Dichlorido(η6-p-cymene)[tris(4-methoxyphenyl)phosphane]ruthenium(II)

The title compound, [RuCl2(C10H14)(C21H21O3P)], crystallizes with two complex molecules in the asymmetric unit. The RuII atom has a classical three-legged piano-stool environment being coordinated by a cymene ligand [Ru—centroid = 1.707 (2)/1.704 (2) Å], a tris(4-methoxyphenyl)phosphane ligand [Ru—P = 2.3629 (15)/2.3665 (15) Å] and two chloride atoms with the Ru—Cl bonds adopting two distinct values of 2.4068 (16)/2.4167 (16) and 2.4016 (15)/2.4244 (16) Å. The effective cone and solid angles for the phosphane ligands were calculated to be 149.5/150.2° and 25.3/25.6°, respectively. In the crystal, weak C—H...Cl/O/π interactions are observed. The crystal was refined as a two-component twin.

Redistributions of luminance patterns on standard sky types

There is insufficient current knowledge on the sky luminance patterns under ISO/CIE standard sky types in absolute luminance physical units in cd.m−2 and the resulting horizontal skylight illuminance in lux at the ground level. This paper explains research results and formulae to enable determination of computer applications which help in any location and time to calculate the distribution of luminance in any sky element on outdoor surfaces or within window solid angles that cause interior illuminance perception problems and, or discomfort glare.

The Centroid Solid Angle and Probability Models of Square Prism Dice Rolls

Past studies have indicated that the centroid solid angle is related to probabilities of square prism dice rolls. We explain how it is relevant to these probabilities and how to use the spherical projection to calculate the centroid solid angles for the faces on a square prism. These values are then used in a statistical analysis in the quest of constructing a mathematical probability model. The proposed model is based on the principle that the probability of ending up on a particular resting aspect is proportional to the centroid solid angle of that aspect and inversely proportional to a power of the centroid height in that aspect. Using a power of 2.427, this proposed model fits our data of over 60,000 non-symmetrical square prism dice rolls of various sizes (unequal heights and widths) with the largest magnitude Z-score of 1.01. Different powers can potentially describe other situations; e.g. different surfaces, larger dice, heavier dice, etc.

Dimensions of plane and solid angles and their units in the International System of Units (SI)

The situation that has developed in the International System of Units (SI) as a result of adopting the recommendation of the International Committee of Weights and Measures (CIPM) in 1980, which proposed to consider plane and solid angles as dimensionless derived quantities, is analyzed. It is shown that the basis for such a solution was a misunderstanding of the mathematical formula relating the arc length of a circle with its radius and corresponding central angle, as well as of the expansions of trigonometric functions in series. From the analysis presented in the article, it follows that a plane angle does not depend on any of the SI quantities and should be assigned to the base quantities, and its unit, the radian, should be added to the base SI units. A solid angle, in this case, turns out to be a derived quantity of a plane angle. Its unit, the steradian, is a coherent derived unit equal to the square radian.