Log Spiral Tesselations

Steven Dutch, Natural and Applied Sciences, University of Wisconsin - Green Bay

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.

Tiles are coded with respect to "unit cells." 0,0 is the left, inner corner, and 100,100 is the right outer corner. Coordinates are given in integers only. In practice, any errors are not noticeable. Since unit cells are arbitrarily given dimension 100 in both directions, tilings with very unequal unit cell ratios in the plane may be stretched or flattened here. This is especially common with tilings of hexagonal symmetry in the plane. To stretch a pattern tangentially, decrease the number of sectors or the increase in size per revolution. To stretch a pattern radially, increase the number of sectors or the increase in size per revolution.

The default setting is 16 sectors with the appropriate increase ratio calculated. The settings can be overriden by simply entering other values in the "Sectors" and "Increase Factor" windows. "Set Proportions" can be used to reset the increase factor for any desired number of sectors.

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 20 September 2014

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