Spirals

The logarithmic or equiangular spiral is one of the most beautiful forms in nature, and it occurs in a wide variety of systems and at an enormous range of scales. No matter what the size, the shape of a given remains the same. Different spirals may twist at different rates however, and the twistedness is governed by the parameter τ in the following equation that defines the spiral in polar coordinates:

Spirals in Nature

Galaxies are the largest examples of spirals known. A single spiral galaxy may contain a trillion stars. Interestingly, there is a relatively uniform distribution of stars in a spiral galaxy. In other words, the spiral arms do not contain a greater number of stars. Rather, the spiral arms are brighter because they contain many short-lived extremely bright stars, formed by a rotating spiral wave of star formation. The waves of star formation are made visible because they contain many young and very bright stars that only live a short time, perhaps 10 million years, as compared to the more common stars, such as our sun which live for several billion years.


M51 the "Whirlpool Galaxy". Scale approximately 100,000 light years. Image Courtesy Nasa.

Hurricanes or typhoons (also called cyclones in the southern hemisphere) are the largest spirals here on Earth. The largest of these storms on record was Typhoon Tip, which measured 2170 km in diameter. In the north, they spin counterclockwise, while in the south they spin in the clockwise direction.


Hurricane Katrina, 2005. The state of Florida is visible for scale. Image courtesy NWS.

The plant kingdom is also full of spirals, as evidenced in many cacti, flowers, fruit, pinecones, etc.


Agave Cactus. Image courtesy of Francesco Hamamatsu.


Sunflower (Helianthus annuus). Image courtesy of Wikimedia Commons.

A nautilus shell serves to illustrate the simple, repetitive process that creates a spiral. The organism keeps expanding its home by adding sections to its shell. Each section is a little bigger than the one before, and a little bit rotated. The scaling factor and the rotation angle remain the same at every step in the process. It is this simple combination of rotation and expansion that creates the spiral, and accounts for its ubiquity.


Nautilus shell. Image courtesy of Wikimedia Commons.

We will revisit spirals in nature in Chapter 11, when we explore the Fibonacci Sequence, a common and beautiful numeric pattern in nature which creates the Golden Ratio.

Question:
How many segments are there in one full rotation (360 degrees) of this shell? [ ]

What is the angle, in degrees, occupied by one segment? [ ]

Exercise:

Order the following fractals by scale, from smallest to largest:[ ] [ ] [ ] [ ] [ ] [ ]

A

B

C

D

E

F

Remember: Fractals transcend scale. Fractals occur over a wide range of sizes in nature, from the microscopic to the galactic - AND at all time scales, from nanoseconds to millions of years.