# Some Special Radial and Spiral Tilings

Steven Dutch, Natural and Applied Sciences, University of Wisconsin - Green Bay
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## The Voderberg Tiling

 This was the first spiral tiling discovered, by Hans Voderberg in 1936. 1. The unit tile is a bent enneagon. All lettered edges are equal in length. Since two tiles fit together with twofold symmetry (see 3 and 4 below) AB is parallel to EF and DE is parallel to BC. 2. Like most radial tiles, this one can also be derived by deforming a triangle, in this case an isoceles 12-degree triangle. 3. The really interesting property of the Voderberg tile is that two tiles (in blue) can completely enclose a third. Thus Voderberg tilings are not normal because contacts between two adjoining tiles are not connected. 4. Two Voderberg tiles can even enclose a pair of tiles.

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The Voderberg tile, of course, can be used to construct a simple radial tiling.

The spiral Voderberg tilings are spectacular since the hooks in the tiles enhance the spiral appearance and the small apical angle of the tiles makes for very smooth curves.

## The "Bent Wedge" Tiling

This tiling does not seem to have a formal name. It is an isoceles 15-degree triangle distorted into a curved enneagon. The apical angle is 15 degrees, the three successive angles on each side are 165 degrees, and the two end angles are 60 and 105 degrees. The 165 degree angles are also the interior angles of a 24-gon, meaning the tiles can fit together in a myriad of ways.

As seen here, the tiles can form a simple radial pattern. Since the tiles can curve in either direction within each annulus, this alone makes for an infinite number of tilings.

 The tiling can be offset in a variety of ways to create spirals. In addition to the familiar ways of creating spirals, here, a 90-degree sector has been offset to make a spiral. There are other ways to form annuli besides the patterns shown above.

## The Hirschhorn Tiling

 This tiling consists of equilateral pentagons. In order, the vertex angles are 60, 160, 80, 100 and 140 degrees. The pentagon can be considered an equilateral triangle attached to a rhombus with 80-degree acute angles.Beyond the initial rosette, the tiling consists of radial sectors made up of pairs of mirror-image pentagons. The pairs stack in the usual triangular stacking pattern. In this figure bright colors show pentagons of one handedness and dark shades the mirror image pentagons. In this tiling the handedness of the central rosette and the next annulus have been reversed.

Return to Radial and Spiral Tilings.

Logarithmic Spiral Tilings. Logarithmic spirals lend themselves to beautiful tilings in which tiles change in size but are all geometrically similar. This property makes logarithmic spirals common growth patterns in biology.

Created July 29, 1999, Last Update July 29, 1999