Description and Requirements The Book Bibliography Syllabus Introduction The Great Pyramid Music of the Spheres Number Symbolism Polygons and Tilings The Platonic Solids Roman Architecture Number Symbolism in the Middle Ages The Wheel of Fortune Celestial Themes in Art Origins of Perspective What Shape Frame? Piero della Francesca Leonardo Façade measurement by Trigonometry Early Twentieth Century Art Dynamic symmetry & The Spiral The Geometric Art of M.C. Escher Later Twentieth Century Geometry Art Art and the Computer Chaos & Fractals

The PLATONIC SOLIDS "Let no one destitute of geometry enter my doors." Plato (c. 427  347 B.C.E.)
American Catalgo, p. 126, #21061. Fresco, Vatican, Stanza della Signurata, the Pope's Private Library We now move forward in time about 150 years, still staying in Greece, from Pythagoras to Plato, himself a Pythagorean. In our last unit we studied some polygons, and I said that one of these, the triangle, was thought by Plato to be the building block of the universe. He presented that idea and others about creation, such as the universe being created to resemble a geometric progression, in one of his books, the Timaeus. In the Timaeus, we'll see how Plato describes how triangles make up five solids, now called the Platonic Solids, and how these solids make up the four elements and heaven. We'll look at regular polyhedra in general, and see why only five are possible. Finally we'll see how the Platonic solids were used as art motifs even before Plato, how they were used later, and how the served to tie together three Renaissance mathematicians and artists, Piero della Francesca, Luca Pacioli, and Leonardo da Vinci.
Plato
Profile: Plato (c.427347 B.C.E.) was born to an aristocratic family in Athens. As a young man Plato had political ambitions, but he became disillusioned by the political leadership in Athens. He eventually became a disciple of Socrates, accepting his basic philosophy and dialectical style of debate, the pursuit of truth through questions, answers, and additional questions. Plato witnessed the death of Socrates at the hands of the Athenian democracy in 399 BC. In Raphael's School of Athens we see Socrates prone, with cup nearby. Plato's most prominent student was Aristotle, shown here with Plato in Raphael's School of Athens, Aristotle holiding his Ethics and Plato with his Timaeus. Plato's Academy In 387 BCE Plato founded an Academy in Athens, often described as the first university. It provided a comprehensive curriculum, including astronomy, biology, mathematics, political theory, and philosophy. Plato's final years were spent lecturing at his Academy and writing. He died at about the age of 80 in Athens in 348 or 347. Over the doors to his academy were the words meaning, "Let no one destitute of geometry enter my doors." Plato on Art and Geometry Although Plato loved geometry, he would not have been good at teaching a course in Art & Geometry because he had a low opinion of art. He taught that, since the world is a copy or image of the real, then a work of art is a copy of a copy, at third remove from reality. He writes in his Republic(p. 603), But on geometry he wrote in his Republic (p. 527), "[Geometry is] . . . persued for the sake of the knowledge of
what eternally exists, and not of what comes for a moment into existence,
and then perishes, ...
The Timaeus Plato left lots of writings. We've mentioned his Republic in our unit on number symbolism in which he gave the four cardinal virtues, but his love of geometry is especially evident in the Timaeus. Written towards the end of Plato's life, c. 355 BCE, the Timaeus describes a conversation between Socrates, Plato's teacher, Critias, Plato's great grandfather, Hermocrates, a Sicilian statesman and soldier, and Timaeus, Pythagorean, philosopher, scientist, general, contemporary of Plato, and the inventor of the pulley. He was the first to distinguish between harmonic, arithmetic, and geometric progressions. In this book, Timaeus does most the talking, with much homage to Pythagoras and echos of the harmony of the spheres, as he describes the geometric creation of the world. Music of the Spheres Plato, through Timaeus, says that the creator made the world soul out of various ingredients, and formed it into a long strip. The strip was then marked out into intervals.
They give the seven integers; 1, 2, 3, 4, 8, 9, 27. These contain the monad, source of all numbers, the first even and first odd, and their squares and cubes.
Plato's Lambda
These seven numbers can be arranged as two progressions
This is called Plato's Lambda, because it is shaped like the Greek letter lambda.
Divisions of the World Soul as Musical Intervals Relating this to music, if we start at low C and lay off these intervals, we get 4 octaves plus a sixth. It doesn't yet look like a musical scale. But Plato goes on to fill in each interval with an arithmetic mean and a harmonic mean. Taking the first interval, from 1 to 2, for example, Arithmetic mean = (1+2)/2 = 3/2 The Harmonic mean of two numbers is the reciprocal of the arithmetic mean of their reciprocals. For 1 and 2, the reciprocals are 1 and 1/2, whose arithmetic mean is 1+ 1/2 ÷ 2 or 3/4. Thus, Harmonic mean = 4/3 Thus we get the fourth or 4/3, and the fifth or 3/2, the same intervals found pleasing by the Pythagoreans. Further, they are made up of the first four numbers 1, 2, 3, 4 of the tetractys.
Filling in the Gaps He took the interval between the fourth and the fifth as a full tone. It is 3/2 ÷ 4/3 = 3/2 x 3/4 = 9/8 Plato then has his creator fill up the scale with intervals of 9/8, the tone. This leaves intervals of 256/243 as remainders, equal to the half tone. Thus Plato has constructed the scale from arithmetic calculations alone, and not by experimenting with stretched strings to find out what sounded best, as did the Pythagoreans. Project: Repeat Plato's calculations and see if you do indeed get a musical scale. Forming the Celestial Circles After marking the strip into these intervals, the creator then cut it lengthwise into two strips that are placed at an angle to each other and formed into circles. These correspond to the celestial equator and the ecliptic, the start of an armillary sphere.
Recall our quotation from Plato's Republic, where, in the Myth of Er he wrote, ". . . Upon each of its circles stood a siren who was carried round with its movements, uttering the concords of a single scale." [Republic p. 354] This is the origin of the expression, Music of the Spheres. The Elements The idea tthat all things are composed of four primal elements: earth, air, fire, and water, is attributed to Empedocles (circa 493433 BCE), Greek philosopher, statesman, and poet. He was born in Agrigentum (now Agrigento), Sicily, and was a disciple of Pythagoras and Parmenides. Remember the opposite forces, Yin and Yang, male and female, whose interaction created everything in the universe? Empedocles thought that active and opposing forces, love and hate, or affinity and antipathy, act upon these elements, combining and separating them into infinitely varied forms. He believed also that no change involving the creation of new matter is possible; only changes in the combinations of the four existing elements may occur. Empedocles died about 6 years before Plato was born. The Universe as a Geometric Progression Plato deduces the need for the four elements. Timaeus, 31B32C 1. First, we must have fire, to make the world visible, and earth to make it resistant to touch. These are the two extreme elements, fire belonging to heaven and earth to earth. He writes, . . . it is necessary that
nature should be visible and tangible ... 2. But two cannot hold together without a third as a bond. [like glue] . . . But it is impossible for two things to cohere without the intervention of a third ... 3. And the most perfect bond is the connued geometric proportion. ... [and] the most beautiful analogy is when in
three numbers, 4. But the primary bodies
are solids, and must be represented by solid numbers (cubes). But if the universe were to have no depth, one medium would suffice to bind all the natures it contains. But . . . the world should be a solid, and solids are never harmonized by one, but always by two mediums. Hence the Divinity placed water and air in the middle of fire and earth, fabricating them in the same ratio to each other; so that fire might be to air as air is to water and that water is to earth. Thus the ratio is constant between successive elements, giving a geometric progression. The Platonic Solids The Platonic Solids belong to the group of geometric figures called polyhedra. A polyhedron is a solid bounded by plane polygons. The polygons are called faces; they intersect in edges, the points where three or more edges intersect are called vertices. A regular polyhedron is one whose faces are identical regular polygons. Only five regular solids are possible cube tetrahedron octahedron icosahedron dodecahedron These have come to be known as the Platonic Solids The Elements Linked to the Platonic Solids Plato associates four of the Platonic Solid with the four elements. He writes, We must proceed to distribute the figures [the solids] we have just described between fire, earth, water, and air. . . Let us assign the cube to earth, for it is the most immobile of the four bodies and most retentive of shape the least mobile of the remaining figures (icosahedron) to waterthe most mobile (tetrahedron) to fire the intermediate (octahedron) to air Note that earth is associated with the cube, with its six square faces. This lent support to the notion of the foursquaredness of the earth. The Cosmos But there are five regular polyhedra and only four elements. Plato writes, "There still remained a fifth construction, which the god used for embroidering the constellations on the whole heaven." Plato's statement is vague, and he gives no further explanation. Later Greek philosophers assign the dodecahedron to the ether or heaven or the cosmos. The dodecahedron has 12 faces, and our number symbolism associates 12 with the zodiac. This might be Plato's meaning when he writes of "embroidering the constellations" on the dodecahedron. Note that the 12 faces of the dodecahedron are pentagons. Recall that the pentagon contains the golden ratio. Perhaps this has something to do with equating this figure with the cosmos. Other Polyhedra The Archimedian Solids
Slide 64: Archimedian Solids Other sets of solids can be obtained from the Platonic Solids. We can get a set by cutting off the corners of the Platonic solids and get truncated polyhedra. They are no longer regular; they are called semi regular; all faces are regular polygons, but there is more than one polygon in a particular solid, and all vertices are identical. These are also called the Archimedian Solids, named for Archimedes, (287212) the Greek mathematician who lived in Syracusa on the lower right corner of Sicily. MiniProject: Make some Archimedian Solids. Star Polyhedra
Slide 65: The Four KeplerPoinsot Solids
Slide 66: Engraving from Harmonices Mundi, 1619. The second obvious way to get another set of solids is to extend the faces of each to form a star, giving the socalled Star Polyhedra. Two star polyhedra were discovered by Poinsot in 1809. The others were discovered about 200 years before that by Johannes Kepler (15711630), the German astronomer and natural philosopher noted for formulating the three laws of planetary motion, now known as Kepler's laws, including the law that celestial bodies have elliptical, not circular orbits. MiniProject: Make some star polyhedra. Polyhedra in Art & Architecture Polyhedra are Nothing New Polyhedra have served as art motifs from prehistoric times right up to the present.
The Egyptians, of course, knew of the tetrahedron, but also the octahedron, and cube. And there are icosahedral dice from the Ptolomaic dynasty in the British Museum, London.
There were Neolithic solids found in Scotland, and excavations near Padova have unearthed an Etruscan dodecahedron, c. 500 BCE, probably used as a toy. Kepler
In 1596 Kepler published a tract called The Cosmic Mystery in which he envisioned the universe as consisting of nested Platonic Solids whose inscribed spheres determine the orbits of the planets, all enclosed in a sphere representing the outer heaven. Of course, his observations did not fit this scheme. We'll encounter Kepler again in our unit on Celestial Themes in Art. Polyhedra and Plagiarism in the Renaissance
This painting shows Fra Luca Pacioli and his student, Guidobaldo, Duke of Urbino. In the upper left is a rhombicuboctahedron, and on the table is a dodecahedron on top of a copy of Euclid's Elements.
Luca Pacioli wrote a book called Da Divina Proportione (1509) which contained a section on the Platonic Solids and other solids, which has 60 plates of solids by none other than his student Leonardo da Vinci. We'll tell the whole story of how this material was stolen from Luca's teacher Piero della Francesca in our unit on Polyhedra and Plagiarism in the Renaissance. Platonic Solids as Art Motifs
Albrech Durer (14711528) had a keen interest in geometry, as we'll see in a later unit. This famous engraving shows an irregular polyhedron, as well as a sphere, a magic square, and compasses. People who have analyzed this polyhedron have decided that its actually a cube with opposite corners cut off.
This ball has hexagons interspersed with pentagons. Polyhedra in Art in the Twentieth Century
The Swiss artist Alberto Giacometti (19011966) often included polyhedra in his earlier surrealist works such as these two drawings and a sculpture.
We'll talk about M.C. Escher (19021972) in detail when we get to the 20th Century, but lets just peek at his 1948 engraving, Stars. Note the similarity between this polyhedron and Leonardo's illustrations for Pacioli's book.
Escher made a set of nested Platonic Solids. When he moved to a new studio he have away most of his belongings but took his beloved model. Other Twentieth Century artists using polyhedra include Harriet Brisson, Paul Calter, and Lucio Saffaro.
Project: Make a work of art featuring polyhedra. Summary So we've seen the origins of the Platonic Solids, starting even before Plato, and have briefly traced the influence of the polygons in art right up to the present. We've also had a first look at some subjects we'll look at in more detail later. For mathematical topics, we've briefly looked at sequences and series and the geometry of the polyhedra.
© Paul Calter, 1998. All Rights Reserved. Dartmouth College
