# Tag Archives: radial

# If I've tried this once, I've tried it at least fifty-five times….

It’s hard to get regular pentagons, regular star pentagons, regular decagons, and related polygons to tessellate the plane while maintaining radial symmetry. This is my latest attempt.

## A Radial Tessellation of Regular Pentagons and Rhombi

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## Radial Five

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# A Radial Tessellation Using Three Types of Rhombi

The yellow rhombi have angles of 40 and 140 degrees, while the blue rhombi have angles of 80 and 100 degrees, just like in the last post here. However, that post did not include the red rhombi, which have angles of 60 and 120 degrees.

# A Radial Tessellation Using Two Types of Rhombi

The yellow rhombi have angles which measure 40 and 140 degrees, while the blue rhombi’s angles measure 80 and 100 degrees.

## Pentagonal Mandala V

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## A Radial Tessellation, on the Topic of the Difficulty in Tessellating with Regular Pentagons

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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.

# Octagons Can Tile a Plane III

Unlike my previous octagon-tiling discoveries (see previous post), this is a chiral, radial tessellation, with the colors chosen to highlight that fact.