Doyle spiral

These patterns contain spiral arms formed by circles linked through opposite points of tangency, with their centers on logarithmic spirals of three different shapes.

[2] However, their study in phyllotaxis (the mathematics of plant growth) dates back to the early 1900s.

, so that the surrounding circles have radii (in cyclic order) Only certain triples of numbers

come from Doyle spirals; others correspond to systems of circles that eventually overlap each other.

[9][10] Within each arm, the circles have radii in a doubly infinite geometric sequence

When one shape of arm occurs infinitely often, its count is defined as 0, rather than

determines a Doyle spiral, with its third and smallest arm count equal to

The shape of this spiral is determined uniquely by these counts, up to similarity.

[9][7] The self-similarities of a spiral centered on the origin form a discrete group generated by

In other cases, they can be accurately approximated by a numerical search, and the results of this search can be used to determine numerical values for the sizes and positions of all of the circles.

[5][9] Doyle spirals have symmetries that combine scaling and rotation around the central point (or translation and rotation, in the case of the regular hexagonal packing of the plane by unit circles), taking any circle of the packing to any other circle.

[6] Applying a Möbius transformation to a Doyle spiral preserves the shape and tangencies of its circles.

An example is Coxeter's loxodromic sequence of tangent circles, a Doyle spiral of type (2,3), with arm counts 1, 2, and 3, and with multipliers

[9][a] In the photo of a stained glass church window, the two rings of nine circles belong to a Doyle spiral of this form, of type (9,9).

In this case, the two spiraling arm types have the same radius multiplier, and are mirror reflections of each other.

Each straight arm is formed by circles with centers that lie on a ray through the central point.

[5] Because the number of straight arms must be even, the straight arms can be grouped into opposite pairs, with the two rays from each pair meeting to form a line.

A final special case is the Doyle spiral of type (0,0), a regular hexagonal packing of the plane by unit circles.

Its radius multipliers are all one and its arms form parallel families of lines of three different slopes.

[5] The Doyle spirals form a discrete analogue of the exponential function, as part of the more general use of circle packings as discrete analogues of conformal maps.

Indeed, patterns closely resembling Doyle spirals (but made of tangent shapes that are not circles) can be obtained by applying the exponential map to a scaled copy of the regular hexagonal circle packing.

[6] Doyle spirals have been used to study Kleinian groups, discrete groups of symmetries of hyperbolic space, by embedding these spirals onto the sphere at infinity of hyperbolic space and lifting the symmetries of each spiral to symmetries of the space itself.

[9] Spirals of tangent circles, often with Fibonacci numbers of arms, have been used to model phyllotaxis, the spiral growth patterns characteristic of certain plant species, beginning with the work of Gerrit van Iterson in 1907.

[4] In this context, an arm of the Doyle spiral is called a parastichy and the arm counts of the Doyle spiral are called parastichy numbers.

[13] With this application in mind, Arnold Emch in 1910 calculated the positions of circles in Doyle spirals of type

[1][3] For modeling plant growth in this way, spiral packings of tangent circles on surfaces other than the plane, including cylinders and cones, may also be used.

[14] Spiral packings of circles have also been studied as a decorative motif in architectural design.

[8] Tangent circles can form spiral patterns whose local structure resembles a square grid rather than a hexagonal grid, which can be continuously transformed into Doyle packings.

[16] The Doyle spiral should not be confused with a different spiral pattern of circles, studied for certain forms of plant growth such as the seed heads of sunflowers.

In this pattern, the circles are of unit size rather than growing logarithmically, and are not tangent.