I have always been fascinated by the way mathematics can be used to describe the working of the natural world. It is an underlying language of nature. I enjoy finding and showing the many variations on these themes.
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This illustrates how a logarithmic spiral forms by rotating a square as it grows.
Pine cones are a classic example of the logarithmic or equiangular spiral in nature. This is from a Short-leaf pine. These spirals form from processes where there is turning at a constant angle but accelerating growth. This describes the growth of many structures in the plant kingdom.
Spirals as seen in the unfurling petals of a rose. For an astronomical connection to this spiral see the home page.
The leaves of this Hen and Chicks grow from the center, each leaf at an angle with the one before. Many radial patterns like this show 2 sets of spirals, one clockwise and one counter-clockwise. The clockwise ones (opening to the left) are a little more apparent in this example.
This Queen Anne’s Lace shows an example of a fractal pattern. Mathematical fractals show the same structure no matter how small or large a piece you look at. Biological specimens do not perfectly duplicate structure at different scales, but they can come close. From the center of the flower stems radiate out, then at the end of each stem is a smaller radiating pattern of florets. The florets in this case also show some arrangement into clockwise spirals.
The dandelion seed head also shows a fractal-like double-radiating pattern. Many of these have a slight resemblance to geodesic domes.
This Carrion flower shows a burst or explosion pattern. This is a type of branching pattern which fills the space around a central point. It also represents the shortest path to supply each developing seed with nutrients. This pattern is also seen in the Queen Anne’s Lace and Dandelion.
The veins on a leaf also represent a branching pattern that efficiently supplies the leaf with nutrients. An explosion pattern would provide a short route to each part of the leaf, but at the cost of a high total length of all the veins. The shortest total length of vein would be a single vein that traverses around the leaf near the perimeter. However, this would mean too long a supply route to some parts of the leaf. This pattern on a Slippery Elm leaf shows one compromise between the two extremes.
This Slippery Elm branch shows a nice alternating symmetry as well as curvature of the twigs. The curve is probably a type called the elastic curve, made by a flexible stem under the weight of the leaves.
These Hay-scented ferns show a variety of patterns. Ferns are fractal-like, with each leaflet representing a miniature of the whole frond. The fronds and leaflets are showing graceful curvature and symmetry. The symmetry between these 2 fronds is also quite nice.
This portion of bark on the lower section of red oak shows a repeating angular pattern.
The Morning Glory has a fountain-like curve in another pattern that radiates out from a central point.
Spirals expressed in cactus spines.
The tip of the frond has not quite finished unfurling, giving a hint of its former spiral form. The upper leaflets have a nice angular symmetry while the lower ones begin to gain a curve under their developing weight.
Spirals in the unfurling petals of a morning glory.
A radiating flow pattern, probably a type of burst, shown by Gazania. These patterns remind me a little of fountains, although the curve is not the same as the parabolic trajectory of water.
This also shows a bit of a fractal pattern as the burst repeats many times at a smaller scale.
There is also a nice triangular shape to the overall flower head. It displays a slight fractal pattern, but it is hard to visualize because the smaller branches droop and so the orientation of the pattern changes.
Here is a classic example of clockwise and counter-clockwise spirals. The number of each type of spiral is usually a Fibonacci number, and the 2 numbers are next to each other in the sequence 1-2-3-5-8-13-21-34-55-89-144… Sunflowers can have 34 spirals one way and 55 the other, or 55 and 89, or 89 and 144. Pine cones have 3 and 5, 5 and 8, or 8 and 13 sets of spirals. The clockwise spirals are in the pine cone picture, but not as pronounced as the counter-clockwise ones. Here both sets are equally prominent.