Spirals

 

Ox-eye Daisy (Leucanthemum vulgare). This flowering head is made of 
           hundreds of small flowers.  The disc flowers in the center 
are arranged in Fibonacci numbers of spirals. 
Rowan County, North Carolina. 

Spirals abound in nature. They appear in spiral galaxies and DNA molecules, pine cones and snail shells, vines and sheep horns.  Biological spirals are all related to growth and have a deep mathematical basis.

 

Logarithmic spiral.
https://simple.wikipedia.org/wiki/Logarithmic_spiral

White-lipped Globe Snail (Mesodon thyroidus).  This snail's
shell is in the form of a logarithmic spiral.
Rowan County, North Carolina. 

One spiral that appears again and again is the logarithmic spiral.  This spiral starts at a central point and increases in diameter with the distance between turnings of the spiral increasing exponentially.  Snail shells are classic examples of logarithmic spirals.   The oldest part of the shell is at the center of the spiral and as the snail grows it increases in size and adds more shell to spiral.  Another logarithmic spiral occurs in the horns of wild sheep.  Dall Sheep (Ovis dalli) are white sheep of the subarctic mountains of North America.  The rams have large logarithmic spiral horns they use in batter each other in the mating season.  The winners of these battles get to mate with the ewes. The oldest part of the horn is the tip and new material is added at the base where the horn contacts the head of the sheep. 

Dall Sheep (Ovis dalli) ram with
 its logarithmic spiral horns. 
Chugach National Forest, Alaska. 

A mathematical concept related to the logarithmic spiral is the Fibonacci sequence.  Each number of the Fibonacci sequence is produced by adding the value of the two previous number.  The start of the Fibonacci sequence is 1, 1, 2, 3, 5, 8, 13…

 

Common Sunflower (Helianthus annuus).  The flowers are
arranged in Fibonacci numbers of spirals.
Rowan County, North Carolina. 

The flowering head of sunflowers and daisies and are made up of hundreds of closely packed flowers and later fruits.  These flowering heads show many logarithmic spirals.  The number of spirals in the head is one of the Fibonacci sequence of numbers.  There are spirals that run both clockwise and counter-clockwise.   

Pine cones also show the Fibonacci sequence in the rows of scales.  These logarithmic spirals allow efficient close packing of flowers and scales. 

Shortleaf Pine (Pinus echinata) cone.  The scales of the cone are
                                                 arranged in clockwise and counter-clockwise spirals.  This cone has 8                                                       counter-clockwise spirals, a Fibonacci number. 
Rowan County, North Carolina. 

The same Shortleaf Pine cone as above showing 13 clockwise
spirals.  13 is another Fibonacci number. 
Rowan County, North Carolina. 

Leaf arrangement in many plants also exhibit an aspect of the Fibonacci sequence.  The ratio of Fibonacci numbers approaches the golden ratio, 1:1.618... This ratio appears in many natural objects, in art and in architecture.  This ratio, designated phi (φ), is an irrational number and was used by the ancient Greeks in the design of the Parthenon, by Leonardo da Vinci in the composition of the Mona Lisa and in many other instances by artists both ancient and modern.  The Fibonacci spacing allows the plant to efficiently collect light for photosynthesis by spacing each successive leaf at an angle of about 137^o.  This angle is derived from φ and is called the golden angle.    

Haircap Moss (Polytrichum strictum).  The leaves are arranged at the
golden angle to allow for optimal photosynthesis. 
Denali State Park, Alaska. 

Agave sp. showing the golden angle arrangement of leaves. 
Biltmore House, Buncombe County, North Carolina. 

Some spirals in nature do not increase in diameter but take the form of a helix.  The tendrils of Passionflower (Passiflora sp.) allow this herbaceous vine to ramble over other plants.  A number of woody vines like Japanese Honeysuckle (Lonicera japonica) and Trumpet Creeper (Campsis radicans) grow in a helical pattern on a supporting tree.  The vine attaches to a trunk and spirals up, seeking more light higher in the tree.  As the vine wraps around the trunk it applies force to the tree and distorts its growth into a complimentary helical form.  Later, the vine may die and leave behind a helical tree trunk that is a favorite of walking stick makers. 

Yellow Passionflower (Passiflora lutea) with a helical tendril.
Rowan County, North Carolina. 

A tree trunk with a helical vine wrapped in a spiral.
Georgetown County, South Carolina. 

Helical tree trunk after the spiral vine has died.
Rowan County, North Carolina. 

Spirals are all examples of evolutionary solutions to problems faced by living things. Spirals solve problems of growth, of optimal space filling and of energy collection. These mathematical solutions are achieved by organisms that cannot think. I find it interesting these mathematical adaptations are also so aesthetically pleasing.