Master degree in Materials Science

Bridges between fine art and natural sciences: the case of fullerenes, polyhedra and symmetry

Bridges between fine art and natural sciences: the case of fullerenes, polyhedra and symmetry

Academic year 2023/2024

Sommario del corso

• Course objectives
• Results of learning outcomes
• Program
• Course delivery
• Learning assessment methods
• Suggested readings and bibliography
• Teaching tools
• Course card

Course objectives

The specialization in art and science led to the appearance of new disciplines and
sub disciplines. It became impossible for an artist or for a scholar to comprehend
many fields, although the links between these could help the developments
significantly. T his is also a problem in university education, since reality is not
divided into departments like the universities. As a result, we have “two cultures”:
art and humanities vs. science and technology. This course is aimed to link the art
and science by a number of bridges.

Eventually the students and researchers should not to stay on these bridges: they should remain at their own respective disciplines, but it is important to visit the other side of the gap(s), to have new impressions, and then to go back and use these.

One of the bridges is the discovery of C60, a third variety of carbon, in addition to the more familiar diamond and graphite forms, that has generated enormous interest in many areas of physics, chemistry and material science. Furthermore, it turns out that C60 is only the first of an entire class of closed-cage polyhedral molecules consisting of only carbon atoms - the fullerenes (C20, C24, C26, … C60, …C70, … carbon nanotubes). The lecturer will explain modern scientific concepts and terms in a popular manner, with main emphasis to morphological aspects of fullerene molecules and fullerene-like structures or, in simple words, to particular characteristics of their chemical and physical properties that are dictated by their form. It will be discussed how Nature uses fullerene-like structures to minimize energy and matter resources in molecules and nanoclusters, viruses and living organisms. Examples of achievement of such goals in architecture will also be presented.

Other bridges are based on history. The lecturer will review scientific achievements of the past on which foundation the modern fullerene science is based, and first of all – long history of exploration of polyhedra in mathematics and art. The students will “travel” in depth of time until the time of Renaissance and even classical Antiquity. The course will review the contribution of the great Renaissance artists to development of polyhedra geometry as well as analysis of their art (mostly from the points of view of mathematics and theory of perspective). Examples of fullerene-like polyhedra in other cultures (Arabic/Persian, Chinese) and modern art will be presented.

Some of these historical excursuses will include surface touching to the problems of philosophy of science such as: role of scientific conjecture, the importance of personality in art and science, other sides of interrelationships and differences of art and science as two different ways of cognition, two parts of one whole – humanity culture.

Results of learning outcomes

On successful completion of the course the
students should be able to: (1) Understand the following scientific concepts with
corresponding analogs in art and architecture, in the context of history of science
and art: Polyhedra; Regular (Platonic) and semiregular (Archimedean) polyhedral;
Dual polyhedra ; Symmetry and Symmetry operations; Asymmetry Antisymmetry
Dissymmetry Symmetry parado x; Crystals ; Fractals ; Chirality; Full er e nes; Carbon
nanotubes; Graphene; Fullerene-like structures in animated nature and architecture
(2) Know the contribution of great Renaissance artists to development of polyhedra
geometry and be able to analys e their art from the points of view of mathematics
and theory of perspective. Among them are Piero della Francesca , Leonardo da
Vinci , Albrecht Durer , Paolo Uccello , Rafael Fra Giovanni de Verona , Bramante,
Baldassare Peruzzi, Giuliano da Sangallo.

Program

Module Content /schedule and outlines:
Fullerenes, carbon nanotubes, graphene (3 hours)
Fullerene-like structures in architecture (3 hours)
Introduction to geometry of polyhedral. Concepts of symmetry (6 hours)
Long history of exploration of polyhedral in science and fine art (15 hours)
Euler relation for convex polyhedra and molecular structure of fullerenes (9 hours)
Fullerene-Like Structures in Animated Nature (3 hours)
Reports of students’ projects (3 hours)

Course delivery

• Frontal lessons by Professor Eugene Katz.
• Each student will deliver a written topical essay and finally present it in class.

Learning assessment methods

Grading is 10% according to class participation, 90% for topical projects delivered
literally and as presentation to the class.
Name of the module

1. J. Wenninger, Polyhedron Models, 1971, Cambridge University Press.
2. Symmetry in Science and Art, Shubnikov, A. (Ed.), Springer, 1974.
3. M. A. Peterson, The geometry of Piero della Francesca, Mathematical Intelligencer, Vol. 19, No. 3, pp. 33-40, 1997.
4. K. Williams, Plagiary in the Renaissance, Mathematical Intelligencer, Vol. 24, No. 2, pp. 45-;57, 2002.
5. J.V. Field, Rediscovering the Archimedean Polyhedra: Piero della Francesca, Luca Pacioli, Leonardo da Vinci, Albrecht Durer, Daniele Barbaro, and Johannes Kepler, Archive for History of Exact Sciences, vol. 50, pp.241-283, 1997.
6. Piero della Francesca, “Libbelus de quinque corporibus regularibus” Vatican library, ttps://digi.vatlib.it/mss/detail/Urb.lat.632.
7. P. della Francesca, “Trattato d'Abaco”, ed. G. Arrighi, Pisa: Domus Galilaeana (1970).
8. E. A. Katz. Leonardo’s Polyhedra with Solid Edges, Fullerenes and Skeletal Nanocages. The Israeli Chemist and Chemical Engineer, Issue 5, September 2019, p. 34-36.
9. G. Vasari, The lives of the artists, Oxford University Press, 1991.
10. J J. Gray and P.R. Cromwell. Kepler’s Work on Polyhedra, Mathematical Intelligencer, Vol. 17, No. 3, pp. 23-;33, 1995.
11. E. A. Katz and B. Y. Jin, Fullerenes, Polyhedra, and Chinese Guardian Lions, Mathematical Intelligencer, Vol. 38, No. 3, pp. 61-68.
12. E. A. Katz. Geometrical analysis of fullerene and radiolaria structures: Who gets the credit? Mathematical Intelligencer, v.36, No.1, p. 34-36 (2014).

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