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Anne Arundel 100
The Poetry of Math
by Katherine Socha, Assistant Professor of Mathematics
In medieval times, the four central subjects taught at universities were geometry, music, astronomy, and arithmetic. The harmonious interplay of these four themes (each embodying concepts of number and of symmetry) led to the classical liberal arts curriculum. Throughout the subsequent centuries, humans have made many efforts to blend music, art, physics, and mathematics, often in attempts to emphasize perceptions of the divine or the heavenly in these fields.
The human notion of the divine naturally is associated with perfection and completion. Throughout the ages, the struggle to embody or portray divinity and perfection – to bring the divine to the human world – has paralleled the struggle to understand or describe geometry and number. The subject of geometry is an exemplar of the human power of abstraction, an extraordinary capability: although much of the world is not perfect, humans can create perfection through thought. Geometry is one instance of Plato’s idea of creating the ideal form and ideal, perfected shapes. Humans envision the divine through envisioning perfect abstract forms. Mathematics, geometry in particular, is the study of the abstractions that are more perfect than the shapes in the real world around us. The sun is round but not perfectly round; in contemplating mathematics, we are able to study the world, perfected.
Perfection is smooth, endless, symmetric, and the same from all possible views. The same is true of a circle: smooth, endless, perfectly symmetric, and the same from all sides. Rotate a circle even a tiny amount and there is no apparent change because the rotated circle lies perfectly atop the original circle. Rotate a circle a lot and there is still no apparent change. Reflect a circle (make a mirror image) across any line through its center, and the reflection will lie perfectly atop the original circle. Nearly perfect, pure circles visible to the eye do appear in nature: in every lunar cycle, in the shape of rings made by raindrops on a pond, or in the iris of a beloved’s eye, and so the perfectly round shape is very special, lending its form to saints’ haloes, wedding rings, and the perfect symmetry of Dante’s rose of heaven, among others.
Historical attempts to describe the motions of planets and stars in the heavens relied on human notions of perfection and symmetry A religious-based line of thought suggested that the very skies, being that much closer to the divine, should exhibit perfect motion, meaning that the planets should move in perfect circles at constant speed. The geocentric (earth-at-the-center) model of the universe placed mankind and earth at the center of the universe, with stars and planets sweeping out perfect circles around us. Unfortunately for geocentrists, it was possible to observe that the planets do not always obey the beautiful requirements of perfect symmetry. Rebellious Mars wandered all around the night sky, defying the assumption of perfect circular motion and a constant (perfect) speed. Thus, an alternative approach that preserved circles and constant speed was developed: the Ptolemaic system used a combination of perfect circles to create a more complex description of the heavens, a description that attempted to capture the retrograde motion of Mars while preserving the Earth’s centrality to the universe. In later times, astronomers such as the great Copernicus and, later, Kepler recognized that the system is heliocentric (sun-centered) and the actual planet paths around the sun are elliptical. However, for mystical reasons, Kepler believed that the night sky contained symmetry beyond that of the perfect sphere or perfect circle. Kepler incorporated a collection of mathematical shapes called the Platonic solids into his own model of the heavens, nesting these beautifully symmetric three-dimensional solids in between layers of thickened spheres that contained the elliptical orbits of the planets. The Platonic solids are also known as the regular polyhedra or the regular solids.
The remarkable Platonic solids are only five in number: the four-faced tetrahedron; the six-faced cube; the eight-faced octahedron; the 12-faced dodecahedron; and the 20-faced icosahedron. Excepting the cube, which has square faces, and the dodecahedron, which has pentagonal faces, the other three Platonic solids have equilateral triangles for faces. The difference between a solid and a Platonic solid is the difference between some regularity and perfect regularity: a buckyball, most familiar to children in its soccer ball form, is nicely shaped but, being made up of both pentagons and hexagons, it is not a Platonic solid. Platonic solids are perfectly regular, by which we mean that every face has exactly the same shape, which is as symmetric as possible (pentagons, squares, or equilateral triangles), and every vertex has exactly the same number of edges coming out from it.
The Platonic solids are not as symmetrical as the perfectly symmetrical sphere, which is a three-dimensional cousin of the perfectly symmetrical two-dimensional circle. However, the Platonic solids have their own symmetry. Rotate a cube halfway around a diagonal line that passes through its center, and the rotated cube will coincide exactly with the original cube. Slice a plane through the exact center of the cube and the cube will appear as if in mirror image across the plane. Similar patterns of reflection and rotation may be discovered for all of the Platonic solids.
Other kinds of beautiful solids may be constructed, but only the tetrahedron, cube, octahedron, dodecahedron, and icosahedron are Platonic solids. Why are there only five such Platonic solids? There are certainly more than five regular polygons (two-dimensional shapes): equilateral triangle, square, pentagon, hexagon, heptagon, octagon, nonagon, decagon, and so on. Every positive natural number (greater than two) is the number of sides for some two-dimensional regular polygon. In fact, the two-dimensional regular polygons show the same sort of reflection and rotation symmetry that are part of the appeal of the Platonic (regular, three-dimensional) solids.
The fact that this infinite number of regular polygons in the plane does not translate to the same situation of infinitely many Platonic solids in three dimensions was a great surprise to mathematicians and philosophers, but it is an absolute fact. That mathematicians are able to prove this fact (that there are only five Platonic solids, not infinitely many) is an extraordinary accomplishment, a taming of infinity.
Humanity has always grasped at any instances of the divine, but perfection and infinity have long appeared to be the province of religion, philosophy, art, literature. The power, precision, and predictability of mathematics have offered all of us, poets, philosophers and dreamers alike, the opportunity to grasp our own bit of the divine through the perfect symmetry of geometric shapes.