Sources page biographical material

6.AT. POLYHEDRA AND TESSELLATIONS
These have been extensively studied, so I give only the major works. See 6.AA for nets of polyhedra.
6.AT.1. REGULAR POLYHEDRA
Gwen White. Antique Toys and Their Background. Batsford, 1971. (Reprinted by Chancellor Press, London, nd [c1989].) P. 9 has a sketch of "Ball of stone, Scotland", which seems to be tetrahedral and she says: "... one of the earliest toys known is a stone ball. Perhaps it is not a plaything, no one knows why it was made, but it is a convenient size to hold in the hand."

Dorothy N. Marshall. Carved stone balls. Proc. Soc. of Antiquaries of Scotland 108 (1976-7) 40-72. Survey of the Scottish neolithic carved stone balls. Lists 387 examples in 36 museums and private collections, mostly of 70mm diameter and mostly from eastern Scotland. Unfortunately Marshall is not interested in the geometry and doesn't clearly describe the patterns -- she describes balls with 3, 4, 5, 6, 7, 8, 9, 10 - 55 and 70 - 160 knobs, but emphasises the decorative styles. From the figures, there are clearly tetrahedral, cubical, dodecahedral(?) and cubo-octahedral shapes. Many are in the National Museum of Antiquities of Scotland (= Royal Museum, see below), but the catalogue uses a number of unexplained abbreviations of collections.

Royal Museum of Scotland, Queen Street, Edinburgh, has several dozen balls on display, showing cubical, tetrahedral, octahedral and dodecahedral symmetry, and one in the form of the dual of the pentagonal prism. [This museum has now moved to a new building beside its other site in Chambers Street and has been renamed the Museum of Scotland. When I visited in 1999, I was dismayed to find that only three of the carved stone balls were on display, in a dimly lit case and some distance behind the glass so that it was difficult to see them. Admittedly, the most famous example, the tetrahedral example with elaborate celtic decorative spirals, NMA AS10 from Glasshill, Towie, Aberdeenshire, is on display -- photo in [Jenni Calder; Museum of Scotland; NMS Publishing, 1998, p. 21]. They are on Level 0 in the section called In Touch with the Gods.]

Ashmolean Museum, Oxford, has six balls on display in case 13a of the John Evans Room. One is tetrahedral, three are cubical, one is dodecahedral and one is unclear.

Keith Critchlow. Time Stands Still -- New Light on Megalithic Science. Gordon Fraser, London, 1979. Chap. 7: Platonic spheres -- a millennium before Plato, pp. 131 149. He discusses and depicts Neolithic Scottish stones carved into rounded polyhedral shapes. All the regular polyhedra and the cubo octahedron occur. He is a bit vague on locations -- a map shows about 50 discovery sites and he indicates that some of these stones are in the Ashmolean Museum, Dundee City Museum and 'in Edinburgh'. Likewise, the dating is not clear -- he only says 'Neolithic' -- and there seem to be no references.

D. V. Clark. Symbols of Power at the Time of Stonehenge. National Museum of Antiquities, Edinburgh, 1985. Pp. 56-62 & 171. ??NYS -- cited by the Christie's Catalogue, below.

Robert Dixon. Mathographics. Blackwell, 1987, fig. 5.1B, p. 130, is a good photo of the Towie example.

Anna Ritchie. Scotland BC. HMSO, Edinburgh, for Scottish Development Department -- Historic Buildings and Monuments, 1988.

P. 8 has a colour photo of a neolithic cubical ball from the Dark Age fort of Dunadd, Argyll.

P. 14 has a colour photo of a cubical and a tetrahedral ball from Skara Brae, Orkney Islands, c-2800.

Simant Bostock of Glastonbury has made a facsimile of the Towie example, casts of which are available from Glastonbury Film Office, 3 Market Place, Glastonbury, Somerset, BA6 9HD; tel: [phone]. You can also contact him at 24 Northload Street, Glastonbury, Somerset, BA6 9JJ; tel: [phone] and he has a mail order catalogue. Since he worked from photographs, there are some slight differences from the original, and the facsimile is slightly larger.

Three examples of tetrahedral stone balls were in Christie's South Kensington antiquities sale of 12 Apr 2000, lots 124 and 125 (2 balls), p. 62, with colour photo of item 124 and the better example in lot 124 on p. 63. (Thanks to Christine Insley Green for a copy of the catalogue.) The descriptive text says 'their exact use is unclear'. Cites Clark, above.

The British Museum has icosahedral dice from Egypt, dated -700/-500.

Moritz Cantor. Vorlesungen über Geschichte der Mathematik. Vol. I, 4th ed., 1906, pp. 174 176. He feels all the regular solids were known to Pythagoras, with the tetrahedron, cube and octahedron having been known long before. Says to see various notices by Count Leopold Hugo in CR 77 for a bronze dodecahedron, a work by Conze on a Celtic bronze example and the paper of Lindemann, below, for a north Italian example. However, he says the dates of these are not determined and I think these are now all dated to later Roman times -- see below. He also notes that moderately regular dodecahedra and icosahedra occur in mineral deposits on Elba and in the Alps and wonders if Pythagoras could have known of these.

Thomas L. Heath. Note about Scholium 1 of Book XIII of Euclid. The Thirteen Books of Euclid's Elements; trans. & ed. by Thomas L. Heath; (1908?); 2nd ed., (1926); Dover, 3 vols., 1956, vol. 3, p. 438. "And it appears that dodecahedra have been found, of bronze or other material, which may belong to periods earlier than Pythagoras' time by some centuries (for references see Cantor's Geschichte der Mathematik I₃, pp. 175-6)."

HGM I 160 cites Hugo and Lindemann, dating the Monte Loffa example as -1000/-500.

HGM I 162 discusses the Scholium, giving it as: "the five so-called Platonic figures, which, however, do not belong to Plato, three of the five being due to the Pythagoreans, namely the cube, the pyramid, and the dodecahedron, while the octahedron and icosahedron are due to Theaetetus". He cites Heiberg's Euclid, vol. v., p. 654.

Thomas, SIHGM I 223 says "A number of objects of dodecahedral form have survived from pre-Pythagorean days." But he gives no details or references. Cf Heath's note to Euclid, above.

Plato ( 427/ 347). Timaeus. c-350. Page references are based on the 1578 edition of Plato which has been used for all later references: pp. 54-56. I use the version in: Edith Hamilton and Huntington Cairns, eds; The Collected Dialogues of Plato including the Letters; Bollingen Series LXXI, Pantheon Books (Random House), (1961), corrected 2nd ptg, 1963, which is the translation of Benjamin Jowett in his The Dialogues of Plato, OUP, (1872), 4th revised ed, 1953. Discusses the regular polyhedra, describing the construction of the tetrahedron, octahedron, icosahedron and cube from triangles since he views the equilateral triangle as made from six 30-60-90 triangles and the square made from eight 45-45-90 triangles. "There was yet a fifth combination which God used in the delineation of the universe with figures of animals." He then relates the first four to the elements: tetrahedron -- fire; octahedron -- air; icosahedron -- water; cube -- earth. [These associations are believed to derive from the Pythagoreans.] However, these associations contribute very little to the rest of the dialogue. The incidental appearance of the dodecahedron lends support to the belief that it was discovered or became known after the initial relation between regular polyhedra and the elements had been established and had to be added in some ad hoc manner. [It is believed that the later Pythagoreans related it to the universe as a whole.]

Scholium 1 of Book XIII of Euclid. Discussed in: The Thirteen Books of Euclid's Elements; op. cit. above, vol. 3, p. 438. Heath's discussion of the Scholia in vol. 1, pp. 64-74, indicates this may be c600. The Scholium asserts that only the tetrahedron, cube and dodecahedron were known to the Pythagoreans and that the other two were due to Theaetetus. Heath thinks the Pythagoreans had all five solids (cf his note to IV.10, vol. 2, pp. 97-100) and the Scholium is taken from Geminus, who may have been influenced by the fact that Theaetetus was the first to write about all five solids and hence the first to write much about the latter two polyhedra.

Stefano de'Stefani. Intorno un dodecaedro quasi regolare di pietra a facce pentagonali scolpite con cifre scoperto nelle antichissime capanne di pietra del Monte Loffa. Atti del Reale Istituto Veneto di Scienze e Lettere, (Ser. 6) 4 (1885) 1437-1459 + plate 18. Separately reprinted by G. Antonelli, Venezia, 1886, which has pp. 1-25 and Tavola 18. This describes perhaps the oldest known reasonably regular dodecahedron, in the Museo Civico di Storia Naturale, Palazzo Pompei, Largo Porta Vittoria 9, Verona, Italy, in the central case of Sala XIX. This is discussed by Herz-Fischler [op. cit. below, p. 61], I have been to see it and the Director, Dr. Alessandra Aspes, has kindly sent me a slide and a photocopy of this article.

The dodecahedron was discovered in 1886 at Monte Loffa, NE of Verona, and has been dated as far back as -10C, but is currently considered to be -3C or -2C [Herz Fischler, p. 61]. Dr. Aspes said the site was inhabited by tribes who had retreated into the mountains when the Romans came to the area, c-3C. These tribes were friendly with the Romans and were assimilated over a few centuries, so it is not possible to know if this object belongs to the pre-Roman culture or was due to Roman influence. She dates it as  4C/ 1C. The stone apparently was cut with a bronze saw and these existed before the Roman incursion (stated in Lindemann, below). It is clearly not perfectly regular -- some of the face angles appear to be 90^o and some edges are clearly much shorter than others. But it also seems clear that it is an attempt at a regular dodecahedron -- the faces are quite flat. Its faces are marked with holes and lines, but their meaning and the function of the object are unknown. de'Stefani conjectures it is a kind of die. Lindemann notes that the symbols are not Etruscan nor Greek, but eventually gets to an interpretation of them, which seems not too fanciful, using the values: 3, 6, 9, 10, 12, 15, 16, 20, 21, 24, 60, 300. (Are there any ancient Greek models of the regular polyhedra?) But see also the carved stone balls above.

About 90 examples of a Roman dodecahedron have been found at Roman sites, north of the Alps, from Britain to the Balkans, dating about 200-400. These are bronze and hollow, but also each face has a hole in it, almost always circular, and each corner has a knob at it, making it look like it could be used for Hamilton's Icosian game! The shape is quite precise.

The Society of Antiquaries, London, has the largest extant specimen, dug up on the north side of the Church of St. Mary, Carmarthen, in 1768 and presented to the Society about 1780. The edge length is 2 1/5 in (= 56mm) and, unusually, has plain faces -- almost all examples have some incised decoration on the faces. [Rupert Bruce-Mitford; The Society of Antiquaries of London Notes on Its History and Possessions; The Society, 1951, p. 75, with photograph as pl. XXIV (b) on p. 74.

The Gallo-Romeins Museum (Kielenstraat 15, B-3700 Tongeren, Belgium; Tel: [phone]) has an example which is the subject of an exhibition and they have produced facsimiles for sale. "The precise significance and exact use of this object have never been explained and remain a great mystery." Luc de Smet says the bronze facsimile is slightly smaller than the original and that the Museum also sells a tin and a bronzed version of the original size.

Other examples are in the Newcastle University Museum (only about half present) and the Hunt Museum, Limerick.

C. W. Ceram. Gods, Graves and Scholars. Knopf, New York, 1956, pp. 26-29. 2nd ed., Gollancz, London, 1971, pp. 24-25. In the first edition, he illustrated this as an example of the mysterious objects which archaeologists turn up and said that it had been described as a toy, a die, a model for teaching measurement of cylinders, a candleholder. His picture shows one opening as being like a key-hole. In the second edition, he added that he had over a hundred suggestions as to what it was for and thinks the most probable answer is that it was a musical instrument.

Jacques Haubrich has recently sent an example of a hollow cubical stone object with different size holes in the faces, apparently currently made in India, sold as a candleholder.

See Thomas, SIHGM I 216 225, for brief references by Philolaus, Aëtius, Plato, Iamblichus.

Euclid. Elements. c-300. Book 13, props. 13 18 and following text. (The Thirteen Books of Euclid's Elements, edited by Sir Thomas L. Heath. 2nd ed., (CUP, 1925??), Dover, vol. 3, pp. 467 511.) Constructs the 5 regular polyhedra in a sphere, compares them. In Prop. 18, he continues "I say next that no other figure, besides the said five figures, can be constructed which is contained by equilateral and equiangular figures equal to one another."

Leonardo Pisano, called Fibonacci (c1170->1240). La Practica Geometriae. 1221. As: La Practica Geometriae di Leonardo Pisano secondo la lezione del Codice Urbinate n^o. 202 della Biblioteca Vaticana. In: Scritti di Leonardo Pisano; vol. II, ed. and pub. by B. Boncompagni; Tipografia delle Scienze Matematiche e Fisiche, Rome, 1862, pp. 1 224. On p. 159, he says there are many polyhedra and mentions there are ones with 8, 12 and 20 faces which Euclid constructs in a sphere in his book XIIII. On pp. 161-162, he describes division in mean and extreme ratio and the construction of the regular pentagon in a circle, then says you can construct, in a sphere, a solid with 20 equilateral triangular faces or with 12 pentagonal faces. After some discussion, he says you can also construct solids with 4, 6, 8, 12, 20 faces, in a sphere. Division in mean and extreme ratio and the construction of the icosahedron are later covered in detail on pp. 196-202. His only drawings of solids are of cubes and pyramids.

Drawings of all the regular polyhedra are included in works, cited in 6.AA, 6.AT.2 and 6.AT.3, by della Francesca (c1480 & c1487), Pacioli (1494), Pacioli & da Vinci (1498), Dürer (1525), Jamnitzer (1568), and Kepler (1619).

F. Lindemann. Zur Geschichte der Polyeder und der Zahlzeichen. Sitzungsber. der math. phys. Classe k. b. Akademie der Wissenschaften zu München 26 (1896) 625 758 & plates I-IX. Discusses and illustrates many ancient polyhedra. Unfortunately, most of these are undated and/or without provenance. He generally dates them as -7C/5C.

A bronze rhombic triacontahedron, which he dates as first half of the first millennium AD.

Roman knobbed dodecahedra, which he describes as Celtic, going back to the La Tène period (Bronze Age) -- these are now dated to late Roman times. He lists 26 examples listed from the works of Conze and Hugo (cf Cantor, above).

A dodecahedral die; an irregular rhombi-cubo-octahedral die; a bronze dodecahedral die (having two 1s, three 2s, two 3s, one 4 and four 5s).

The Verona dodecahedron (from de'Stefani), which he dates as -1000/-500.

An enamelled icosahedron in Turin with Greek letters on the faces.

An octagonal bipyramid (elongated) from Meclo, South Tyrol, marked with a form of Roman numerals in a somewhat irregular order. It is dated to before the Barbarian migrations.

Three bronze cubo-octahedra.

He then does a long analysis of north-Italian culture and its relations to other cultures and of their number symbols, eventually obtaining an interpretation of the symbols on the Monte Loffa dodecahedron, which he then justifies further with Pythagorean number relations.

Roger Herz-Fischler. A Mathematical History of Division in Extreme and Mean Ratio. Wilfrid Laurier University Press, Waterloo, Ontario, 1987. Corrected and extended as: A Mathematical History of the Golden Number; Dover, 1998. P. 61 discusses the history of the dodecahedron and refers to the best articles on the history of polyhedra. Discusses the Verona dodecahedron, see above.

Judith V. Field. Kepler's Geometrical Cosmology. Athlone Press, London, 1988. This gives a good survey of the work of Kepler and his predecessors. In particular, Appendix 4: Kepler and the rhombic solids, pp. 201-219, is most informative. Kepler described most of his ideas several times and this book describes all of them and the relationships among the various versions.

The regular polyhedra in four dimensions were described by Ludwig Schläfli, c1850, but this was not recognised and in the 1880s, several authors rediscovered them.

H. S. M. Coxeter. Regular skew polyhedra. Proc. London Math. Soc. (2) 43 (1937) 33-62. ??NYS -- cited and discussed by Gott, qv.

J. R. Gott III. Pseudopolyhedrons. AMM 74:5 (May 1967) 497-504. Regular polyhedra have their sum of face angles at a vertex being less than 360^o and approximate to surfaces of constant positive curvature, while tessellations, with angle sum equal to 360^o, correspond to surfaces of zero curvature. The pseudopolyhedra have angle sum greater than 360^o and approximate to surfaces of negative curvature. There are seven regular pseudopolyhedra. Each is a periodic structure. He subsequently discovered that J. F. Petrie and Coxeter had discovered three of these in 1926 and had shown that they were the only examples satisfying an additional condition that arrangement of polygons at any vertex have rotational symmetry, and hence that the dihedral angles between adjacent faces are all equal. Coxeter later refers to these structures as regular honeycombs. Some of Gott's examples have some dihedral angles of 180^o. Two of these consist of two planes, with a regular replacement of pieces in the planes by pieces joining the two planes. The other five examples go to infinity in all directions and divide space into two congruent parts. He makes some remarks about extending this to general and Archimedean pseudopolyhedra.