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| {{Semireg dual polyhedra db|Semireg dual polyhedron stat table|dCO}}
| | Storm Water Damages due to plenty and various factors such as breakdown of appliances aftermath of storms, breakdown of appliances, blocked empties, damaged pipe joints and other mishaps. This can also occur when your gutter system does not get water away from the house allowing it to run down the basement walls causing water damage. For most homes, the most effective (and sometimes the lowest cost) way for keeping your carpets mold free is hiring a Houston carpet cleaning company. They eventually nestle in the small little porous holes inside our body and begin to reproduce. But water restoration service for personal items ensures that your valuables are cleaned and dried. <br><br>Everything that has been polluted requires directattention to avert the spread of bio-contaminants including bacteria, fungi, viruses and parasites. So it's not just physical damage to the wood itself than can cause problems, but biological growth, too. Hurricanes and also surges can easily eliminate the home, yet specialist water damage service can easily bring back your premises to be able to pre-incident problem. Any building materials that are dried out completely within 48 hours should be removed to at least 24 inches above the level of saturation. In a natural forest or woodland, the trees are heterogeneous. <br><br>This home improvement project will take several days, so don't get in a rush to finish the job. Data loggers can be set to record the temperature, relative humidity, grains, and even dew point as often as you would like. It is often necessary to work on all of the endocrine system at once, giving nutrients for the adrenals, thyroid and sex hormones altogether. Toxic forms of mold can lead to all kinds of allergic reactions for people or pets. Professional companies use technologically advanced devices, processes and equipment to speed up the cleanup and restoration process. <br><br>Before you buy replacement hardwood, check for used floor boards at a salvage store. Also if the installer chips a corner or scratches the surface you will generally need a replacement piece. Now I’m not talking about table salt, which is simply sodium chloride, a chemical, which can cause high blood pressure among other things and offers nothing good for your body. Observation is only about what is, not what you know or don't know about a situation. It is the absence of these plants which causes erosion in the first place. <br><br>Here is more on [http://micahbnll.iloveblog.com recruit] look at our own webpage. Some of the grain will be raised, so don't walk on the flooring barefoot. Divert the water: Long before flood strikes, look for ways to divert the flow of water so that it never makes it into your basement. Category 3 water damage requires swift assessment and treatment. Don't go more than 20 minutes or you may feel groggy. Cattle kept in specific areas can trample the soil, leaving it vulnerable to runoff. |
| In [[geometry]], the '''rhombic dodecahedron''' is a [[convex polyhedron]] with 12 [[congruence (geometry)|congruent]] [[rhombus|rhombic]] [[face (geometry)|faces]]. It has 24 [[edge (geometry)|edges]], and 14 [[vertex (geometry)|vertices]] of two types. It is a [[Catalan solid]], and the [[dual polyhedron]] of the [[cuboctahedron]].
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| ==Properties==
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| The rhombic dodecahedron is a [[zonohedron]]. Its polyhedral [[Dual polyhedron|dual]] is the [[cuboctahedron]]. The long diagonal of each face is exactly [[Square root of two|√2]] times the length of the short diagonal, so that the [[Angle#Types of angles|acute]] angles on each face measure arccos(1/3), or approximately 70.53°.
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| Being the dual of an [[Archimedean solid|Archimedean polyhedron]], the rhombic dodecahedron is [[face-transitive]], meaning the [[symmetry group]] of the solid acts [[transitive action|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 (mathematics)|reflection]] of the solid that leaves it occupying the same region of space while moving face A to face B.
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| The rhombic dodecahedron is one of the nine [[edge-transitive]] convex polyhedra, the others being the five [[Platonic solid]]s, the [[cuboctahedron]], the [[icosidodecahedron]] and the [[rhombic triacontahedron]].
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| [[File:Grenat pyrope 1 .jpg|thumb|left|120px|A [[garnet]] crystal]]
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| The rhombic dodecahedron can be used to [[Tessellation|tessellate]] three-dimensional space. It can be stacked to fill a space much like [[hexagon]]s fill a plane.
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| This tessellation can be seen as the [[Voronoi tessellation]] of the [[Crystal structure|face-centered cubic lattice]]. Some minerals such as [[garnet]] form a rhombic dodecahedral [[crystal habit]]. [[Honeybee]]s 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 [[diamondoid]]s. In these cases, four vertices (alternate threefold ones) are absent, but the chemical bonds lie on the remaining edges.<ref name=cryst1>[http://www.khulsey.com/jewelry/crystal_habit.html#h-5. Dodecahedral Crystal Habit]</ref>
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| The graph of rhombic dodecahedron is [[Hamiltonian path|nonhamiltonian]].
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| ==Dimensions==
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| If the edge length of a rhombic dodecahedron is ''a'', the [[radius]] of an [[inscribed sphere]] ([[tangent]] to each of the rhombic dodecahedron's faces) is
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| :<math>r_i = \frac{\sqrt{6}}{3}a \approx 0.8164965809a,</math> {{OEIS2C|A157697}}
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| the radius of the [[midsphere]] is | |
| :<math>r_m = \frac{2\sqrt{2}}{3}a \approx 0.94280904158a,</math> {{OEIS2C|A179587}}.
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| and the radius of the [[circumscribed sphere]] is
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| :<math>r_o = \frac{2\sqrt{3}}{3}a \approx 1.154700538a,</math> {{OEIS2C|A020832}}.
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| ==Area and volume==
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| The area ''A'' and the volume ''V'' of the rhombic dodecahedron of edge length ''a'' are:
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| :<math>A = 8\sqrt{2}a^2 \approx 11.3137085a^2</math>
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| :<math>V = \frac{16}{9} \sqrt{3}a^3 \approx 3.07920144a^3</math>
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| ==Cartesian coordinates==
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| The eight vertices where three faces meet at their obtuse angles have [[Cartesian coordinates]]:
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| :(±1, ±1, ±1)
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| The coordinates of the six vertices where four faces meet at their acute angles are the permutations of:
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| :(0, 0, ±2)
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| ==Related polyhedra==
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| {{Octahedral truncations}}
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| This polyhedron is a part of a sequence of rhombic polyhedra and tilings with [''n'',3] [[Coxeter group]] symmetry. The cube can be seen as a rhombic hexahedron where the rhombi are squares.
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| {{Quasiregular figure table}}
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| {{Quasiregular4 table}}
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| Similarly it relates to the infinite series of tilings with the [[face configuration]]s V3.2''n''.3.2''n'', the first in the Euclidean plane, and the rest in the hyperbolic plane.
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| {|class="wikitable"
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| |align=center valign=top|[[Image:Rhombicdodecahedron net2.png|210px]]<br>V3.4.3.4<br>(Drawn as a [[Net (polyhedron)|net]])
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| |align=center valign=top|[[Image:Tile V3636.svg|210px]]<br>V3.6.3.6<br>Euclidean plane tiling<br>[[Rhombille tiling]]
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| |align=center valign=top|[[Image:Uniform dual tiling 433-t01.png|210px]]<br>V3.8.3.8<br>Hyperbolic plane tiling<br>(Drawn in a [[Poincaré disk model]])
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| |}
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| ===Stellations===
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| Like many convex polyhedra, the rhombic dodecahedron can be [[stellation|stellated]] by extending the faces or edges until they meet to form a new polyhedron. Several such stellations have been described by Dorman Luke <ref>Luke, D.; Stellations of the rhombic dodecahedron, ''Math. Gaz.'' '''337''' (1957), pp. 189-194.</ref>
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| The first stellation, often simply called the [[stellated rhombic dodecahedron]], is well known. It can be seen as a rhombic dodecahedron with each face augmented by attaching a rhombic-based pyramid to it, with a pyramid height such that the sides lie in the face planes of the neighbouring faces:
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| :[[File:Three flattened octahedra compound.png|240px]]
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| Luke describes four more stellations: the second and third stellations (expanding outwards), one formed by removing the second from the third, and another by adding the original rhombic dodecahedron back to the previous one.
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| ==Honeycomb==
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| The rhombic dodecahedron can [[Rhombic dodecahedral honeycomb|tessellate space by translational copies of itself]]:
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| :[[File:Rhombic dodecahedra.png|none|300px]]
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| ==Related polytopes==
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| [[Image:Hypercubeorder.svg|thumb|In a perfect vertex-first projection two of the [[tesseract]]'s vertices (marked in green) are projected exactly in the center of the rhombic dodecahedron]]
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| The rhombic dodecahedron forms the hull of the vertex-first projection of a [[tesseract]] to three dimensions. There are exactly two ways of decomposing a rhombic dodecahedron into four congruent [[parallelepiped]]s, giving eight possible parallelepipeds. The eight cells of the tesseract under this projection map precisely to these eight parallelepipeds.
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| The rhombic dodecahedron forms the maximal cross-section of a [[24-cell]], and also forms the hull of its vertex-first parallel projection into three dimensions. The rhombic dodecahedron can be decomposed into six congruent (but non-regular) [[square dipyramid]]s meeting at a single vertex in the center; these form the images of six 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.
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| This decomposition gives an interesting method for constructing the rhombic dodecahedron: cut a [[cube]] into six 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 [[tesseract]]s.
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| ==See also==
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| *[[Dodecahedron]]
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| *[[Rhombic triacontahedron]]
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| *[[Rhombille tiling]]
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| *[[Truncated rhombic dodecahedron]]
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| *[[24-cell]] – 4D analog of rhombic dodecahedron
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| *[[Rhombic dodecahedral honeycomb]]
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| ==References==
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| {{reflist}}
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| *{{The Geometrical Foundation of Natural Structure (book)}} (Section 3-9)
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| *{{Citation |last1=Wenninger |first1=Magnus |author1-link=Magnus Wenninger |title=Dual Models |publisher=Cambridge University Press |url=http://books.google.com/?id=mfmzUjhs-_8C&pg=PA19 |isbn=978-0-521-54325-5 |doi=10.1017/CBO9780511569371 |id={{MathSciNet|id=730208}} |year=1983}} (The thirteen semiregular convex polyhedra and their duals, Page 19, Rhombic dodecahedron)
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| *[http://books.google.com/books?id=EtQCk0TNafsC&pg=PA345 ''The Symmetries of Things''] 2008, John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, ISBN 978-1-56881-220-5 (Chapter 21, Naming the Archimedean and Catalan polyhedra and tilings, page 285, Rhombic dodecahedron )
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| ==External links==
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| *{{Mathworld2 |urlname=RhombicDodecahedron |title=Rhombic dodecahedron |urlname2=CatalanSolid |title2=Catalan solid}}
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| *[http://www.georgehart.com/virtual-polyhedra/vp.html Virtual Reality Polyhedra] – The Encyclopedia of Polyhedra
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| ===Computer models===
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| *[http://polyhedra.org/poly/show/33/rhombic_dodecahedron Rhombic Dodecahedron] – interactive 3-d model
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| *[http://demonstrations.wolfram.com/RelatingARhombicTriacontahedronAndARhombicDodecahedron/ Relating a Rhombic Triacontahedron and a Rhombic Dodecahedron], [http://demonstrations.wolfram.com/RhombicDodecahedron5Compound/ Rhombic Dodecahedron 5-Compound] and [http://demonstrations.wolfram.com/RhombicDodecahedron5Compound/ Rhombic Dodecahedron 5-Compound] by Sándor Kabai, [[The Wolfram Demonstrations Project]].
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| ===Paper projects===
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| *[http://www.ii.uib.no/~arntzen/kalender/ Rhombic Dodecahedron Calendar] – make a rhombic dodecahedron calendar without glue
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| *[http://www.southernct.edu/~fields/PlaitedCalendar/ Another Rhombic Dodecahedron Calendar] – made by plaiting paper strips
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| ===Practical applications===
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| *[http://archinstitute.blogspot.com Archimede Institute] Examples of actual housing construction projects using this geometry
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| {{Polyhedron navigator}}
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| [[Category:Catalan solids]]
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| [[Category:Quasiregular polyhedra]]
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| [[Category:Space-filling polyhedra]]
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| [[Category:Zonohedra]]
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Storm Water Damages due to plenty and various factors such as breakdown of appliances aftermath of storms, breakdown of appliances, blocked empties, damaged pipe joints and other mishaps. This can also occur when your gutter system does not get water away from the house allowing it to run down the basement walls causing water damage. For most homes, the most effective (and sometimes the lowest cost) way for keeping your carpets mold free is hiring a Houston carpet cleaning company. They eventually nestle in the small little porous holes inside our body and begin to reproduce. But water restoration service for personal items ensures that your valuables are cleaned and dried.
Everything that has been polluted requires directattention to avert the spread of bio-contaminants including bacteria, fungi, viruses and parasites. So it's not just physical damage to the wood itself than can cause problems, but biological growth, too. Hurricanes and also surges can easily eliminate the home, yet specialist water damage service can easily bring back your premises to be able to pre-incident problem. Any building materials that are dried out completely within 48 hours should be removed to at least 24 inches above the level of saturation. In a natural forest or woodland, the trees are heterogeneous.
This home improvement project will take several days, so don't get in a rush to finish the job. Data loggers can be set to record the temperature, relative humidity, grains, and even dew point as often as you would like. It is often necessary to work on all of the endocrine system at once, giving nutrients for the adrenals, thyroid and sex hormones altogether. Toxic forms of mold can lead to all kinds of allergic reactions for people or pets. Professional companies use technologically advanced devices, processes and equipment to speed up the cleanup and restoration process.
Before you buy replacement hardwood, check for used floor boards at a salvage store. Also if the installer chips a corner or scratches the surface you will generally need a replacement piece. Now I’m not talking about table salt, which is simply sodium chloride, a chemical, which can cause high blood pressure among other things and offers nothing good for your body. Observation is only about what is, not what you know or don't know about a situation. It is the absence of these plants which causes erosion in the first place.
Here is more on recruit look at our own webpage. Some of the grain will be raised, so don't walk on the flooring barefoot. Divert the water: Long before flood strikes, look for ways to divert the flow of water so that it never makes it into your basement. Category 3 water damage requires swift assessment and treatment. Don't go more than 20 minutes or you may feel groggy. Cattle kept in specific areas can trample the soil, leaving it vulnerable to runoff.