Orbital resonance: Difference between revisions

Revision as of 09:16, 9 January 2014

28 year-old Painting Investments Worker Truman from Regina, usually spends time with pastimes for instance interior design, property developers in new launch ec Singapore and writing. Last month just traveled to City of the Renaissance. Template:Reg polyhedra db In geometry, an octahedron (plural: octahedra) is a polyhedron with eight faces. A regular octahedron is a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex.

A regular octahedron is the dual polyhedron of a cube. It is a rectified tetrahedron. It is a square bipyramid in any of three orthogonal orientations. It is also a triangular antiprism in any of four orientations.

An octahedron is the three-dimensional case of the more general concept of a cross polytope.

Regular octahedron

Dimensions

If the edge length of a regular octahedron is a, the radius of a circumscribed sphere (one that touches the octahedron at all vertices) is

r_{u} = \frac{a}{2} \sqrt{2} \approx 0.7071067 \cdot a

and the radius of an inscribed sphere (tangent to each of the octahedron's faces) is

r_{i} = \frac{a}{6} \sqrt{6} \approx 0.4082482 \cdot a

while the midradius, which touches the middle of each edge, is

r_{m} = \frac{a}{2} = 0.5 \cdot a

Orthogonal projections

The octahedron has four special orthogonal projections, centered, on an edge, vertex, face, and normal to a face. The second and third correspond to the B₂ and A₂ Coxeter planes.

Orthogonal projections
Centered by	Edge	Face Normal	Vertex	Face
Image	File:Cube t2 e.png	File:Cube t2 fb.png	File:3-cube t2 B2.svg	File:3-cube t2.svg
Projective symmetry	[2]	[2]	[4]	[6]

Cartesian coordinates

An octahedron with edge length sqrt(2) can be placed with its center at the origin and its vertices on the coordinate axes; the Cartesian coordinates of the vertices are then

( ±1, 0, 0 );

( 0, ±1, 0 );

( 0, 0, ±1 ).

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

| x - a | + | y - b | + | z - c | = r .

Area and volume

The surface area A and the volume V of a regular octahedron of edge length a are:

A = 2 \sqrt{3} a^{2} \approx 3.46410162 a^{2}

V = \frac{1}{3} \sqrt{2} a^{3} \approx 0.471404521 a^{3}

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 vs. 4 triangles).

If an octahedron has been stretched so that it obeys the equation:

| \frac{x}{x_{m}} | + | \frac{y}{y_{m}} | + | \frac{z}{z_{m}} | = 1

The formula for the surface area and volume expand to become:

A = 4 x_{m} y_{m} z_{m} \times \sqrt{\frac{1}{x_{m}^{2}} + \frac{1}{y_{m}^{2}} + \frac{1}{z_{m}^{2}}}

V = \frac{4}{3} x_{m} y_{m} z_{m}

Additionally the inertia tensor of the stretched octahedron is:

I = [\begin{matrix} \frac{1}{10} m (y_{m}^{2} + z_{m}^{2}) & 0 & 0 \\ 0 & \frac{1}{10} m (x_{m}^{2} + z_{m}^{2}) & 0 \\ 0 & 0 & \frac{1}{10} m (x_{m}^{2} + y_{m}^{2}) \end{matrix}]

These reduce to the equations for the regular octahedron when:

x_{m} = y_{m} = z_{m} = a \frac{\sqrt{2}}{2}

Geometric relations

File:Compound of two tetrahedra.png

The octahedron represents the central intersection of two tetrahedra

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.^[1]

Uniform colorings and symmetry

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_h, of order 48, the three dimensional hyperoctahedral group. This group's subgroups include D_3d (order 12), the symmetry group of a triangular antiprism; D_4h (order 16), the symmetry group of a square bipyramid; and T_d (order 24), the symmetry group of a rectified tetrahedron. These symmetries can be emphasized by different colorings of the faces.

Name	Octahedron	Rectified tetrahedron (Tetratetrahedron)	Triangular antiprism	Square bipyramid	Rhombic fusil
Image (Face coloring)	File:Uniform polyhedron-43-t2.png (1111)	File:Uniform polyhedron-33-t1.png (1212)	File:Trigonal antiprism.png (1112)	File:Square bipyramid.png (1111)
Coxeter-Dynkin	Template:CDD	Template:CDD	Template:CDD Template:CDD	Template:CDD	Template:CDD
Schläfli symbol	{3,4}	t₁{3,3}	s{2,6} sr{2,3}	fs{2,4} { } + {4}	fsr{2,2} { } + { } + { }
Wythoff symbol	4 \| 3 2	2 \| 4 3	2 \| 6 2 \| 2 3 2
Symmetry	O_h, [4,3], (*432)	T_d, [3,3], (*332)	D_3d, [2⁺,6], (2*3) D₃, [2,3]⁺, (322)	D_4h, [2,4], (*422)	D_2h, [2,2], (*222)
Symmetry order	48	24	12 6	16	8

Dual

The octahedron is the dual polyhedron to the cube.

File:Dual Cube-Octahedron.svg

Nets

It has eleven arrangements of nets.

Irregular octahedra

The following polyhedra are combinatorially equivalent to the regular polyhedron. They all have six vertices, eight triangular faces, and twelve edges that correspond one-for-one with the features of a regular octahedron.

Triangular antiprisms: Two faces are equilateral, lie on parallel planes, and have a common axis of symmetry. The other six triangles are isosceles.
Tetragonal bipyramids, in which at least one of the equatorial quadrilaterals lies on a plane. The regular octahedron is a special case in which all three quadrilaterals are planar squares.
Schönhardt polyhedron, a nonconvex polyhedron that cannot be partitioned into tetrahedra without introducing new vertices.

Other convex octahedra

More generally, an octahedron can be any polyhedron with eight faces. The regular octahedron has 6 vertices and 12 edges, the minimum for an octahedron; irregular octahedra may have as many as 12 vertices and 18 edges.^[2] There are 257 topologically distinct convex octahedra, excluding mirror images. More specifically there are 2, 11, 42, 74, 76, 38, 14 for octahedra with 6 to 12 vertices respectively.^[3]^[4] (Two polyhedra are "topologically distinct" if they have intrinsically different arrangements of faces and vertices, such that it is impossible to distort one into the other simply by changing the lengths of edges or the angles between edges or faces.)

Some better known irregular octahedra include the following:

Hexagonal prism: Two faces are parallel regular hexagons; six squares link corresponding pairs of hexagon edges.
Heptagonal pyramid: One face is a heptagon (usually regular), and the remaining seven faces are triangles (usually isosceles). It is not possible for all triangular faces to be equilateral.
Truncated tetrahedron: The four faces from the tetrahedron are truncated to become regular hexagons, and there are four more equilateral triangle faces where each tetrahedron vertex was truncated.
Tetragonal trapezohedron: The eight faces are congruent kites.

Related polyhedra

A regular octahedron can be augmented into a tetrahedron by adding 4 tetrahedra on alternated faces. Adding tetrahedra to all 8 faces creates the stellated octahedron.

tetrahedron	stellated octahedron
File:Triangulated tetrahedron.png	File:Compound of two tetrahedra.png

The octahedron is one of a family of uniform polyhedra related to the cube.

Template:Octahedral truncations

The octahedron is topologically related as a part of sequence of regular polyhedra with Schläfli symbols {3,n}, continuing into the hyperbolic plane. Template:Triangular regular tiling

Tetratetrahedron

The regular octahedron can also be considered a rectified tetrahedron – and can be called a tetratetrahedron. This can be shown by a 2-color face model. With this coloring, the octahedron has tetrahedral symmetry.

Compare this truncation sequence between a tetrahedron and its dual: Template:Tetrahedron family

The above shapes may also be realized as slices orthogonal to the long diagonal of a tesseract. If this diagonal is oriented vertically with a height of 1, then the first five slices above occur at heights r, 3/8, 1/2, 5/8, and s, where r is any number in the range (0,1/4], and s is any number in the range [3/4,1).

The tetratetrahedron can be seen in a sequence of quasiregular polyhedrons and tilings: Template:Quasiregular figure table

Trigonal antiprism

As a trigonal antiprism, the octahedron is related to the hexagonal dihedral symmetry family. Template:Hexagonal dihedral truncations

Template:UniformAntiprisms

Square bipyramid

Template:Bipyramids

Tetrahemihexahedron

The regular octahedron shares its edges and vertex arrangement with one nonconvex uniform polyhedron: the tetrahemihexahedron, with which it shares four of the triangular faces.

File:Octahedron.png Octahedron	File:Tetrahemihexahedron.png Tetrahemihexahedron

Tetrahedral Truss

A framework of repeating tetrahedrons and octahedrons was invented by Buckminster Fuller in the 1950s, known as a space frame, commonly regarded as the strongest structure for resisting cantilever stresses.

Octahedra in the physical world

Octahedra in nature

File:Fluorite octahedron.jpg

Fluorite octahedron.

Natural crystals of diamond, alum or fluorite are commonly octahedral, as the space-filling tetrahedral-octahedral honeycomb.
The plates of kamacite alloy in octahedrite meteorites are arranged paralleling the eight faces of an octahedron.
Many metal ions coordinate six ligands in an octahedral or distorted octahedral configuration.
Widmanstätten patterns in nickel-iron crystals

Octahedra in art and culture

File:Rubiks snake octahedron.jpg

Two identically formed rubik's snakes can approximate an octahedron.

Especially in roleplaying games, this solid is known as a "d8", one of the more common non-cubical dice.
If each edge of an octahedron is replaced by a one ohm resistor, the resistance between opposite vertices is 1/2 ohms, and that between adjacent vertices 5/12 ohms.^[5]
Six musical notes can be arranged on the vertices of an octahedron in such a way that each edge represents a consonant dyad and each face represents a consonant triad; see hexany.

References

↑ One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
↑ [1]
↑ Counting polyhedra
↑ http://www.uwgb.edu/dutchs/symmetry/poly8f0.htm
↑ One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang

External links

Template:Wikisource1911Enc

22 year-old Systems Analyst Rave from Merrickville-Wolford, has lots of hobbies and interests including quick cars, property developers in singapore and baking. Always loves visiting spots like Historic Monuments Zone of Querétaro.

Here is my web site - cottagehillchurch.com
Template:KlitzingPolytopes
Editable printable net of an octahedron with interactive 3D view
Paper model of the octahedron
K.J.M. MacLean, A Geometric Analysis of the Five Platonic Solids and Other Semi-Regular Polyhedra
The Uniform Polyhedra
Virtual Reality Polyhedra The Encyclopedia of Polyhedra

Template:Polyhedra Template:Polyhedron navigator Template:Polytopes 30 year-old Entertainer or Range Artist Wesley from Drumheller, really loves vehicle, property developers properties for sale in singapore singapore and horse racing. Finds inspiration by traveling to Works of Antoni Gaudí.

[1] One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang

[2] [1]

[3] Counting polyhedra

[4] ttp://www.uwgb.edu/dutchs/symmetry/poly8f0.htm

[5] One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang

[1]

[2]

[3]

[4]

[5]

Orbital resonance: Difference between revisions

en>WolfmanSF

Revision as of 09:16, 9 January 2014

Contents

Regular octahedron

Dimensions

Orthogonal projections

Cartesian coordinates

Area and volume

Geometric relations

Uniform colorings and symmetry

Dual

Nets

Irregular octahedra

Other convex octahedra

Related polyhedra

Tetratetrahedron

Trigonal antiprism

Square bipyramid

Tetrahemihexahedron

Tetrahedral Truss

Octahedra in the physical world

Octahedra in nature

Octahedra in art and culture

See also

References

External links

Navigation menu

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+{{For|the album by The Mars Volta|Octahedron (album)}}
+{{Reg polyhedra db|Reg polyhedron stat table|O}}
+In [[geometry]], an '''octahedron''' (plural: octahedra) is a [[polyhedron]] with eight faces.  A '''regular''' octahedron is a [[Platonic solid]] composed of eight [[equilateral triangle]]s, four of which meet at each [[wikt:vertex|vertex]].
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+A regular octahedron is the [[dual polyhedron]] of a [[cube]]. It is a [[Rectification (geometry)|rectified]] [[tetrahedron]]. It is a square [[bipyramid]] in any of three [[orthogonal]] orientations. It is also a triangular [[antiprism]] in any of four orientations.
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+An octahedron is the three-dimensional case of the more general concept of a [[cross polytope]].
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+==Regular octahedron==
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-== On April 15 Hollister Melbourne Fl ==
+===Dimensions===
+If the edge length of a regular octahedron is ''a'', the [[radius]] of a circumscribed [[sphere]] (one that touches the octahedron at all vertices) is
+:<math>r_u = \frac{a}{2} \sqrt{2} \approx 0.7071067 \cdot a</math>
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+and the radius of an inscribed sphere ([[tangent]] to each of the octahedron's faces) is
+:<math>r_i = \frac{a}{6} \sqrt{6}  \approx 0.4082482\cdot a</math>
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+while the midradius, which touches the middle of each edge, is
+:<math>r_m = \frac{a}{2} = 0.5\cdot a</math>
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+===Orthogonal projections===
+The ''octahedron'' has four special [[orthogonal projection]]s, centered, on an edge, vertex, face, and normal to a face. The second and third correspond to the B<sub>2</sub> and A<sub>2</sub> [[Coxeter plane]]s.
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+{|class=wikitable width=480
+|+ Orthogonal projections
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+|-
+!Centered by
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+!Edge
+!Face<BR>Normal
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+!Vertex
+!Face
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+|-
+!Image
- </ul>
+|[[File:Cube t2 e.png|100px]]
+|[[File:Cube t2 fb.png|100px]]
+|[[File:3-cube t2 B2.svg|100px]]
+|[[File:3-cube t2.svg|100px]]
+|- align=center
+!Projective<BR>symmetry
+|[2]
+|[2]
+|[4]
+|[6]
+|}
-== at that. Finally New Balance 993 Custom ==
+===Cartesian coordinates===
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+An octahedron with edge length sqrt(2) can be placed with its center at the origin and its vertices on the coordinate axes; the [[Cartesian coordinates]] of the vertices are then
+: ( ±1, 0, 0 );
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+: ( 0, ±1, 0 );
+: ( 0, 0, ±1 ).
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+In an ''x''–''y''–''z'' [[Cartesian coordinate system]], the octahedron with center [[Coordinate system|coordinates]] (''a'', ''b'', ''c'') and radius ''r'' is the set of all points (''x'', ''y'', ''z'') such that
+:<math>\left|x - a\right| + \left|y - b\right| + \left|z - c\right| = r.</math>
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+===Area and volume===
+The surface area ''A'' and the [[volume]] ''V'' of a regular octahedron of edge length ''a'' are:
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+:<math>A=2\sqrt{3}a^2 \approx 3.46410162a^2</math>
+:<math>V=\frac{1}{3} \sqrt{2}a^3 \approx 0.471404521a^3</math>
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+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 vs. 4 triangles).
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+If an octahedron has been stretched so that it obeys the equation:
+:<math>\left|\frac{x}{x_m}\right|+\left|\frac{y}{y_m}\right|+\left|\frac{z}{z_m}\right| = 1</math>
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- </ul>
-== 1 Nike Jordan 3 ==
+The formula for the surface area and volume expand to become:
+:<math>A=4 \, x_m \, y_m \, z_m \times \sqrt{\frac{1}{x_m^2}+\frac{1}{y_m^2}+\frac{1}{z_m^2}}</math>
+:<math>V=\frac{4}{3}\,x_m\,y_m\,z_m</math>
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+Additionally the inertia tensor of the stretched octahedron is:
+:<math>
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+I =
+\begin{bmatrix}
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+   \frac{1}{10} m (y_m^2+z_m^2) & 0 & 0 \\
+& \frac{1}{10} m (x_m^2+z_m^2) & 0 \\
-   <li>[http://xtu.shutong.me/forum.php?mod=viewthread&tid=676675 http://xtu.shutong.me/forum.php?mod=viewthread&tid=676675]</li>
+& 0 & \frac{1}{10} m (x_m^2+y_m^2)
+\end{bmatrix}
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+</math>
-   <li>[http://enseignement-lsf.com/spip.php?article65#forum18225503 http://enseignement-lsf.com/spip.php?article65#forum18225503]</li>
- </ul>
-== because they are. Celine Handbags Online ==
+These reduce to the equations for the regular octahedron when:
+:<math>x_m=y_m=z_m=a\,\frac{\sqrt{2}}{2}</math>
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+===Geometric relations===
+[[File:Compound of two tetrahedra.png|left|thumb|The octahedron represents the central intersection of two tetrahedra]]
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+The interior of the [[polyhedral compound|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. [[Rectification (geometry)|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 ratio|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''.
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-== NJ Barbour Stockists Perthshire ==
+[[Tetrahedral-octahedral honeycomb|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 [[cube]]s, and is one of the 28 [[convex uniform honeycomb]]s. Another is a tessellation of octahedra and [[cuboctahedra]].
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+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.
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+Using the standard nomenclature for [[Johnson solid]]s, an octahedron would be called a ''square bipyramid''. Truncation of two opposite vertices results in a [[square bifrustum]].
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+The octahedron is [[k-vertex-connected graph|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 polytope|simplicial]] [[well-covered graph|well-covered]] polyhedra, meaning that all of the [[maximal independent set]]s 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.<ref>{{Cite journal
+|last1=Finbow |first1=Arthur S.
-   <li>[http://blacktr.co.vu/bbs/forum.php?mod=viewthread&tid=750399 http://blacktr.co.vu/bbs/forum.php?mod=viewthread&tid=750399]</li>
+|last2=Hartnell |first2=Bert L.
+|last3=Nowakowski |first3=Richard J.
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+|last4=Plummer |first4=Michael D. | author4-link = Michael D. Plummer
+|doi=10.1016/j.dam.2009.08.002
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+|issue=8
+|journal=Discrete Applied Mathematics
- </ul>
+|mr=2602814
+|pages=894–912
+|title=On well-covered triangulations. III
+|volume=158
+|year=2010}}</ref>
+===Uniform colorings and symmetry===
+There are 3 [[uniform coloring]]s of the octahedron, named by the triangular face colors going around each vertex: 1212, 1112, 1111.
+The octahedron's [[symmetry group]] is O<sub>h</sub>, of order 48, the three dimensional [[hyperoctahedral group]].  This group's [[subgroup]]s include D<sub>3d</sub> (order 12), the symmetry group of a triangular [[antiprism]]; '''D<sub>4h</sub>''' (order 16), the symmetry group of a square [[bipyramid]]; and T<sub>d</sub> (order 24), the symmetry group of a [[Octahedron#Tetratetrahedron|rectified tetrahedron]]. These symmetries can be emphasized by different colorings of the faces.
+{| class=wikitable
+!Name
+!Octahedron
+![[Rectification (geometry)|Rectified]] [[tetrahedron]]<BR>(Tetratetrahedron)
+!Triangular [[antiprism]]
+!Square [[bipyramid]]
+!Rhombic fusil
+|- align=center
+!Image<BR>(Face coloring)
+|[[File:Uniform polyhedron-43-t2.png|100px]]<BR>(1111)
+|[[File:Uniform polyhedron-33-t1.png|100px]]<BR>(1212)
+|[[File:Trigonal antiprism.png|100px]]<BR>(1112)
+|[[File:Square bipyramid.png|100px]]<BR>(1111)
+|
+|- align=center
+![[Coxeter-Dynkin diagram|Coxeter-Dynkin]]
+|{{CDD|node_1|3|node|4|node}}
+|{{CDD|node|3|node_1|3|node}}
+|{{CDD|node_h|2x|node_h|6|node}}<BR>{{CDD|node_h|2x|node_h|3|node_h}}
+|{{CDD|node_f1|2|node_f1|4|node}}
+|{{CDD|node_f1|2|node_f1|2|node_f1}}
+|- align=center
+![[Schläfli symbol]]
+|{3,4}
+|t<sub>1</sub>{3,3}
+|s{2,6}<BR>sr{2,3}
+|fs{2,4}<BR>{ } + {4}
+|fsr{2,2}<BR>{ } + { } + { }
+|- align=center
+![[Wythoff symbol]]
+| 4 &#124; 3 2
+| 2 &#124; 4 3
+| 2 &#124; 6 2 <BR> &#124; 2 3 2
+| ||
+|- align=center
+![[List of spherical symmetry groups|Symmetry]]
+|O<sub>h</sub>, [4,3], (*432)
+|T<sub>d</sub>, [3,3], (*332)
+|D<sub>3d</sub>, [2<sup>+</sup>,6], (2*3)<BR>D<sub>3</sub>, [2,3]<sup>+</sup>, (322)
+|D<sub>4h</sub>, [2,4], (*422)
+|D<sub>2h</sub>, [2,2], (*222)
+|- align=center
+!Symmetry order
+|48
+|24
+|12<BR>6
+|16
+|8
+|}
+===Dual===
+The octahedron is the [[dual polyhedron]] to the [[cube]].
+:[[File:Dual Cube-Octahedron.svg|240px]]
+===Nets===
+It has eleven arrangements of [[net (polyhedron)|nets]].
+==Irregular octahedra==
+The following polyhedra are combinatorially equivalent to the regular polyhedron. They all have six vertices, eight triangular faces, and twelve edges that correspond one-for-one with the features of a regular octahedron.
+*''Triangular [[antiprism]]s'': Two faces are equilateral, lie on parallel planes, and have a common axis of symmetry. The other six triangles are isosceles.
+*Tetragonal [[bipyramid]]s, in which at least one of the equatorial quadrilaterals lies on a plane. The regular octahedron is a special case in which all three quadrilaterals are planar squares.
+*[[Schönhardt polyhedron]], a nonconvex polyhedron that cannot be partitioned into tetrahedra without introducing new vertices.
+=== Other convex octahedra ===
+More generally, an octahedron can be any polyhedron with eight faces. The regular octahedron has 6 vertices and 12 edges, the minimum for an octahedron; irregular octahedra may have as many as 12 vertices and 18 edges.<ref>[http://www.uwgb.edu/dutchs/symmetry/polynum0.htm]</ref>
+There are 257 topologically distinct ''convex'' octahedra, excluding mirror images. More specifically there are 2, 11, 42, 74, 76, 38, 14 for octahedra with 6 to 12 vertices respectively.<ref>[http://www.numericana.com/data/polycount.htm Counting polyhedra]</ref><ref>http://www.uwgb.edu/dutchs/symmetry/poly8f0.htm</ref> (Two polyhedra are "topologically distinct" if they have intrinsically different arrangements of faces and vertices, such that it is impossible to distort one into the other simply by changing the lengths of edges or the angles between edges or faces.)
+Some better known irregular octahedra include the following:
+*[[Hexagonal prism]]: Two faces are parallel regular hexagons; six squares link corresponding pairs of hexagon edges.
+*Heptagonal [[Pyramid (geometry)|pyramid]]: One face is a heptagon (usually regular), and the remaining seven faces are triangles (usually isosceles). It is not possible for all triangular faces to be equilateral.
+*[[Truncated tetrahedron]]: The four faces from the tetrahedron are truncated to become regular hexagons, and there are four more equilateral triangle faces where each tetrahedron vertex was truncated.
+*[[Tetragonal trapezohedron]]: The eight faces are congruent [[kite (geometry)|kites]].
+==Related polyhedra==
+A regular octahedron can be augmented into a [[tetrahedron]] by adding 4 tetrahedra on alternated faces. Adding tetrahedra to all 8 faces creates the [[stellated octahedron]].
+{| class=wikitable
+|[[File:Triangulated tetrahedron.png|120px]]
+|[[File:Compound_of_two_tetrahedra.png|120px]]
+|-
+![[tetrahedron]]
+![[stellated octahedron]]
+|}
+The octahedron is one of a family of uniform polyhedra related to the cube.
+{{Octahedral truncations}}
+The octahedron is topologically related as a part of sequence of regular polyhedra with [[Schläfli symbol]]s {3,''n''}, continuing into the [[Hyperbolic space|hyperbolic plane]].
+{{Triangular regular tiling}}
+===Tetratetrahedron===
+The regular octahedron can also be considered a ''[[rectification (geometry)|rectified]] tetrahedron'' – and can be called a ''tetratetrahedron''. This can be shown by a 2-color face model. With this coloring, the octahedron has [[tetrahedral symmetry]].
+Compare this truncation sequence between a tetrahedron and its dual:
+{{Tetrahedron family}} <!-- This template shows too many figures. It needs replacing with the simple set described in the text -->
+The above shapes may also be realized as slices orthogonal to the long diagonal of a [[tesseract]]. If this diagonal is oriented vertically with a height of 1, then the first five slices above occur at heights ''r'', 3/8, 1/2, 5/8, and ''s'', where ''r'' is any number in the range (0,1/4], and ''s'' is any number in the range [3/4,1).
+The tetratetrahedron can be seen in a sequence of [[quasiregular polyhedron]]s and tilings:
+{{Quasiregular figure table}}
+===Trigonal antiprism===
+As a [[antiprism|trigonal antiprism]], the octahedron is related to the hexagonal dihedral symmetry family.
+{{Hexagonal dihedral truncations}}
+{{UniformAntiprisms}}
+===Square bipyramid===
+{{Bipyramids}}
+===Tetrahemihexahedron===
+The regular octahedron shares its edges and vertex arrangement with one [[nonconvex uniform polyhedron]]: the [[tetrahemihexahedron]], with which it shares four of the triangular faces.
+{| class="wikitable" width="400" style="vertical-align:top;text-align:center"
+| [[Image:octahedron.png|100px]]<br>Octahedron
+| [[Image:tetrahemihexahedron.png|100px]]<br>[[Tetrahemihexahedron]]
+|}
+===Tetrahedral Truss===
+A framework of repeating tetrahedrons and octahedrons was invented by [[Buckminster Fuller]] in the 1950s, known as a [[space frame]], commonly regarded as the strongest structure for resisting [[cantilever]] stresses.
+==Octahedra in the physical world==
+===Octahedra in nature===
+[[Image:Fluorite octahedron.jpg|thumb|[[Fluorite]] octahedron.]]
+*Natural crystals of [[diamond]], [[alum]] or [[fluorite]] are commonly octahedral, as the space-filling [[tetrahedral-octahedral honeycomb]].
+*The plates of [[kamacite]] alloy in [[octahedrite]] [[meteorites]] are arranged paralleling the eight faces of an octahedron.
+*Many metal ions [[Coordination chemistry|coordinate]] six ligands in an octahedral or [[Jahn-Teller effect|distorted]] octahedral configuration.
+* [[Widmanstätten pattern]]s in [[nickel]]-[[iron]] [[crystal]]s
+===Octahedra in art and culture===
+[[Image:Rubiks snake octahedron.jpg|thumb|Two identically formed [[rubik's snake]]s can approximate an octahedron.]]
+*Especially in [[roleplaying game]]s, this solid is known as a "d8", one of the more common [[dice#Non-cubical dice|non-cubical dice]].
+*If each edge of an octahedron is replaced by a one [[ohm (unit)|ohm]] [[resistor]], the resistance between opposite vertices is 1/2 ohms, and that between adjacent vertices 5/12 ohms.<ref>{{cite journal |last= Klein |first=Douglas J. |year=2002 |title=Resistance-Distance Sum Rules |journal=Croatica Chemica Acta |volume=75 |issue=2 |pages=633–649 |url=http://jagor.srce.hr/ccacaa/CCA-PDF/cca2002/v75-n2/CCA_75_2002_633_649_KLEIN.pdf |format=PDF |accessdate=2006-09-30}}</ref>
+*Six musical notes can be arranged on the vertices of an octahedron in such a way that each edge represents a consonant dyad and each face represents a consonant triad; see [[hexany]].
+==See also==
+*[[:Image:Octahedron.gif|Spinning octahedron]]
+*[[Stella octangula]]
+*[[Triakis octahedron]]
+*[[Disdyakis dodecahedron|Hexakis octahedron]]
+*[[Truncated octahedron]]
+*[[Octahedral molecular geometry]]
+*[[Octahedral symmetry]]
+*[[Octahedral graph]]
+==References==
+<!--See [[Wikipedia:Footnotes]] for instructions.-->
+<references/>
+==External links==
+{{Wikisource1911Enc}}
+*{{mathworld |urlname=Octahedron |title=Octahedron}}
+*{{KlitzingPolytopes|polyhedra.htm|3D convex uniform polyhedra|x3o4o - oct}}
+*[http://www.dr-mikes-math-games-for-kids.com/polyhedral-nets.html?net=1S2GJRuqXD7blH9ixbn1mPoTSPo6vkjWddm7xoNxFj&name=Octahedron#applet Editable printable net of an octahedron with interactive 3D view]
+*[http://www.software3d.com/Octahedron.php Paper model of the octahedron]
+*[http://www.kjmaclean.com/Geometry/GeometryHome.html K.J.M. MacLean, A Geometric Analysis of the Five Platonic Solids and Other Semi-Regular Polyhedra]
+*[http://www.mathconsult.ch/showroom/unipoly/ The Uniform Polyhedra]
+*[http://www.georgehart.com/virtual-polyhedra/vp.html Virtual Reality Polyhedra] The Encyclopedia of Polyhedra
+{{Polyhedra}}
+{{Polyhedron navigator}}
+{{Polytopes}}
+{{Use dmy dates|date=December 2010}}
+[[Category:Deltahedra]]
+[[Category:Platonic solids]]
+[[Category:Prismatoid polyhedra]]
+[[Category:Pyramids and bipyramids]]

Orbital resonance: Difference between revisions

en>WolfmanSF

Revision as of 09:16, 9 January 2014

Regular octahedron

Dimensions

Orthogonal projections

Cartesian coordinates

Area and volume

Geometric relations

Uniform colorings and symmetry

Dual

Nets

Irregular octahedra

Other convex octahedra

Related polyhedra

Tetratetrahedron

Trigonal antiprism

Square bipyramid

Tetrahemihexahedron

Tetrahedral Truss

Octahedra in the physical world

Octahedra in nature

Octahedra in art and culture

See also

References

External links

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