In geometry, the rhombic dodecahedron is a convex polyhedron with 12 congruent rhombic faces. It has 24 edges, 14 vertices of two types, it is a Catalan solid, the dual polyhedron of the cuboctahedron. The rhombic dodecahedron is a zonohedron, its polyhedral dual is the cuboctahedron. The long diagonal of each face is √2 times the length of the short diagonal, so that the acute angles on each face measure arccos, or 70.53°. Being the dual of an Archimedean polyhedron, the rhombic dodecahedron is face-transitive, meaning the symmetry group of the solid acts transitively on the set of faces. In elementary terms, this means that for any two faces A and B there is a rotation or reflection of the solid that leaves it occupying the same region of space while moving face A to face B; the rhombic dodecahedron can be viewed as the convex hull of the union of the vertices of a cube and an octahedron. The 6 vertices where 4 rhombi meet correspond to the vertices of the octahedron, while the 8 vertices where 3 rhombi meet correspond to the vertices of the cube.
The rhombic dodecahedron is one of the nine edge-transitive convex polyhedra, the others being the five Platonic solids, the cuboctahedron, the icosidodecahedron and the rhombic triacontahedron. The rhombic dodecahedron can be used to tessellate three-dimensional space, it can be stacked to fill a space. This polyhedron in a space-filling tessellation can be seen as the Voronoi tessellation of the face-centered cubic lattice, it is the Brillouin zone of body centered cubic crystals. Some minerals such as garnet form a rhombic dodecahedral crystal habit. Honey bees use the geometry of rhombic dodecahedra to form honeycombs from a tessellation of cells each of, a hexagonal prism capped with half a rhombic dodecahedron; the rhombic dodecahedron appears in the unit cells of diamond and diamondoids. In these cases, four vertices are absent; the graph of the rhombic dodecahedron is nonhamiltonian. A rhombic dodecahedron can be dissected with its center into 4 trigonal trapezohedra; these rhombohedra are the cells of a trigonal trapezohedral honeycomb.
This is analogous to the dissection of a regular hexagon dissected into rhombi, tiled in the plane as a rhombille. If the edge length of a rhombic dodecahedron is a, the radius of an inscribed sphere is r i = 6 3 a ≈ 0.816 496 5809 a, OEIS: A157697and the radius of the midsphere is r m = 2 2 3 a ≈ 0.942 809 041 58 a, OEIS: A179587. The area A and the volume V of the rhombic dodecahedron of edge length a are: A = 8 2 a 2 ≈ 11.313 7085 a 2 V = 16 3 9 a 3 ≈ 3.079 201 44 a 3 The rhombic dodecahedron has four special orthogonal projections along its axes of symmetry, centered on a face, an edge, the two types of vertex and fourfold. The last two correspond to the B2 and A2 Coxeter planes; the eight vertices where three faces meet at their obtuse angles have Cartesian coordinates: The coordinates of the six vertices where four faces meet at their acute angles are:, The rhombic dodecahedron can be seen as a degenerate limiting case of a pyritohedron, with permutation of coordinates and with parameter h = 1.
The rhombic dodecahedron is a parallelohedron, a space-filling polyhedron, being the dual to the tetroctahedrille or half cubic honeycomb, described by two Coxeter diagrams: and. With D3d symmetry, it can be seen as an elongated trigonal trapezohedron. Other symmetry constructions of the rhombic dodecahedron are space-filling, as parallelotopes they are similar to variations of space-filling truncated octahedra. For example, with 4 square faces, 60-degree rhombic faces, D4h dihedral symmetry, order 16, it can be seen as a cuboctahedron with square pyramids augmented on the bottom. In 1960 Stanko Bilinski discovered a second rhombic dodecahedron with 12 congruent rhombus faces, the Bilinski dodecahedron, it has the same different geometry. The rhombic faces in this form have the golden ratio. Another topologically equivalent variation, sometimes called a deltoidal dodecahedron or trapezoidal dodecahedron, is isohedral with tetrahedral symmetry order 24, distorting rhombic faces into kites, it has 8 vertices adjusted in or out in alternate sets of 4, with the limiting case a tetrahedral envelope.
Variations can be parametrized by. Is the rhombic solution; as approaches 1/2, approaches infinity. When projected onto a s
Zerbe Run is a tributary of Mahanoy Creek in Northumberland County, Pennsylvania, in the United States. It is 8.3 miles long and flows through Coal Township, Zerbe Township, Little Mahanoy Township. The watershed of the stream has an area of 13.1 square miles. Part of the stream is impaired by abandoned mine drainage, but its upper reaches are not impacted by mining. Several mine drainage discharges occur within the watershed; the stream is not far from the Western Middle Anthracite Field. Zerbe Run is one of the major tributaries of Mahanoy Creek, its watershed makes up 8.1 percent of the Mahanoy Creek drainage basin. A number of bridges have been constructed over the stream, its watershed is designated as a Coldwater Fishery and a Migratory Fishery, but there are no fish in the stream. However, the Zerbe Run Rod And Gun Club Pond is stocked with trout and numerous macroinvertebrate taxa inhabit the stream. An area in its upper reaches is on the Northumberland County Natural Areas Inventory. Zerbe Run begins in Coal Township.
It flows west-southwest for a few tenths of a mile, entering Zerbe Township and the census-designated place of Trevorton. The stream continues flowing west-southwest through Zerbe Township and Trevorton for a few miles passing through a pond and crossing Pennsylvania Route 890, it turns west-northwest for a few tenths of a mile, receiving an unnamed tributary from the left before turning west-southwest, crossing Pennsylvania Route 225, which it continues to flow alongside for several miles. After several tenths of a mile, the stream receives another unnamed tributary from the left and continues flowing west-southwest for several tenths of a mile, it exits Trevorton and Zerbe Township and enters Little Mahanoy Township. Here, the stream turns southwest for more than a mile, receiving two more unnamed tributaries from the left as its valley broadens, it reaches its confluence with Mahanoy Creek. Zerbe Run joins Mahanoy Creek 10.74 miles upstream of its mouth. Zerbe Run has no named tributaries. However, it does have four unnamed tributaries.
Some stream reaches in the watershed of Zerbe Run are designated as impaired waterbodies, but others are not. Abandoned mine drainage is the source of impairment of impaired streams in the watershed; the stream's watershed has four abandoned mine drainage sites: the North Franklin Mine drift and Borehole, the North Franklin Mine seepage, the North Franklin Mine bank seepage, the North Franklin Mine Sunshine Mine overflow. Passive treatment wetlands may be suitable for treating the North Franklin Mine drift and Borehole discharge. There are large silt piles along the stream near Trevorton; the effect that the abandoned mine drainage discharges have on the water quality of the stream depends on how much unimpacted water is flowing from the stream's upper reaches. In 2001, Zerbe Run was found to be acidic during low base-flow conditions, but was nearly neutral during high base-flow conditions. In March 2001 and August 2001, the discharge of the stream was 7.99 and 0.62 cubic feet per second while 4.6 miles further downstream, the discharges were 30.2 and 4.36 cubic feet per second, respectively.
In March and August 2001, the net alkalinity of the stream near Trevorton was 11 and 49 milligrams per liter, while the net alkalinity 4.6 miles further downstream was −3 and −20 milligrams per liter. In March and August 2001, the net pH of the stream near Trevorton was 7.0 to 7.2, while the net alkalinity 4.6 miles further downstream was 4.3 to 6.0. In March 2001, the concentration of aluminum in Zerbe Run near Trevorton was 0.04 milligrams per liter, while in August 2001, it was 0.02 milligrams per liter. 4.6 miles downstream, the concentration was 0.10 milligrams per liter in March 2001 and 2.3 milligrams per liter in August 2001. The manganese concentration of the stream near Trevorton was 0.06 and 0.02 milligrams per liter in March and August 2001, respectively. The manganese concentration 4.6 miles further downstream was 0.93 and 2.7 milligrams per liter in March and August 2001, respectively. The concentration of dissolved oxygen in the stream near Trevorton was 11.6 and 9.5 milligrams per liter in March and August 2001, while further downstream, it was 11.5 and 8.8 milligrams per liter.
In March 2001, the nitrate concentration in Zerbe Run near Trevorton was 0.51 and 0.72 milligrams per liter in March and August 2001, while 4.6 miles the concentrations were 0.50 and 0.60 milligrams per liter. The phosphorus concentration near Trevorton was 0.01 milligrams per liter both times, but 4.6 miles further downstream, it was 0.04 milligrams per liter in March and 0.02 milligrams per liter in August. The sulfate concentration in March was 10 milligrams per liter near Trevorton and 99 milligrams per liter further downstream, while it was 8 milligrams per liter near Trevorton in August and 264 milligrams per liter 4.6 miles further downstream. In the early 1900s, the channel of Zerbe Run was colored yellow by sulfur pollution. A 1909 report by the Commissioner of Health of the Commonwealth of Pennsylvania stated that "there will be no
Jacob Frankfort was the first known Jew to come to Los Angeles. He immigrated from Poland in 1841, he would be joined by other Central European Jews. By 1855, there were 60 Jews living in Los Angeles. Frankfort arrived in Los Angeles as a member of the Rowland-Workman exploratory party; the party had come to the city from New Mexico. Jacob's position in the team was bolstered by skills of tailoring and ownership of a rifle. Frankfort was a wealthy man; this fact is reflected in Rafael Gallardo's declaration of bankruptcy, which states that Jacob Frankfort was owed $400 in 1845. Frankfort started his business with a tailoring and men's apparel store in Bell's Row, an adobe building; when Mr. Mellus bought Bell's Row from Mr. Bell, it was Frankfort. Subsequently, Bell's Row name was changed to Mellus' Row. Bell's Row was fantastically located: all the traffic coming in from the L. A. River arrived at the corner of Aliso & Los Angeles Streets, where the Bell's Row sat. "L. A. Scene / The City Then and Now".
CECILIA RASMUSSEN. Los Angeles Times, March 21, 1994