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The Five Platonic Solids



Pythagorus of Samos, the famous mathematician and mystic, lived in the time of Buddha and Confucius, around 550 BC. But even then, three of the five regular polyhedra were known. Historians now seem to agree that the cube, tetrahedron, and dodecahedron were know to the Greeks of this period. Archiologists have unearthed a stone dodecahedron near Padua in Italy dating back to the Etruscans before 500 BC. By the time of Plato, around 400 BC all five regular polyhedra were known. Some evidence suggests that Plato learned of them on a visit to Sicily. What he learned, he put into a dialogue entitled Timaeus.

In this dialogue he also associated the five solids with the known elements. His cosmic scheme associated the tetrahedron with fire, the earth with a cube, air with an octahedron, and water with an icosahedron. The most favored of the Pythagoreans, the dodecagon, was given the status of representing the universe. It is becuause of this dialogue that they are referred to as the Platonic Solids. By 300 BC when Euclid's Elements were written, he was able to show that these were the only five possible. Euclid credits the discovery of the octahedron and icosahedron to Theaetetus. Each of the regular polyhedra have certain common characteristics.
The faces are all identical regular polygons. The tetrahedron, octahedron, and icosahedron are made up of regular triangles. The cube or hexahedron is made up of squares, and the dodecahedron is made up of regular pentagons.
The same number of faces meet at each vertex so that a regular polygon looks the same when viewed from any vertex.

The dates of mans discovery (creation) of the Platonic solids is made more complicated by the existance of some neolithic Scottish stones that have been unearthed dating back to about 2000 BC. I was made aware of these by discussions on the Historia Matematica discussion group, and quote here from a posting of Dick Tahta:

On neolithic carved stone balls: There are nearly 400 of these objects found in various sites in Scotland and now in various museums and private collections. They include various regular and semi-regular solids, alternatively they can be seen as arrangements of knobs - from 3 to 10 and then various numbers up to 160!
Some of them are decorated, notably the tetrahedral Towie stone [see image here] now in the Edinburgh museum (which stocks an excellent coloured postcard). . I have a baked clay model of this stone, bought from a shop in Avebury, Wiltshire.
J Frazer is quoted as taking the grooves to be meant for thongs so that the balls could be hurled through the air "uttering oracles in a whistling voice which a wizard was able to interpret". I have been unable to trace this quotation, and the author was unable to help me at the time I inquired.
The Ashmolean museum, Oxford, has a number of balls, kept in a drawer - I have handled these and they are certainly as remarkable as Keith Critchlow has pointed out. "These neolithic objects display the regular mathematical symmetries normally associated with the Platonic solids, yet appear to be at least a thousand years before the time of either Pythagoras or Plato." (K Critchlow, Time stands still, London - Gordon Fraser, 1979, p133 - this book has some splendid photos of various stone balls.)
There have been various attempts to guess at what the balls might have beenmade for. The nineteenth century archeologists who excavated them thought they might be weapons - whether as pike heads, or hurled from slings, or used in games or perhaps for divination. It seems that the balls were never found in personal graves, so it has been suggested they were a sort of ceremonial conch, a prized possession of the tribe. Contemporary archeologists tend to be more cautious. According to Dorothy Marshall, "there is so little hard fact to be extracted from the evidence available about the carved stone balls that postulation as to their evolution and use if very difficult." ( D Marshall, Carved stone balls, Proc Soc Antiq Scotland, 108 (1976-7) 40-72 - this is the most up-to-date and authoritative account. Some previous papers in the same journal are to be found in 11 (1874-6) 29-62 and 48 (1913-4) 407-20.)


The names of the five solids come from the number of faces each has, tetra for four, hexa for six, octa for eight, dodeca for twelve, and icosa for twenty. Here is a table showing the characteristic parts of each solid. You may also click on the link at the right of each row to see a skelaton of the solid
Name
VerticesEdgesFacesImage source
Tetrahedron464Pic
Cube8126Pic
Octahedron6128Pic
Dodecahedron203012Pic
Icosahedron123020Pic

For a really great look at the five platonic solids, you can click on this link to see a Java script animated 3-D version of the figures. The site has an excellent presentation of all the platonic solids. Java 3D viewer
Teachers and students who wish to make their own models of the platonic solids can print off the nets from this link to copy on tag board or other appropriate material and fold into solids.
Here is a site where you can unload an interesting screen saver that I was sent by Li Huo. The Screen saver is simple to download and install. There are also some other interseting creations on this same site including 3-D rotatable graphics of the platonic solids. My thanks to Li Huo for providing this excellent resource.