented to Mr Roger Penrose last year, who appreciated it. In 2008 in the proceedings of the the BRIDGES Conference (Leeuwarden, NE) we published a paper with Walt van Ballegooijen, where we presented Marc's idea, the spidronized Penorse tiling.
Dániel Erdély and Walt van Ballegooijen
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