page_title: Doyle spiral
section_title: Uniqueness and related patterns
hierarchical_section_title: Doyle spiral / Uniqueness and related patterns
caption_attribution_description: English: Illustration of pattern of florets in a sunflower following Vogles model. Made with SingSurf and JavaView, post processed with gimp/inkscape. Author: R Morris
context_page_description: In the mathematics of circle packing, a Doyle spiral is a pattern of non-crossing circles in the plane, each tangent to six others. The sequences of circles linked to each other through opposite points of tangency lie on logarithmic spirals having, in general, three different shapes of spirals.
  These patterns are named after mathematician Peter G. Doyle, who made an important contribution to their mathematical construction in the late 1980s or early 1990s. However, their study in phyllotaxis dates back to the early 20th century.
context_section_description: The Doyle spirals (and the hexagonal packing of the plane) are the only possible "coherent hexagonal circle packings" in the plane, where "coherent" means that no two circles overlap and "hexagonal" means that each circle is tangent to six others that surround it by a ring of tangent circles. Applying a Möbius transformation to a Doyle spiral can produce a related pattern of non-crossing tangent circles, each tangent to six others, with a double-spiral pattern in which the connected sequences of circles spiral out of one center point and into another; however, some circles in this pattern will not be surrounded by their six neighboring circles.
  Additional patterns are possible with six circles surrounding each interior circle but only covering a partial subset of the plane and with circles on the boundary of that region not completely surrounded by other circles. It is also possible to form spiral patterns of tangent circles whose local structure resembles a square grid rather than a hexagonal grid, or to continuously transform these patterns into Doyle packings or vice versa. However, the space of realizations of locally-square spiral packings is infinite-dimensional, unlike the Doyle spirals which can be determined only by a constant number of parameters.
  It is also possible to describe spiraling systems of overlapping circles that cover the plane, rathar than non-crossing circles that pack the plane, with each point of the plane covered by at most two circles except for points where three circles meet at  angles, and with each circle surrounded by six others. These have many properties in common with the Doyle spirals.
  The Doyle spiral, in which the circle centers lie on logarithmic spirals and their radii increase geometrically in proportion to their distance from the central limit point, should be distinguished from a different spiral pattern of disjoint but non-tangent unit circles, also resembling certain forms of plant growth such as the seed heads of sunflowers. This different pattern can be obtained by placing the centers of unit circles on an appropriately scaled Fermat's spiral, at angular offsets of  from each other relative to the center of the spiral, where again  is the golden ratio. For more, see Fermat's spiral § The golden ratio and the golden angle.
caption: Non-Doyle spiral patterns obtained by placing unit circles at equal angular offsets on Fermat's spiral; the central image is the one with golden-ratio angular offsets
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