Log Spiral Tesselations

Steven Dutch, Natural and Applied Sciences, University of Wisconsin - Green Bay
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Number of Sectors Increase Factor per Revolution Minimum Radius

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A logarithmic spiral has the polar equation r=exp(ka) where r is the radius and a is the azimuth. It has the property that the curve makes a constant angle with the radius. Thus, a logarithmic spiral divided into equal radial sectors is a tesselation of geometrically similar tiles, differing only in size

Any plane tesselation has a logarithmic spiral counterpart. The radial cells formed by the spiral and its radii correspond to the unit cells or period parallelograms of the plane tesselation. Basically a point (x,y) in the plane is mapped to point (r,a) in logarithmic spiral space. However, note that this is not a one-to-one mapping. If the radius increases by q per revolution, then point (kx,y) is mapped to (qr,a). But so is point (x,y+2pi). Adjust the increase per revolution and number of sectors to get the most pleasing proportions.

In a few cases, interesting results come from using non-integral values for the number of sectors. Simple box patterns with sectors equal to an integer plus one half result in a brick pattern. Two different hexagon-triangle patterns can be created by selecting integral or integer plus one half sectors. Using integer plus 1/3 or 2/3 with the hexagon pattern results in hexagons with bow-tie polygons.

Since logarithmic spiral tesselations consist of repeated motifs that grow in size, they are common growth forms in biology (sunflower heads, sea shells, etc.).


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Created 29 November 2010, Last Update 09 February 2012

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