A Mineralogical Study of a Dravitic Tourmaline from the Grenville Province

Tourmaline from a gem-quality deposit in the Grenville province has been studied with X-ray diffraction, visible-near infrared spectroscopy, Fourier transform infrared spectroscopy, scanning electron microscopy, electron microprobe and optical measurements. The tourmaline is found within tremolite-rich calc-silicate pods hosted in marble of the Central Metasedimentary Belt. The crystals are greenish-greyish-brown and have yielded facetable material up to 2.09 carats in size. Using the classification of Henry et al. 2011 the tourmaline is classified as a dravite, with a representative formula shown to be (Na0.73Ca0.2380.032)(Mg2+2.913Fe2+0.057Ti4+0.030) (Al3+5.787Fe3+0.017Mg2+0.14)(Si6.013O18)(BO3)3(OH)3((OH,O)0.907F0.093). Rietveld analysis of powder diffraction data gives a = 15.9436(8) Å, c = 7.2126(7) Å and a unit cell volume of 1587.8 Å3. A polished thin section was cut perpendicular to the c-axis of one tourmaline crystal, which showed zoning from a dark brown core into a lighter rim into a thin darker rim and back into lighter zonation. Through the geochemical data, three key stages of crystal growth can be seen within this thin section. The first is the core stage which occurs from the dark core to the first colourless zone; the second is from this colourless zone increasing in brown colour to the outer limit before a sudden absence of colour is noted; the third is a sharp change from the end of the second and is entirely colourless. These events are the result of metamorphism and hydrothermal fluids resulting from nearby felsic intrusive plutons. Scanning electron microscope, and electron microprobe traverses across this cross-section revealed that the green colour is the result of iron present throughout the system while the brown colour is correlated with titanium content. Crystal inclusions in the tourmaline of chlorapatite, and zircon were identified by petrographic analysis and confirmed using scanning electron microscope data and occur within the third stage of formation.

Efficient algorithms for beacon routing in polygons

In a paper by Biro et al. [7], a novel twist on guarding in art galleries is introduced. A beacon is a fixed point with an attraction pull that can move points within the polygon. Points move greedily to monotonically decrease their Euclidean distance to the beacon by moving straight towards the beacon or sliding on the edges of the polygon. The beacon attracts a point if the point eventually reaches the beacon. Unlike most variations of the art gallery problem, the beacon attraction has the intriguing property of being asymmetric, leading to separate definitions of attraction region and inverse attraction region. The attraction region of a beacon is the set of points that it attracts. For a given point in the polygon, the inverse attraction region is the set of beacon locations that can attract the point. We first study the characteristics of beacon attraction. We consider the quality of a "successful" beacon attraction and provide an upper bound of $\sqrt{2}$ on the ratio between the length of the beacon trajectory and the length of the geodesic distance in a simple polygon. In addition, we provide an example of a polygon with holes in which this ratio is unbounded. Next we consider the problem of computing the shortest beacon watchtower in a polygonal terrain and present an $O(n \log n)$ time algorithm to solve this problem. In doing this, we introduce $O(n \log n)$ time algorithms to compute the beacon kernel and the inverse beacon kernel in a monotone polygon. We also prove that $\Omega(n \log n)$ time is a lower bound for computing the beacon kernel of a monotone polygon. Finally, we study the inverse attraction region of a point in a simple polygon. We present algorithms to efficiently compute the inverse attraction region of a point for simple, monotone, and terrain polygons with respective time complexities $O(n^2)$, $O(n \log n)$ and $O(n)$. We show that the inverse attraction region of a point in a simple polygon has linear complexity and the problem of computing the inverse attraction region has a lower bound of $\Omega(n \log n)$ in monotone polygons and consequently in simple polygons.