Number of Sectors
Increase Factor per Revolution
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.).
Created 29 November 2010, Last Update 09 February 2012
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