This is an excerpt from Master Fibonacci: The Man Who Changed Math. All citations are catalogued on the Citations page. (Previous Section: Fibonacci in Nature)
Nature’s language is empirically mathematical. As Salvador Rueda says in “La Nitidula,” the insect’s “dos vuelos descurbrn/sus hermeticas palabras” (two wings disclose/its hermetic words). Simple observation of the body sections of ants and millipedes, the wing dimensions and location of eye-like spots on moths, and the beautiful design of butterfly wings reveal shapes related to the Golden Ratio (Meisner). But even more intricate biological aspects of some insects illustrate properties of the Fibonacci sequence and the Golden Ratio. For example, from studies of the sensory reaction and attendance frequency of plant pollinators and their relationships with flowers, Leppik found that most pollinating insects have the ability to distinguish angular-form and radial- symmetry in flowers (Leppik). Even more remarkable is the fact that some insects have powers not limited to the recognition of mathematical shape and structure; they are capable of creating such harmonious structure, as well. Indeed, Rueda poetically describes the inhabitants of the beehive as geometers (skilled in geometry):
Llenas de logaritmos sus celulas obscuras, encierran de un misterio la gran filosofia, y rie entre sus mallas armonicaas y puras el cuerpo rubio y ritmico que encierra la ambrosia.
(Filled with logarithms its dark cells enclose in mystery the great philosophy, and the golden rhythmic army that surround the ambrosia laugh away their harmonious pure mazes.) (Spooner)
Honey bees, like certain fish, construct their honeycombs in the form of a hexagonal lattice (Spooner 38). Their hives as a whole are elliptical in shape (Stakhov 27). Perhaps honeybees instinctually know that “the geometrical correlations of the golden ellipse give optimal conditions for the attainability by photons …with minimal energetic losses,” as Polish scientist Jan Grzedzielski (Energetic and Geometric Nature Code) found when studying the golden ellipse. Before he was murdered by the Gestapo in 1941, the physician and professor observed that the golden ellipse can be used as a geometric model for the spreading of the light in optic crystals (Stakhov 27).
Speaking of bees, there are between 20,000-30,000 species of bees but the one most familiar is the honeybee, which lives in a colony called a hive. Honeybees have an unusual family tree; this is due to the fact that male honeybees have only one parent. Dr. Knott explains why:
The queen produces the eggs in a colony of honeybees. Worker honeybees are female and produce no eggs. Drone honeybees are male; their job is to mate with the queen. The queen’s unfertilized eggs produce males, so male honeybees have a mother but no father! All female honeybees are produced from fertilized eggs, when the queen has mated with a male, so they have two parents. Females most often end up as worker honeybees; only a few are fed with a special substance called royal jelly which makes them grow into queens who will leave the hive to start new colonies by taking swarms of honeybees with them to build new nests.
So female honeybees have two parents, a male and a female, whereas male honeybees have just one parent, a female. Dr. Knott portrays the Honeybee Family Tree as showing parents above their children, so newer generations are at the bottom and older generations are higher. Such trees are valuable because they show the lineage of ancestors (predecessors, forebears, antecedents) of the creature at the bottom of the diagram. This is different from the family tree charts of the rabbit problem, which show descendants of the original pair (progeny or offspring) at the bottom.
He had 1 parent, a female. Since his mother had two parents, he has 2 grandparents, a male and a female. Since his grandmother had two parents but his grandfather had only one, he has 3 great-grand-parents. The numbers of ancestors in each generation of the honeybee are Fibonacci numbers:
|Number of . . .||Male Honeybee||Female Honeybee|
|Great, Great, Great-Grandparents||8||13|
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Paperback: 128 pages
Author: Shelley Allen, M.A.Ed.
Publisher: Fibonacci Inc.; 1st edition (2019)