Medieval Islamic Architecture, Quasicrystals, and Penrose and Girih Tiles: Questions from the Classroom

Tiling Theory studies how one might cover the plane with various shapes. Medieval Islamic artisans developed intricate geometric tilings to decorate their mosques, mausoleums, and shrines. Some of these patterns, called girih tilings, first appeared in the 12 th Century AD. Recent investigations show these medieval tilings contain symmetries similar to those found in aperiodic Penrose tilings first investigated in the West in the 1970\u27s. These intriguing discoveries may suggest that the mathematical understanding of these artisans was much deeper than originally thought. Connections like these, made across the centuries, provide a wonderful opportunity for students to discover the beauty of Islamic architecture in a mathematical and historical context. This paper describes several geometric constructions for Islamic tilings for use in the classroom along with projects utilizing girih tiles as construction templates. Open questions, observations, and conjectures raised in seminars across the United Arab Emirates are described including what the medieval artisans may have known as well as how girih tiles might have been used as tools in the actual construction of intricate patterns

A Novel Longitudinal Mode in the Coupled Quantum Chain Compound KCuF3

Inelastic neutron scattering measurements are reported that show a new longitudinal mode in the antiferromagnetically ordered phase of the spin-1/2 quasi-one-dimensional antiferromagnet KCuF3. This mode signals the cross-over from one-dimensional to three-dimensional behavior and indicates a reduction in the ordered spin moment of a spin-1/2 antiferromagnet. The measurements are compared with recent quantum field theory results and are found to be in excellent agreement. A feature of the data not predicted by theory is a damping of the mode by decay processes to the transverse spin-wave branches.Comment: 9 pages of text plus 4 postscript figures (1 color

Energy separation of single-particle and continuum states in a S=1/2 weakly-coupled chains antiferromagnet

Inelastic neutron scattering is used to study transverse-polarized magnetic excitations in the quasi-one-dimensional S=1/2 antiferromagnet BaCu_2Si_2O_7, where the saturation value for the N\'eel order parameter is $m_0=0.12 \mu_{\rm B}$ per spin. At low energies the spectrum is totally dominated by resolution-limited spin wave-like excitations. An excitation continuum sets in above a well-defined threshold frequency. Experimental results are discussed in the context of current theories for weakly-interacting quantum half-integer spin chains.Comment: 4 pages 4 figure

Stephan's Quintet: The X-ray Anatomy of a Multiple Galaxy Collision

Chandra observations of the compact galaxy group known as Stephan's Quintet (SQ) are presented. The major morphological features that were discovered with the ROSAT HRI are now imaged with higher resolution and S/N. The large scale shock (1.5', ~40kpc if at 85 Mpc) is resolved into a narrow NS feature embedded in more extended diffuse emission (D>=3'). The NS structure is somewhat clumpy, more sharply bounded on the W side and prominent only in the soft band (energies below ~2 keV). Its observational properties are best explained as a shock produced by a high velocity encounter between NGC7318b, a ``new intruder'', and the intergalactic medium in SQ. The shock conditions near the high speed intruder suggest that a bow shock is propagating into a pre-existing HI cloud and heating the gas to a temperature of ~0.5 keV. The low temperature in the shock is a problem unless we postulate an oblique shock. One member, NGC7319, hosts a Seyfert 2 nucleus, with an intrinsic luminosity Lx=10^43 erg/s, embedded in a region of more diffuse emission with 10'' radius extent. The nuclear spectrum can be modeled with a strongly absorbed power-law typical of this class of sources. Several additional compact sources are detected including three in foreground NGC7320. Some of these sources are very luminous and could be related to the ultraluminous X-ray sources found in nearby galaxies.Comment: Accepted for publication in Astronomy & Astrophysics Quality of figures has been degraded to fit in the astroph requirements; fig12 could not be inserted in the tex and is given as a jpe

A microscopic model for a class of mixed-spin quantum antiferromagnets

We propose a microscopic model that describes the magnetic behavior of the mixed-spin quantum systems R$_2$BaNiO$_5$ (R= magnetic rare earth). An evaluation of the properties of this model by Quantum Monte Carlo simulations shows remarkable good agreement with the experimental data and provides new insight into the physics of mixed-spin quantum magnets.Comment: revised version to be published in Phys. Rev.