The rhombic dodecahedron can be used to tessellate 3-dimensional space. It can be stacked to fill a space much like hexagons fill a plane.

This tessellation can be seen as the Voronoi tessellation of theface-centred cubic lattice. Some minerals such as garnet form a rhombic dodecahedral crystal habit. Honeybees use the geometry of rhombic dodecahedra to form honeycomb from a tessellation of cells each of which is a hexagonal prism capped with half a rhombic dodecahedron. The rhombic dodecahedron also appears in the unit cells of diamond and diamondoids. In these cases, four vertices are absent, but the chemical bonds lie on the remaining edges

The rhombic dodecahedron forms the hull of the vertex-first projection of a tesseract to 3 dimensions. There are exactly two ways of decomposing a rhombic dodecahedron into 4 congruent parallelepipeds, giving 8 possible parallelepipeds. The 8 cells of the tesseract under this projection map precisely to these 8 parallelepipeds.

The rhombic dodecahedron forms the maximal cross-section of a 24-cell, and also forms the hull of its vertex-first parallel projection into 3 dimensions. The rhombic dodecahedron can be decomposed into 6 congruent (but non-regular) square dipyramids meeting at a single vertex in the center; these form the images of 6 pairs of the 24-cell’s octahedral cells. The remaining 12 octahedral cells project onto the faces of the rhombic dodecahedron. The non-regularity of these images are due to projective distortion; the facets of the 24-cell are regular octahedra in 4-space.

This decomposition gives an interesting method for constructing the rhombic dodecahedron: cut a cube into 6 congruent square pyramids, and attach them to the faces of a second cube. The triangular faces of each pair of adjacent pyramids lie on the same plane, and so merge into rhombuses. The 24-cell may also be constructed in an analogous way using two tesseracts.

The tesseract represents 4 th dimension and is the projection or shadow of the rhombic dodecahedron

A 3D projection of an 8-cell performing adouble rotation about two orthogonal planes

1111 familiar ?

The rhombic dodecahedron forms the convex hull of the tesseracts vertex-first parallel-projection. The number of vertices in the layers of this projection is 1 4 6 4 1 – the fourth row in Pascal’s triangle.

Pascal triangle

Each number in the triangle is the sum of the two directly above it.

Patterns and properties

Pascal’s triangle has many properties and contains many patterns of numbers.

[edit]Rows

• When adding all the digits in a single row, each successive row has twice the value of the row preceding it. For example, row 1 has a value of 1, row 2 has a value of 2, row 3 has a value of 4, and so forth.
• The value of a row, if each entry is considered a decimal place (and numbers larger than 9 carried over accordingly) is a power of 11 ( 11ⁿ, for row n). Thus, in row two, ’1,2,1′ becomes 11², while ’1,5,10,10,5,1′ in row six becomes (after carrying) 161,051, which is 11⁵. This property is explained by setting x = ’10′ in the binomial expansion of (x + 1)^row=n, and adjusting values to the decimal system. But x can be chosen to allow rows to represent values in any base – such as base 3; 1 2 1₃['1,2,1'] = 4² (16), 2 1 0 1₃ ['1,3,3,1'] = 4³ (64) – or base 9; 1 2 1₉ = 10² (100), 1 3 3 1₉ = 10³ (1000) and 1 6 2 1 5 1₉ ['1,5,10,10,5,1'] = 10⁵ (100,000). In particular (see next property), for x = 1 place value remains constant(1^place=1). Thus entries can simply be added in interpreting the value of a row.
• The sum of the elements of row m is equal to 2^m−1. For example, the sum of the elements of row 5 is 1 + 4 + 6 + 4 + 1 = 16, which is equal to 2⁴ = 16. This follows from the binomial theorem proved above, applied to (1 + 1)^m−1.
• If rows are numbered starting with n = 0, the sum of the elements in the row is simply 2ⁿ, so row 0 adds to 2⁰ = 1, row 1 adds to 2¹ = 2, etc.
• Some of the numbers in Pascal’s triangle correlate to numbers in Lozanić’s triangle.
• The sum of the squares of the elements of row m equals the middle element of row (2m − 1). For example, 1² + 4² + 6² + 4² + 1² = 70. In general form:

• Another interesting pattern is that on any row m, wherem is odd, the middle term minus the term two spots to the left equals a Catalan number, specifically the (m + 1)/2 Catalan number. For example: on row 5, 6 − 1 = 5, which is the 3^rd Catalan number, and (5 + 1)/2 = 3.
• Another interesting property of Pascal’s triangle is that in rows where the second number (immediately following ’1′) is prime, all the terms in that row except the 1s are multiples of that prime.

A particularly popular polyhedron is the pyramid. If we restrict ourselves to regular polygons for faces, there are three possible pyramids: the triangle-based tetrahedron, the square pyramid, and the pentagonal pyramid. Being bounded by regular polygons, these last two fall within the class of Johnson solids. One interesting property of pyramids is that like the tetrahedron, their duals are also pyramids. (Incidentally, the Egyptian pyramids have square bases but the triangular side faces are not quite equilateral; they are very close to half a golden rhombus.)

Cygnus (constellation)

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Cygnus

Constellation
List of stars in Cygnus
Abbreviation	Cyg
Genitive	Cygni
Pronunciation	/ˈsɪɡnəs/, genitive /ˈsɪɡnaɪ/
Symbolism	the Swan or The Northern Cross
Right ascension	20.62 h
Declination	+42.03°
Quadrant	NQ4
Area	804 sq. deg. (16th)
Main stars	9
Bayer/Flamsteed stars	84
Stars with planets	57
Stars brighter than 3.00m	4
Stars within 10.00 pc (32.62 ly)	1
Brightest star	Deneb (α Cyg) (1.25m)
Nearest star	61 Cyg (11.36 ly, 3.48 pc)
Messier objects	2
Meteor showers	October Cygnids Kappa Cygnids
Bordering constellations	Cepheus Draco Lyra Vulpecula Pegasus Lacerta
Visible at latitudes between +90° and −40°. Best visible at 21:00 (9 p.m.) during the month of September.

Cygnus (pron.: /ˈsɪɡnəs/) is a northern constellation lying on the plane of the Milky Way. Its name is the Latinized Hellenic (Greek) word for swan. One of the most recognizable constellations of the northern summer and autumn, it features a prominent asterism known as the Northern Cross (in contrast to the Southern Cross). Cygnus was among the 48 constellations listed by the 2nd century astronomer Ptolemy, and it remains one of the 88 modern constellations.

Contents  [hide]  1 Notable features 1.1 Stars 1.2 Deep-sky objects 1.3 Extrasolar planets 2 History and mythology 3 Equivalents 4 Musical references 5 Namesakes 6 See also 7 Notes 8 References 9 External links

[edit] Notable features

The constellation Cygnus as it can be seen by the naked eye.

[edit] Stars

See also: List of stars in Cygnus

There are several bright stars in Cygnus. Alpha Cygni, called Deneb, is the brightest star in Cygnus. It is a blue-white hued supergiant star of magnitude 1.3, 3200 light-years away. The traditional name of Alpha Cygni means "tail" and refers to its position in the constellation. Albireo, designated Beta Cygni, is a celebrated binary star among amateur astronomers for its contrasting hues. The primary is an orange-hued giant star of magnitude 3.1 and the secondary is a blue-green hued star of magnitude 5.1. The system is 380 light-years away and is divisible in large binoculars and all amateur telescopes. Gamma Cygni, traditionally named Sadr, is a yellow-tinged supergiant star of magnitude 2.2, 1500 light-years away. Its traditional name means "breast" and refers to its position in the constellation. Delta Cygni is another bright binary star in Cygnus, 171 light-years with a period of 800 years. The primary is a blue-white hued giant star of magnitude 2.9, and the secondary is a star of magnitude 6.6. The two components are divisible in a medium-sized amateur telescope. The fifth star in Cygnus above magnitude 3 is Gienah, designated Epsilon Cygni. It is an orange-hued giant star of magnitude 2.5, 72 light-years from Earth.^[1]

There are several other dimmer double and binary stars in Cygnus. Mu Cygni is a binary star with an optical tertiary component. The binary system has a period of 790 years and is 73 light-years from Earth. The primary and secondary, both white stars, are of magnitude 4.8 and 6.2, respectively. The unrelated tertiary component is of magnitude 6.9. Though the tertiary component is divisible in binoculars, the primary and secondary currently require a medium-sized amateur telescope to split, as they will through the year 2020. The two stars will be closest between 2043 and 2050, when they will require a telescope with larger aperture to split. Omicron Cygni is a contrasting double star similar to the brighter Albireo. The components, 30 Cygni and 31 Cygni, are divisible in binoculars. The primary, 31 Cygni, is an orange-hued star of magnitude 3.8, 1400 light-years from Earth. The secondary, 30 Cygni, is a blue-green hued star of magnitude 4.8, 720 light-years from Earth. 31 Cygni itself is a binary star; the tertiary component is a blue star of magnitude 7.0. Psi Cygni is a binary star divisible in small amateur telescopes, with two white components. The primary is of magnitude 5.0 and the secondary is of magnitude 7.5. 61 Cygni is a binary star divisible in large binoculars or a small amateur telescope. It is 11.4 light-years from Earth and has a period of 650 years. Both components are orange-hued dwarf (main sequence) stars; the primary is of magnitude 5.2 and the secondary is of magnitude 6.1. 61 Cygni is significant because Friedrich Wilhelm Bessel determined its parallax in 1838, the first star to have a known parallax. 16 Cygni is a binary star 70 light-years from Earth. Both components are white and of the 6th magnitude.^[1]

Cygnus is also home to several variable stars. Chi Cygni is a red giant and the second-brightest Mira variable star at its maximum. Its maximum magnitude is 3.3 and its minimum magnitude is 14; it has a diameter of 300 solar diameters. Chi Cygni has a period of 400 days and is 350 light-years from Earth. P Cygni is a large, unstable blue supergiant that is evolving into a red supergiant. In 1600, it brightened suddenly to 3rd magnitude, though its normal magnitude is approximately 5. It began to brighten gradually in the 1700s and has continued to brighten to the present day; this is part of the process of stellar evolution. W Cygni is a semi-regular variable red giant star, 618 light-years from Earth. It has a maximum magnitude of 5 and a minimum magnitude 8; its period of 130 days.^[1]

There are several asterisms in Cygnus. Patchik 56 is one less prominent, a small asterism that measures 3.3 by 2.2 arcminutes. Named for its discoverer, Dana Patchik, an amateur astronomer, it appears in small amateur telescopes as a group of six stars described as resembling a dolphin.^[2]

 

http://en.wikipedia.org/wiki/Cygnus_(constellation)