In this tessellation, golden rectangles are shown in yellow. The orange darts are each made of two golden gnomons, joined at a leg — while the blue rhombi are each made of two golden triangles, sharing a base.

# Tag Archives: tiling

# Stars, Pentagons, and Rhombi #2

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

# 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.

# Beginning the Fractiles-7 Refrigerator Experiment

To begin this experiment, I first purchased two refrigerator-sized Fractiles-7 sets (available at http://fractiles.com/), and then, early on a Sunday, quietly arranged these rhombus-shaped magnets on the refrigerator in our apartment (population: 4, which includes two math teachers and two teenagers), using a very simple pattern.

Here’s a close-up of the center. There are 32 each, of three types of rhombus., in this double-set, for a total of 96 rhombic magnets, all with the same edge length.

The number of possible arrangements of these rhombi is far greater than the population of Earth.

The next step of the experiment is simple. I wait, and see what happens.

It should be noted that there is a limit on how long I *can* wait before my inner mathematical drives compel me to play with these magnets more, myself — but I do not yet know the extent of that limit.