Tetrahedron
In geometry, a tetrahedron known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, four vertex corners. The tetrahedron is the simplest of all the ordinary convex polyhedra and the only one that has fewer than 5 faces; the tetrahedron is the three-dimensional case of the more general concept of a Euclidean simplex, may thus be called a 3-simplex. The tetrahedron is one kind of pyramid, a polyhedron with a flat polygon base and triangular faces connecting the base to a common point. In the case of a tetrahedron the base is a triangle, so a tetrahedron is known as a "triangular pyramid". Like all convex polyhedra, a tetrahedron can be folded from a single sheet of paper, it has two such nets. For any tetrahedron there exists a sphere on which all four vertices lie, another sphere tangent to the tetrahedron's faces. A regular tetrahedron is one, it is one of the five regular Platonic solids. In a regular tetrahedron, all faces are the same size and shape and all edges are the same length.
Regular tetrahedra alone do not tessellate, but if alternated with regular octahedra in the ratio of two tetrahedra to one octahedron, they form the alternated cubic honeycomb, a tessellation. The regular tetrahedron is self-dual; the compound figure comprising two such dual tetrahedra form a stellated octahedron or stella octangula. The following Cartesian coordinates define the four vertices of a tetrahedron with edge length 2, centered at the origin, two level edges: and Expressed symmetrically as 4 points on the unit sphere, centroid at the origin, with lower face level, the vertices are: v1 = v2 = v3 = v4 = with the edge length of sqrt. Still another set of coordinates are based on an alternated cube or demicube with edge length 2; this form has Coxeter diagram and Schläfli symbol h. The tetrahedron in this case has edge length 2√2. Inverting these coordinates generates the dual tetrahedron, the pair together form the stellated octahedron, whose vertices are those of the original cube. Tetrahedron:, Dual tetrahedron:, For a regular tetrahedron of edge length a: With respect to the base plane the slope of a face is twice that of an edge, corresponding to the fact that the horizontal distance covered from the base to the apex along an edge is twice that along the median of a face.
In other words, if C is the centroid of the base, the distance from C to a vertex of the base is twice that from C to the midpoint of an edge of the base. This follows from the fact that the medians of a triangle intersect at its centroid, this point divides each of them in two segments, one of, twice as long as the other. For a regular tetrahedron with side length a, radius R of its circumscribing sphere, distances di from an arbitrary point in 3-space to its four vertices, we have d 1 4 + d 2 4 + d 3 4 + d 4 4 4 + 16 R 4 9 = 2.
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
Hyperbolic geometry
In mathematics, hyperbolic geometry is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: For any given line R and point P not on R, in the plane containing both line R and point P there are at least two distinct lines through P that do not intersect R. Hyperbolic plane geometry is the geometry of saddle surfaces and pseudospherical surfaces, surfaces with a constant negative Gaussian curvature. A modern use of hyperbolic geometry is in the theory of special relativity Minkowski spacetime and gyrovector space; when geometers first realised they were working with something other than the standard Euclidean geometry they described their geometry under many different names. In the former Soviet Union, it is called Lobachevskian geometry, named after one of its discoverers, the Russian geometer Nikolai Lobachevsky; this page is about the 2-dimensional hyperbolic geometry and the differences and similarities between Euclidean and hyperbolic geometry. Hyperbolic geometry can be extended to three and more dimensions.
Hyperbolic geometry is more related to Euclidean geometry than it seems: the only axiomatic difference is the parallel postulate. When the parallel postulate is removed from Euclidean geometry the resulting geometry is absolute geometry. There are two kinds of absolute geometry and hyperbolic. All theorems of absolute geometry, including the first 28 propositions of book one of Euclid's Elements, are valid in Euclidean and hyperbolic geometry. Propositions 27 and 28 of Book One of Euclid's Elements prove the existence of parallel/non-intersecting lines; this difference has many consequences: concepts that are equivalent in Euclidean geometry are not equivalent in hyperbolic geometry. Further, because of the angle of parallelism, hyperbolic geometry has an absolute scale, a relation between distance and angle measurements. Single lines in hyperbolic geometry have the same properties as single straight lines in Euclidean geometry. For example, two points uniquely define a line, lines can be infinitely extended.
Two intersecting lines have the same properties as two intersecting lines in Euclidean geometry. For example, two lines can intersect in no more than one point, intersecting lines have equal opposite angles, adjacent angles of intersecting lines are supplementary; when we add a third line there are properties of intersecting lines that differ from intersecting lines in Euclidean geometry. For example, given 2 intersecting lines there are infinitely many lines that do not intersect either of the given lines; these properties all are independent of the model used if the lines may look radically different. Non-intersecting lines in hyperbolic geometry have properties that differ from non-intersecting lines in Euclidean geometry: For any line R and any point P which does not lie on R, in the plane containing line R and point P there are at least two distinct lines through P that do not intersect R; this implies that there are through P an infinite number of coplanar lines that do not intersect R.
These non-intersecting lines are divided into two classes: Two of the lines are limiting parallels: there is one in the direction of each of the ideal points at the "ends" of R, asymptotically approaching R, always getting closer to R, but never meeting it. All other non-intersecting lines have a point of minimum distance and diverge from both sides of that point, are called ultraparallel, diverging parallel or sometimes non-intersecting; some geometers use parallel lines instead of limiting parallel lines, with ultraparallel lines being just non-intersecting. These limiting parallels make an angle θ with PB. For ultraparallel lines, the ultraparallel theorem states that there is a unique line in the hyperbolic plane, perpendicular to each pair of ultraparallel lines. In hyperbolic geometry, the circumference of a circle of radius r is greater than 2 π r. Let R = 1 − K, where K is the Gaussian curvature of the plane. In hyperbolic geometry, K is negative, so the square root is of a positive number.
The circumference of a circle of radius r is equal to: 2 π R sinh r R. And the area of the enclosed disk is: 4 π R 2 sinh 2 r 2 R = 2 π R 2. Therefore, in hyperbolic geometry the ratio of a circle's circumference to its radius is always greater than 2 π, though
Octahedron
In geometry, an octahedron is a polyhedron with eight faces, twelve edges, six vertices. The term is most used to refer to the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex. A regular octahedron is the dual polyhedron of a cube, it is a rectified tetrahedron. It is a square bipyramid in any of three orthogonal orientations, it is a triangular antiprism in any of four orientations. An octahedron is the three-dimensional case of the more general concept of a cross polytope. A regular octahedron is a 3-ball in the Manhattan metric. If the edge length of a regular octahedron is a, the radius of a circumscribed sphere is r u = a 2 2 ≈ 0.707 ⋅ a and the radius of an inscribed sphere is r i = a 6 6 ≈ 0.408 ⋅ a while the midradius, which touches the middle of each edge, is r m = a 2 = 0.5 ⋅ a The octahedron has four special orthogonal projections, centered, on an edge, vertex and normal to a face. The second and third correspond to A2 Coxeter planes.
The octahedron can be represented as a spherical tiling, projected onto the plane via a stereographic projection. This projection is conformal, preserving angles but not lengths. Straight lines on the sphere are projected as circular arcs on the plane. An octahedron with edge length √2 can be placed with its center at the origin and its vertices on the coordinate axes. In an x–y–z Cartesian coordinate system, the octahedron with center coordinates and radius r is the set of all points such that | x − a | + | y − b | + | z − c | = r; the surface area A and the volume V of a regular octahedron of edge length a are: A = 2 3 a 2 ≈ 3.464 a 2 V = 1 3 2 a 3 ≈ 0.471 a 3 Thus the volume is four times that of a regular tetrahedron with the same edge length, while the surface area is twice. If an octahedron has been stretched so that it obeys the equation | x x m | + | y y m | + | z z m | = 1, the formulas for the surface area and volume expand to become A = 4 x m y m z m × 1 x m 2 + 1 y m 2 + 1 z m 2, V = 4 3 x m y m z m.
Additionally the inertia tensor of the stretched octahedron is I =. These reduce to the equations for the regular octahedron when x m = y m = z m = a 2 2. {\displaystyle x_=y_=z_=
Apollonian gasket
In mathematics, an Apollonian gasket or Apollonian net is a fractal generated starting from a triple of circles, each tangent to the other two, successively filling in more circles, each tangent to another three. It is named after Greek mathematician Apollonius of Perga. An Apollonian gasket can be constructed. Start with three circles C1, C2 and C3, each one of, tangent to the other two. Apollonius discovered that there are two other non-intersecting circles, C4 and C5, which have the property that they are tangent to all three of the original circles – these are called Apollonian circles. Adding the two Apollonian circles to the original three, we now have five circles. Take one of the two Apollonian circles – say C4, it is tangent to C1 and C2, so the triplet of circles C4, C1 and C2 has its own two Apollonian circles. We know one of these – it is C3 – but the other is a new circle C6. In a similar way we can construct another new circle C7, tangent to C4, C2 and C3, another circle C8 from C4, C3 and C1.
This gives us 3 new circles. We can construct another three new circles from C5. Together with the circles C1 to C5, this gives a total of 11 circles. Continuing the construction stage by stage in this way, we can add 2·3n new circles at stage n, giving a total of 3n+1 + 2 circles after n stages. In the limit, this set of circles is an Apollonian gasket; the sizes of the new circles are determined by Descartes' theorem. Let ki denote the curvatures of four mutually tangent circles. Descartes' Theorem states The Apollonian gasket has a Hausdorff dimension of about 1.3057. The curvature of a circle is defined to be the inverse of its radius. Negative curvature indicates; this is bounding circle. Zero curvature gives a line. Positive curvature indicates; this circle is in the interior of circle with negative curvature. An Apollonian gasket can be constructed by replacing one of the generating circles by a straight line, which can be regarded as a circle passing through the point at infinity. Alternatively, two of the generating circles may be replaced by parallel straight lines, which can be regarded as being tangent to one another at infinity.
In this construction, the additional circles form a family of Ford circles. The three-dimensional equivalent of the Apollonian gasket is the Apollonian sphere packing. If two of the original generating circles have the same radius and the third circle has a radius, two-thirds of this the Apollonian gasket has two lines of reflective symmetry; these lines are perpendicular to one another, so the Apollonian gasket has rotational symmetry of degree 2. If all three of the original generating circles have the same radius the Apollonian gasket has three lines of reflective symmetry; each mutual tangent passes through the centre of the third circle and the common centre of the first two Apollonian circles. These lines of symmetry are at angles of 60 degrees to one another, so the Apollonian gasket has rotational symmetry of degree 3; the three generating circles, hence the entire construction, are determined by the location of the three points where they are tangent to one another. Since there is a Möbius transformation which maps any three given points in the plane to any other three points, since Möbius transformations preserve circles there is a Möbius transformation which maps any two Apollonian gaskets to one another.
Möbius transformations are isometries of the hyperbolic plane, so in hyperbolic geometry all Apollonian gaskets are congruent. In a sense, there is therefore only one Apollonian gasket, up to isometry; the Apollonian gasket is the limit set of a group of Möbius transformations known as a Kleinian group. If any four mutually tangent circles in an Apollonian gasket all have integer curvature all circles in the gasket will have integer curvature. Since the equation relating curvatures in an Apollonian gasket, integral or not, is a 2 + b 2 + c 2 + d 2 = 2 a b + 2 a c + 2 a d + 2 b c + 2 b d + 2 c d, it follows that one may move from one quadruple of curvatures to another by Vieta jumping, just as when finding a new Markov number; the first few of these integral Apollonian gaskets are listed in the following table. The table lists the curvatures of the largest circles in the gasket. Only the first three curvatures are needed to describe each gasket – all other curvatures can be derived from these three.
If none of the curvatures are repeated within the first five, the gasket contains no symmetry, represented by symmetry group C1. Whenever two of the largest five circles in the gasket have the same curvature, that gasket will have D1 symmetry, which corresponds to a reflection along a diameter of the bounding circl
Truncated triapeirogonal tiling
In geometry, the truncated triapeirogonal tiling is a uniform tiling of the hyperbolic plane with a Schläfli symbol of tr. The dual of this tiling represents the fundamental domains of, *∞32 symmetry. There are 3 small index subgroup constructed from by mirror alternation. In these images fundamental domains are alternately colored black and white, mirrors exist on the boundaries between colors. A special index 4 reflective subgroup, is, its direct subgroup +, semidirect subgroup. Given with generating mirrors its index 4 subgroup has generators. An index 6 subgroup constructed as, becomes; this tiling can be considered a member of a sequence of uniform patterns with vertex figure and Coxeter-Dynkin diagram. For p < 6, the members of the sequence are omnitruncated polyhedra, shown below as spherical tilings. For p > 6, they are tilings of the hyperbolic plane, starting with the truncated triheptagonal tiling. List of uniform planar tilings Tilings of regular polygons Uniform tilings in hyperbolic plane
Isotoxal figure
In geometry, a polytope, or a tiling, is isotoxal or edge-transitive if its symmetries act transitively on its edges. Informally, this means that there is only one type of edge to the object: given two edges, there is a translation, rotation and/or reflection that will move one edge to the other, while leaving the region occupied by the object unchanged; the term isotoxal is derived from the Greek τοξον meaning arc. An isotoxal polygon is an equilateral polygon; the duals of isotoxal polygons are isogonal polygons. In general, an isotoxal 2n-gon will have Dn dihedral symmetry. A rhombus is an isotoxal polygon with D2 symmetry. All regular polygons are isotoxal, having double the minimum symmetry order: a regular n-gon has Dn dihedral symmetry. A regular 2n-gon is an isotoxal polygon and can be marked with alternately colored vertices, removing the line of reflection through the mid-edges. Regular polyhedra are isohedral and isotoxal. Quasiregular polyhedra are not isohedral. Not every polyhedron or 2-dimensional tessellation constructed from regular polygons is isotoxal.
For instance, the truncated icosahedron has two types of edges: hexagon-hexagon and hexagon-pentagon, it is not possible for a symmetry of the solid to move a hexagon-hexagon edge onto a hexagon-pentagon edge. An isotoxal polyhedron has the same dihedral angle for all edges. There are nine convex isotoxal polyhedra formed from the Platonic solids, 8 formed by the Kepler–Poinsot polyhedra, six more as quasiregular star polyhedra and their duals. There are at least 5 polygonal tilings of the Euclidean plane that are isotoxal, infinitely many isotoxal polygonal tilings of the hyperbolic plane, including the Wythoff constructions from the regular hyperbolic tilings, non-right groups. Table of polyhedron dihedral angles Vertex-transitive Face-transitive Cell-transitive Peter R. Cromwell, Cambridge University Press 1997, ISBN 0-521-55432-2, p. 371 Transitivity Grünbaum, Branko. C.. Tilings and Patterns. New York: W. H. Freeman. ISBN 0-7167-1193-1. CS1 maint: Multiple names: authors list Coxeter, Harold Scott MacDonald.
"Uniform polyhedra", Philosophical Transactions of the Royal Society of London. Series A. Mathematical and Physical Sciences, 246: 401–450, doi:10.1098/rsta.1954.0003, ISSN 0080-4614, JSTOR 91532, MR 0062446