Pentagram

Abstract: Klyachko and coworkers consider an orthogonality graph in the form of a pentagram, and in this way derive a Kochen-Specker inequality for spin 1 systems. In some low-dimensional situations Hilbert spaces are naturally organised, by a magical choice of basis, into SO(N) orbits. Combining these ideas some very elegant results emerge. We give a careful discussion of the pentagram operator, and then show how the pentagram underlies a number of other quantum “paradoxes”, such as that of Hardy.

Abstract: The pentagram map was introduced by R. Schwartz more than 20 years ago. In 2009, V. Ovsienko, R. Schwartz and S. Tabachnikov established Liouville complete integrability of this discrete dynamical system. In 2011, M. Glick interpreted the pentagram map as a sequence of cluster transformations associated with a special quiver. Using compatible Poisson structures in cluster algebras and Poisson geometry of directed networks on surfaces, we generalize Glick's construction to include the pentagram map into a family of discrete integrable maps and we give these maps geometric interpretations. This paper expands on our research announcement arXiv:1110.0472

Abstract: Klyachko and coworkers consider an orthogonality graph in the form of a pentagram, and in this way derive a Kochen-Specker inequality for spin 1 systems. In some low-dimensional situations Hilbert spaces are naturally organised, by a magical choice of basis, into SO(N) orbits. Combining these ideas some very elegant results emerge. We give a careful discussion of the pentagram operator, and then show how the pentagram underlies a number of other quantum "paradoxes", such as that of Hardy.

Abstract: It is not unanimous among scientists if there is beauty in science. Some deny it. Mental clarity of conclusions when captured in simple looking equations is mathematical beauty. This we also find in the Euclidian geometry when performing the Golden Section and by deriving the Golden or Devine Number in golden rectangles, spirals and the Golden Angle. The Golden Section is considered as most beautiful and used in architecture and art. It is found everywhere in nature, e.g. in the pentagram of flowers, in the spirals of the shells of snails and Nautilus and even in galaxies of space. The Golden Angle in plants is realized in the phyllotaxis of spirals of leaf rosettes, in fruit stands and in the cones of conifers and cycads. It optimizes packing of modules such as seeds and fruits as well as the capture of light by leaves for photosynthesis and the fitness of productivity. Although we can mathematically deduce it and scientifically explain its role in organization and formation of patterns of structure and function, we cannot explain why we find it beautiful. In a methodological dualism esthetics and beauty are transcendental categories besides science. Or are the pleasant sensations of the Golden Section elicited by different stimuli to which our brain is adapted? Perhaps the Golden Section found everywhere in the entire universe is a link between natural science and the transcendental dimension, while a flower of a rose remains both a complex scientific system and an object of overwhelming beauty.