Mathematical Spirals

According to Wikipedia, a spiral is a curve which emanates from a central point, getting progressively farther away as it revolves around the point (similar to helices [plural for helix!] which are three-dimensional). Pictured above are some of the most important spirals of mathematics.

Logarithmic Spiral: Equation: r=ae^bθ. I must admit that these are my favorite! Logarithmic spirals are self-similar, basically meaning that the spiral maintains the same shape even as it grows. There are many examples of approximate logarithmic spirals in nature: the spiral arms of galaxies, the shape of nautilus shells, the approach of an insect to a light source, and more. Additionally, the awesome Mandelbrot set features some logarithmic spirals. Fun fact: the Fibonacci spiral is an approximation of the Golden spiral which is only a special case of the Logarithmic spiral. 

Fermat’s Spiral: Equation: r= ±θ^(½). This is a type of Archimedean spiral and is also known as the parabolic spiral. Fermat’s spiral plays a role in disk phyllotaxis (the arrangement of leaves in a plant system). 

Archimedean Spiral: Equation: r=a+bθThe Archimedean spiral has the property that the distance between each successive turning of the spiral remains constant. This kind of spiral can have two arms (like in the Fermat’s spiral image), but pictured above is the one-armed version. 

Hyperbolic Spiral: Equation: r=a/θ. It is also know as the reciprocal spiral and is the opposite of an Archimedian spiral. It begins at an infinite distance from the pole in the center (for θ starting from zero r = a/θ starts from infinity), and it winds faster and faster around as it approaches the pole; the distance from any point to the pole, following the curve, is infinite. 

Like standard jigsaw puzzles, this puzzle has only one solution, but instead of every piece being a different shape and approximately the same size, every piece is the same shape and a different size. The placement of the pieces is based on the golden angle (≈137.5º), and results in a pattern frequently found in nature, for example on pine cones or sunflowers. The puzzle has 8 spirals in one direction, and 13 in the other.

Golden ratio in phyllotaxis

The appearance of Fibonacci numbers and the golden ratio everywhere in nature is mostly a persistent myth. Famous (but most presumably accidental) examples are the ratios between the phalanges of your fingers, or the position of the belly button in the human body. One example where there is an explanation for the golden ratio’s involvement is phyllotaxis, the arrangement of leaves on a plant stem. In sunflowers, pineapples, romanesco, aloe plants, pine cones, artichokes and numerous other flowers, the number of leaves or seed spirals frequently equals a Fibonacci number.

Let’s consider a mathematical model. We define a “flower” starting from a central growing point, producing a new leaf or seed after each α turns (α being a fixed parameter) and constantly growing outward. For simple rational numbers we get the following patterns:

So we see this results in some radial spokes (as many as the denominator in α's irreducible fraction). For a flower this is very unappealing, since this arrangements waste a lot of space. Instead a plant wants to maximize its exposure to sunlight, dew or carbon dioxide.

More interesting patterns occur when we choose an irrational α:

You notice this arrangements fill space more evenly but also stagnate into spiraling patterns, resembling the rational case. These spirals correspond with the best rational approximations for α. Centrally in the left example you clearly notice three spirals, because 1/π ≈ 1/3. After a while they break apart into 22 spirals: 1/π ≈ 7/22. One can show that the “best rational approximations” or convergents of an irrational number are precisely the fractions resulting from keeping only a limited number of terms in the continued fraction expansion.

So if a flower wants to distribute its seeds optimally, it needs an α which is “hard to approximate” with fractions, and the golden ratio is essentially the hardest one because its continued fraction consists only of 1’s. The convergents of the golden ratio have only Fibonacci numbers as denominators (in lower terms), which helps explain their ubiquitous occurrence in nature.

Indeed, if we run the model with the inverse golden ratio, we get a marvelous uniform pattern, not exhibiting any obvious spirals, resembling for instance a sunflower’s face:

Finally, the parameter α appears to be rather sensitive: even a small deviation in the angle of rotation quickly spoils the delicate balance achieved by the golden ratio.


What you are viewing in each of the above videos is a solid 3D printed sculpture spinning at 550 RPMs while being videotaped at 24 frames-per-second with a very fast shutter speed (1/2000 sec). The rotation speed is carefully synchronized to the camera’s frame rate so that one frame of video is captured every time the sculpture turns ~137.5º—the golden angle*. Each petal on the sculpture is placed at a unique distance from the top-center of the form. If you follow what appears to be a single petal as it works its way out and down the sculpture, what you are actually seeing is all the petals on the sculpture in the order of their respective distances from the top-center. Read on to learn more about how these were made, and why the golden angle is such an important angle.

*Note: the exact value for the golden angle is irrational. Here it is to five decimal places: 137.50776º

© edmark, 2014

Plant architecture of the Spiral Aloe

The Spiral Aloe, Aloe polyphylla (Asparagales - Xanthorrhoeaceae), is a rare and beautiful aloe from the high Maluti Mountains of Lesotho. The most striking feature of this aloe is the perfect spiral in which the leaves are arranged. This may be clockwise or anti-clockwise. The spiral is formed by five ranks of leaves which contain between 15 and 30 leaves each.

Plant architecture is species specific, indicating that it is under strict genetic control. Although it is also influenced by environmental conditions such as light, temperature, humidity and nutrient status. 

During vegetative development, plants continuously form new leaves that are arranged in regular patterns (phyllotaxis), with defined divergence angles between successive leaves. There are several phyllotactic patterns; one of these patterns is the Spiral phyllotaxis, best represented by Aloe polyphylla, in which successive leaves are initiated with a divergence angle of 137°, and the apparent spirals are due to dense packing rather than the sequence of leaf formation. 

References: [1] - [2]

Photo credit: ©J. Brew (CC BY-SA 2.0) | Locality: cultivated, Wanganui, Manawatu Wanganui, New Zealand (2005)

Made with Flickr

Haworthia truncata is in the family Xanthorrhoeaceae. Like many other species of Haworthia, it is native to the Little Karoo region of South Africa. The specific epithet, “truncata”, refers to the shape of the tips of the leaves, which give the leaves a truncated appearance. This species is stemless with the leaves being born from an underground meristem and arranged in two parallel rows, known botanically as distichous phyllotaxy. Haworthia truncata is popularly cultivated as a house plant, and is quite hardy in cultivation. This is due to the fact that this plant naturally occurs in dry, arid deserts with extreme temperature fluctuations, making this plant amenable to indoor growing conditions.


Logarithmic Spirals

You see logarithmic spirals every day. They are the natural growth curves of plants and seashells, the celebrated golden curve of ancient Greek mathematics and architecture, the optimal curve for highway turns. Peer into a flower or look down at a cactus and you will see a pattern of logarithmic spirals criss-crossing each other like so:

This elegant spiral pattern is called phyllotaxis and it has a mathematics that is equally lovely. One reason why the logarithmic spiral appears in nature is that it is the result of very simple growth programs such as
grow 1 unit, bend 1 unit
grow 2 units, bend 1 unit
grow 3 units, bend 1 unit
and so on…

Any process which turns or twists at a constant rate but grows or moves with constant acceleration will generate a single logarithmic spiral. An equally similar cellular automata program will generate phyllotaxis.