Polyhedra

In the opening scene of [68 Scorpion, Part One], in the top righthand part of the screenshot shown below there is a three-dimensional shape (presumably made of wood). The close-up of the relevant part, shown left, is shown bright to show detail. This recalls Leonardo's interest in mathematics and the relationship between art, mathematics and, especially, perspective.

[68 Scorpion, Part One]

Leonardo's famous painting "The Annunciation" can be seen, standing on end with only part showing, in the above screenshot.

above: [75 Scientific Method] - the 3-dimensionsional geometric shape by Leonardo is in the middle of the picture sitting on the barrel, and looks distorted and with a green Borg hue because this is what Seven is seeing after the Doctor has adjusted her Borg sensory nodes to a phase variance of .15 using a type-4 micro-inducer

Luca Pacioli (1445-1514) (sometimes called Luca Paciolo) epitomises the deep Renaissance connection between art and mathematics. He is the main character in Jacopo de Barbari's painting, 1495, shown left.

In this famous painting, Pacioli, who is a Franciscan friar and shown in his robes, stands at a table on which there are geometrical tools such as slate, chalk, compass, dodecahedron model, illustrating a theorem from Euclid (the ancient Greek mathematician and philosopher of the 3rd century BC) while examining a beautiful glass rhombicuboctahedron half-filled with water. (The identity of the character at the righthand side of the picture remains a mystery.) The painting is the earliest known image of a rhombicuboctahedron.

Leonardo, Pacioli's friend, drew the first illustration to be printed of that polyhedron. He illustrated Pacioli's landmark book published in 1509 called "De Divina Proportione" ("the divine proportion"). Pacioli coined the term "the divine proportion" for what is otherwise known as "the golden ratio". The book was very influential in circulating information about geometry and about polyhedra in particular.

As well Leonardo's duodecehedron (shown left), the book contained the first printed illustration of the icosidodecahedron, again drawn by the Maestro. In the printed version of the book there are also shown woodcuts based on Leonardo's drawings. Pacioli plagarised parts from unpublished manuscripts of Piero della Francesca, whose mathematical accomplishments thereby remained uncredited until the manuscripts were rediscovered in the early 20th century.

Below: more illustrations by Leonardo are shown below. The Latin term "vacuus" indicates that the models are hollow.

  

What made Leonardo's drawings so influential was that the solidity of the edges lets one see easily which edges belong to the front and which ones belong to the back, unlike simple line drawings where the front and back surfaces might be visually confused, and yet the hollow faces allow one to see through to the structure of the rear surface. At the time this was a brilliant new form of geometric illustration, a new form of graphic information display. However, it is not known whether Leonardo invented this new form, or whether he was simply drawing a series of wooden models with solid edges designed by Pacioli.

There are approximately 60 similar illustrations in the book, mostly in pairs as contrasting models with solid faces and models with this solid edge technique, for instance:

Leonardo's illustrations of wooden models were influential, for instance, appearing in a number of intarsia. Intarsia are mosaics made of pieces of inlaid wood, and developed into an art form which peaked in northern Italy in the late 15th and early 16th centuries. Many outstanding examples of this period feature polyhedra. Below are two intarsia panels by Fra Giovanni da Verona, constructed around 1520, respectively from the Monastery of Monte Olivetto Maggiore near Siena and from the church of Santa Maria in Organo in Verona.

The popularity of these three-dimensional geometrical shapes can be seen in the fact that the City of Florence at that time purchased a set of Pacioli's wooden models for public display in the Council Hall.

drawings by Leonardo of fortifications, and at the centre bottom he has doodled a dodecahedron in the "solid edge" form he employed in Pacioli's book

Leonardo's interest in perspective.

Sources - for a website with more information and an exhibition for 2009

Janet's Star Trek Voyager Site