Fibonacci numbers and the Pascal Triangle

 

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Leonardo Da Vinci and Platonic Solids

        

The constructions of the Platonic solids are included in Book XIII of Euclid's Elements. Propositions 13 through 17 describe the construction of the tetrahedron, octahedron, cube, icosahedron, and dodecahedron in that order. Plato was mightily impressed by these five definite shapes that constitute the only perfectly symmetrical arrangements of a set of (non-planar) points in space, and late in life he expounded a complete "theory of everything" (in the treatise called Timaeus) based explicitly on these five solids. It's uncertain who first described all five of these shapes - it may have been the early Pythagoreans - but some sources (including Euclid) indicate that Theaetetus (another friend of Plato's) wrote the first complete account of the five regular solids. Plato conceived the four classical elements as atoms with the geometrical shapes of four of the five platonic solids . These are, of course, not the true shapes of atoms; but it turns out that they are some of the true shapes of packed atoms and molecules, namely crystals: The mineral salt sodium chloride occurs in cubic crystals, fluorite (calcium fluoride) in octahedra, and pyrite in dodecahedra. This concept linked fire with the tetrahedron, earth with the cube, air with the octahedron and water with the icosahedron. There was intuitive justification for these associations: the heat of fire feels sharp and stabbing (like little tetrahedra).


Air is made of the octahedron; its minuscule components are so smooth that one can barely feel it. Water, the icosahedron, flows out of one's hand when picked up, as if it is made of tiny little balls. By contrast, a highly un-spherical solid, the hexahedron (cube) represents earth. These clumsy little solids cause dirt to crumble and breaks when picked up, in stark difference to the smooth flow of water. Plato made the following identifications:

    tetrahedron = plasma ("fire")
    octahedron = gas ("air")
    icosahedron = liquid ("water")
    hexahedron = solid ("earth")

The fifth Platonic Solid, the dodecahedron, Plato obscurely remarks, ...the god used for arranging the constellations on the whole heaven (Timaeus 55).
Interestingly, almost 2000 years later, Johannes Kepler was similarly fascinated by these five Platonic shapes, and developed his own cosmology from them..
Newton's comments about the sides of light particles are very reminiscent of Plato's language in Timaeus. It's also interesting to compare some passages in Timaeus, such as
And so all these things were taken in hand, their natures being determined by necessity in the way we've described, by the craftsman of the most perfect and excellent among things that come to be...
with phrases in Newton's Principia, such as
...All the diversity of created things, each in its place and time, could only have arisen from the ideas and the will of a necessarily existing being... ...all phenomena may depend on certain forces by which the particles of bodies...either are impelled toward one another and cohere in regular figures, or are repelled from one another and recede... ...if anyone could work with perfect exactness, he would be the most perfect mechanic of all...



Leonardo da Vinci (1451-1519) was one of the greatest inventor-scientist of recorded history. His genius was unbounded by time and technology, and was driven by his insatiable curiosity, and his intuitive sense of the laws of nature. Da Vinci was dedicated to discovery of truth and the mysteries of nature, and his insightful contributions to science and technology were legendary. As the archetypical Renaissance man, Leonardo helped set an ignorant and superstitous world on a course of reason, science, learning, and tolerance. He was an internationally renowned inventor, scientists, engineer, architect, painter, sculptor, musician, mathematician, anatomist, astronomer, geologists, biologist, and philosopher in his time. He researched the natural code ( Da Vinci code ). He connected it with his knowledge about the Golden Section code and Platonic solids.



Leonardo was a great lover of geometry, and devoted much time to it starting in his early forties.
In 1496 the Franciscan monk, theorist, mathematician, and writer Fra Luca Pacioli (ca. 1445-ca. 1514) arrives in Milan, where he stays until 1499 and collaborates with Leonardo on studies of proportion, geometry, and mathematics.
Leonardo most outstanding polyhedral accomplishment is the illustrations for Luca Pacioli's book The Divine Proportion.
1498, February 9, Fra Luca Pacioli dedicates the treatise De divina proportione to Ludovico "Il Moro" (published in Venice, 1509.). (He coined the term "the divine proportion" for what is otherwise known as "the golden ratio".) The book was very influential in circulating information about geometry, and polyhedra in particular.



These are the first illustrations of polyhedra ever in the form of "solid edges." The solidity of the edges lets one easily see which edges belong to the front and which to the back, unlike simple line drawings where the front and back surfaces may be visually confused. Yet the hollow faces allow one to see through to the structure of the rear surface. This is a brilliant new form of geometric illustration, one worthy of Leonardo's genius for insightful graphic display of information.





There are roughly sixty similar illustrations in the book, mostly in pairs contrasting models with solid faces and models with this solid edge technique, such as this version of the dodecahedron:





  2001-2005 Radoslav Jovanovic                 created:  november, 2005.