Abstract
In La Terza Parte of the treatise La pratica della perspettiva, published in 1568, Daniele Barbaro describes 39 solid bodies between Chapters I and XXXIV through edge unfolding, orthographic projections and/or perspective drawings, some of these in positions that are not the easiest to draw. Barbaro begins with a triangular pyramid and the Platonic Solids before describing the irregular bodies that are born from the regular bodies that he obtains from successive truncations of other bodies. Besides eleven Archimedeans, Barbaro obtains nine convex non-uniform polyhedra most of which are found nowhere before La pratica della perspettiva and are currently identifiable as symmetrohedra, near-miss Johnson Solids and even a Goldberg polyhedra, between others. We will analyse how these and other non-uniform polyhedra were conceived and illustrate them to unveil what Barbaro had in mind and the depth of Barbaro’s research. In spite of the imprecisions recognizable in some of the results, the originality and complexity of Barbaro’s systematic studies on solid geometry deserve our reanalysis, not only because they are intriguingly accurate for his time but because there is still much to learn from his scientific creativity.
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Notes
- 1.
La Terza Parte, che tratta del modo di levare i corpi dalle piante (Barbaro 1568, p. 43).
- 2.
In this we are not considering the mazzocchio and its variations (Chapters XXXIV, XXXVIII and XXXIX). The polyhedra described between Chapters I and XXXIV are:
-
16 convex uniform: tetrahedron (Chapter II), cube (Chapter III), octahedron (Chapter IV), dodecahedron (Chapter V), icosahedron (Chapter VI), truncated tetrahedron (Chapter VII), cuboctahedron (Chapter VIII), truncated cube (Chapter IX), rhombicuboctahedron (Chapter X), truncated octahedron (Chapter XI), icosidodecahedron (Chapter XII), truncated dodecahedron (Chapter XIII), truncated icosahedron (Chapter XI), rhombitruncated cuboctahedron (Chapter XV), rhombicosidodecahedron (Chapter XVII), and the rhombitruncated icosidodecahedron (Chapter XXI);
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10 convex non-uniform (Chapters I, XVI, XVIII, XIX, XX, XXII, XXV plus three in Chapter XXXIV);
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10 concave non-uniform with regular faces (Chapters XXIII, XXIV, XXVI, XXVII, XXVIII, XXIX, XXX, XXXI, XXXII and XXXIII);
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3 augmented versions of previous bodies (Chapter XXXIV).
-
- 3.
corpi sodi or corpo sodo (Barbaro 1568, pp. 45, 57).
- 4.
Interestingly, Barbaro describes each solid body without specifically mentioning the volume within it, for instance, in: “The cube is formed of 6 surfaces, all perfect squares, and 8 right solid angles, 24 plane angles, and 12 edges, as its unfolding demonstrates in figure H; this, when closed into a body, will represent the true cube” (Williams and Monteleone 2021, p. 240). From similar occurrences in the treatise, we may infer that Barbaro understands that the polyhedral surface completely defines the solid body, a conception close to Leonard Euler’s premise, almost a century after: “The consideration of solid bodies therefore must be directed to their boundary; for when the boundary which encloses a solid body on all sides is known, that solid is known […]. Therefore, three kinds of bounds are to be considered in any solid body; namely 1) points, 2) lines, 3) surfaces, or, with the names specially used for this purpose: 1) solid angles, 2) edges and 3) faces. These three kinds of bounds completely determine the solid” (Euler 1758; Federico 1982, p. 66). According to Williams (2021, p. 1) the term polyhedron only appears in English in the Billingsley edition of Euclid’s Elements published in 1570 and solidorum polyedrum in Christopher Clavius’ edition of six books of Euclid, published in 1574.
- 5.
le descrittioni delle figure aperte, dellequali si fanno i corpi sodi piegando le insieme per dimostratione del vero (Barbaro 1568, p. 45).
- 6.
le loro piante perfette & digradate, & finalmente i dritti & le loro adombrationi (Barbaro 1568, p. 45).
- 7.
From the truncated polyhedra, Chapter VII is the only body that Barbaro does not explain how to obtain from another body.
- 8.
Anthony Pugh (1976, pp. 76–83) interestingly describes “The truncation of existing polyhedra” and describes three of Barbaro’s solids.
- 9.
An augmentation is the operation through which a polyhedron, such as a pyramid or a cupola, is adjoined to a face or faces of a base polyhedron (Weisstein n.d.-a).
- 10.
The numbers between parenthesis that follow are notations for semiregular tessellations. They stand for the regular polygons that outline the vertex of each tessellation.
- 11.
Grünbaum and Johnson (1965) demonstrated that convex polyhedra with regular faces are limited to the triangular, square, pentagonal, hexagonal, octagonal, and decagonal configurations.
- 12.
& si acconcera l’errore dello intagliatore nella figura 12 con le regole dette (Barbaro 1568, p. 63).
- 13.
Regarding the convex non-uniform polyhedra: in Chapter XVI, Barbaro says that the twice-truncated octahedron has squared faces and the edges are equal (of the hexagons or the whole polyhedron, see footnote 10); in Chapter XX, Barbaro only says that the dodecagons are equal-sided, after stating they derive from the division of the edges of the truncated icosahedron into three equal parts (which would not yield regular dodecagons). The same happens in Chapter XXII but, in this case, Barbaro does not say that the dodecagons’ edges are equal.
- 14.
Dal corpo di sei quadrati, & otto essagoni, partendo i suoi lati in due parti eguali, & levati gli anguli sodi dove finisceno le dette parti, si formerá un altro corpo, la cui superficie e di 24 trianguli, sei quadrati, & otto essagoni. Anguli stretti 72, dritti 24, larghi 58, sodi 36, & lati 72 come si vede nella spiegatura alla figura 32. Il detto corpo e alquanto irregulato, perche posa con tre anguli soli della base essagona se bene tutti i suoi lati sono eguali (Barbaro 1568, pp. 89–90).
- 15.
partendo i suoi lati egualmente (Barbaro 1568, p. 97).
- 16.
e bellissima forma, benche alquanto irregulare, per la sua giacitura (Barbaro 1568, p. 97).
- 17.
Dal corpo di 60, trianguli, 20 essagoni, & dodici pentagoni, mutando i trianguli in essagoni, si formera il corpo di 12 pentagoni e 80 essagoni & havera anguli sodi 180, lati 270, anguli larghi 540 (Barbaro 1568, p. 98).
- 18.
The pentagonal faces are separated by two hexagons and to get from any pentagon to another (for instance, with your fingers) we take 3 steps: the first from the pentagon to a hexagon, the second from this to another hexagon, and the third from the latter to another pentagon (hence the 3), without moving to another sequence of hexagons on the side (hence the 0). See: Hart (2012); Senechal (2012, pp. 125–138).
- 19.
Dal sopra scritto corpo, partendo i suoi lati in tre parti eguali, & levati gli anguli sodi ove termina la parte dimezzo, nasce il corpo formato di 60 trianguli, 12 diecianguli e 20 soperficie di dodici lati equali (Barbaro 1568, p. 99).
- 20.
Dal corpo di sei quadrati, & otto essagoni, partendo i suoi lati in tre parti eguali (Barbaro 1568. P. 101).
- 21.
Dal corpo sopranominato di trianguli 24, octanguli 6, dodicianguli otto, mutanto i trianguli 24 in essagoni 24 & interponendo tra uno dodiciangulo & l’altro uno quadrato, egli si forma uno corpo di 36 quadrati 24 essagoni, ottanguli sei & dodicianguli otto (Barbaro 1568, p. 104).
- 22.
If we decided to maintain the right angles in the quadrangular faces, the hexagonal faces would not be planar. Replacing the 24 hexagons for 48 plane trapezoids would ensure that every face was planar, but the resulting body would not be convex.
- 23.
In addition to these versions of the truncated rectified truncated octahedron, we modelled a third from the division of the edges of the rectified truncated octahedron in three equal parts and found two other versions online, slightly different from ours. See: https://en.wikipedia.org/wiki/Rectified_truncated_octahedron
- 24.
To be sure that it was possible to obtain each of the models from these planar nets, we built each of them with Polyhdron Frameworks and folded them according to Barbaro’s instructions. The faces in each planar net were correctly drawn and numbered, except for Chapter XXX, in which one face is missing drawn and Chapter XXXI that has an extra face.
- 25.
rinchiusa, & posta insieme (Barbaro 1568, p. 105).
- 26.
andarebbe in infinito, però lasciando la noia di molto scrivere (Barbaro 1568, p. 111).
- 27.
Egli si potrebbe formare molti altri corpi simili, come sarebbe uno di sei quadrati, & dodici essagoni, & un altro di 32 essagoni, & quadrati sei, & un altro di 90 quadrati, 60 essagoni, 12 dieci anguli, & 20 dodicianguli (Barbaro 1568, p. 111).
- 28.
In this case ‘elongated’ does not comply with Johnson’s terminology, in which ‘elongated’ applies to: “A pyramid, cupola, or rotunda is elongated if it is adjoined to an appropriate prism (a pentagonal pyramid to a pentagonal prism, a pentagonal cupola or rotunda to a decagonal prism, etc.) or gyroelongated if it is adjoined to an antiprism” (Johnson 1966, p. 182).
- 29.
per avvertimento di chi legge (Barbaro 1568, p. 111).
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Viana, V. (2023). Non-Uniform Polyhedra Described by Daniele Barbaro. In: Monteleone, C., Williams, K. (eds) Daniele Barbaro and the University of Padova. DBSPA 2022. Trends in the History of Science. Birkhäuser, Cham. https://doi.org/10.1007/978-3-031-29483-9_6
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