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| [[Image:Dual Cube-Octahedron.svg|thumb|The dual of a [[cube]] is an [[octahedron]], shown here with vertices at the cube face centers.]]
| | BMI is brief for "body mass index". It's calculated from a person's fat in kilograms plus height in meters. The easiest method to determine BMI is to employ an online calculator where you connect your weight and height. A regular BMI is between 18.5 and 24.9, whilst 25 to 29.9 is considered to be obese. When you hit a BMI of 30, you're fat. On the additional hand, these values are less exact in athletic persons whom have more lean body mass plus people who have large frames.<br><br>People following the 3 day cardiac diet may probably feel hunger pangs plus have lower vitality degrees. This really is specifically true by the 3rd day. But, keep in your mind that the diet is of limited length plus you need to be fine unless we have stamina based bodily feats to perform.<br><br>As of 2010 the percentage of Americans with obesity is over 35% plus steadily growing. In 1985 lower than 15% of Americans were obese. Obesity is defined has having a BMI (Body Mass Index) of 30 or higher. For someone that is 5'9" which is 203 pounds or higher. For someone which is 5'5" which is regarding 180 pounds or higher.<br><br>It is significant to note which fat is regarded as the most crucial factors associated to many diseases. Other important factors that ought to be taken into consideration whilst assessing the dangers of chronic illnesses include exercise, blood stress, blood sugar level, and diet to name only a few. BMI indicates the total body fat of an individual, which is calculated by a [http://safedietplans.com/bmi-calculator bmi calculator men]. The calculator requires 2 pieces of information - fat in kilograms or pounds and height inside feet or centimeters.<br><br>However possibly the researches have noticed anything significant. In the past, big was breathtaking. Overweight folks were considered more attractive, more affluent, and healthier. Since the advent of the camera, which supposedly "adds ten pounds," skinny folks have become the modern idols to emulate. In Quentin Tarantino's Pulp Fiction, a female says, "it's unfortunate what we find pleasing to the touch plus pleasing to the eye is seldom the same." Is which regarding to change?<br><br>We can additionally calculate the waist-hip ratio (WHR) by dividing your waist measurement by hip measuring. Studies show which the WHR of women of any build strongly affects the perception of bodily attractiveness over all creeds and cultures. Studies show that the most attractive females (according to men!) are those who have 0.7 WHR (waist circumference is 70% of the cool circumference). Do we understand that super stars Marilyn Monroe, Sophia Loren and the Venus de Milo had WHRs about 0.7?<br><br>If you don't fall inside the general range, then receive yourself checked with alternative (omit) methods to figure out the amount of body fat. This usually provide a greater perspective and enable we to do or keep up a ideal weight. |
| [[Image:Birectified cube sequence.png|thumb|[[Truncation (geometry)|Truncation]] sequence from a [[cube]] to its dual [[octahedron]]. A polyhedral dual is called a ''face-rectification'' or a ''[[Rectification (geometry)|birectification]]''.]]
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| In [[geometry]], '''[[polyhedron|polyhedra]]''' are associated into pairs called '''''duals''''', where the [[wikt:vertex|vertices]] of one correspond to the [[face (geometry)|face]]s of the other. The dual of the dual is the original polyhedron. The dual of a polyhedron with equivalent vertices is one with equivalent faces, and of one with equivalent edges is another with equivalent edges. So the regular polyhedra — the [[Platonic solid]]s and [[Kepler-Poinsot polyhedra]] — are arranged into dual pairs, with the exception of the regular [[tetrahedron]] which is [[#Self-dual polyhedra|self-dual]].
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| Duality is also sometimes called ''reciprocity'' or ''polarity''.
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| ==Kinds of duality==
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| There are many kinds of duality. The kinds most relevant to polyhedra are:
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| *Polar reciprocity
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| *Topological duality
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| *Abstract duality
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| ===Polar reciprocation===
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| {{see also|Polar reciprocation}}
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| Duality is most commonly defined in terms of [[polar reciprocation]] about a concentric sphere. Here, each vertex (pole) is associated with a face plane (polar plane or just polar) so that the ray from the center to the vertex is perpendicular to the plane, and the product of the distances from the center to each is equal to the square of the radius. In coordinates, for reciprocation about the sphere
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| :<math>x^2 + y^2 + z^2 = r^2,</math>
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| the vertex
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| :<math>(x_0, y_0, z_0)</math>
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| is associated with the plane | |
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| :<math>x_0x + y_0y + z_0z = r^2</math>.
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| The vertices of the dual, then, are the poles reciprocal to the face planes of the original, and the faces of the dual lie in the polars reciprocal to the vertices of the original. Also, any two adjacent vertices define an edge, and these will reciprocate to two adjacent faces which intersect to define an edge of the dual.
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| Notice that the exact form of the dual will depend on what sphere we reciprocate with respect to; as we move the sphere around, the dual form distorts. The choice of center (of the sphere) is sufficient to define the dual up to similarity. If multiple symmetry axes are present, they will necessarily intersect at a single point, and this is usually taken to be the center. Failing that a circumscribed sphere, inscribed sphere, or midsphere (one with all edges as tangents) can be used.
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| If a polyhedron has an element passing through the center of the sphere, the corresponding element of its dual will go to infinity. Since traditional "Euclidean" space never reaches infinity, the projective equivalent, called extended Euclidean space, must be formed by adding the required 'plane at infinity'. Some theorists prefer to stick to Euclidean space and say that there is no dual. Meanwhile Wenninger (1983) found a way to represent these infinite duals, in a manner suitable for making models (of some finite portion!).
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| The concept of ''duality'' here is closely related to the [[duality (projective geometry)|duality]] in [[projective geometry]], where lines and edges are interchanged; in fact it is often mistakenly taken to be a particular version of the same. Projective polarity works well enough for convex polyhedra. But for non-convex figures such as star polyhedra, when we seek to rigorously define this form of polyhedral duality in terms of projective polarity, various problems appear. See for example Grünbaum & Shepherd (1988), and Gailiunas & Sharp (2005). Wenninger (1983) also discusses some issues on the way to deriving his infinite duals.
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| ====Canonical duals====
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| Any convex polyhedron can be distorted into a canonical form, in which a [[midsphere]] (or intersphere) exists tangent to every edge, such that the average position of these points is the center of the sphere, and this form is unique up to congruences.
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| If we reciprocate such a polyhedron about its intersphere, the dual polyhedron will share the same edge-tangency points and so must also be canonical; it is the canonical dual, and the two together form a canonical dual compound.
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| ===Topological duality===
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| We can distort a dual polyhedron such that it can no longer be obtained by reciprocating the original in any sphere; in this case we can say that the two polyhedra are still topologically dual.
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| It is worth noting that the vertices and edges of a convex polyhedron can be projected to form a [[graph theory|graph]] (sometimes called a [[Schlegel diagram]]) on the sphere or on a flat plane, and the corresponding graph formed by the dual of this polyhedron is its [[dual graph]].
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| ===Abstract duality===
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| An [[Abstract polytope|abstract polyhedron]] is a certain kind of [[partially ordered set]] (poset) of elements, such that adjacencies, or connections, between elements of the set correspond to adjacencies between elements (faces, edges, etc.) of a polyhedron. Such a poset may be represented in a [[Hasse diagram]]. Any such poset has a dual poset. The Hasse diagram of the dual polyhedron is obtained very simply, by turning the original diagram upside-down.
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| ==Dorman Luke construction==
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| For a [[uniform polyhedron]], the face of the dual polyhedron may be found from the original polyhedron's [[vertex figure]] using the '''Dorman Luke''' construction. This construction was originally described by Cundy & Rollett (1961) and later generalised by Wenninger (1983).
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| As an example, here is the vertex figure (red) of the [[cuboctahedron]] being used to derive a face (blue) of the [[rhombic dodecahedron]].
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| [[Image:DormanLuke.svg]]
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| Before beginning the construction, the [[vertex figure]] ''ABCD'' is obtained by cutting each connected edge at (in this case) its midpoint.
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| Dorman Luke's construction then proceeds:
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| :#Draw the vertex figure ''ABCD''
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| :#Draw the circumcircle (tangent to every corner ''A'', ''B'', ''C'' and ''D'').
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| :#Draw lines tangent to the circumcircle at each corner ''A'', ''B'', ''C'', ''D''.
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| :#Mark the points ''E'', ''F'', ''G'', ''H'', where each tangent line meets the adjacent tangent.
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| :#The polygon ''EFGH'' is a face of the dual polyhedron.
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| In this example the size of the vertex figure was chosen so that its circumcircle lies on the [[intersphere]] of the cuboctahedron, which also becomes the intersphere of the dual rhombic dodecahedron.
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| Dorman Luke's construction can only be used where a polyhedron has such an intersphere and the vertex figure is cyclic, i.e. for [[uniform polyhedron|uniform polyhedra]].
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| == Self-dual polyhedra ==
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| A '''self-dual polyhedron''' is a [[polyhedron]] whose dual is a congruent figure, though not necessarily the identical figure: for example, the dual of a regular tetrahedron is a regular tetrahedron "facing the opposite direction" ([[reflection through the origin|reflected through the origin]]).
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| A self-dual polyhedron must have the same number of vertices as faces. We can distinguish between structural ([[Topology|topological]]) duality and [[Geometry|geometrical]] duality. The topological structure of a self-dual polyhedron is also self-dual. Whether or not such a polyhedron is also geometrically self-dual will depend on the particular geometrical duality being considered. For example, every polygon is ''topologically'' self-dual (it has the same number of vertices as edges, and these are switched by duality), but will not in general be ''geometrically'' self-dual (up to rigid motion, for instance) – regular polygons are geometrically self-dual (all angles are congruent, as are all edges, so under duality these congruences swap), but irregular polygons may not be geometrically self-dual.
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| The most common geometric arrangement is where some convex polyhedron is in its canonical form, which is to say that the all its edges must be tangent to a certain sphere whose centre coincides with the centre of gravity (average position) of the tangent points. If the polar reciprocal of the canonical form in the sphere is congruent to the original, then the figure is self-dual.
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| There are infinitely many self-dual polyhedra. The simplest infinite family are the [[Pyramid (geometry)|pyramids]] of ''n'' sides and of canonical form. Another infinite family, [[elongated pyramid]]s, consists of polyhedra that can be roughly described as a pyramid sitting on top of a [[prism (geometry)|prism]] (with the same number of sides). Add a frustum (pyramid with the top cut off) below the prism and you get another infinite family, and so on.
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| There are many other convex, self-dual polyhedra. For example, there are 6 different ones with 7 vertices, and 16 with 8 vertices. <ref>3D [[Java]] models at [http://dmccooey.com/polyhedra/SymmetricSelfDuals.html Symmetries of Canonical Self-Dual Polyhedra], based on paper by Gunnar Brinkmann, Brendan D. McKay, ''Fast generation of planar graphs'' [[PDF]] [http://cs.anu.edu.au/~bdm/papers/plantri-full.pdf]</ref>
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| Non-convex self-dual polyhedra can also be found, such as the [[excavated dodecahedron]].
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| {| class=wikitable
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| |+ [[Pyramid (geometry)|Family of pyramids]]
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| |- align=center
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| |[[Image:Tetrahedron.jpg|100px]]<BR>[[Tetrahedron|3]]
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| |[[Image:Square pyramid.png|100px]]<BR>[[Square pyramid|4]]
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| |[[Image:Pentagonal pyramid.png|100px]]<BR>[[Pentagonal pyramid|5]]
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| |[[Image:Hexagonal pyramid.png|100px]]<BR>[[Hexagonal pyramid|6]]
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| |}
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| {| class=wikitable
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| |+ Family of [[elongated pyramid]]s
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| |- align=center
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| |[[Image:Elongated triangular pyramid.png|100px]]<BR>[[Elongated triangular pyramid|3]]
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| |[[Image:Elongated square pyramid.png|100px]]<BR>[[Elongated square pyramid|4]]
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| |[[Image:Elongated pentagonal pyramid.png|100px]]<BR>[[Elongated pentagonal pyramid|5]]
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| |}
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| {| class="wikitable"
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| |+ Family of [[diminished trapezohedron|diminished trapezohedra]]
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| |- align=center
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| | [[Image:Diminished trigonal trapezohedron.png|80px]]<BR>3
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| | [[Image:Diminished square trapezohedron.png|80px]]<BR>4
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| | [[Image:Diminished pentagonal trapezohedron.png|80px]]<BR>5
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| | [[Image:Diminished hexagonal trapezohedron.png|80px]]<BR>6
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| | [[Image:Diminished heptagonal trapezohedron.png|80px]]<BR>7
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| |}
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| === Self dual compound polyhedra ===
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| The [[Stella octangula]], being a compound of two [[tetrahedron|tetrahedra]] is also self-dual, as well as four other [[Polyhedral compound#Dual-regular_compounds|regular-dual compounds]].
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| == Dual polytopes and tessellations ==
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| Duality can be generalized to ''n''-dimensional space and '''dual [[polytope]]s;''' in 2-dimensions these are called [[dual polygon]]s.
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| The vertices of one polytope correspond to the (''n'' − 1)-dimensional elements, or facets, of the other, and the ''j'' points that define a (''j'' − 1)-dimensional element will correspond to ''j'' hyperplanes that intersect to give a (''n'' − ''j'')-dimensional element. The dual of a [[Honeycomb (geometry)|honeycomb]] can be defined similarly.
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| In general, the facets of a polytope's dual will be the topological duals of the polytope's vertex figures. For [[regular polytope|regular]] and [[Uniform polytope|uniform]] polytopes, the dual facets will be the polar reciprocals of the original's facets. For example, in four dimensions, the vertex figure of the [[600-cell]] is the [[icosahedron]]; the dual of the 600-cell is the [[120-cell]], whose facets are [[dodecahedron|dodecahedra]], which are the dual of the icosahedron. | |
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| === Self-dual polytopes and tessellations ===
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| [[File:Kah 4 4.png|thumb|The [[square tiling]] is self-dual, as shown by these red and blue tilings]]
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| The primary class of self-dual polytopes are [[regular polytope]]s with [[palindromic]] [[Schläfli symbol]]s. All regular polygons, {a} are self-dual, [[polyhedron|polyhedra]] of the form {a,a}, [[polychoron|4-polytopes]] of the form {a,b,a}, [[5-polytope]]s of the form {a,b,b,a}, etc.
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| The self-dual regular polytopes are:
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| * All [[regular polygon]]s, {a}.
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| * Regular [[tetrahedron]]: {3,3}
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| * In general, all regular ''n''-[[simplex]]es, {3,3,...,3}
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| * The regular [[24-cell]] in 4 dimensions, {3,4,3}.
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| The self-dual (infinite) regular Euclidean [[Honeycomb (geometry)|honeycombs]] are:
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| * [[Apeirogon]]: {∞}
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| * [[Square tiling]]: {4,4}
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| * [[Cubic honeycomb]]: {4,3,4}
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| * In general, all regular ''n''-dimensional Euclidean [[hypercubic honeycomb]]s: {4,3,...,3,4}.
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| The self-dual (infinite) regular [[Coxeter_diagram#Hyperbolic_Coxeter_groups|hyperbolic]] honeycombs are:
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| * Hyperbolic tilings: {p,p}, p>4.
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| * Noncompact hyperbolic tiling: {∞,∞}
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| * Compact hyperbolic honeycombs: [[Icosahedral honeycomb|{3,5,3}]], [[Order-5 dodecahedral honeycomb|{5,3,5}]], and {5,3,3,5}
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| * Noncompact hyperbolic honeycombs: {3,6,3}, {6,3,6}, {4,4,4}, and {3,3,4,3,3}
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| ==See also==
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| *[[Conway polyhedron notation]]
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| *[[Dual polygon]]
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| *[[Self-dual graph]]
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| *[[Self-dual polygon]]
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| ==References==
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| {{reflist}}
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| <div class="references-small"> | |
| * H.M. Cundy & A.P. Rollett, ''Mathematical models'', Oxford University Press (1961).
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| * {{cite book | first=Magnus | last=Wenninger | authorlink=Magnus Wenninger | title=Dual Models | publisher=Cambridge University Press | year=1983 | isbn=0-521-54325-8 }}
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| * B. Grünbaum & G. Shephard, Duality of polyhedra, ''Shaping space – a polyhedral approach'', ed. Senechal and Fleck, Birkhäuser (1988), pp. 205–211.
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| * P. Gailiunas & J. Sharp, Duality of polyhedra, ''Internat. journ. of math. ed. in science and technology'', Vol. '''36''', No. 6 (2005), pp. 617–642.
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| </div>
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| ==External links==
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| {{commonscat}}
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| * {{mathworld | urlname = DualPolyhedron | title = Dual polyhedron}}
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| * {{MathWorld|urlname=DualTessellation|title=Dual tessellation}}
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| * {{mathworld | urlname = Self-DualPolyhedron | title = Self-dual polyhedron}}
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| * {{GlossaryForHyperspace | anchor=Duality | title=Duality }}
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| *[http://www.software3d.com/Stella.html Software for displaying duals]
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| *[http://www.mathconsult.ch/showroom/unipoly/ The Uniform Polyhedra]
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| *[http://www.georgehart.com/virtual-polyhedra/vp.html Virtual Reality Polyhedra] The Encyclopedia of Polyhedra
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| {{Polyhedron_operators}}
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| {{DEFAULTSORT:Dual Polyhedron}}
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| [[Category:Polyhedra]]
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| [[Category:Duality theories|Polyhedron]]
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| [[Category:Self-dual polyhedra| ]]
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| [[Category:Polytopes]]
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