Deltahedra a la Leonardo, 16th of November 2009.

The only collection of drawings Leonardo managed to publish in his lifetime were the illustrations for Luca Pacioli's book, De divina proportione (On divine proportions). Luca Pacioli was a Franciscan friar and Leonardo picked up a lot of mathematical knowledge from him in the period 1496-1501 when both of them worked for the Duke of Milan, Ludovico Sforza. Leonardo made for Pacioli's book, whose theme were proportions in geometry and architecture, about sixty illustrations that showed polyhedra. The illustrations were printed using the woodcut techique and that could be the reason that Leonardo in his Notebooks wrote down that all that should be published from his work should be made in the copperplate technique, and not woodcuts.

Polyhedra were shown by Leonardo as frames so that one can simultaneously see their front and back sides. The two shown in the image above were called exacedron elevatus vacuus (left) and septuaginta duarum basium vacuum, where the words vacuus and vacuum should I guess mean that the polyhedra are empty, i.e. it should emphasize the way the Leonardo depicted them. I wanted to make something similar to his illustrations.

I, for certain reasons, needed exactly 8 polyhedra, so I thought about how to logicaly choose a group of 8 polyhedra from an infinite number of them. There are five Platonic bodies, thirteen Archimedean bodies, but what group contains eight polyhedra? There are eight strictly convex deltahedra. Deltahedron is a polyhedron whose all sides are equilateral triangles. Upon hearing such definition, one immediately thinks about tetrahedron, octahedron and icosahedron which are, besides being convex deltahedra, also platonic bodies, but there are still five deltahedra that have quite complicated names.

It is interesting that the definition I specified above sounds like the definition of platonic body, so one wonders how can there be eight deltahedra instead of only three already mentioned (having triangles for sides)? The thing is that in the definition of a platonic body one needs to add a condition that all the vertices are the same regarding the number of edges that meet in them. In five deltahedra, there are different types of vertices, depending on the number of edges that meet in them. Therefore, these deltahedra are not platonic bodies. Incidentally, this is the condition that is often forgotten in the definition of the platonic body (I think I forgot it too several times, but I am in a good company since Euclid also forgot it). In the image that opens this post one finds tetrahedron and triangular dipyramid (!) that has six sides - it is in fact two tetrahedra glued together along one (common) side. In the image above there are octahedron and pentagonal dipyramid i.e. two pentagonal pyramids with their bases glued together.

In the image above there are dodekadeltahedron and tetracaidecadeltahedron (!) that have 12 and 14 faces. Both polyhedra have a plane of mirror symmetry. The cylinder that the polyhedra "float" above is beautified with the ornament developed earlier and described in the post Leonardo's knotiness. My "hollow" polyhedra are made of silver and gold.

In the image above are heocaidecadeltahedron and icosahedron. Heocaidecadeltahedron is in fact two four-sided pyramids linked through the square antiprism (heh, thih is the language of the polyhedra addicts which are not that rare at all). The surrounding of the polyhedra is the interior of St. Peter basilica, made from a precise photograph by Patrick Landy. This photograph was also used as ambiental source of light, i.e. there are no additional sources of light besides the one that is consistent with the lights on the photo (it would take me a long time to explain all that in details... the technology is similar to projecting the HDRI image on the "ambiental" sphere that was spectacularly used by Paul Debevec in his movie Fiat Lux).

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Last updated on 16th of November 2009.