Welcome to GRAZIOTTI ON POLYHEDRA
Presentation Forty years ago Ugo Adriano Graziotti published his magnificent book on Polyhedra. Geometric beauty had fascinated his mind for a long time, and it is no wonder that he subtitled his book The Realm of Geometric Beauty. This subtitle is quite indicative in the sense that Geometry's hidden beauty is here "uncovered' and made apparent. For, if it is true that Beauty arises from sensory manifestation—forms, shape, color, etc.— here, in this book, it is represented for us by the expert and refined artistic hand of the author.
Forty years ago Ugo Adriano Graziotti published his magnificent book on Polyhedra. Geometric beauty had fascinated his mind for a long time, and it is no wonder that he subtitled his book The Realm of Geometric Beauty. This subtitle is quite indicative in the sense that Geometry's hidden beauty is here "uncovered' and made apparent. For, if it is true that Beauty arises from sensory manifestation—forms, shape, color, etc.— here, in this book, it is represented for us by the expert and refined artistic hand of the author.
But Graziotti on Polyhedra offers also a dimension pertaining to science. In fact, against the odds of previous geometricians, as the author says, the book "shows for the first time in history the construction of the main thirteen Archimedean duals and their solution by geometrical means" (p. 6). To be sure, the author's scientific accomplishment is the original constructions for these polyhedra.
To the above, a further note relative to the physical structure of the book must be added. The book contains 38 "pages" or—better—folds, because it opens like an accordion. So, in reality, the book is only one page. And, when it is totally opened out or "unfolded", we are in front of a very, very wide page—over 17', almost 6 m., in width. Graziotti's book "unfolded" would be like watching all of it on your computer.... if you had a screen over 17', or about 6 m. wide! The decision to adopt this unusual format was not due to chance. Graziotti had already experimented with "unfoldings of polyhedra" (1960): or, better, he was well versed in, and had already used extensively the art of stereography. Thus the paginal setting of Polyhedra and Graziotti's very beautiful rendering of these polyhedric bodies are not only a homage to Lady Geometry but, above all and more importantly, exquisite forms of art.†
Pisano's Lady Geometria -- Perugia, Fontana Maggiore [click on Diptych 23]-- year 1278.
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Ugo and I met in San Francisco in the late fifties and soon became friends. I was a frequent visitor of his rather large Studio on Market Street. It was located on the top floor of a building with a high ceiling. But one would not notice the height of the ceiling because it was totally covered with perfectly executed balsa-wood models of polyhedra hanging from it.
Graziotti was not a professional mathematician. He was an artist, in the style of the masters of the Italian Renaissance: Piero della Francesca, Paolo Uccello, Leonardo da Vinci and many others, big and small. In fact Graziotti was so knowledgeable about Italian Renaissance art and artists, that he served as an official authenticator of Renaissance art works that were brought from Italy to the USA during and after the Second World War. In the vein of the Renaissance masters, two-dimentional and three-dimentional shapes had been the very foundation of Graziotti's academic education. Among other works, he had studied Piero della Francesca's treatise on the Five regular bodies, De Quinque Corporibus Regularibus, Leonardo Da Vinci's Codice Atlantico, Luca Pacioli's De Divina Proportione, etc. He knew profoundly De Prospectiva Pingendi of Piero della Francesca for whom the laws of pictorial representation are identical with the rigorous laws of mathematical perspective. And as in Piero's, in the drawings, paintings and sculptures of Ugo Adriano, man and his world are seen as subject to the perspective essence of geometric forms. No wonder then that Graziotti was so fascinated with the design and construction of polyhedra. This passion of his goes back many years. Even before coming to the United States Graziotti had constructed a number of polyhedra in wood which for many years were on exhibition in Milan. Once in the U. S., the Pacific Science Center of Seattle, Washington, commissioned him to undertake a research on the plane partition of Euclidean space. The result was a series of publications (including the very beautiful Historical Chart of Polyhedra of 1966), and the design and execution of several balsa-wood models of polyhedra. Upon his return to Italy in 1970, Graziotti established his residence in Rome. There he became a member of the A.I.A.M., Accademia Internazionale dell'Arte Moderna and President of its Co.S.P.A., or Commissione di Studi per la Programmazione Artistica. Once in Rome, he decided to donate his balsa-wood polyhedra models to the Sovraintendenza Antichità e Belle Arti of Rome. Now the 116 models are in the permanent collection of the Museo della Matematica of the Commune of Rome.
Star dodecahedron, inlaid in marble. Floor of St. Mark's Basilica, Venice. Attributed to Paolo Uccello.
These 116 models include the Platonic polyhedra and their duals (10), The Archimedean polyhedra and their duals (26), Leonardo’s polyhedra(20) and their complementary star polyhedra (10), Stellated polyhedra and their duals (22), Graziotti’s polyhedra (3), Archimedean prisms (10), Archimedean antiprisms (5), Stellated prisms and antiprisms (10). It may be of interest to note that the above-mentioned groups of polyhedra are further distinguished by color. Graziotti left 24 of these models in the natural color of the balsa-wood, and painted the remaining 92 models in ten different colors: yellow (31), red (20 -- and exclusively for Leonardo's polyhedra--), green (15), gold (8), etc. This "classification" is meant to subgroup the models into mathematical classes and, at the same time, to serve as historical referent.
Photographs of these 116 balsa-wood models have been published in a very fine catalogue by the title: Polyhedra. I Poliedri della donazione Adriano Graziotti, edited by Wilma di Palma, Àrgos Edizioni, Roma 1994, 144 pp.
One of Graziotti's balsa-wood models: the Small stellated rhombicosidodecahedron, number 65 in the Catalogue. For a view of its solid counterpart, see Polyhedra: The Realm of Geometric Beauty, Plate 11, central figure.
I consider myself very fortunate to have been his friend, to have one of his drawings, Kathi, and also to have a dedicated copy of his book—number 900 of the 1000 autographed copies published. And today, in remembrance, I would like to pay homage to him with this web site so that many more people can become acquainted with his name, his work on Polyhedra and his great artistic talent.
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