Sources page biographical material

6.AT.3. ARCHIMEDEAN POLYHEDRA
Archimedes discovered the Archimedean solids, -3C, but his work is lost. Heron quotes some of it and Pappus summarises it. See HGM II 98-101.

Hero of Alexandria (c150). Definitiones. IN: Heronis Alexandrini Opera quae supersunt omnia; Vol. IV, Heronis Definitiones Cum Variis Collectionibus Heronis Quae Feruntur Geometrica, ed. by J. L. Heiberg, Teubner, 1912, pp. 64-67. Heath, HGM I 294-295 has a translation, but it doesn't give the complete text which seems open to two interpretations. The German goes: Archimedes aber sagt, es gebe in ganzen dreizehn Körper, die in einer Kugel eingeschreiben werden können, indem er ausser den genannten fünf noch acht hinzufügt; von diesen habe auch Platon das Tessareskaidekaeder gekannt, dies aber sei ein zweifaches, das eine aus acht Dreiecken und sechs Quadraten zusammengesetzt, aus Erde und Luft, welches auch einige von den Alten gekannt hätten, das andere umgekehrt aus acht Quadraten und sechs Dreiecken, welches schwieriger zu sein scheint. My translation: But Archimedes said, there are in total 13 bodies, which can be inscribed in a sphere, as he added eight beyond the named five [regular solids, which he had just defined]; but of these Plato knew the 14-hedron, however this is a double, one is composed of eight triangles and six squares, from Earth and Air, which some of the ancient also knew, the other conversely [is composed] of eight squares and six triangles, which seems to be more difficult.

Note that Hero has got the numbers wrong - Archimedes found 13 more than the 5 regular solids. Secondly, the 'more difficult' solid does not exist! Heath notes this and suggests that either the truncated cube or the truncated octahedron was intended. The question of interpretation arises at the first semicolon -- is this continuing the statement of Archimedes or is Hero commenting on Archimedes' results? Heath seems to say Archimedes is making the attribution to Plato, but see below. MacKinnon, below, seems to be say this is being made by Hero. Heath's discussion on HGM II 100 says "We have seen that, according to Heron, two of the semi-regular [i.e. Archimedean] solids had already been discovered by Plato" undoubtedly using the method of truncation. However, I don't see that Heron is saying that Plato discovered the cubo-octahedron and the other solid, only that he knew it. Mackinnon says "Plato is said by Heron to have discovered the cuboctahedron by making a model of it from a net." But I don't see that Heron says this.

Pappus. Collection. c290. Vol. 19. In: SIHGM II 194 199. Describes the 13 Archimedean solids. "..., but also the solids, thirteen in number, which were discovered by Archimedes and are contained by equilateral and equiangular, but not similar, polygons." He then describes each one. Pappus' work has survived in a single MS (Vat. gr. 218) of the 10C in the Vatican and was not copied until 1550, but see Mackinnon, pp. 175-177, on whether it had been seen by Piero. For the history of this MS, see also: Noel M. Swerdlow; The recovery of the exact sciences of antiquity: mathematics, astronomy, geography; IN: Anthony Grafton, ed.; Rome Reborn The Vatican Library and Renaissance Culture Catalog of an exhibition at the Library of Congress, Washington, D.C., Jan. 6   Apr. 30, 1993; Library of Congress, Washington & Yale University Press, New Haven & London; in association with the Biblioteca Apostolica Vaticana; 1993; pp. 137-139. [This exhibition is on-line at www.ibiblio.org/expo/vatican.exhibit/vatican.exhibit.html.]

R. Ripley. Believe It Or Not. 18th series, Pocket Books, NY, 1971. P. 116 asserts the Romans used dice in the shape of cubo octahedra.

The British Museum, Room 72, Case 9, has two Roman cubo-octahedral dice on display.

F. Lindemann, op. cit. in 6.AT.1, 1896, describes and illustrates an antique rhombic triacontahedron, possibly a die, possibly from the middle of the Byzantine era.

Nick Mackinnon. The portrait of Fra Luca Pacioli. MG 77 (No. 479) (Jul 1993) plates 1-4 & pp. 129-219. Discusses the various early authors, but has mistakes.

della Francesca. Trattato. c1480. Ff. 105r - 117v (224-250) treats solid bodies, discussing all the regular polyhedra, with figures, though Arrighi gives only a projection of the octahedron. Discusses and gives good diagrams of the truncated tetrahedron and cubo octahedron, apparently the first drawings of any Archimedean polyhedra. Jayawardene refers to the cubo-octahedron as a truncated cube.

Davis notes that Pacioli's Summa, Part II, ff. 68v - 73v, prob. 1-56, are essentially identical to della Francesca's Trattato, ff. 105r - 120r.

Piero della Francesca. Libellus de Quinque Corporibus Regularibus. c1487 [Davis, p. 44, dates it to 1482-1492]. Piero would have written this in Italian and it is believed to have been translated into Latin by Matteo da Borgo [Davis, p. 54], who improved the style. First post-classical discussion of the Archimedean polyhedra, but it was not published until an Italian translation (probably by Pacioli) was printed in Pacioli & da Vinci, qv, in 1509, as: Libellus in tres partiales tractatus divisus quae corpori regularium e depēdentiū actine perscrutatiōis ..., ff. 1 27. A Latin version was discovered by J. Dennistoun, c1850, and rediscovered by Max Jordan, 1880, in the Urbino manuscripts in the Vatican -- MS Vat.Urb.lat. 632; the Duke of Urbino was a patron of Piero and in the MS, Piero asks that it be placed by his De Prospectiva Pingendi in the Duke's library. This was published by Girolamo Mancini in: L'opera "De corporibus regularibus" di Pietro Franceschi detto della Francesca usurpata da Fra Luca Pacioli, Memorie della Reale Accademia dei Lincei, Classe di Scienze Morali, Storiche e Filologiche (5) 14:7B (1915) 441-580 & 8 plates, also separately published by Tipografia della Reale Accademia dei Lincei, Rome, 1916. Davis identifies 139 problems in this, of which 85 (= 61%) are taken from the Trattato. There is debate as to how much of this work is due to Piero and how much to Pacioli. The Latin text differs a bit from the Italian. See the works of Taylor and Davis in Section 1 under Pacioli and the discussion on della Francesca's Trattato and Pacioli's Summa in the common references.

He describes a sphere divided into 6 zones and 12 sectors. Mackinnon says Piero describes seven of the Archimedean polyhedra, but without pictures, namely: cuboctahedron, truncated tetrahedron, truncated cube, truncated octahedron, truncated dodecahedron, truncated icosahedron, rhombi-cuboctahedron. Field, op. cit. in 6.AT.1, p. 107, says Piero gives six of the Archimedean polyhedra. In recent lectures Field has given a table showing which Archimedean polyhedra appear in Piero, Pacioli, Dürer and Barbaro and this lists just the first six of the above as being in Piero. I find just the five truncated regular polyhedra -- see above for the cubo-octahedron -- and there is an excellent picture of the truncated tetrahedron on f. 22v of the printed version. Mancini gives different diagrams than in the 1509 printed version, including clear pictures of the truncated icosahedron and the truncated dodecahedron. della Francesca clearly has the general idea of truncation. An internet biographical piece, apparently by, or taken from, J. V. Field, (http://www history.mcs.st andrews.ac.uk/history/Mathematicians.Francesca.html), shows that the counting is confused by the presence of the cubo-octahedron in the Trattato but not in the Libellus. So della Francesca rediscovered six Archimedean polyhedra, but only five appear in the Libellus. The work of Pappus was not known at this time.

Pacioli. Summa. 1494.

f. 4. Brief descriptions of the cubo-octahedron, truncated tetrahedron, icosidodecahedron, truncated icosahedron. No drawings.

Part II, ff. 68v - 72r, sections 2 (unlabelled) - 35. Discussion and some crude drawings of the regular polyhedra, the truncated tetrahedron and the cubo-octahedron. Mackinnon says these are the first printed illustrations of any Archimedean polyhedra. Davis notes that Part II, ff. 68v - 73v, prob. 1-56, are essentially identical to della Francesca's Trattato, ff. 105r - 120r.

Jacopo de'Barbari or Leonardo da Vinci. Portrait of Fra Luca Pacioli. 1495. In the Museo Nazionale di Capodimonte, Naples. The upper left shows a glass rhombi-cuboctahedron half filled with water. Discussed by Mackinnon, with colour reproduction on the cover. Colour reproduction in Pacioli, Summa, 1994 reprint supplement.

Luca Pacioli & Leonardo da Vinci. Untitled MS of 1498, beginning: Tavola dela presente opera e utilissimo compendio detto dela divina proportione dele mathematici discipline e lecto -- generally called De divina proportione. Three copies of this MS were made. One is in the Civic Library of Geneva, one is the Biblioteca Ambrosiana in Milan and the third is lost. Three modern versions of this exist.

Transcription published as Fontes Ambrosiani XXXI, Bibliothecae Ambrosianae, Milan, 1956. This was sponsored by Mediobanca as a private edition. There is a copy at University College London.

Colour facsimile of the Milan copy, Silvana Editoriale, Milano, (1982), 2nd ptg, 1986. With a separate booklet giving bibliographical details and an Introduzione di Augusto Marinoni, 20pp + covers. The booklet indicates this is Fontes Ambrosiani LXXII.

Printed version: [De] Divina proportione Opera a tutti glingegni perspicaci e curiosi necessaria Ove ciascun studioso di Philosophia: Prospectiva Pictura Sculptura: Architectura: Musica: e altre Mathematice: suavissima: sottile: e admirabile doctrina consequira: e delectarassi: cōvarie questione de secretissima scientia. Ill. by Leonardo da Vinci. Paganino de Paganini, Brescia, 1509. Facsimile in series Fontes Ambrosiana, no. XXXI, Milan, 1956; also by Editrice Dominioni, Maslianico (Como), 1967. (On f. 23r, the date of completion of the original part is printed as 1497, but both MSS have 1498.)

The printed version was assembled from three codices dating from 1497 1498 and contains the above MS with several additional items. However, the diagrams in the text are simplified and the plates are in a different order. The MS has 60 coloured plates, double sided; the printed version has 59 B&W plates, single sided. There are errors of pagination and plate numbering in both versions. On f. 3 of the printed version is a list of plates and one sees that plate LXI should be numbered LVIIII and that plates LX, LXI were omitted and were to have been a hexagonal pyramid in solid and framework views (the framework view is in the MS, but the solid pyramid is not).

NOTE. Simon Finch's Catalogue 48, item 4, describes the copy that was in the Honeyman Collection and says it has 59 printed plates of geometric figures and is unique in having two contemporary additional MS plates showing the hexagonal pyramid (numbered LX and LXI), which are given in the list of plates, but which do not appear in any other known copy. It seems that these figures were overlooked in printing and that the owner of the Honeyman copy decided to make his own versions, or, more likely, got someone to make versions in the original style. There is a framework hexagonal pyramid in the MS, and this makes it seem likely that these figures had been prepared and were omitted in printing -- indeed the Honeyman leaves could be the overlooked drawings. That leaves the question of whether there was a solid hexagonal pyramid in the MS?

Most pictures come in pairs -- a solid figure and then a framework figure. There are the five regular polyhedra, the following six Archimedean polyhedra: truncated tetrahedron, cubo-octahedron, truncated octahedron, truncated icosahedron, icosi-dodecahedron, rhombi-cubo-octahedron and also the stella octangula. There are raised or elevated versions of the tetrahedron, cube, cubo-octahedron, icosahedron, dodecahedron, icosi-dodecahedron, rhombi-cubo-octahedron. Also triangular, square, pentagonal and hexagonal prisms and tall triangular, square and pentagonal pyramids. Also a triangular pyramid not quite regular and a sphere divided into 12 sectors and 6 zones. There are also a solid sphere, a solid cylinder, a solid cone and a framework hexagonal pyramid (the last is not in the printed version). Mackinnon says they give the same seven Archimedean polyhedra as Piero, but Piero gives five or six and Pacioli & da Vinci gives six, with only four common polyhedra. Pacioli & da Vinci assert that the rhombi-cuboctahedron arises by truncating a cuboctahedron, but this is not exactly correct.

Part of the printed version is Libellus in tres partiales tractatus divisus quae corpori regularium e depēdentiū actine perscrutatiōis ..., which is an Italian translation (probably by Luca Pacioli) of Piero della Francesca's Libellus de quinque corporibus regularibus. There is debate as to whether this was actually written by Pacioli or whether Pacioli plagiarized it and whether it actually appeared in the 1509 printing or was added to a later reprinting, etc.

Davis [p. 65] says the drawings were made from models prepared by Da Vinci. Davis [p. 74] cites Summa, Part II, f. 68v, and she quotes part of it on pp. 100-101. This is also referred to by MacKinnon [p. 170] and Taylor [p. 344], neither giving details and no two of the three agreeing on what the passage means. I have not been able to make complete sense of the passage, but it seems clearly to say that in Apr 1489, Pacioli presented models of at least the regular solids to the Duke of Urbino at the palace of Pacioli's protector [Cardinal Giuliano della Rovere, later Pope Julius II] in Rome. He then says many other dependants [= variations] of the regular solids can be made, and models were made for Pietro Valletari, Bishop of Carpentras. There is no reference to the number of models, nor their material, nor to a set being given to the Cardinal, nor whether the Cardinal was present when the models were given to the Duke. Due to a missing right parenthesis, ), the sense of one statement involving 'his own hands' could mean either that Pacioli presented the models to the Duke's own hands or that Pacioli had decorated the models himself. I doubt whether there were many other solids at this time, otherwise he would have mentioned them in the Summa -- the Summa only describes four of the Archimedean solids and only two of them are in Part II. I suggest that he didn't start developing the other shapes until about 1494, or later, in 1496 when he went to Milan and met Leonardo.

However, on f. 28v of De Divina Proportione, Pacioli is clearer and says that he arranged, coloured and decorated with his own hands 60 models in Milan and two other sets for Galeazzo Sanseverino in Milan and for Piero Soderini in Florence. This refers to his time in Milan, which was 1496-1499, though the Soderini set might have been made after Pacioli and da Vinci moved to Florence.

Albrecht Dürer. Underweysung der messung .... 1525 & 1538. Op. cit. in 6.AA. Figures 29 43 (erroneously printed 34) (pp. 316-347 in The Painter's Manual, Dürer's 1525 ff. M-iii-v - N-v-r) show a net of each of the regular polyhedra, an approximate sphere (16 sectors and 8 zones), truncated tetrahedron, truncated cube, cubo-octahedron, truncated octahedron, rhombi-cubo-octahedron, snub cube, great rhombi-cubo-octahedron, polyhedron of six dodecagons and thirty-two triangles (having a pattern of four triangles replacing each triangle of the cubo-octahedron, so a sort of truncated cubo-octahedron -- not an Archimedean solid and not correctly drawn) and an elongated hexagonal bipyramid (not even regular faced). This gives 7 of the 13 non-regular Archimedean polyhedra. Mackinnon says Figures 29 41 show a net of each of the regular polyhedra and the same seven Archimedean ones as given by Pacioli & da Vinci, but they give six and there are only four common ones. In the revised version of 1538, figure 43 is replaced by the truncated icosahedron and icosi-dodecahedron (figures 43 & 43a, pp. 414 419 in The Painter's Manual), giving 9 of the 13 non-regular Archimedean solids. P. 457 shows the remaining four Archimedean cases from an 1892 edition.

Albrecht Dürer. Elementorum Geometricorum (?). 1534. Op. cit. in 6.AA. Liber quartus, fig. 29-43, pp. 145-158 shows the same material as in the 1525 edition.

See Barbaro, 1568, in 6.AT.2, pp. 45-104 for drawings and nets of 11 of the 13 Archimedean solids - he omits the two snub solids.

See Jamnitzer, 1568, in 6.AT.2 for drawings of eight of the 13 Archimedean solids.

J. Kepler. Letter to Maestlin (= Mästlin). 22 Nov 1599. In: Johannes Kepler Gesammelte Werke, ed. by M. Caspar, Beck, Munich, 1938. Vol 14, p. 87, letter 142, lines 21-22. ??NYS. Described by Field, p. 202. Describes both rhombic solids.

J. Kepler. Strena seu De Nive Sexangula [A New Year's Gift or The Six Cornered Snowflake]. Godfrey Tampach, Frankfurt am Main, 1611. (Reprinted in: Johannes Kepler Gesammelte Werke; ed. by M. Caspar, Beck, Munich, 1938, vol. 4, ??NYS.) Reprinted, with translation by C. Hardie and discussion by B. J. Mason & L. L. Whyte, OUP, 1966. I will cite the pages from Kepler (and then the OUP pages). P. 7 (10 11). Mentions 'the fourteen Archimedean solids' [sic!]. Describes the rhombic dodecahedron and mentions the rhombic triacontahedron. The translator erroneously adds that the angles of the rhombi of the dodecahedron are 6O^o and 120^o. Kepler adds that the rhombic dodecahedron fills space. Kepler's discussion is thorough and gives no references, so he seems to feel it was his own discovery.

J. Kepler. Harmonices Mundi, 1619. Book II, opp. cit. above. Prop. XXVII, p. 61. Proves that there are just two rhombic 'semi regular' solids, the rhombic dodecahedron and the rhombic triacontahedron, though the cube and the 'baby blocks' tessellation can also be considered as limiting cases. He illustrates both polyhedra. Def. XIII, p. 50 & plate (missing in facsimile). Mentions prisms and antiprisms. Prop. XXVIII, pp. 61 65. Finds the 13 Archimedean solids and illustrates them -- the first complete set -- but he does not formally show existence.

J. Kepler. Epitome Astronomiae Copernicanae. Linz, 1618-1621. Book IV, 1620. P. 464. = Johannes Kepler Gesammelte Werke; ed. by M. Caspar, Beck, Munich, 1938, vol. 7, p. 272. Shows both rhombic solids.

The following table shows which Archimedean polyhedra appear in the various early books from della Francesca to Kepler.
dF = della Francesca, c1487, folio

T indicates the object appears in the Trattato of c1480.

P = Pacioli, Summa, Part II, 1494, folio.

P&dV = Pacioli & da Vinci, 1498, plate number of the solid version; the framework version is

the next plate

D = Dürer, 1525, plate

+ indicates the object is added in the 1538 edition.

B = Barbaro, 1568, page

J = Jamnitzer, 1568, plate

K = Kepler, 1619, figure