Pythagoreanism and Theories of the Structure of Matter

| Pythagoreanism and Theories Nathan Feifer, California State University at San Francisco, San Francisco, Calif. 94132 Geometry has long molded concepts in chemistry. We speak of spherically shaped s-orbitals, tetrahedral carbon atoms, trigonal planar boron fluoride molecules, octahedral ferricyanide ions, and face-centered cubic lattices of sodium chloride crystals. These present-day configurations have roots in the geometry of antiquity. a 1 4etrahecrox à. ION # f €~ i H | ai TT n°7; Pythagoras (fl. latter half —6th century) and his followers made up a religious brotherhood that believed in an arithmetically and geometrically organized universe, ruled by simple number ratios and revealed in perfect figures such as the sphere, circle, and cube. In Pythagorean cosmology, stars were small spheres attached to a large, hollow, rotating sphere and carried along by it. The spherical TA Earth remained stationary in this celestial USFnrtainacvti.nsec,o,6(E-fin1g0).andtgoe0hrmetrymfdtoedvherlesoTpolehycsoer.d-ra SdotNshrufrocteufÈsngaen ces:fbehao0rsnwiesa Tmhatreh-.ebyrg eairntrdh,cryestralnog. atnerd, opmfent oftirfse, ddfeoalrimzesdsearivsemd omisacusrfidon Pythagoreanism C1h4e97m(is34t)r,ypbryemdiecytred psaolpnheidsxtiEtKhmoe ple aiseorsvedthcoenscepts oclean 1 9635c Fmater. Calif.).amsnodnyPlat mcorem respondin FENER,N, vus lay H sphere's center (8). This arrangement explained the apparent circular paths of stars, the position of the North Star, the fixed formation of the constellations, and the circumpolar motion of northern stars, because the large celestial sphere rotated uniformly and carried the fixed stars in precisely circular Figure 1. Regular polyhedra and corresponding Empedoclean elements orbits. Pythagoras himself is credited with discovering the relationship between string length and musical intervals. He showed that the sound made by two vibrating strings differs by an octave if one is twice the length of the other. Musical harmony was even compatible with the features of a cube, considered an especially beautiful figure having all solid angles identical and six identical faces, each being a perfect square with all sides and angles equal. The ratio of the cube’s features—12 edges to 8 points to 6 faces—was also mathematically beautiful. The harmony of this ratio was also found in music. Three strings differing in lengths of 12 to 8 to 6 produced a harmonious chord when plucked (8). This Pythagorean view of a mathematically oriented world has influenced man’s attempts to explain nature throughout the ages. Plato (—428 to —347) was a mathematical idealist who saw much beauty in the numbers of the Pythagoreans. He adopted the four-element (fire, earth, air, water) theory of Empedocles (fl. about —450)—that objects in this world consist of varying proportions of two or more of those four elements —and related it to the regular solids. Plato's idealism meant that he believed this world is a reflection of ideal forms and so he tried to relate his idealism to objects of this world. 6 CHEMISTRY VOL 47 NO From among the five regular solids (Theaitetos, a contemporary of Plato, proved there could be only five), Plato selected four to represent the four Empedeclean elements —the tetrahedron for fire, octahedron for air, icosahedron for water, and the cube fer earth (Figure 1). He argued that fire's penetrability and painful effects were a consequence of the tetrahedron's sharp angles. Earth's stability and solidity were ascribed to its cubic form, and the motion of water was related to the icosahedron. Air was represented by the octahedron because it was a figure more penetrating than the icosahedron but less penetrating than the tetrahedron (6). In Plato's theory of matter, the elements were related to triangles: In the first place, then, as is evident to all, fire and earth and water and air are bodies. And every sort of body possesses solidity, and every solid must necessarily be contained in planes; and every plane rectilinear figure is composed of triangles ... (6). The most beautiful triangular form was the right angle triangle, the basic unit from which the four figures of the elements could be constructed. Plato says single particles, which he calls the original elements or seeds

f T he a PEt CLES EL ELE OS Ol BT EE Tes Via LISCI LAT À * Ca sert ro Plato's geometric figures arc a monument to man's mathematical approach to understanding nature. It was echoed through the ages by such giants as Johannes Kepler (1571-1630) and Galileo Galilei (1564-1642). Kepler’s theory of the solar was system based upon the mathematical relationships of the five regular solids. Johannes Kepler also worked out a theory to explain why snowflakes should be hexagonal and sym- Figure 2, Water becomes two parts air and one part fire in Empedocles’ system metrical (Chemistry, Sept. 1971, page 19). Galileo said, “we cannot understand it [nature] if we do not learn the language and grasp the svmbols in which it is written. This book is written in the mathematical language, and the symbols are triangles, circles, and other geometric figures.” Certainly, to the modern scientist, mathematics is an indispensable tool. But there are differences between the Platonic and the modern role of mathematics in science. To Plato, ideal forms were more important than real things. He and the Greek civilization of antiquity left to succeeding generations the of any of the four solids, are so small they cannot be seen by us. In aggregates, however, they can be seen. Changes occurred in nature when elements acted on cach other. The role of motion is very important: Earth, when meeting with fire and disideas from which today’s efforts to explain natural phenomena evolved. The Platonic theory in its specifics did not survive, but the Pythagorean principles of mathematical harmony and symmetry in nature underlying Plato's ideas have had a pronounced influence on modern science. Investigators of the properties of crystals solved by its sharpness, whether the dishave found solution takes place in the fire itself or forms very useful. For perhaps in some mass of air or water, is stances borne hither and thither until its parts, meeting together and mutually harmonsium bromide (KBr), lithium iodide (Lil), magnesium oxide (MgO), and a host of other izing, again become earth; for they can solids have a cubic form, externally and inas Plato's sodium and the Pythagorean instance, chloride such sub- (NaCl), potasnever take any other form. But water, ternallv. when divided by fire or by air, on rehedra in the solid crystalline state are alum forming, may become one part fire and [KAI(SO,)-12H.O], two parts air (Figure 2); and a single (SbCl,), and antimony trifluoride (SbF.), for volume of air divided becomes example, which occur as regular octahedra. two of fire (6). Other examples of regular antimony _ polytrichloride Copper(I) chloride (Cu.Cl.) exists either as The evaporation of water could be explained a cube or as a regular tetrahedron in the In terms of its geometrical components (9): l part — 2 parts + 1part water air fire l icosa- 2 octa- 1 tetrahedron | 40 right triangles hedra | — 32 right triangles hedron | + 8 right triangles Figure 3. Forms made by packing spheres In the method of Robert Hooke MARCH a 1974 CHEMISTRY

Figure 4. Crystals of calcite, left and center, shown with Haüy’s idea of the relation of natural faces to p ——, Figure 6. Setup of Friedrich and Knipping to examine crystals with X-rays the stack of tiny rhombohedral units crystalline state, Dodecahedra and icosahedra are virtually unknown in the world of over to the fundamental of crystallography that crystals exhibit some form of external crystals. regularity resulting from a regular internal Although the number of crystalline substances occurring as Plato’s regular polyhedra is quite small, there is a preponderance of symmetrically shaped crystals in nature that are not regular polyhedra. But Platonic and Pythagorean idealism carry arrangement of component atoms or ions. Robert Hooke in 1665 (2 millennia after Plato) proposed a theory of crystal structure embodying such notions of regularity. He noted the regular cubical shapes found in such solids as rock salt, NaCl, and explained their formation by assuming that crvstals were built of uniform, spherical particles arranged in a regular pattern; he did not select a miniature cubical particle as his basic building block. He demonstrated the reasonableness of his model by building a cube of spherically shaped lead shot (3). He also showed how other crystalline forms could be obtained by varying the method of packing spheres (Figure 3). Using the perfect sphere as the basic unit of structure would have been approved by the Pythagoreans. Hooke was proposing a simple explanation, an idealized model consistent with his observed facts. The difference in approach between Hooke's selection of the sphere as the ultimate particle in the sodium chloride crystal and Plato’s choice of the cube as the ultimate particle in all solids is indicative of the two different intellectual climates in which each lived. Plato was a mathematical idealist, Hooke lived in an era where the role of observation had grown much stronger. After observing the macroscopic cubic structure of rock salt, Hooke proposed a model to fit his idea of the facts. Figure 5. Bravals lattices 8 CHEMISTRY

Sir William H. Bragg and his son, William L. Bragg, developed an improved technique for studying effects of crystals on X-rays (2). They used the crystal as a reflector of a beam of X-rays and noted how a reflected beam varied in intensity as the angle between the incident beam and the crystal surface varied (Figure 7). From their data they concluded that certain crystals, such as diamond, were composed of a lattice network of carbon atoms (Figure 8). The Braggs’ basic assumption was that the Figure 7. Bragg equipment for studying crystals with X-rays structural units of crystals are spheres arranged in symmetrical geometric patterns, a very Pythagorean opinion. Modern theory suggests that geometric arrangement of ions in an ionic crystalline solid depends mainly From 1784 to 1822, René Haiiy addressed on the charges of the ions and the ratio of himself to the study of crystal structure (7). the ionic radii. In nonionic crystals the geo- He introduced the experimental technique of cleaving crystals repeatedly until no change metric structure may depend on in shape resulted in the pieces upon further cleavage. His extensive observations indiing, or van der Waals forces. The mobility of directed localized covalent bonding or hydrogen bondthe electrons in metals, and the nondireccrystalline substances were tional nature of the bonding enable atoms to aggregates of one of three basic polyhedral assume close-packed arrangements. Thus, a building blocks—the rhombohedron (of which the cube is an example), the tetragreat variety of crystalline forms exist; hedron, and the triangular prism. The calcite known (3). crystal (Figure 4, left) was postulated by Haüy as an aggregate of cubic building than crystallography (/), the Platonic figures cated that all fact, about 20000 crystalline species in are In the realm of molecular geometry, rather blocks (Figure 4, right). In 1848, Auguste Bravais modified Haüy’s have become the architectural models of a model by proposing an open lattice of spheraccepted as the structural configuration of ical units; each sphere occupied the center of the methane (CH,) modecule and the carbon each polyhedron in Haüy's model (Figure 5). tetrachloride (CCl,) molecule, for example. number of molecules (3). The tetrahedron is Over a hundred crystalline varieties were The octahedral shape has been assigned to recognized and organized into 14 basic space the cobaltic lattices (7). and the sulfur hexafluoride molecule, SF,, Evidence to support the Bravais model was provided by Max von Laue in 1912. He first assumed that the atoms in a cubic as well as hexammine to such ion, ions Co(NH,),3+, as PbCl,?- and FeCN,?*-. Elementary boron and carborane (B;,5C:H;:) are recognized icosahedral struc- Ciathrates are inclusion crystal such as NaCl were spheres arranged tures. of compounds in which an in a lattice framework just as postulated by water are classed as dodecahedrons. Despite Bravais. He predicted that passing a beam of X-rays through these crystals, and allowing the emergent rays to fall on a photographic atom or molecule is trapped in a crystalline modern geometry of matter, the number of Some of the hydrate clathrates these examples of regular polyhedra in the nonregular polyhedra serving as molecular plate, would result in a symmetrical diffracmodels far exceeds the regular polyhedra, tion Walter and the modern scientist is not constrained Friedrich and Paul Knipping performed a to limit his theories to Platonic polyhedra. series pattern. of The following experiments by year, sending X-rays William Hyde Wollaston in 1808 main- (Figure 6) through a variety of crystals and they obtained the patterns which had been when he wrote: “If we suppose the limit to predicted by von Laue (2). the approach of particles to be the same in tained the Pythagorean definition of beauty MARCH 1974 CHEMISTRY cage

their forms are geometrically symmetrical. Thus, in the BF, molecule the ratio of 1 to 3 gives rise to a trigonal planar figure, as Wollaston predicted. The NH, molecule has SA \ , ZA —"\ / Figure 8. Structure of sodium chloride (NaCl) crystal composed of sodium and chloride ions (smaller balls represent sodium ions; larger ones, chloride ions) all directions, and hence their virtual extent to be spherical (which is the most simple hypothesis), when different sorts combine . .. in the proportion of two to one, the two particles will naturally arrange themselves at opposite poles. ... If there be three, they might be arranged with regularity, at the angles of an equilateral triangle ... when the number of one set of particles exceeds in the proportion of four to one ... a stable equilibrium may take place if the four particles are situated at the angles of four equilateral triangles composing a regular tetrahedron” (1). On extending Wollaston's reasoning, we find that the geometric figure formed by the symmetrical and stable arrangement of one species of spherical particles around a single centrally placed spherical particle would depend mainly upon the ratio between the two species of particles, assuming the deranging force of other adjacent particles is negligible. Four such combinations could and do give rise to regular polyhedra: A ratio of 1:4 would give a tetrahedron with one central particle and four particles at the vertices (CH,, CCl,); 1:6, the octahedron (S7,, PCl,—); 1:8, the cube: (CsCl); 1:12 the icosahedron (MoAI,.). Many other ratios are possible that are not expressed as regular polyhedra, although a ratio of one N atom to three H atoms, but it forms a trigonal pyramid. A ratio of 1:5 is found in the trigonal bipyramid molecule of PCI.. Like the overwhelming majority of molecules or complex ions that have been investigated, BF,, NH,, and PCI, have geometrical symmetry but they are not the Platonic regular polyhedra. To determine the possible geometrical form of a molecule, one must today consider the electronic configuration of the central atom in a molecule and that of the peripheral atoms that surround it as well as the number of electron pairs that can form. Directed valence, bond distances between atomic nuclei, internal motion in molecules, such as rotation about covalent bonds, and repulsions among electron pairs, bonding and nonbonding in the valence shell of the central atom also must be considered. Consequently, it turns out that conditions for the formation of regular polyhedral molecules are very unfavorable. Thus, the kind of mathematical harmony and symmetry predicted by Pythagoras and postulated by Plato clearly does not govern the geometry of all matter. Nevertheless, their idealized forms and their notions of symmetry served as simple and convenient models from which our more complex and sophisticated theories were developed. 1 Suggested Reading (1) Benfey, T., “Geometry and Chemical Bonding,” Chemistry, 1967, 40 (5), 20-6. (2) Bragg, W., “Concerning the Nature of Things,” 1924, Harper € Brothers, New York, N.Y.; Bragg, L., "The Start of X-ray Analysis,” Chemistry, 1967, 40 (11), 8. (3) Bunn, C., “Crystals: Their Role in Nature and in Science,” 1964, Academic Press, New York, N.Y. (4) Holden, A., Singer, P., “Crystals and Crystal Growing,” 1960, Anchor Books, Garden City, New York, N.Y. (5) Pauling, L., Hayward, R., “The Architecture of rt ind] 1964, W. H. Freeman, San Francisco, alif. NATHAN FEIFER, associate professor in the department of interdisciplinary physical sciences, has undertaken field studies in the history of science and has done research on transfer inks. 10 CHEMISTRY VOL 47 NO 3 (6) Plato, "Timaeus,"” Benjamin Jowett, trans., 1949, Liberal Arts Press, New York, N.Y. (7) Ryba, E., “The History of Crystallography,” Earth Min. Sci., 1968, 38 (5), 49-54. (8) Sarton, G., “A History of Science: Ancient Science through the Golden Age of Greece,” 1952, John Wiley and Sons, New York, N.Y. (9) Toulmin, S., Goodfield, J., "The Architecture of Matter,” 1962, Harper and Row, New York, N.Y.