This is an expansion of the last post here. It may be possible to continue this tiling outward indefnitely, forming an aperioidic tiling — or it may not. I am simply uncertain about this

# Tag Archives: tiling

# Floral Shield #2

This is one of the aperiodic tilings made famous by Roger Penrose.

## Tiles

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## Tessellation Featuring Regular Enneagons, Regular Hexagons, and Isosceles Triangles

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## Tessellation Featuring Regular Enneagons and Hexaconcave, Equilateral Dodecagons

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## A Tiling of a Plane, Using Diconcave Octagons, Rhombi, and Two Types of Kites

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# Fractiles’ Mandala, Based on Angles of Pi/7 Radians

Although this was based on something I constructed using the *Fractiles-7* magnetic tiling toy, I did not have enough magnetic pieces to finish this. The idea was, therefore, converted into a (non-Euclidean) construction using *Geometer’s Sketchpad*, and then refined using *MS-Paint*. The reason I describe this as a non-Euclidean construction is that an angle of pi/7 radians, such as the acute angles in the red rhombi, cannot be constructed using compass and unmarked straight edge: antiquity’s Euclidean tools. The other angles used are whole-number multiples of pi/7 radians, up to and including 6pi/7 radians for the obtuse angles of the red rhombi.

The yellow rhombi have angles measuring 2pi/7 and 5pi/7 radians, while the blue rhombi’s angles measures 3pi/7 and 4pi/7 radians. None of these angles have degree measures which are whole numbers. It is no coincidence that 7 is not found among the numerous factors of 360. It is, in fact, the smallest whole number for which this is true.

I have a conjecture that this aperiodic radial tiling-pattern could be continued, using these same three rhombi, indefinitely, but this has not yet been tested beyond the point shown.