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Some Polyhedra Dodecahedron (Regular polyhedron) Small stellated dodecahedron (Regular star) Icosidodecahedron (Uniform/Quasiregular) Great cubicuboctahedron (Uniform star) Rhombic triacontahedron (Uniform/Quasiregular dual) Elongated pentagonal cupola (Convex regular-faced) File:Octagonal prism.png Octagonal prism (Uniform prism) File:Square antiprism.png Square antiprism (Uniform antiprism)

In geometry, a polyhedron (plural polyhedra or polyhedrons) is a solid in three dimensions with flat faces and straight edges. The word polyhedron comes from the Classical Greek πολύεδρον, as poly- (stem of πολύς, "many") + -hedron (form of έδρα, "base", "seat", or "face").

A polyhedron is a 3-dimensional example of the more general polytope in any number of dimensions.

Basis for definition

File:UniversumUNAM19.JPG

Polyhedron blocks on display at the Universum museum in Mexico City

Defining a polyhedron as a solid bounded by flat faces and straight edges is not very precise and, to a modern mathematician, quite unsatisfactory, for example it is difficult to reconcile with star polyhedra. Grünbaum (1994, p. 43) observed, "The Original Sin in the theory of polyhedra goes back to Euclid, and through Kepler, Poinsot, Cauchy and many others ... [in that] at each stage ... the writers failed to define what are the 'polyhedra' ...." Many definitions of "polyhedron" have been given within particular contexts, some more rigorous than others. For example definitions based on the idea of a boundary surface rather than a solid are common. However such definitions are not always compatible in other mathematical contexts.

One modern approach treats a geometric polyhedron as a realisation of some abstract polyhedron. Any such polyhedron can be built up from different kinds of element or entity, each associated with a different number of dimensions:

• 3 dimensions: The body is bounded by the faces, and is usually the volume enclosed by them.
• 2 dimensions: A face is a polygon bounded by a circuit of edges, and usually including the flat (plane) region inside the boundary. These polygonal faces together make up the polyhedral surface.
• 1 dimension: An edge joins one vertex to another and one face to another, and is usually a line segment. The edges together make up the polyhedral skeleton.
• 0 dimensions: A vertex (plural vertices) is a corner point.
• −1 dimension: The null polytope is a kind of mathematical convention required by abstract set-based definitions, and may be represented geometrically as the empty set of points. From an inductive dimension approach, the boundary of a point is the empty set and it has dimension one less than a point, leading to a dimension of −1.

More generally in mathematics and other disciplines, "polyhedron" is used to refer to a variety of related constructs, some geometric and others purely algebraic or abstract.

Characteristics

Polyhedral surface

A defining characteristic of almost all kinds of polyhedra is that just two faces join along any common edge. This ensures that the polyhedral surface is continuously connected and does not end abruptly or split off in different directions.

Edges

Edges have two important characteristics (unless the polyhedron is complex):

• An edge joins just two vertices.
• An edge joins just two faces.

These two characteristics are dual to each other.

Euler characteristic

The Euler characteristic χ relates the number of vertices V, edges E, and faces F of a polyhedron:

$\chi =V-E+F.$

For a convex polyhedron or more generally for any simply connected polyhedron whose faces are also simply connected and whose boundary is a manifold, χ = 2. For a detailed discussion, see Proofs and Refutations by Imre Lakatos.

Orientability

Some polyhedra, such as all convex polyhedra, have two distinct sides to their surface, for example one side can consistently be coloured black and the other white. We say that the figure is orientable.

But for some polyhedra this is not possible, and the figure is said to be non-orientable. All polyhedra with odd-numbered Euler characteristic are non-orientable. A given figure with even χ < 2 may or may not be orientable.

Vertex figure

For every vertex one can define a vertex figure, which describes the local structure of the figure around the vertex. If the vertex figure is a regular polygon, then the vertex itself is said to be regular.

Duality

File:Dual Cube-Octahedron.svg

For every polyhedron there exists a dual polyhedron having:

• faces in place of the original's vertices and vice versa,
• the same number of edges
• the same Euler characteristic and orientability

The dual of a convex polyhedron can be obtained by the process of polar reciprocation.

Volume

Elementary calculation

Any regular polyhedron can be divided up into congruent pyramids, with each pyramid having a face of the polyhedron as its base and the centre of the polyhedron as its apex. The height of a pyramid is equal to the inradius of the polyhedron. If the area of a face is $A$ and the in-radius is $r$ then the volume of the pyramid is one-third of the base times the height, or $Ar/3$. For a regular polyhedron with $n$ faces, its volume is then simply

$\text{volume}=nAr/3$.

For instance, a cube with edges of length $L$ has six faces, each face being a square with area $A=L^2$. The inradius from the center of the face to the center of the cube is $r=L/2$. Then the volume is given by

$\text{volume}=\frac{6\cdot L^2\cdot \frac{L}{2}}{3}=L^3,$

the usual formula for the volume of a cube.

Advanced calculation

The volume of any orientable polyhedron can be calculated using the divergence theorem. Consider the vector field $\overrightarrow{F}(\overrightarrow{x})=\frac{1}{3}\overrightarrow{x}=(\frac{x_1}{3},\frac{x_2}{3},\frac{x_3}{3})$, whose divergence is identically 1. The divergence theorem implies that the volume is equal to a surface integral of $F(x)$:

$\text{volume}(\Omega )={\int }_{\Omega }\mathrm{\nabla }\cdot \overrightarrow{F}d\Omega ={\oint }_S\overrightarrow{F}\cdot \hat{n}dS.$

When Ω is the region enclosed by a polyhedron, since the faces of a polyhedron are planar and have piecewise constant normal vectors, this simplifies to

$\text{volume}=\frac{1}{3}\sum _{\text{face }i}{\overrightarrow{x}}_i\cdot {\hat{n}}_i{A}_i$

where for the i'th face, ${\overrightarrow{x}}_i$ is the face barycenter, ${\hat{n}}_i$ is its normal vector, and $A_i$ is its area. Once the faces are decomposed in a set of non-overlapping triangles with surface normals pointing away from the volume, the volume is a sixths of the sum over the triple products of the nine Cartesian vertex coordinates of the triangles.

Since it may be difficult to enumerate the faces, volume computation may be challenging, and hence there exist specialized algorithms to determine the volume (many of these generalize to convex polytopes in higher dimensions).

Names of polyhedra

Polyhedra are often named according to the number of faces. The naming system is again based on Classical Greek, for example tetrahedron (4), pentahedron (5), hexahedron (6), heptahedron (7), triacontahedron (30), and so on. Sometimes this is qualified by a description of the kinds of faces present, for example the Rhombic dodecahedron vs. the Pentagonal dodecahedron.

Some polyhedra have gained common names, for example the regular hexahedron is commonly known as the cube. Others are named after their discoverer, such as Miller's monster or the Szilassi polyhedron.

Other common names indicate that some operation has been performed on a simpler polyhedron, for example the truncated cube looks like a cube with its corners cut off, and has 14 faces (so it is also an example of a tetrakaidecahedron or tetradecahedron).

Traditional polyhedra

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A dodecahedron

In geometry, a polyhedron is traditionally a three-dimensional shape that is made up of a finite number of polygonal faces which are parts of planes; the faces meet in pairs along edges which are straight-line segments, and the edges meet in points called vertices. Cubes, prisms and pyramids are examples of polyhedra. The polyhedron surrounds a bounded volume in three-dimensional space; sometimes this interior volume is considered to be part of the polyhedron, sometimes only the surface is considered, and occasionally only the skeleton of edges.

A polyhedron is said to be convex if its surface (comprising its faces, edges and vertices) does not intersect itself and the line segment joining any two points of the polyhedron is contained in the interior or surface.

Symmetrical polyhedra

Many of the most studied polyhedra are highly symmetrical.

Of course it is easy to distort such polyhedra so they are no longer symmetrical. But where a polyhedral name is given, such as icosidodecahedron, the most symmetrical geometry is almost always implied, unless otherwise stated.

Some of the most common names in particular are often used with "regular" in front or implied because for each there are different types which have little in common except for having the same number of faces. These are the triangular pyramid or tetrahedron, cube or hexahedron, octahedron, dodecahedron and icosahedron:

File:Tetrahedron.svg

File:Hexahedron.svg

File:Octahedron.svg

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File:Icosahedron.svg

There are several types of highly-symmetric polyhedron, classified by which kind of element - faces, edges and/or vertices - belong to a single symmetry orbit:

• Isogonal or Vertex-transitive if all vertices are the same, in the sense that for any two vertices there exists a symmetry of the polyhedron mapping the first isometrically onto the second.
• Isotoxal or Edge-transitive if all edges are the same, in the sense that for any two edges there exists a symmetry of the polyhedron mapping the first isometrically onto the second.
• Isohedral or Face-transitive if all faces are the same, in the sense that for any two faces there exists a symmetry of the polyhedron mapping the first isometrically onto the second.
• Regular if it is vertex-transitive, edge-transitive and face-transitive (this implies that every face is the same regular polygon; it also implies that every vertex is regular).
• Quasi-regular if it is vertex-transitive and edge-transitive (and hence has regular faces) but not face-transitive. A quasi-regular dual is face-transitive and edge-transitive (and hence every vertex is regular) but not vertex-transitive.
• Semi-regular if it is vertex-transitive but not edge-transitive, and every face is a regular polygon. (This is one of several definitions of the term, depending on author. Some definitions overlap with the quasi-regular class). A semi-regular dual is face-transitive but not vertex-transitive, and every vertex is regular.
• Uniform if it is vertex-transitive and every face is a regular polygon, i.e. it is regular, quasi-regular or semi-regular. A uniform dual is face-transitive and has regular vertices, but is not necessarily vertex-transitive).
• Noble if it is face-transitive and vertex-transitive (but not necessarily edge-transitive). The regular polyhedra are also noble; they are the only noble uniform polyhedra.

A polyhedron can belong to the same overall symmetry group as one of higher symmetry, but will have several groups of elements (for example faces) in different symmetry orbits.

Uniform polyhedra and their duals

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Uniform polyhedra are vertex-transitive and every face is a regular polygon. They may be regular, quasi-regular, or semi-regular, and may be convex or starry.

The uniform duals are face-transitive and every vertex figure is a regular polygon.

Face-transitivity of a polyhedron corresponds to vertex-transitivity of the dual and conversely, and edge-transitivity of a polyhedron corresponds to edge-transitivity of the dual. The dual of a regular polyhedron is also regular. The dual of a non-regular uniform polyhedron (called a Catalan solid if convex) has irregular faces.

Each uniform polyhedron shares the same symmetry as its dual, with the symmetries of faces and vertices simply swapped over. Because of this some authorities regard the duals as uniform too. But this idea is not held widely: a polyhedron and its symmetries are not the same thing.

The uniform polyhedra and their duals are traditionally classified according to their degree of symmetry, and whether they are convex or not.

	Convex uniform	Convex uniform dual	Star uniform	Star uniform dual
Regular	Platonic solids		Kepler-Poinsot polyhedra
Quasiregular	Archimedean solids	Catalan solids	(no special name)	(no special name)
Semiregular			(no special name)	(no special name)
	Prisms	Dipyramids	Star Prisms	Star Dipyramids
	Antiprisms	Trapezohedra	Star Antiprisms	Star Trapezohedra

Noble polyhedra

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A noble polyhedron is both isohedral (equal-faced) and isogonal (equal-cornered). Besides the regular polyhedra, there are many other examples.

The dual of a noble polyhedron is also noble.

Symmetry groups

The polyhedral symmetry groups (using Schoenflies notation) are all point groups and include:

• T - chiral tetrahedral symmetry; the rotation group for a regular tetrahedron; order 12.
• T_d - full tetrahedral symmetry; the symmetry group for a regular tetrahedron; order 24.
• T_h - pyritohedral symmetry; order 24. The symmetry of a pyritohedron.
• O - chiral octahedral symmetry;the rotation group of the cube and octahedron; order 24.
• O_h - full octahedral symmetry; the symmetry group of the cube and octahedron; order 48.
• I - chiral icosahedral symmetry; the rotation group of the icosahedron and the dodecahedron; order 60.
• I_h - full icosahedral symmetry; the symmetry group of the icosahedron and the dodecahedron; order 120.
• C_nv - n-fold pyramidal symmetry
• D_nh - n-fold prismatic symmetry
• D_nv - n-fold antiprismatic symmetry.

Those with chiral symmetry do not have reflection symmetry and hence have two enantiomorphous forms which are reflections of each other. The snub Archimedean polyhedra have this property.

Other polyhedra with regular faces

Equal regular faces

A few families of polyhedra, where every face is the same kind of polygon:

• Deltahedra have equilateral triangles for faces.

• With regard to polyhedra whose faces are all squares: if coplanar faces are not allowed, even if they are disconnected, there is only the cube. Otherwise there is also the result of pasting six cubes to the sides of one, all seven of the same size; it has 30 square faces (counting disconnected faces in the same plane as separate). This can be extended in one, two, or three directions: we can consider the union of arbitrarily many copies of these structures, obtained by translations of (expressed in cube sizes) (2,0,0), (0,2,0), and/or (0,0,2), hence with each adjacent pair having one common cube. The result can be any connected set of cubes with positions (a,b,c), with integers a,b,c of which at most one is even.

• There is no special name for polyhedra whose faces are all equilateral pentagons or pentagrams. There are infinitely many of these, but only one is convex: the dodecahedron. The rest are assembled by (pasting) combinations of the regular polyhedra described earlier: the dodecahedron, the small stellated dodecahedron, the great stellated dodecahedron and the great icosahedron.

There exists no polyhedron whose faces are all identical and are regular polygons with six or more sides because the vertex of three regular hexagons defines a plane. (See infinite skew polyhedron for exceptions with zig-zagging vertex figures.)

Deltahedra

A deltahedron (plural deltahedra) is a polyhedron whose faces are all equilateral triangles. There are infinitely many deltahedra, but only eight of these are convex:

• 3 regular convex polyhedra (3 of the Platonic solids)
  • Tetrahedron
  • Octahedron
  • Icosahedron
• 5 non-uniform convex polyhedra (5 of the Johnson solids)
  • Triangular dipyramid
  • Pentagonal dipyramid
  • Snub disphenoid
  • Triaugmented triangular prism
  • Gyroelongated square dipyramid.

Johnson solids

Mining Engineer (Excluding Oil ) Truman from Alma, loves to spend time knotting, largest property developers in singapore developers in singapore and stamp collecting. Recently had a family visit to Urnes Stave Church. Norman Johnson sought which convex non-uniform polyhedra had regular faces. In 1966, he published a list of 92 such solids, gave them names and numbers, and conjectured that there were no others. Victor Zalgaller proved in 1969 that the list of these Johnson solids was complete.

Other important families of polyhedra

Pyramids

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Stellations and facettings

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File:First stellation of octahedron.png

File:First stellation of dodecahedron.png

File:Second stellation of dodecahedron.png

File:Third stellation of dodecahedron.png

File:Sixteenth stellation of icosahedron.png

File:First stellation of icosahedron.png

File:Seventeenth stellation of icosahedron.png

Stellation of a polyhedron is the process of extending the faces (within their planes) so that they meet to form a new polyhedron.

It is the exact reciprocal to the process of facetting which is the process of removing parts of a polyhedron without creating any new vertices.

Zonohedra

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Toroidal polyhedra

Mining Engineer (Excluding Oil ) Truman from Alma, loves to spend time knotting, largest property developers in singapore developers in singapore and stamp collecting. Recently had a family visit to Urnes Stave Church. A toroidal polyhedron is a polyhedron with an Euler characteristic of 0 or smaller, equivalent to a Genus of 1 or greater, representing a torus surface having one or more holes through the middle.

Compounds

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These compounds often share the same vertices as other polyhedra and are often formed by stellation. Some are listed in the list of Wenninger polyhedron models.

Orthogonal polyhedra

An orthogonal polyhedron is one all of whose faces meet at right angles, and all of whose edges are parallel to axes of a Cartesian coordinate system. Aside from a rectangular box, orthogonal polyhedra are nonconvex. They are the 3D analogs of 2D orthogonal polygons, also known as rectilinear polygons. Orthogonal polyhedra are used in computational geometry, where their constrained structure has enabled advances on problems unsolved for arbitrary polyhedra, for example, unfolding the surface of a polyhedron to a polygonal net.

Generalisations of polyhedra

The name 'polyhedron' has come to be used for a variety of objects having similar structural properties to traditional polyhedra.

Apeirohedra

A classical polyhedral surface comprises finite, bounded plane regions, joined in pairs along edges. If such a surface extends indefinitely it is called an apeirohedron. Examples include:

• Tilings or tessellations of the plane.
• Sponge-like structures called infinite skew polyhedra.

See also: Apeirogon - infinite regular polygon: {∞}

Complex polyhedra

A complex polyhedron is one which is constructed in complex Hilbert 3-space. This space has six dimensions: three real ones corresponding to ordinary space, with each accompanied by an imaginary dimension. See for example Coxeter (1974).

Curved polyhedra

Some fields of study allow polyhedra to have curved faces and edges.

Spherical polyhedra

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The surface of a sphere may be divided by line segments into bounded regions, to form a spherical polyhedron. Much of the theory of symmetrical polyhedra is most conveniently derived in this way.

Spherical polyhedra have a long and respectable history:

• The first known man-made polyhedra are spherical polyhedra carved in stone.
• Poinsot used spherical polyhedra to discover the four regular star polyhedra.
• Coxeter used them to enumerate all but one of the uniform polyhedra.

Some polyhedra, such as hosohedra and dihedra, exist only as spherical polyhedra and have no flat-faced analogue.

Curved spacefilling polyhedra

Two important types are:

• Bubbles in froths and foams, such as Weaire-Phelan bubbles.
• Spacefilling forms used in architecture. See for example Pearce (1978).

General polyhedra

More recently mathematics has defined a polyhedron as a set in real affine (or Euclidean) space of any dimensional n that has flat sides. It may alternatively be defined as the union of a finite number of convex polyhedra, where a convex polyhedron is any set that is the intersection of a finite number of half-spaces. It may be bounded or unbounded. In this meaning, a polytope is a bounded polyhedron.

Analytically, such a convex polyhedron is expressed as the solution set for a system of linear inequalities. Defining polyhedra in this way provides a geometric perspective for problems in Linear programming.

Many traditional polyhedral forms are general polyhedra. Other examples include:

• A quadrant in the plane. For instance, the region of the cartesian plane consisting of all points above the horizontal axis and to the right of the vertical axis: { ( x, y ) : x ≥ 0, y ≥ 0 }. Its sides are the two positive axes.
• An octant in Euclidean 3-space, { ( x, y, z ) : x ≥ 0, y ≥ 0, z ≥ 0 }.
• A prism of infinite extent. For instance a doubly infinite square prism in 3-space, consisting of a square in the xy-plane swept along the z-axis: { ( x, y, z ) : 0 ≤ x ≤ 1, 0 ≤ y ≤ 1 }.
• Each cell in a Voronoi tessellation is a convex polyhedron. In the Voronoi tessellation of a set S, the cell A corresponding to a point c∈S is bounded (hence a traditional polyhedron) when c lies in the interior of the convex hull of S, and otherwise (when c lies on the boundary of the convex hull of S) A is unbounded.

Hollow-faced or skeletal polyhedra

It is not necessary to fill in the face of a figure before we can call it a polyhedron. For example Leonardo da Vinci devised frame models of the regular solids, which he drew for Pacioli's book Divina Proportione. In modern times, Branko Grünbaum (1994) made a special study of this class of polyhedra, in which he developed an early idea of abstract polyhedra. He defined a face as a cyclically ordered set of vertices, and allowed faces to be skew as well as planar.

Non-geometric polyhedra

Various mathematical constructs have been found to have properties also present in traditional polyhedra.

Topological polyhedra

A topological polytope is a topological space given along with a specific decomposition into shapes that are topologically equivalent to convex polytopes and that are attached to each other in a regular way.

Such a figure is called simplicial if each of its regions is a simplex, i.e. in an n-dimensional space each region has n+1 vertices. The dual of a simplicial polytope is called simple. Similarly, a widely studied class of polytopes (polyhedra) is that of cubical polyhedra, when the basic building block is an n-dimensional cube.

Abstract polyhedra

An abstract polyhedron is a partially ordered set (poset) of elements whose partial ordering obeys certain rules. Theories differ in detail, but essentially the elements of the set correspond to the body, faces, edges and vertices of the polyhedron. The empty set corresponds to the null polytope, or nullitope, which has a dimensionality of −1. These posets belong to the larger family of abstract polytopes in any number of dimensions.

Polyhedra as graphs

Any polyhedron gives rise to a graph, or skeleton, with corresponding vertices and edges. Thus graph terminology and properties can be applied to polyhedra. For example:

• Due to Steinitz theorem convex polyhedra are in one-to-one correspondence with 3-connected planar graphs.
• The tetrahedron gives rise to a complete graph (K₄). It is the only polyhedron to do so.
• The octahedron gives rise to a strongly regular graph, because adjacent vertices always have two common neighbors, and non-adjacent vertices have four.
• The Archimedean solids give rise to regular graphs: 7 of the Archimedean solids are of degree 3, 4 of degree 4, and the remaining 2 are chiral pairs of degree 5.

History

Prehistory

Stones carved in the shape of a cluster of spheres or similar objects have been found in Scotland and may be as much as 4,000 years old. These stones show the symmetries of various polyhedra, but have curved surfaces. Examples of these stones are on display in the John Evans room of the Ashmolean Museum at Oxford University. It is not known why these objects were made, or how the sculptor gained the inspiration for them.

Polyhedra appeared in early architectural forms such as cubes and cuboids, with the earliest four-sided pyramids of ancient Egypt also dating from the Stone Age.

The Etruscans preceded the Greeks in their awareness of at least some of the regular polyhedra, as evidenced by the discovery near Padua (in Northern Italy) in the late 19th century of a dodecahedron made of soapstone, and dating back more than 2,500 years (Lindemann, 1987).

Greeks

The earliest known written records of these shapes come from Classical Greek authors, who also gave the first known mathematical description of them. The earlier Greeks were interested primarily in the convex regular polyhedra, which came to be known as the Platonic solids. Pythagoras knew at least three of them, and Theaetetus (circa 417 B. C.) described all five. Eventually, Euclid described their construction in his Elements. Later, Archimedes expanded his study to the convex uniform polyhedra which now bear his name. His original work is lost and his solids come down to us through Pappus.

Chinese

By 236 AD, in China Liu Hui was describing the dissection of the cube into its characteristic tetrahedron (orthoscheme) and related solids, using assemblages of these solids as the basis for calculating volumes of earth to be moved during engineering excavations.

Islamic

After the end of the Classical era, scholars in the Islamic civilisation continued to take the Greek knowledge forward (see Mathematics in medieval Islam).

The 9th century scholar Thabit ibn Qurra gave formulae for calculating the volumes of polyhedra such as truncated pyramids.

Then in the 10th century Abu'l Wafa described the convex regular and quasiregular spherical polyhedra.

Renaissance

As with other areas of Greek thought maintained and enhanced by Islamic scholars, Western interest in polyhedra revived during the Italian Renaissance. Artists constructed skeletal polyhedra, depicting them from life as a part of their investigations into perspective. Several appear in marquetry panels of the period. Piero della Francesca gave the first written description of direct geometrical construction of such perspective views of polyhedra. Leonardo da Vinci made skeletal models of several polyhedra and drew illustrations of them for a book by Pacioli. A painting by an anonymous artist of Pacioli and a pupil depicts a glass rhombicuboctahedron half-filled with water.

As the Renaissance spread beyond Italy, later artists such as Wenzel Jamnitzer, Dürer and others also depicted polyhedra of various kinds, many of them novel, in imaginative etchings.

Star polyhedra

For almost 2,000 years, the concept of a polyhedron as a convex solid had remained as developed by the ancient Greek mathematicians.

During the Renaissance star forms were discovered. A marble tarsia in the floor of St. Mark's Basilica, Venice, depicts a stellated dodecahedron. Artists such as Wenzel Jamnitzer delighted in depicting novel star-like forms of increasing complexity.

Johannes Kepler realized that star polygons, typically pentagrams, could be used to build star polyhedra. Some of these star polyhedra may have been discovered before Kepler's time, but he was the first to recognize that they could be considered "regular" if one removed the restriction that regular polytopes be convex. Later, Louis Poinsot realized that star vertex figures (circuits around each corner) can also be used, and discovered the remaining two regular star polyhedra. Cauchy proved Poinsot's list complete, and Cayley gave them their accepted English names: (Kepler's) the small stellated dodecahedron and great stellated dodecahedron, and (Poinsot's) the great icosahedron and great dodecahedron. Collectively they are called the Kepler-Poinsot polyhedra.

The Kepler-Poinsot polyhedra may be constructed from the Platonic solids by a process called stellation. Most stellations are not regular. The study of stellations of the Platonic solids was given a big push by H. S. M. Coxeter and others in 1938, with the now famous paper The 59 icosahedra. This work has recently been re-published (Coxeter, 1999).

The reciprocal process to stellation is called facetting (or faceting). Every stellation of one polytope is dual, or reciprocal, to some facetting of the dual polytope. The regular star polyhedra can also be obtained by facetting the Platonic solids. Template:Harvnb listed the simpler facettings of the dodecahedron, and reciprocated them to discover a stellation of the icosahedron that was missing from the famous "59". More have been discovered since, and the story is not yet ended.

See also:

• Regular polyhedron: History
• Regular polytope: History of discovery.

Polyhedra in nature

For natural occurrences of regular polyhedra, see Regular polyhedron: Regular polyhedra in nature.

Irregular polyhedra appear in nature as crystals.

See also

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Property developer and residential landlord for flats and homes for lease and sale. Most property firms share the same database of property listings in Singapore. Due to this fact it is best to solely use only ONE agent at a time. In case you approach many agents at the similar time, very likely that they'll present you the same property. A lot confusion and embarrassment will arise should you engage many brokers. One of the best, and most of the time only, strategy to discover a good property agent in Singapore is phrase-of-mouth. Ask your friends and colleagues for reference. It is very simple to provide you with a couple of candidates since a lot of the expatriates dwelling in Singapore for a long time can have several good agent contacts to guide you. Toa Payoh, Singapore Singapore 319378 Estate

This is precisely what happened to me and my husband at the moment, to not point out a very unscrupulous developer operating in a really unprofessional manner. I need to share this story with everyone here, and please pass the message round particularly among expats communities, beware while you want to purchase property developed by VicLand Pte Ltd and if developer's agent is ECG property. There was only one unit left on the market by developer, 03-09, a 3 bed room flat. On the time my husband was out of town, and initially I liked what I saw so I instructed the developer's agent and my agent we should come back with my husband in two weeks to view it once more and make a decision after ward. Complaint / Suggestions about lousy property agent Darren Ng from Dennis Wee

This bought me thinking and I started to surprise – how much does a property agent really earn? We often hear or read about sure brokers making million dollar commissions, however is that the exception or the norm? That piqued my curiosity. Like any job, those who put in time and effort will do well and rise to the top. The ethics of exhausting work apply to the true property market as nicely. For individuals who are pondering of making a career change to develop into a property agent, you should be ready to invest the trouble to do properly. Otherwise you may just add to the statistic of brokers who eventually drop out of the realtor game. Properties that do not fall within the definition of residential properties stated above are non-residential properties Web site - www.riaschool.com.sg

Ought to you are on the lookout for new properties for investment or for own stay, we offer property recommendation and search services tailored to your needs. We have represented many together with worldwide and local buyers in efficiently finishing their property purchases. We work with main builders to bring you the latest and one of the best prime properties in Singapore. We are a one-cease service that may full your property cycle from purchase to sale. Property agents for Singapore Land Authority protecting among the government colonial properties for rent. Property leases for expatriates and foreigners. Also helps expats to purchase and promote their properties as well as property investment opportunities in Singapore and China. The Restaurant Affiliation of Singapore

• Antiprism
• Archimedean solid
• Bipyramid
• Conway polyhedron notation (a notation for describing construction of polyhedra)
• Defect
• Deltahedron
• Deltohedron
• Escher
• Flexible polyhedra
• Extension of a polyhedron
• Images of polyhedra
• Johnson solid
• Kepler–Poinsot polyhedra
• Near-miss Johnson solid
• Net (polyhedron)
• Platonic solid
• Polychoron (4 dimensional analogues to polyhedra)
• Polyhedral compound
• Polyhedron models
• Prism
• Schlegel diagram
• Semiregular polyhedron
• Spidron
• Tessellation
• Trapezohedron
• Uniform polyhedron
• Waterman polyhedron
• Zonohedron

References

Template:More footnotes 43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

• Coxeter, H.S.M.; Regular complex Polytopes, CUP (1974).
• Cromwell, P.;Polyhedra, CUP hbk (1997), pbk. (1999).
• Grünbaum, B.; Polyhedra with Hollow Faces, Proc of NATO-ASI Conference on Polytopes ... etc. (Toronto 1993), ed T. Bisztriczky et al., Kluwer Academic (1994) pp. 43–70.
• Grünbaum, B.; Are your polyhedra the same as my polyhedra? Discrete and comput. geom: the Goodman-Pollack festschrift, ed. Aronov et al. Springer (2003) pp. 461–488. (pdf)
• Pearce, P.; Structure in nature is a strategy for design, MIT (1978)

Books on polyhedra

Mining Engineer (Excluding Oil ) Truman from Alma, loves to spend time knotting, largest property developers in singapore developers in singapore and stamp collecting. Recently had a family visit to Urnes Stave Church.

External links

Property Brokers and Team Managers – Looking for good Actual Estate Agency to join or contemplating which is the Finest Property Agency to join in Singapore? Join Leon Low in OrangeTee Singapore! In OrangeTee, we've much more attractive commission structure than before, enrichment courses, 10 most vital components to hitch OrangeTee and 1 motive to join Leon Low and his Workforce. 1. Conducive working environment

Via PropNex International, we continually construct on our fame in the international property enviornment. Click here for more of our abroad initiatives. Instances have modified. We don't see those unlawful hawkers anymore. Instead, nicely dressed property brokers were seen reaching out to people visiting the market in the morning. Real estate can be a lonely enterprise and it is straightforward to really feel demoralised, especially when there are no enquiries despite your greatest effort in advertising your shopper's property. That is the place having the fitting assist from fellow associates is essential. Our firm offers administration services for condominiums and apartments. With a crew of qualified folks, we assist to make your estate a nicer place to stay in. HDB Flat for Hire 2 Rooms

Achievers are all the time the first to check new technologies & providers that can help them enhance their sales. When property guru first began, many brokers didn't consider in it until they began listening to other colleagues getting unbelievable outcomes. Most brokers needs to see proof first, before they dare to take the first step in attempting. These are often the late comers or late adopters. There is a purpose why top achievers are heading the wave or heading the best way. Just because they try new properties in singapore issues ahead of others. The rest just observe after!

Firstly, a Fraudulent Misrepresentation is one that is made knowingly by the Representor that it was false or if it was made without belief in its fact or made recklessly without concerning whether or not it is true or false. For instance estate agent A told the potential consumers that the tenure of a landed property they are considering is freehold when it is really one with a ninety nine-yr leasehold! A is responsible of constructing a fraudulent misrepresentation if he is aware of that the tenure is the truth is a ninety nine-yr leasehold instead of it being freehold or he didn't consider that the tenure of the house was freehold or he had made the assertion with out caring whether or not the tenure of the topic property is in fact freehold.

I such as you to be, am a brand new projects specialist. You've got the conception that new tasks personnel should be showflat certain. Should you're eager, let me train you the right way to master the entire show flats island vast as a substitute of getting to stay just at 1 place. Is that attainable you may ask, well, I've achieved it in 6 months, you can too. Which company is well-recognized and is actually dedicated for developing rookie within the industry in venture sales market with success? Can a rookie join the company's core group from day one? I wish to propose a third class, which I have been grooming my agents in the direction of, and that is as a Huttons agent, you will be able to market and have knowledge of ALL Huttons projects, and if essential, projects exterior of Huttons as properly.

GPS has assembled a high workforce of personnel who are additionally well-known figures in the native actual property scene to pioneer this up-and-coming organization. At GPS Alliance, WE LEAD THE WAY! Many people have asked me how I managed to earn S$114,000 from my sales job (my third job) at age 24. The reply is easy. After graduation from NUS with a Historical past diploma, my first job was in actual estate. Within the ultimate part of this series, I interview one of the top agents in ERA Horizon Group and share with you the secrets to his success! Learn it RIGHT HERE

Notice that the application must be submitted by the appointed Key Government Officer (KEO) such as the CEO, COO, or MD. Once the KEO has submitted the mandatory paperwork and assuming all documents are in order, an email notification shall be sent stating that the applying is permitted. No hardcopy of the license might be issued. A delicate-copy could be downloaded and printed by logging into the CEA website. It takes roughly four-6 weeks to course of an utility.

General theory

• 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
• Polyhedra Pages
• uniform solution for uniform polyhedra by Dr. Zvi Har'El
• Symmetry, Crystals and Polyhedra
• Symmetrohedra: Polyhedra from Symmetric Placement of Regular Polygons

Lists and databases of polyhedra

• The Uniform Polyhedra
• Virtual Reality Polyhedra - The Encyclopedia of Polyhedra
  • Polyhedra and Pyramids
• Interactive 3D polyhedra in Java
• Electronic Geometry Models - Contains a peer reviewed selection of polyhedra with unusual properties.
• Origami Polyhedra - Models made with Modular Origami
• Polyhedra Collection - Various virtual and physical polyhedra models.
• Polyhedra plaited with paper strips - Polyhedra models constructed without use of glue.
• Rotatable polyhedron models (uses Javascript)
• Rotatable self-intersecting polyhedra (uses Java)
• Polyhedron Models - Virtual polyhedra
• Java Applets for Visualizing Polyhedra - Systematically explicit formulae given!

Software

• A Plethora of Polyhedra – An interactive and free collection of polyhedra in Java. Features includes nets, planar sections, duals, truncations and stellations of more than 300 polyhedra.
• Stella: Polyhedron Navigator - Software for exploring polyhedra and printing nets for their physical construction. Includes uniform polyhedra, stellations, compounds, Johnson solids, etc.
• World of Polyhedra - Comprehensive polyhedra in Flash applet, showing vertices and edges (but not shaded faces)
• Hyperspace Star Polytope Slicer - Explorer java applet, includes a variety of 3d viewer options.
• HEDRON - Polyhedron modelling software
• Uniform Polyhedra Java Applets with sources
• openSCAD - Free cross-platform software for programmers. Polyhedra are just one of the things you can model. The openSCAD User Manual is also available.
• OpenVolumeMesh - An open source cross-platform C++ library for handling polyhedral meshes. Developed by the Aachen Computer Graphics Group, RWTH Aachen University.
• Polyhedronisme - Web based tool for generating polyhedra models using Conway Polyhedron Notation. Models can be exported as 2D PNG images, or as 3D OBJ or VRML2 files. The 3D files can be opened in CAD software, or uploaded for 3D printing at services such as Shapeways.

Resources for making physical models, and models for sale

• Making Polyhedra
• Paper Models of Polyhedra Free nets of polyhedra
• Paper Models of Uniform (and other) Polyhedra
• Kits and instructions for making models of polyhedra out of bamboo
• Polyhedra software, die-cast models, & posters
• Simple instructions for building over 30 paper polyhedra
• Kits to print out, cut, and fold into various polyhedra

Miscellaneous

• PictureSpice - A site that lets you make polyhedra with your own uploaded pictures.
• Lattice Polyhedra

Real Estate Agent Renaldo Lester from Saint-Jean-Chrysostome, has several hobbies which include leathercrafting, property developers in singapore apartment for sale, this contact form, and crochet. Loves to see new cities and places like Ruins of Loropéni. Real Estate Agent Renaldo Lester from Saint-Jean-Chrysostome, has several hobbies which include leathercrafting, property developers in singapore apartment for sale, this contact form, and crochet. Loves to see new cities and places like Ruins of Loropéni.