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{{Semireg dual polyhedra db|Semireg dual polyhedron stat table|dCO}}
Presently Aupie Australia Sheepskin UGG boots have come to be fashionable footwears. Made of one hundred% sheepskin, these Aupie UGG boots retain your feet cozy and dry. The Australians also employed to don them in home during mild winters. Even so, now Uggs are a craze all around the earth, and has proved to be a hit in both warm and cold nations around the world alike.<br>Higher top quality sheepskin acknowledged as 'twin-faced' is made use of in production Ugg boots. This materials aids to continue to keep the toes dry. Considering the fact that sheepskin insulates the boot will help retain bare ft warm even in temperatures beneath 30F. Sturdy stitches and sole of these boots along with the h2o resistant property of sheepskin make this boot long lasting. These boots despite the fact that appears to look weighty but are quite gentle. Strolling all around in an Aupie ugg boot will make a person really feel like as if he/she is in slippers or just the socks. No make any difference, what your foot size is, you can get a pair of Aupie Australia Sheepskin Ugg boots for yourself.<br>Aupie Australia Sheepskin Ugg boots commonly has a various look from other sorts of boots. It is in fact this unique glance that can make it stand out in the crowd. Therefore 'ugg boot' in simple fact meant 'ugly boot'. But today these boots are no extended regarded as unsightly, they are additionally regarded as as one particular of the most fashionable and comfy footwear in the manner field. Just imagine strolling miles on an undulated terrain in a pair of higher heels! What would be the end result? You would have blisters in the dimension of baseball. No subject how much you walk, you will generally experience comfortable putting on an Aupie Australia Sheepskin Ugg boots.<br>Individuals in particular females and teen women have grow to be extremely style acutely aware at present. Not only do they interesting clothing but also loves placing on modern and relaxed Aupie Australia Sheepskin UGG Boots. This is a point the explanation why women have turned in direction of uggs for boots. A single can obtain a great wide range of types and colors when choosing the appropriate size shoes for you. Aupie Australia Sheepskin Ugg Boots for females arrive in classic colours, unique sizes and styles. Children also look extremely sensible in Aupie Australia Sheepskin Uggs. That is to say, no subject you are a male, a woman or a boy or girl, Aupie UGG boots can be your most effective close friends.If you are an Aupie Australia Sheepskin UGG boots admirer, you should know that Aupie Australian Sheepskin Ug boots have usually been identified as Ug, Ugh or Ugg boots. This sort of types of UGG boots give individuals various feelings ahead of and following persons try them on.<br>The authentic Aupie UGG Boot is not a model title but an age previous generic time period for this kind of model of Australian-created sheepskin boots. In Aussie slang, the Ug title is brief for "hideous." In conditions of consolation, nonetheless, the Aupie Ug Boot is a matter of beauty. The softness of the Australian Merino sheepskin makes a boot with a cosy, cozy, sort-fitting truly feel thats extra like a sock than a shoe, yet its rugged adequate for out of doors wear. The fleece lining has the astonishing property of furnishing year-spherical comfort. In cold temperature, the plush fleece delivers an insulating layer of heat by trapping your system warmth, significantly like goose down does. But in the heat of summer months, the normal fibers of the fleece in fact great your toes by wicking away perspiration.<br>From the seashores of Western Australia, Ugs had been shortly observed on the toes of East Coast surfers from Brisbane to Sydney. And it wasnt prolonged ahead of some of these surfers -- the ones who traded their surfboards for skis in the winter -- observed that their Ug Boots ended up just as at dwelling in the ski vacation resort parts of the Snowy Mountains as they were on the heat sands of Sydneys Bondi Seashore.<br>Today the Ug "secret" is out. Youll locate these flexible boots to be in trend on beaches from San Diego to Santa Cruz and in ski resorts from Tahoe to Vail. And right now, you dont have to "know somebody" in Australia who will deliver you a pair, due to the fact there are now a selection of companies importing boots of this variety. But, if you want the legitimate post, you do have to know what youre searching for for the reason that, no make any difference how they spell it -- Ug, Ugg or Ugh -- there are quite a few importers with glimpse-alike <br>This is a superb tale about Aupie Sheepskin Australia UGG boots. After we know its heritage of Aupie UGG boots, I assume we will like Aupie UGG boots far more and much more.<br><br>When you loved this information and you want to receive details with regards to [http://tinyurl.com/l2bb8j2 http://tinyurl.com/l2bb8j2] kindly visit the web-page.
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]].
 
==Properties==
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°.
 
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.
 
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]].
 
[[File:Grenat pyrope 1 .jpg|thumb|left|120px|A [[garnet]] crystal]]
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.
 
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>
 
The graph of rhombic dodecahedron is [[Hamiltonian path|nonhamiltonian]].
 
==Dimensions==
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
:<math>r_i = \frac{\sqrt{6}}{3}a \approx 0.8164965809a,</math> {{OEIS2C|A157697}}
 
the radius of the [[midsphere]] is
:<math>r_m = \frac{2\sqrt{2}}{3}a \approx 0.94280904158a,</math> {{OEIS2C|A179587}}.
 
and the radius of the [[circumscribed sphere]] is
:<math>r_o = \frac{2\sqrt{3}}{3}a \approx 1.154700538a,</math> {{OEIS2C|A020832}}.
 
==Area and volume==
The area ''A'' and the volume ''V'' of the rhombic dodecahedron of edge length ''a'' are:
:<math>A = 8\sqrt{2}a^2 \approx 11.3137085a^2</math>
:<math>V = \frac{16}{9} \sqrt{3}a^3 \approx 3.07920144a^3</math>
 
==Cartesian coordinates==
The eight vertices where three faces meet at their obtuse angles have [[Cartesian coordinates]]:
:(±1, ±1, ±1)
 
The coordinates of the six vertices where four faces meet at their acute angles are the permutations of:
:(0, 0, ±2)
 
==Related polyhedra==
 
{{Octahedral truncations}}
 
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.
{{Quasiregular figure table}}
{{Quasiregular4 table}}
 
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.
{|class="wikitable"
|align=center valign=top|[[Image:Rhombicdodecahedron net2.png|210px]]<br>V3.4.3.4<br>(Drawn as a [[Net (polyhedron)|net]])
|align=center valign=top|[[Image:Tile V3636.svg|210px]]<br>V3.6.3.6<br>Euclidean plane tiling<br>[[Rhombille tiling]]
|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]])
|}
 
===Stellations===
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>
 
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:
:[[File:Three flattened octahedra compound.png|240px]]
 
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.
 
==Honeycomb==
The rhombic dodecahedron can [[Rhombic dodecahedral honeycomb|tessellate space by translational copies of itself]]:
:[[File:Rhombic dodecahedra.png|none|300px]]
 
==Related polytopes==
[[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]]
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.
 
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.
 
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.
 
==See also==
*[[Dodecahedron]]
*[[Rhombic triacontahedron]]
*[[Rhombille tiling]]
*[[Truncated rhombic dodecahedron]]
*[[24-cell]] – 4D analog of rhombic dodecahedron
*[[Rhombic dodecahedral honeycomb]]
 
==References==
{{reflist}}
 
*{{The Geometrical Foundation of Natural Structure (book)}} (Section 3-9)
*{{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)
*[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 )
 
==External links==
*{{Mathworld2 |urlname=RhombicDodecahedron |title=Rhombic dodecahedron |urlname2=CatalanSolid |title2=Catalan solid}}
*[http://www.georgehart.com/virtual-polyhedra/vp.html Virtual Reality Polyhedra] – The Encyclopedia of Polyhedra
 
===Computer models===
*[http://polyhedra.org/poly/show/33/rhombic_dodecahedron Rhombic Dodecahedron] – interactive 3-d model
*[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]].
 
===Paper projects===
*[http://www.ii.uib.no/~arntzen/kalender/ Rhombic Dodecahedron Calendar] – make a rhombic dodecahedron calendar without glue
*[http://www.southernct.edu/~fields/PlaitedCalendar/ Another Rhombic Dodecahedron Calendar] – made by plaiting paper strips
 
===Practical applications===
*[http://archinstitute.blogspot.com Archimede Institute] Examples of actual housing construction projects using this geometry
 
{{Polyhedron navigator}}
 
[[Category:Catalan solids]]
[[Category:Quasiregular polyhedra]]
[[Category:Space-filling polyhedra]]
[[Category:Zonohedra]]

Revision as of 07:26, 22 February 2014

Presently Aupie Australia Sheepskin UGG boots have come to be fashionable footwears. Made of one hundred% sheepskin, these Aupie UGG boots retain your feet cozy and dry. The Australians also employed to don them in home during mild winters. Even so, now Uggs are a craze all around the earth, and has proved to be a hit in both warm and cold nations around the world alike.
Higher top quality sheepskin acknowledged as 'twin-faced' is made use of in production Ugg boots. This materials aids to continue to keep the toes dry. Considering the fact that sheepskin insulates the boot will help retain bare ft warm even in temperatures beneath 30F. Sturdy stitches and sole of these boots along with the h2o resistant property of sheepskin make this boot long lasting. These boots despite the fact that appears to look weighty but are quite gentle. Strolling all around in an Aupie ugg boot will make a person really feel like as if he/she is in slippers or just the socks. No make any difference, what your foot size is, you can get a pair of Aupie Australia Sheepskin Ugg boots for yourself.
Aupie Australia Sheepskin Ugg boots commonly has a various look from other sorts of boots. It is in fact this unique glance that can make it stand out in the crowd. Therefore 'ugg boot' in simple fact meant 'ugly boot'. But today these boots are no extended regarded as unsightly, they are additionally regarded as as one particular of the most fashionable and comfy footwear in the manner field. Just imagine strolling miles on an undulated terrain in a pair of higher heels! What would be the end result? You would have blisters in the dimension of baseball. No subject how much you walk, you will generally experience comfortable putting on an Aupie Australia Sheepskin Ugg boots.
Individuals in particular females and teen women have grow to be extremely style acutely aware at present. Not only do they interesting clothing but also loves placing on modern and relaxed Aupie Australia Sheepskin UGG Boots. This is a point the explanation why women have turned in direction of uggs for boots. A single can obtain a great wide range of types and colors when choosing the appropriate size shoes for you. Aupie Australia Sheepskin Ugg Boots for females arrive in classic colours, unique sizes and styles. Children also look extremely sensible in Aupie Australia Sheepskin Uggs. That is to say, no subject you are a male, a woman or a boy or girl, Aupie UGG boots can be your most effective close friends.If you are an Aupie Australia Sheepskin UGG boots admirer, you should know that Aupie Australian Sheepskin Ug boots have usually been identified as Ug, Ugh or Ugg boots. This sort of types of UGG boots give individuals various feelings ahead of and following persons try them on.
The authentic Aupie UGG Boot is not a model title but an age previous generic time period for this kind of model of Australian-created sheepskin boots. In Aussie slang, the Ug title is brief for "hideous." In conditions of consolation, nonetheless, the Aupie Ug Boot is a matter of beauty. The softness of the Australian Merino sheepskin makes a boot with a cosy, cozy, sort-fitting truly feel thats extra like a sock than a shoe, yet its rugged adequate for out of doors wear. The fleece lining has the astonishing property of furnishing year-spherical comfort. In cold temperature, the plush fleece delivers an insulating layer of heat by trapping your system warmth, significantly like goose down does. But in the heat of summer months, the normal fibers of the fleece in fact great your toes by wicking away perspiration.
From the seashores of Western Australia, Ugs had been shortly observed on the toes of East Coast surfers from Brisbane to Sydney. And it wasnt prolonged ahead of some of these surfers -- the ones who traded their surfboards for skis in the winter -- observed that their Ug Boots ended up just as at dwelling in the ski vacation resort parts of the Snowy Mountains as they were on the heat sands of Sydneys Bondi Seashore.
Today the Ug "secret" is out. Youll locate these flexible boots to be in trend on beaches from San Diego to Santa Cruz and in ski resorts from Tahoe to Vail. And right now, you dont have to "know somebody" in Australia who will deliver you a pair, due to the fact there are now a selection of companies importing boots of this variety. But, if you want the legitimate post, you do have to know what youre searching for for the reason that, no make any difference how they spell it -- Ug, Ugg or Ugh -- there are quite a few importers with glimpse-alike
This is a superb tale about Aupie Sheepskin Australia UGG boots. After we know its heritage of Aupie UGG boots, I assume we will like Aupie UGG boots far more and much more.

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