Sources page biographical material

6.AT.2 STAR AND STELLATED POLYHEDRA
I have heard it stated that Kircher was the first to draw star polygons.

Paolo Uccello (1397 1475). Mosaic square on the floor at the door of San Pietro in San Marco, Venice. 1425 1430. (This doorway is not labelled on the maps that I have seen -- it is the inner doorway corresponding to the outer doorway second from the left, i.e. between Porta di Sant'Alipio and Porta di San Clemente, which are often labelled.) This seems to show the small stellated dodecahedron {5/2, 5}. This mosaic has only recently (1955 & 1957) been attributed to Uccello, so it can only be found in more recent books on him. See, e.g., Ennio Flaiano & Lucia Tongiorgi Tomase; L'Opera Completa di Paolo Uccello; Rizzoli, Milan, 1971 (and several translations). The mosaic is item 5.A: Rombo con elementi geometrici in the Catalogo delle Opere, with description and a small B&W picture on p. 85. [Bokowski & Wills, below, give the date 1420.]

Coxeter [Elem. der Math. 44 (1989) 25 36] says it "is evidently intended to be a picture of this star polyhedron."

However, J. V. Field tells me that the shape is not truly the small stellated dodecahedron, but just a 'spiky' dodecahedron. She has examined the mosaic and the 'lines' of the pentagrams are not straight. [The above cited photo is too small to confirm this.] She says it appears to be a direct copy of a drawing in Daniele Barbaro; La Practica della Perspettiva; Venice, 1568, 1569, see below, and is most unlikely to be by Uccello. See Field, Appendix 4, for a discussion of early stellations.

In 1998, I examined the mosaic and my photos of it and decided that the 'lines' are pretty straight, to the degree of error that a mason could work, and some are dead straight, so I agree with Coxeter that it is intended to be the small stellated dodecahedron. I now have a postcard of this. However, I have recently seen a poster of a different mosaic of the same shape which is distinctly irregular, so the different opinions may be based on seeing different mosaics!

Both mosaics are viewed directly onto a pentagonal pyramid, but the pyramids are distinctly too short in the poster version. The only spiky dodecahedron in Barbaro is on p. 111, fig. 52, and this is viewed looking at a common edge of two of the pyramids and the pyramids are distinctly too tall, so this is unlikely to be the source of the mosaics. The 'elevated dodecahedron' in Pacioli & da Vinci, plate XXXI, f. CVI-v, has short pyramids and looks quite like the second mosaic, but it is viewed slightly at an angle so the image does not have rotational symmetry. If anyone is in Venice, perhaps they could check whether there are two (or more?) mosaics and get pictures of them.

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. Ill. by Leonardo da Vinci. See the entry in 6.AT.3 for fuller details of the facsimiles and details about which plates are in which of the editions.

Discussed by Mackinnon (see in 6.AT.3 below) and Field, pp. 214-215. Clearly shows the stella octangula in one of the superb illustrations of Leonardo, described as a raised or elevated octahedron (plates XVIIII & XX). Field, p. 214, gives the illustration. None of the other raised shapes is a star, but the raised icosahedron is close to a star shape.

Barbaro, Daniele (1514-1570). La Practica della Perspettiva. Camillo & Rutilio Borgominieri, Venice, 1568, 2nd ptg, 1569. (Facsimile from a 1569 copy, Arnaldo Forni, Milan, 1980. The facsimile's TP doesn't have the publication details, but they are given in the colophon. Various catalogues say there are several versions with dates on the TP and colophon varying independently between 1568 and 1569. A version has both dates being 1568, so this is presumed to be the first appearance. Another version has an undated title in an elaborate border and this facsimile must be from that version.) P. 111 has a dodecahedron with pyramids on each face, close to, but clearly not the stellated dodecahedron. P. 112 has an icosahedron with pyramids on each face, again close to, but clearly not the stellated icosahedron. I would have expected a reasonably accurate drawing, but in both drawings, several of the triples of segments which should lie on a single straight line clearly do not. P. 113 shows an icosi-dodecahedron with pyramids on the triangular faces. If the pyramids extended the edges of the pentagons, this would produce the dodecahedron! But here the pyramids distinctly point much further out and the overall perspective seems wrong. [Honeyman, no. 207, observing that some blocks come from the 1566 edition of Serlio which was dedicated to Barbaro.]

Wentzel Jamnitzer (or Jamitzer). Perspectiva Corporum Regularum. With 50 copper plates by Jost Amman. (Nürnberg, 1568.) Facsimile by Akademische Druck- u. Verlagsanstalt, Graz, 1973. [Facsimiles or reprints have also been issued by Alain Brieux, Paris, 1964 and Verlag Biermann und Boukes, Frankfurt, 1972.]

This includes 164 drawings of polyhedra in various elaborations, ranging from the 5 regular solids through various stellations and truncations, various skeletal versions, pseudo-spherical shapes and even rings. Some polyhedra are shown in different views on different pages. Nameable objects, sometimes part of larger drawings, include: tetrahedron, cubo-octahedron, truncated tetrahedron, stella octangula, octahedron, cube, truncated octahedron, rhombi-cuboctahedron, compound of a cube and an octahedron (not quite correct), great rhombi-cuboctahedron, icosahedron, great dodecahedron, dodecahedron, icosi-dodecahedron, rhombi-icosi-dodecahedron, truncated cube, and skeletal versions of: stella octangula, octahedron, cube, icosahedron, dodecahedron, icosi-dodecahedron. There are probably some uniform polyhedra, but I haven't tried to identify them, and some of the truncated and stellated objects might be nameable with some effort.

J. Kepler. Letter to Herwart von Hohenberg. 6 Aug 1599. In: Johannes Kepler Gesammelte Werke, ed. by M. Caspar, Beck, Munich, 1938. Vol 14, p. 21, letter 130, line 457. ??NYS. Cited by Field, op. cit. below. Refers to (small??) stellated dodecahedron.

J. Kepler. Letter to Maestlin (= Mästlin). 29 Aug 1599. Ibid. Vol. 14, p. 43, letter 132, lines 142-145. ??NYS. Cited by Field, below, and in [Kepler's Geometrical Cosmology; Athlone Press, London, 1988, p. 202]. Refers to (small??) stellated dodecahedron.

J. Kepler. Harmonices Mundi. Godfrey Tampach, Linz, Austria, 1619; facsimile: (Editions) Culture et Civilization, Brussels, 1968 (but my copy is missing three plates!) [Editions probably should have É, but my only text which uses the word Editions is a leaflet in English.] = Joannis Kepleri Astronomi Opera Omnia; ed. Ch. Frisch, Heyder & Zimmer, Frankfurt & Erlangen, 1864, vol. 5. = Johannes Kepler Gesammelte Werke; ed. by M. Caspar, Beck, Munich, 1938, vol. 6, ??NYS. Book II. Translated by J. V. Field; Kepler's star polyhedra; Vistas in Astronomy 23 (1979) 109 141.

Prop. XXVI, p. 60 & figs. Ss & Tt on p. 53. Describes both stellated dodecahedra, {5/2, 5} and {5/2, 3}. This is often cited as the source of the stella octangula, but the translation is referring to an 'eared cube' with six octagram faces and the stella octangula is clearly shown by Pacioli & da Vinci and by Jamnitzer.

Louis Poinsot. Mémoire sur les polygones et les polyèdres. J. de L'École Polytechnique 4 (1810) 16 48 & plate opp. p. 48. Art. 33 40, pp. 39 42, describe all the regular star polyhedra. He doesn't mention Kepler here, but does a few pages later when discussing Archimedean polyhedra.

A. L. Cauchy. Recherches sur les polyèdres. J. de L'École Polytechnique 16 (1813) 68 86. ??NYS. Shows there are no more regular star polyhedra and this also shows there are no more stellations of the dodecahedron.

H. S. M. Coxeter, P. Du Val, H. T. Flather & J. F. Petrie. The Fifty-Nine Icosahedra. Univ. of Toronto Press, 1938; with new Preface by Du Val, Springer, 1982. Shows that there are just 59 stellations of the icosahedron. They cite earlier workers: M. Brückner (1900) found 12; A. H. Wheeler (1924) found 22.

Dorman Luke. Stellations of the rhombic dodecahedron. MG 41 (No. 337) (Oct 1957) 189 194. With a note by H. M. Cundy which says that the first stellation is well known (see 6.W.4) and that the second and third are in Brückner's Vielecke und Vielfläche, but that the new combinations shown here complete the stellations in the sense of Coxeter et al.

J. D. Ede. Rhombic triacontahedra. MG 42 (No. 340) (May 1958) 98 100. Discusses Coxeter et al. and says the main process generates 8 solids for the icosahedron. He finds that the main process gives 13 for the rhombic triacontahedron, but makes no attempt to find the analogues of Coxeter et al.'s 59.