Uniform polyhedron
A uniform polyhedron is a polyhedron which has regular polygons as faces and is vertex-transitive. It follows. Uniform polyhedra may be quasi-regular or semi-regular; the faces and vertices need not be convex, so many of the uniform polyhedra are star polyhedra. There are two infinite classes of uniform polyhedra together with 75 others. Infinite classes prisms antiprisms Convex exceptional 5 Platonic solids – regular convex polyhedra 13 Archimedean solids – 2 quasiregular and 11 semiregular convex polyhedra Star exceptional 4 Kepler–Poinsot polyhedra – regular nonconvex polyhedra 53 uniform star polyhedra – 5 quasiregular and 48 semiregularhence 5 + 13 + 4 + 53 = 75. There are many degenerate uniform polyhedra with pairs of edges that coincide, including one found by John Skilling called the great disnub dirhombidodecahedron. Dual polyhedra to uniform polyhedra are face-transitive and have regular vertex figures, are classified in parallel with their dual polyhedron; the dual of a regular polyhedron is regular, while the dual of an Archimedean solid is a Catalan solid.
The concept of uniform polyhedron is a special case of the concept of uniform polytope, which applies to shapes in higher-dimensional space. Coxeter, Longuet-Higgins & Miller define uniform polyhedra to be vertex-transitive polyhedra with regular faces, they define a polyhedron to be a finite set of polygons such that each side of a polygon is a side of just one other polygon, such that no non-empty proper subset of the polygons has the same property. By a polygon they implicitly mean a polygon in 3-dimensional Euclidean space. There are some generalizations of the concept of a uniform polyhedron. If the connectedness assumption is dropped we get uniform compounds, which can be split as a union of polyhedra, such as the compound of 5 cubes. If we drop the condition that the realization of the polyhedron is non-degenerate we get the so-called degenerate uniform polyhedra; these require a more general definition of polyhedra. Grunbaum gave a rather complicated definition of a polyhedron, while McMullen & Schulte gave a simpler and more general definition of a polyhedron: in their terminology, a polyhedron is a 2-dimensional abstract polytope with a non-degenerate 3-dimensional realization.
Here an abstract polytope is a poset of its "faces" satisfying various condition, a realization is a function from its vertices to some space, the realization is called non-degenerate if any two distinct faces of the abstract polytope have distinct realizations. Some of the ways they can be degenerate are as follows: Hidden faces; some polyhedra have faces that are hidden, in the sense that no points of their interior can be seen from the outside. These are not counted as uniform polyhedra. Degenerate compounds; some polyhedra have multiple edges and their faces are the faces of two or more polyhedra, though these are not compounds in the previous sense since the polyhedra share edges. Double covers. There are some non-orientable polyhedra that have double covers satisfying the definition of a uniform polyhedron. There double covers have doubled faces and vertices, they are not counted as uniform polyhedra. Double faces. There are several polyhedra with doubled faces produced by Wythoff's construction.
Most authors do not remove them as part of the construction. Double edges. Skilling's figure has the property that it has double edges but its faces cannot be written as a union of two uniform polyhedra; the Platonic solids date back to the classical Greeks and were studied by the Pythagoreans, Theaetetus, Timaeus of Locri and Euclid. The Etruscans discovered the regular dodecahedron before 500 BC; the cuboctahedron was known by Plato. Archimedes discovered all of the 13 Archimedean solids, his original book on the subject was lost, but Pappus of Alexandria mentioned Archimedes listed 13 polyhedra. Piero della Francesca rediscovered the five truncation of the Platonic solids: truncated tetrahedron, truncated octahedron, truncated cube, truncated dodecahedron, truncated icosahedron. Luca Pacioli republished Francesca's work in De divina proportione in 1509, adding the rhombicuboctahedron, calling it a icosihexahedron for its 26 faces, drawn by Leonardo da Vinci. Johannes Kepler was the first to publish the complete list of Archimedean solids, in 1619, as well as identified the infinite families of uniform prisms and antiprisms.
Kepler discovered two of the regular Kepler–Poinsot polyhedra and Louis Poinsot discovered the other two. The set of four were named by Arthur Cayley. Of the remaining 53, Edmund Hess discovered two, Albert Badoureau discovered 36 more, Pitsch independently discovered 18, of which 3 had not been discovered. Together these gave 41 polyhedra; the geometer H. S. M. Coxeter did not publish. M. S. Longuet-Higgins and H. C. Longuet-Higgins independently discovered eleven of these. Lesavre and Mercier rediscovered five of them in 1947. Coxeter, Longuet-Higgins & Miller published the list of uniform polyhedra. Sopov (19
Dual polyhedron
In geometry, any polyhedron is associated with a second dual figure, where the vertices of one correspond to the faces of the other and the edges between pairs of vertices of one correspond to the edges between pairs of faces of the other. Such dual figures remain combinatorial or abstract polyhedra, but not all are geometric polyhedra. Starting with any given polyhedron, the dual of its dual is the original polyhedron. Duality preserves the symmetries of a polyhedron. Therefore, for many classes of polyhedra defined by their symmetries, the duals belong to a symmetric class. Thus, the regular polyhedra – the Platonic solids and Kepler–Poinsot polyhedra – form dual pairs, where the regular tetrahedron is self-dual; the dual of an isogonal polyhedron, having equivalent vertices, is one, isohedral, having equivalent faces. The dual of an isotoxal polyhedron is isotoxal. Duality is related to reciprocity or polarity, a geometric transformation that, when applied to a convex polyhedron, realizes the dual polyhedron as another convex polyhedron.
There are many kinds of duality. The kinds most relevant to elementary polyhedra are polar reciprocity and topological or abstract duality; the duality of polyhedra is defined in terms of polar reciprocation about a concentric sphere. Here, each vertex is associated with a face plane so that the ray from the center to the vertex is perpendicular to the plane, 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 x 2 + y 2 + z 2 = r 2, the vertex is associated with the plane x 0 x + y 0 y + z 0 z = r 2; the vertices of the dual are the poles reciprocal to the face planes of the original, the faces of the dual lie in the polars reciprocal to the vertices of the original. Any two adjacent vertices define an edge, these will reciprocate to two adjacent faces which intersect to define an edge of the dual; this dual pair of edges are always orthogonal to each other. If r 0 is the radius of the sphere, r 1 and r 2 the distances from its centre to the pole and its polar, then: r 1.
R 2 = r 0 2 For the more symmetrical polyhedra having an obvious centroid, it is common to make the polyhedron and sphere concentric, as in the Dorman Luke construction described below. However, it is possible to reciprocate a polyhedron about any sphere, the resulting form of the dual will depend on the size and position of the sphere; the choice of center for the sphere is sufficient to define the dual up to similarity. If multiple symmetry axes are present, they will intersect at a single point, this is taken to be the centroid. Failing that, a circumscribed sphere, inscribed sphere, or midsphere is used. If a polyhedron in Euclidean space has an element passing through the center of the sphere, the corresponding element of its dual will go to infinity. Since Euclidean space never reaches infinity, the projective equivalent, called extended Euclidean space, may be formed by adding the required'plane at infinity'; some theorists prefer to say that there is no dual. Meanwhile, Wenninger found a way to represent these infinite duals, in a manner suitable for making models.
The concept of duality here is related to the duality in projective geometry, where lines and edges are interchanged. 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. Because of the definitional issues for geometric duality of non-convex polyhedra, Grünbaum argues that any proper definition of a non-convex polyhedron should include a notion of a dual polyhedron. Any convex polyhedron can be distorted into a canonical form, in which a unit midsphere exists tangent to every edge, such that the average position of the points of tangency is the center of the sphere; this form is unique up to congruences. If we reciprocate such a canonical polyhedron about its midsphere, the dual polyhedron will share the same edge-tangency points and so must be canonical, it is the canonical dual, the two together form a canonical dual pair.
When a pair of polyhedra cannot be obtained by reciprocation from each other, they may be called duals of each other as long as the vertices of one correspond to the faces of the other, the edges of one correspond to the edges of the other, in an incidence-preserving way. Such pairs of polyhedra are abstractly dual; the vertices and edges of a convex polyhedron form a graph, embedded on a topological sphere, the surface of the polyhedron. The same graph can be projected to form
Dihedral group
In mathematics, a dihedral group is the group of symmetries of a regular polygon, which includes rotations and reflections. Dihedral groups are among the simplest examples of finite groups, they play an important role in group theory and chemistry; the notation for the dihedral group differs in abstract algebra. In geometry, Dn or Dihn refers to the symmetries of a group of order 2n. In abstract algebra, D2n refers to this same dihedral group; the geometric convention is used in this article. A regular polygon with n sides has 2 n different symmetries: n rotational symmetries and n reflection symmetries. We take n ≥ 3 here; the associated rotations and reflections make up the dihedral group D n. If n is odd, each axis of symmetry connects the midpoint of one side to the opposite vertex. If n is there are n/2 axes of symmetry connecting the midpoints of opposite sides and n / 2 axes of symmetry connecting opposite vertices. In either case, there are 2 n elements in the symmetry group. Reflecting in one axis of symmetry followed by reflecting in another axis of symmetry produces a rotation through twice the angle between the axes.
The following picture shows the effect of the sixteen elements of D 8 on a stop sign: The first row shows the effect of the eight rotations, the second row shows the effect of the eight reflections, in each case acting on the stop sign with the orientation as shown at the top left. As with any geometric object, the composition of two symmetries of a regular polygon is again a symmetry of this object. With composition of symmetries to produce another as the binary operation, this gives the symmetries of a polygon the algebraic structure of a finite group; the following Cayley table shows the effect of composition in the group D3. R0 denotes the identity. For example, s2s1 = r1, because the reflection s1 followed by the reflection s2 results in a rotation of 120°; the order of elements denoting the composition is right to left, reflecting the convention that the element acts on the expression to its right. The composition operation is not commutative. In general, the group Dn has elements r0, …, rn−1 and s0, …, sn−1, with composition given by the following formulae: r i r j = r i + j, r i s j = s i + j, s i r j = s i − j, s i s j = r i − j.
In all cases and subtraction of subscripts are to be performed using modular arithmetic with modulus n. If we center the regular polygon at the origin elements of the dihedral group act as linear transformations of the plane; this lets us represent elements of Dn with composition being matrix multiplication. This is an example of a group representation. For example, the elements of the group D4 can be represented by the following eight matrices: r 0 =, r 1 =, r 2 =, r 3 =, s 0 =, s 1 =, s 2 =
Cartesian product
In set theory, a Cartesian product is a mathematical operation that returns a set from multiple sets. That is, for sets A and B, the Cartesian product A × B is the set of all ordered pairs where a ∈ A and b ∈ B. Products can be specified using e.g.. A × B =. A table can be created by taking the Cartesian product of a set of columns. If the Cartesian product rows × columns is taken, the cells of the table contain ordered pairs of the form. More a Cartesian product of n sets known as an n-fold Cartesian product, can be represented by an array of n dimensions, where each element is an n-tuple. An ordered pair is a couple; the Cartesian product is named after René Descartes, whose formulation of analytic geometry gave rise to the concept, further generalized in terms of direct product. An illustrative example is the standard 52-card deck; the standard playing card ranks form a 13-element set. The card suits form a four-element set; the Cartesian product of these sets returns a 52-element set consisting of 52 ordered pairs, which correspond to all 52 possible playing cards.
Ranks × Suits returns a set of the form. Suits × Ranks returns a set of the form. Both sets are distinct disjoint; the main historical example is the Cartesian plane in analytic geometry. In order to represent geometrical shapes in a numerical way and extract numerical information from shapes' numerical representations, René Descartes assigned to each point in the plane a pair of real numbers, called its coordinates; such a pair's first and second components are called its x and y coordinates, respectively. The set of all such pairs is thus assigned to the set of all points in the plane. A formal definition of the Cartesian product from set-theoretical principles follows from a definition of ordered pair; the most common definition of ordered pairs, the Kuratowski definition, is =. Under this definition, is an element of P, X × Y is a subset of that set, where P represents the power set operator. Therefore, the existence of the Cartesian product of any two sets in ZFC follows from the axioms of pairing, power set, specification.
Since functions are defined as a special case of relations, relations are defined as subsets of the Cartesian product, the definition of the two-set Cartesian product is prior to most other definitions. Let A, B, C, D be sets; the Cartesian product A × B is not commutative, A × B ≠ B × A, because the ordered pairs are reversed unless at least one of the following conditions is satisfied: A is equal to B, or A or B is the empty set. For example: A =. × C ≠ A × If for example A = × A = ≠ = A ×. The Cartesian product behaves nicely with respect to intersections. × = ∩. × ≠ ∪ In fact, we have that: ∪ = ∪ ∪ [ ( B
Geometry
Geometry is a branch of mathematics concerned with questions of shape, relative position of figures, the properties of space. A mathematician who works in the field of geometry is called a geometer. Geometry arose independently in a number of early cultures as a practical way for dealing with lengths and volumes. Geometry began to see elements of formal mathematical science emerging in the West as early as the 6th century BC. By the 3rd century BC, geometry was put into an axiomatic form by Euclid, whose treatment, Euclid's Elements, set a standard for many centuries to follow. Geometry arose independently in India, with texts providing rules for geometric constructions appearing as early as the 3rd century BC. Islamic scientists expanded on them during the Middle Ages. By the early 17th century, geometry had been put on a solid analytic footing by mathematicians such as René Descartes and Pierre de Fermat. Since and into modern times, geometry has expanded into non-Euclidean geometry and manifolds, describing spaces that lie beyond the normal range of human experience.
While geometry has evolved throughout the years, there are some general concepts that are more or less fundamental to geometry. These include the concepts of points, planes, surfaces and curves, as well as the more advanced notions of manifolds and topology or metric. Geometry has applications to many fields, including art, physics, as well as to other branches of mathematics. Contemporary geometry has many subfields: Euclidean geometry is geometry in its classical sense; the mandatory educational curriculum of the majority of nations includes the study of points, planes, triangles, similarity, solid figures and analytic geometry. Euclidean geometry has applications in computer science and various branches of modern mathematics. Differential geometry uses techniques of linear algebra to study problems in geometry, it has applications in physics, including in general relativity. Topology is the field concerned with the properties of geometric objects that are unchanged by continuous mappings. In practice, this means dealing with large-scale properties of spaces, such as connectedness and compactness.
Convex geometry investigates convex shapes in the Euclidean space and its more abstract analogues using techniques of real analysis. It has close connections to convex analysis and functional analysis and important applications in number theory. Algebraic geometry studies geometry through the use of multivariate polynomials and other algebraic techniques, it has applications including cryptography and string theory. Discrete geometry is concerned with questions of relative position of simple geometric objects, such as points and circles, it shares many principles with combinatorics. Computational geometry deals with algorithms and their implementations for manipulating geometrical objects. Although being a young area of geometry, it has many applications in computer vision, image processing, computer-aided design, medical imaging, etc; the earliest recorded beginnings of geometry can be traced to ancient Mesopotamia and Egypt in the 2nd millennium BC. Early geometry was a collection of empirically discovered principles concerning lengths, angles and volumes, which were developed to meet some practical need in surveying, construction and various crafts.
The earliest known texts on geometry are the Egyptian Rhind Papyrus and Moscow Papyrus, the Babylonian clay tablets such as Plimpton 322. For example, the Moscow Papyrus gives a formula for calculating the volume of a truncated pyramid, or frustum. Clay tablets demonstrate that Babylonian astronomers implemented trapezoid procedures for computing Jupiter's position and motion within time-velocity space; these geometric procedures anticipated the Oxford Calculators, including the mean speed theorem, by 14 centuries. South of Egypt the ancient Nubians established a system of geometry including early versions of sun clocks. In the 7th century BC, the Greek mathematician Thales of Miletus used geometry to solve problems such as calculating the height of pyramids and the distance of ships from the shore, he is credited with the first use of deductive reasoning applied to geometry, by deriving four corollaries to Thales' Theorem. Pythagoras established the Pythagorean School, credited with the first proof of the Pythagorean theorem, though the statement of the theorem has a long history.
Eudoxus developed the method of exhaustion, which allowed the calculation of areas and volumes of curvilinear figures, as well as a theory of ratios that avoided the problem of incommensurable magnitudes, which enabled subsequent geometers to make significant advances. Around 300 BC, geometry was revolutionized by Euclid, whose Elements considered the most successful and influential textbook of all time, introduced mathematical rigor through the axiomatic method and is the earliest example of the format still used in mathematics today, that of definition, axiom and proof. Although most of the contents of the Elements were known, Euclid arranged them into a single, coherent logical framework; the Elements was known to all educated people in the West until the middle of the 20th century and its contents are still taught in geometry classes today. Archimedes of Syracuse used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, gave remarkably accurate approximations of Pi.
He studied the sp
Point groups in three dimensions
In geometry, a point group in three dimensions is an isometry group in three dimensions that leaves the origin fixed, or correspondingly, an isometry group of a sphere. It is a subgroup of the orthogonal group O, the group of all isometries that leave the origin fixed, or correspondingly, the group of orthogonal matrices. O itself is a subgroup of the Euclidean group E of all isometries. Symmetry groups of objects are isometry groups. Accordingly, analysis of isometry groups is analysis of possible symmetries. All isometries of a bounded 3D object have one or more common fixed points. We choose the origin as one of them; the symmetry group of an object is sometimes called full symmetry group, as opposed to its rotation group or proper symmetry group, the intersection of its full symmetry group and the rotation group SO of the 3D space itself. The rotation group of an object is equal to its full symmetry group if and only if the object is chiral; the point groups in three dimensions are used in chemistry to describe the symmetries of a molecule and of molecular orbitals forming covalent bonds, in this context they are called molecular point groups.
Finite Coxeter groups are a special set of point groups generated purely by a set of reflectional mirrors passing through the same point. A rank n Coxeter group is represented by a Coxeter -- Dynkin diagram. Coxeter notation offers a bracketed notation equivalent to the Coxeter diagram, with markup symbols for rotational and other subsymmetry point groups. SO is a subgroup of E +, which consists of i.e. isometries preserving orientation. O is the direct product of SO and the group generated by inversion: O = SO × Thus there is a 1-to-1 correspondence between all direct isometries and all indirect isometries, through inversion. There is a 1-to-1 correspondence between all groups of direct isometries H in O and all groups K of isometries in O that contain inversion: K = H × H = K ∩ SOFor instance, if H is C2 K is C2h, or if H is C3 K is S6. If a group of direct isometries H has a subgroup L of index 2 apart from the corresponding group containing inversion there is a corresponding group that contains indirect isometries but no inversion: M = L ∪ where isometry is identified with A.
An example would be C4 for H and S4 for M. Thus M is obtained from H by inverting the isometries in H ∖ L; this group M is as abstract group isomorphic with H. Conversely, for all isometry groups that contain indirect isometries but no inversion we can obtain a rotation group by inverting the indirect isometries; this is clarifying when see below. In 2D the cyclic group of k-fold rotations Ck is for every positive integer k a normal subgroup of O and SO. Accordingly, in 3D, for every axis the cyclic group of k-fold rotations about that axis is a normal subgroup of the group of all rotations about that axis. Since any subgroup of index two is normal, the group of rotations is normal both in the group obtained by adding reflections in planes through the axis and in the group obtained by adding a reflection plane perpendicular to the axis; the isometries of R3 that leave the origin fixed, forming the group O, can be categorized as follows: SO: identity rotation about an axis through the origin by an angle not equal to 180° rotation about an axis through the origin by an angle of 180° the same with inversion, i.e. respectively: inversion rotation about an axis by an angle not equal to 180°, combined with reflection in the plane through the origin perpendicular to the axis reflection in a plane through the originThe 4th and 5th in particular, in a wider sense the 6th are called improper rotations.
See the similar overview including translations. When comparing the symmetry type of two objects, the origin is chosen for each separately, i.e. they need not have the same center. Moreover, two objects are considered to be of the same symmetry type if their symmetry groups are conjugate subgroups of O. For example, two 3D objects have the same symmetry type: if both have mirror symmetry, but with respect to a different mirror plane if both have 3-fold rotational symmetry, but with respect to a different axis. In the case of multiple mirror planes and/or axes of rotation, two symmetry groups are of the same symmetry type if and only if there is a rotation mapping the whole structure of the first symmetry group to that of the second; the conjugacy definition would allow a mirror image of the structure, but this is not needed, the structure itself is achiral. For example, if a symmetry group contains a 3-fold axis of rotation, it contains rotations in two opposite directions. There are many infinite isometry groups.
We may create non-cyclical abelian groups by adding more rotations around the same axis. There are non-abelian groups generated by rotations around different axes; these are free groups. They will be infinite. All the infinite groups mentioned so far are not closed as topological subgroups of O. We now discuss
Prismatic uniform polyhedron
In geometry, a prismatic uniform polyhedron is a uniform polyhedron with dihedral symmetry. They exist in the uniform prisms and the uniform antiprisms. All are therefore prismatoids; because they are isogonal, their vertex arrangement uniquely corresponds to a symmetry group. The difference between the prismatic and antiprismatic symmetry groups is that Dph has the vertices lined up in both planes, which gives it a reflection plane perpendicular to its p-fold axis; each has p reflection planes. The Dph symmetry group contains inversion if and only if p is while Dpd contains inversion symmetry if and only if p is odd. There are: prisms, for each rational number p/q > 2, with symmetry group Dph. If p/q is an integer, i.e. if q = 1, the prism or antiprism is convex. An antiprism with p/q < 2 is crossed or retrograde. If p/q ≤ 3/2 no uniform antiprism can exist, as its vertex figure would have to violate the triangle inequality. Note: The tetrahedron and octahedron are listed here with dihedral symmetry, although if uniformly colored, the tetrahedron has tetrahedral symmetry and the cube and octahedron have octahedral symmetry.
Uniform polyhedron Prism Antiprism Coxeter, Harold Scott MacDonald. P.. "Uniform polyhedra". Philosophical Transactions of the Royal Society of London. Series A. Mathematical and Physical Sciences; the Royal Society. 246: 401–450. Doi:10.1098/rsta.1954.0003. ISSN 0080-4614. JSTOR 91532. MR 0062446. Cromwell, P.. 1997, ISBN 0-521-66432-2. Pbk. ISBN 0-521-66405-5. P.175 Skilling, John, "Uniform Compounds of Uniform Polyhedra", Mathematical Proceedings of the Cambridge Philosophical Society, 79: 447–457, doi:10.1017/S0305004100052440, MR 0397554. Prisms and Antiprisms George W. Hart