polyhedra in the history of mathematics
the Prehistory
Neolithic people in Scotland have built stone models of "polyhedra". Their function is unknown.
two references: Ashmolean Museum - Virtual Polyhedra by George W. Hart
the Antiquity (from 3300 BC to 476 fall of Rome)
The Etruscan preceded the Greeks in their awareness of at least some of the regular polyhedra.
The regular polyhedra had a considerable influence in the Greek Antiquity.
- Pythagoras of Samos (570-476 BC) is considered as the inventor of the regular dodecahedron.
- Theetete of Athena (dead around 360 BC) discovered the regular octahedron and icosahedron; it seems that he was the first to construct the five regular polyhedra.
- Euclide (325-265 BC) studied the regular polyhedra in his work The Elements . He proved that there can't exist more than five of these solids.
- Platon (428-348 BC) used the regular polyhedra (also known as "platonic solids") in his philosophy by associating them with the "elements" (Timaeus ).
- Archimedes (287-212 BC) gives the list of the 13 semi-regular polyhedra of the first kind ("archimedean" polyhedra).
|
roman dodecahedron whose use is unknown |
the Middle Ages (from 476 to 1492 discovery of America by C. Colomb)
This millennium was not really interested in polyhedra...
the artists of the Renaissance (XVth and beginning of the XVIth century)
Piero della Francesca (1412-1492), Pacioli (1445-1517), Leonardo Da Vinci (1452-1519) with his drawings of skeleton (semi)regular polyhedra, Wenzel Jamnitzer (1508-1585), Dürer (1471-1528)...
Star and other non-convex polyhedra become popular; they lead eventually to Kepler's idea of stellation.
the modern times (from 492 to 1789 French Revolution)
- Johannes Kepler (1571-1630) revives the study of polyhedra and discovers two star polyhedra (Kepler's hedgehog / sea urchin) whose regularity (pentagrammic faces) escaped him. He gives in 1619 the first example of facetting, the "stella octangula".
He tries to find a relationship between the "pythagorician" solids and the six planets known at this time (the "copernician" planets).
- Leonhard Euler (1707-1787) proves the formula which links the numbers of vertices, edges and faces for the convex polyhedra.
the contemporary age (from 1789 to these days)
- Louis Poinsot (1777-1859) discovers the non convex regular polyhedra, especially the two last (duals of the two Kepler's polyhedra).
- Augustin Cauchy (1789-1857) proves that there exists only four non convex regular polyhedra (Kepler & Poinsot).
- Eugène Catalan (1814-1894) studies the archimedean polyhedra and their duals (semi-regular polyhedra of the second kind).
- J. Bertrand describes in 1848 the regular star polyhedra as facettings of the regular dodecahedron and icosahedron.
- Henri Poincaré (1854-1912) proves and generalizes Euler's polyhedral formula.
- Ludwig Schläfli (1814-1895) invents a combinatory notation for polyhedra and discovers the polytopes.
- Victor Schlegel (1843-1905) invents a projection of a polyhedron on a plan (Schlegel diagram) which is a visual assistance to comprehend the connectivity of the edges.
- Willem Abraham Wythoff (1865-1939) is behind an new method to construct uniform polyhedra; he also invents a new symbolic notation for these polyhedra.
- Patrick du Val (1903-1987) invents a notation for the stellations of a polyhedron.
- Jeffery C.P. Miller establishes the rules which define a stellation; he also discovers with H.S.M. Coxeter the 12 last non convex semi-regular polyhedra.
- Harold S.M. Coxeter (1907-2003) makes a major contribution to the study of polyhedra and their generalizations to the polytopes.
- John Flinders Petrie (1907-1972) found in 1926 two of the three unbounded regular polyhedral shapes and communicated it to his friend H.S.M. Coxeter who then found a third one.
- Norman W. Johnson establishes in 1966 the list of the 92 non uniform convex polyhedra with regular faces.
- N.J. Bridge enumerates in 1974 the facettings of the regular polyhedra (especially those of the dodecahedron).
- Robert Connelly shows in 1978 that there exists flexible polyhedra.
- John H. Conway proposes, at the end of the XXth century, a new notation for polyhedra (completed by George W. Hart) which indicates the operations to be performed on a "seed" polyhedron to generate the new polyhedron.
- Guy Inchbald describes in 2004 the basic theory of facetting diagrams for polyhedra.
a few general references (books, programs, internet sites) about polyhedra
• Virtual Polyhedra, online polyhedra encyclopaedia by George W. Hart
• Poly (Pro) is a must program to begin with convex polyhedra
• Polyhedron Models and Dual Models by Magnus J. Wenninger, Cambridge University Press, London & New York, 1971/1983
• Polyhedra by Peter R. Cromwell (Cambridge University Press, 1997)
• Great Stella by Robert Webb is a very complete program to explore the polyhedra world and create one's own models
