The octahedron can also be represented as a spherical tiling, and projected onto the plane via a stereographic projection. This projection is conformal, preserving angles but not areas or lengths. Straight lines on the sphere are projected as circular arcs on the plane.
An octahedron with edge length can be placed with its center at the origin and its vertices on the coordinate axes; the Cartesian coordinates of the vertices are then
In an "x"–"y"–"z" Cartesian coordinate system, the octahedron with center coordinates ("a", "b", "c") and radius "r" is the set of all points ("x", "y", "z") such that
The surface area "A" and the volume "V" of a regular octahedron of edge length "a" are:
Thus the volume is four times that of a regular tetrahedron with the same edge length, while the surface area is twice (because we have 8 rather than 4 triangles).
If an octahedron has been stretched so that it obeys the equation
the formulas for the surface area and volume expand to become
Additionally the inertia tensor of the stretched octahedron is
These reduce to the equations for the regular octahedron when
The interior of the compound of two dual tetrahedra is an octahedron, and this compound, called the stella octangula, is its first and only stellation. Correspondingly, a regular octahedron is the result of cutting off from a regular tetrahedron, four regular tetrahedra of half the linear size (i.e. rectifying the tetrahedron). The vertices of the octahedron lie at the midpoints of the edges of the tetrahedron, and in this sense it relates to the tetrahedron in the same way that the cuboctahedron and icosidodecahedron relate to the other Platonic solids. One can also divide the edges of an octahedron in the ratio of the golden mean to define the vertices of an icosahedron. This is done by first placing vectors along the octahedron's edges such that each face is bounded by a cycle, then similarly partitioning each edge into the golden mean along the direction of its vector. There are five octahedra that define any given icosahedron in this fashion, and together they define a "regular compound".
Octahedra and tetrahedra can be alternated to form a vertex, edge, and face-uniform tessellation of space, called the octet truss by Buckminster Fuller. This is the only such tiling save the regular tessellation of cubes, and is one of the 28 convex uniform honeycombs. Another is a tessellation of octahedra and cuboctahedra.
The octahedron is unique among the Platonic solids in having an even number of faces meeting at each vertex. Consequently, it is the only member of that group to possess mirror planes that do not pass through any of the faces.
Using the standard nomenclature for Johnson solids, an octahedron would be called a "square bipyramid". Truncation of two opposite vertices results in a square bifrustum.
The octahedron is 4-connected, meaning that it takes the removal of four vertices to disconnect the remaining vertices. It is one of only four 4-connected simplicial well-covered polyhedra, meaning that all of the maximal independent sets of its vertices have the same size. The other three polyhedra with this property are the pentagonal dipyramid, the snub disphenoid, and an irregular polyhedron with 12 vertices and 20 triangular faces.
There are 3 uniform colorings of the octahedron, named by the triangular face colors going around each vertex: 1212, 1112, 1111.
The octahedron's symmetry group is O, of order 48, the three dimensional hyperoctahedral group. This group's subgroups include D (order 12), the symmetry group of a triangular antiprism; D (order 16), the symmetry group of a square bipyramid; and T (order 24), the symmetry group of a rectified tetrahedron. These symmetries can be emphasized by different colorings of the faces.
It has eleven arrangements of nets.
The octahedron is the dual polyhedron to the cube.
The uniform tetrahemihexahedron is a tetrahedral symmetry faceting of the regular octahedron, sharing edge and vertex arrangement. It has four of the triangular faces, and 3 central squares.
Rounder Records
Rounder Records is an American record label that formed in Cambridge, Massachusetts in 1970. Since then, it has moved to Burlington, Massachusetts. Over the years, some of the artists and bands that have been signed to the label include Alison Krauss, Ween, and Bela Fleck.
Columbia Records
Columbia Records is an American record label that formed in 1888. It is the oldest existing record company in the world.
Its parent company is now Sony BMG. Over the years, some of the artists and bands that have been signed to the label include Aretha Franklin, System of a Down, Johnny Cash, Bob Dylan, Bela Fleck, Janis Joplin, Paul McCartney, John Lennon, Prince, Willie Nelson, Dolly Parton, and hundreds of other artists.
List of Columbia Records artists
A partial list of artists who have recorded for Columbia Records or EMI's Columbia label include:
Origin of life
The origin of life on Earth is a scientific problem which is not yet solved. There are many ideas, but few clear facts.
Most experts agree that all life today evolved by common descent from a single primitive lifeform. It is not known how this early life form evolved, but scientists think it was a natural process which happened about 3,900 million years ago. This is in accord with the philosophy of naturalism: only natural causes are admitted.
It is not known if metabolism came first or genetics. The main hypothesis which supports genetics first is the RNA world hypothesis, and the one which supports metabolism first is the protein world hypothesis.
Another big problem is how cells developed. Melvin Calvin, winner of the Nobel Prize in Chemistry, wrote a book on the subject, and so did Alexander Oparin. What links most of the early work on the origin of life is the idea that before life began there must have been a process of chemical change. Another question which has been discussed by J.D. Bernal and others is the origin of the cell membrane. By concentrating the chemicals in one place, the cell membrane performs a vital function.
What we call life has only been verified in things that include RNA, mechanisms for encoding and decoding RNA, and mechanisms for building proteins from amino acids. The search for a verifiable theory of biogenesis is a separate field of research.
The earliest claimed lifeforms are fossilized microorganisms (or microfossils). They were found in iron and silica-rich rocks which were once hydrothermal vents in the Nuvvuagittuq greenstone belt of Quebec, Canada.
These rocks are as old as 4.28 billion years. The tubular forms they contain are shown in a report. If this is the oldest record of life on Earth, it suggests "an almost instantaneous emergence of life" after oceans formed 4.4 billion years ago. According to Stephen Blair Hedges, "If life arose relatively quickly on Earth… then it could be common in the universe".
A scientific study from 2002 showed that geological formations of stromatolites 3.45 billion years old contain fossilized cyanobacteria. At the time it was widely agreed that stromatolites were the oldest known lifeforms on Earth which had left a record of its existence. Therefore, if life originated on Earth, this happened sometime between 4.4 billion years ago, when water vapor first liquefied, and 3.5 billion years ago. This is the background to the latest discovery discussed above.