A Radial Tessellation Using Three Types of Rhombi

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.

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

fractiles7withblackbackground

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.

Order-Six Radial Tessellations of the Plane, Using Elongated and Equilateral Hexagons, Rendered with Different Coloring-Schemes

I explored radial tessellations of the plane, using only hexagons, in this earlier post. Order-three tessellations of this type are the familiar regular-hexagon tessellations of the plane. With higher-order all-hexagon radial tessellations, though, the hexagons must be elongated, although they can still remain equilateral, and all congruent, with bilateral symmetry. In that previous post, examples were shown of order 4, 5, and 8, in addition to the familiar order-3 regular-hexagon tessellation.

This left out order-6, of which I show many examples below. As it turns out, this particular radial tessellation lends itself particularly well to a variety of coloring-schemes. In the first picture, the construction-circles, -points, and -lines I used are shown; in the rest, they are hidden.

No upper limit exists to the order-number of these all-hexagon radial tessellations — although the larger that number gets, the thinner the hexagons become, relative to their edge length. At some point (which I expect would vary from person to person), as the order-number increases, the hexagons needed will become so thin that they will no longer be recognizable as hexagons.

frequency 6 radial tessellation of hexagons with construction lines

Next, with construction artifacts hidden, are some two-color designs I found.

frequency 6 radial tessellation of hexagons without construction lines 2-color

frequency 6 radial tessellation of hexagons without construction lines 2-color version two

Here are some which use three colors each:

frequency 6 radial tessellation of hexagons without construction lines 3-color version colored by another system

frequency 6 radial tessellation of hexagons without construction lines 3-color version two

frequency 6 radial tessellation of hexagons without construction lines 3-color

frequency 6 radial tessellation of hexagons without construction lines 3-color version colored by rings

I also found some four-color patterns with interesting symmetry:

frequency 6 radial tessellation of hexagons without construction lines

frequency 6 radial tessellation of hexagons without construction lines. four colors version 2png

Finally, here are some which each use six colors.

frequency 6 radial tessellation of hexagons without construction lines 6-color version two

frequency 6 radial tessellation of hexagons without construction lines 6-color version colored by another system

frequency 6 radial tessellation of hexagons without construction lines 6-color