The Mathematics of Beauty

In 1180, an Italian mathematician named Leonardo Fibonacci discovered a strange sequence of numbers that have since attracted the attention of many perceptive observers:

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233…etc.

Each number is the sum of the previous two. It turns out that the ratio of any adjacent numbers approximates (1 + 5(½))/2 = 1.618. It would be several hundred years before these sequences would be broadly recognized in nature. In floral arrangements, the lily has 3 petals; the yellow violet, 5; delphinium, 8; mayweed, 13; aster, 21; pyrethrum, 34; helenium, 55; and the Michaelmas daisy, 89 – all Fibonacci numbers!

In the study of phyllotaxis, the spiral arrangement of leaves around a plant’s stem, the leaves of the elm are arranged at 1/2 circumference; the beech and hazel, 1/3; the apricot and oak, 2/5; the pear and poplar, 3/8; the almond and pussy willow, 5/13; pines, 5/21 or 13/34; etc. In a review of 434 Angiospermae and 44 Gymnospermae, they all involve Fibonacci numbers! It turns out that this maximizes their exposure to sunlight and air without shading or crowding from other leaves.

In the study of seeds, the rows of bracts on pinecones are 8 and 13; pineapples, 8, 13, and 21; etc. The optimum divergence angle of 137.5o produces the best packing. That’s why you see Fibonacci spirals in the seed heads (sunflowers, etc.) But what’s really astonishing is that this peculiar sequence is far more pervasive than in just botany alone.

In art, it has long been recognized that there is a relationship known as “the Golden Rectangle”. This has the peculiar property in that if you remove a square, you still retain the same “ideal” rectangle in the remainder. You find this relationship exploited in the Parthenon in Greece, the Great Pyramid in Egypt, the United Nations Building, credit cards, playing cards, postcards, light switch plates, writing pads, 3×5, 5×8 index cards, etc. In classic art, Leonardo da Vinci, Van Gogh, Vermeer, John Singer Sargent, Monet, Whistler, Renoir, Mary Cassatt, Giotto, Durer and others relied on this “golden rectangle” in their designs.

In music, the various scales are all Fibonacci numbers: most beautiful chords found in music are the major and minor sixths.

Musicians like Bach, Beethoven, Bartok, et al., would divide musical time into periods based on the same “golden” proportions to determine the beginnings and endings of themes, moods, texture, etc.

This same “Golden Rectangle” is the basis for the “Golden Spiral,” which is the only spiral that does not alter its shape as it grows. This is often noticed in the chambered nautilus, but this “Golden Spiral” also appears in hurricanes, spiral seeds, ram’s horns, sea-horse tails, growing fern leaves, the DNA molecule, waves breaking on the beach, tornados, galaxies, the tail of a comet around the sun, whirlpools, seed patterns of sunflowers, daisies, and dandelions; the ears of all mammals; and, the cochlea of the human ear.

What is also surprising is that even the orbits of the planets reveal a relationship suggesting the Fibonacci numbers. Penetrate into nature wherever he [the scientist] may, thought has been there before him. Clearly, the same architect that designed the plants, designed the animals, and the universe itself! His fingerprints seem to define beauty itself. God is, indeed, a mathematician.

Leave a Reply