Disphenoid
In geometry, a disphenoid is a tetrahedron whose four faces are congruent acute-angled triangles. It can be described as a tetrahedron in which every two edges that are opposite each other have equal lengths. Other names for the same shape are sphenoid, isosceles tetrahedron, equifacial tetrahedron regular tetrahedron, tetramonohedron. All the solid angles and vertex figures of a disphenoid are the same, the sum of the face angles at each vertex is equal to two right angles. However, a disphenoid is not a regular polyhedron, because, in general, its faces are not regular polygons, its edges have three different lengths. If the faces of a disphenoid are equilateral triangles, it is a regular tetrahedron with Td tetrahedral symmetry, although this is not called a disphenoid; when the faces of a disphenoid are isosceles triangles, it is called a tetragonal disphenoid. In this case it has D2d dihedral symmetry. A sphenoid with scalene triangles as its faces is called a rhombic disphenoid and it has D2 dihedral symmetry.
Unlike the tetragonal disphenoid, the rhombic disphenoid has no reflection symmetry, so it is chiral. Both tetragonal disphenoids and rhombic disphenoids are isohedra: as well as being congruent to each other, all of their faces are symmetric to each other, it is not possible to construct a disphenoid with right triangle or obtuse triangle faces. When right triangles are glued together in the pattern of a disphenoid, they form a flat figure that does not enclose any volume; when obtuse triangles are glued in this way, the resulting surface can be folded to form a disphenoid but one with acute triangle faces and with edges that in general do not lie along the edges of the given obtuse triangles. Two more types of tetrahedron have similar names; the digonal disphenoid has faces with two different shapes, both isosceles triangles, with two faces of each shape. The phyllic disphenoid has faces with two shapes of scalene triangles. Disphenoids can be seen as digonal antiprisms or as alternated quadrilateral prisms.
A tetrahedron is a disphenoid. We have that a tetrahedron is a disphenoid if and only if the center in the circumscribed sphere and the inscribed sphere coincide. Another characterization states that if d1, d2 and d3 are the common perpendiculars of AB and CD; the disphenoids are the only polyhedra having infinitely many non-self-intersecting closed geodesics. On a disphenoid, all closed; the disphenoids are the tetrahedra in which all four faces have the same perimeter, the tetrahedra in which all four faces have the same area, the tetrahedra in which the angular defects of all four vertices equal π. They are the polyhedra having a net in the shape of an acute triangle, divided into four similar triangles by segments connecting the edge midpoints; the volume of a disphenoid with opposite edges of length l, m and n is given by V = 72. The circumscribed sphere has radius R = l 2 + m 2 + n 2 8 and the inscribed sphere has radius r = 3 V 4 T where V is the volume of the disphenoid and T is the area of any face, given by Heron's formula.
There is the following interesting relation connecting the volume and the circumradius: 16 T 2 R 2 = l 2 m 2 n 2 + 9 V 2. The squares of the lengths of the bimedians are 1 2, 1 2, 1 2. If the four faces of a tetrahedron have the same perimeter the tetrahedron is a disphenoid. If the four faces of a tetrahedron have the same area it is a disphenoid; the centers in the circumscribed and in
Regular dodecahedron
A regular dodecahedron or pentagonal dodecahedron is a dodecahedron, regular, composed of twelve regular pentagonal faces, three meeting at each vertex. It is one of the five Platonic solids, it has 12 faces, 20 vertices, 30 edges, 160 diagonals. It is represented by the Schläfli symbol. If the edge length of a regular dodecahedron is a, the radius of a circumscribed sphere is r u = a 3 4 ≈ 1.401 258 538 ⋅ a OEIS: A179296and the radius of an inscribed sphere is r i = a 1 2 5 2 + 11 10 5 ≈ 1.113 516 364 ⋅ a while the midradius, which touches the middle of each edge, is r m = a 1 4 ≈ 1.309 016 994 ⋅ a These quantities may be expressed as r u = a 3 2 ϕ r i = a ϕ 2 2 3 − ϕ r m = a ϕ 2 2 where ϕ is the golden ratio. Note that, given a regular dodecahedron of edge length one, ru is the radius of a circumscribing sphere about a cube of edge length ϕ, ri is the apothem of a regular pentagon of edge length ϕ; the surface area A and the volume V of a regular dodecahedron of edge length a are: A = 3 25 + 10 5 a 2 ≈ 20.645 728 807 a 2 V = 1 4 a 3 ≈ 7.663 118 9606 a 3 Additionally, the surface area and volume of a regular dodecahedron are related to the golden ratio.
A dodecahedron with an edge length of one unit has the properties: A = 15 φ 3 − φ V = 5 φ 3 6 − 2 φ The regular dodecahedron has two special orthogonal projections, centered, on vertices and pentagonal faces, correspond to the A2 and H2 Coxeter planes. In perspective projection, viewed on top of a pentagonal face, the regular dodecahedron can be seen as a linear-edged Schlegel diagram, or stereographic projection as a spherical polyhedron; these projections are used in showing the four-dimensional 120-cell, a regular 4-dimensional polytope, constructed from 120 dodecahedra, projecting it down to 3-dimensions. The regular dodecahedron can be represented as a spherical tiling; the following Cartesian coordinates define the 20 vertices of a regular dodecahedron centered at the origin and suitably scaled and oriented: where ϕ = 1 + √5/2 is the golden ratio ≈ 1.618. The edge length is 2/ϕ = √5 − 1; the circumradius is √3. Similar to the symmetry of the vertex coordinates, the equations of the twelve facets of the regular dodecahedron display symmetry in their coefficients: x ± ϕy = ±ϕ2 y ± ϕz = ±ϕ2 z ± ϕx = ±ϕ2 The dihedral angle of a regular dodecahedron is 2 arctan or 116.56505117707798935157219372045°.
OEIS: A137218 Note that the tangent of the dihedral angle is −2. If the original regular dodecahedron has edge length 1, its dual icosahedron has edge length ϕ. If the five Platonic solids are built with same volume, the regular dodecahedron has the shortest edges, it has 43,380 nets. The map-coloring number of a regular dodecahedron's faces is 4; the distance between the vertices on the same face not connected by an edge is ϕ times the edge length. If two edges share a common vertex the midpoints of those edges form a 36-72-72 triangle with the body center; the regular dodecahedron is the third in an infinite set of truncated trapezohedra which can be constructed by truncating the two axial vertices of a pentagonal trapezohedron. The stellations of the regular dodecahedron make up three of the four Kepler–Poinsot polyhedra. A rectified regular dodecahedron forms an
Antiprism
In geometry, an n-sided antiprism is a polyhedron composed of two parallel copies of some particular n-sided polygon, connected by an alternating band of triangles. Antiprisms are a type of snub polyhedra. Antiprisms are similar to prisms except the bases are twisted relative to each other, that the side faces are triangles, rather than quadrilaterals. In the case of a regular n-sided base, one considers the case where its copy is twisted by an angle 180°/n. Extra regularity is obtained when the line connecting the base centers is perpendicular to the base planes, making it a right antiprism; as faces, it has the two n-gonal bases and, connecting 2n isosceles triangles. A uniform antiprism has, apart from 2n equilateral triangles as faces; as a class, the uniform antiprisms form an infinite series of vertex-uniform polyhedra, as do the uniform prisms. For n = 2 we have as degenerate case the regular tetrahedron as a digonal antiprism, for n = 3 the non-degenerate regular octahedron as a triangular antiprism.
The dual polyhedra of the antiprisms are the trapezohedra. Their existence was discussed and their name was coined by Johannes Kepler, though it is possible that they were known to Archimedes as they satisfy the same conditions on vertices as the Archimedean solids. Cartesian coordinates for the vertices of a right antiprism with n-gonal bases and isosceles triangles are with k ranging from 0 to 2n − 1. Let a be the edge-length of a uniform antiprism; the volume is V = n 4 cos 2 π 2 n − 1 sin 3 π 2 n 12 sin 2 π n a 3 and the surface area is A = n 2 a 2. There are an infinite set of truncated antiprisms, including a lower-symmetry form of the truncated octahedron; these can be alternated to create snub antiprisms, two of which are Johnson solids, the snub triangular antiprism is a lower symmetry form of the icosahedron. The symmetry group of a right n-sided antiprism with regular base and isosceles side faces is Dnd of order 4n, except in the case of a tetrahedron, which has the larger symmetry group Td of order 24, which has three versions of D2d as subgroups, the octahedron, which has the larger symmetry group Oh of order 48, which has four versions of D3d as subgroups.
The symmetry group contains inversion if and only. The rotation group is Dn of order 2n, except in the case of a tetrahedron, which has the larger rotation group T of order 12, which has three versions of D2 as subgroups, the octahedron, which has the larger rotation group O of order 24, which has four versions of D3 as subgroups. Uniform star antiprisms are named by their star polygon bases, exist in prograde and retrograde solutions. Crossed forms have intersecting vertex figures, are denoted by inverted fractions, p/ instead of p/q, e.g. 5/3 instead of 5/2. In the retrograde forms but not in the prograde forms, the triangles joining the star bases intersect the axis of rotational symmetry; some retrograde star antiprisms with regular star polygon bases cannot be constructed with equal edge lengths, so are not uniform polyhedra. Star antiprism compounds can be constructed where p and q have common factors. Apeirogonal antiprism Rectified antiprism Grand antiprism – a four-dimensional polytope One World Trade Center, a building consisting of an elongated square antiprism Skew polygon Anthony Pugh.
Polyhedra: A visual approach. California: University of California Press Berkeley. ISBN 0-520-03056-7. Chapter 2: Archimedean polyhedra and antiprisms Weisstein, Eric W. "Antiprism". MathWorld. Nonconvex Prisms and Antiprisms Paper models of prisms and antiprisms
Parity (mathematics)
In mathematics, parity is the property of an integer's inclusion in one of two categories: or odd. An integer is if it is divisible by two and odd if it is not even. For example, 6 is because there is no remainder when dividing it by 2. By contrast, 3, 5, 7, 21 leave a remainder of 1 when divided by 2. Examples of numbers include −4, 0, 82 and 178. In particular, zero is an number; some examples of odd numbers are −5, 3, 29, 73. A formal definition of an number is that it is an integer of the form n = 2k, where k is an integer, it is important to realize that the above definition of parity applies only to integer numbers, hence it cannot be applied to numbers like 1/2 or 4.201. See the section "Higher mathematics" below for some extensions of the notion of parity to a larger class of "numbers" or in other more general settings; the sets of and odd numbers can be defined as following: Even = Odd = A number expressed in the decimal numeral system is or odd according to whether its last digit is or odd.
That is, if the last digit is 1, 3, 5, 7, or 9 it is odd. The same idea will work using any base. In particular, a number expressed in the binary numeral system is odd if its last digit is 1 and if its last digit is 0. In an odd base, the number is according to the sum of its digits – it is if and only if the sum of its digits is even; the following laws can be verified using the properties of divisibility. They are a special case of rules in modular arithmetic, are used to check if an equality is to be correct by testing the parity of each side; as with ordinary arithmetic and addition are commutative and associative in modulo 2 arithmetic, multiplication is distributive over addition. However, subtraction in modulo 2 is identical to addition, so subtraction possesses these properties, not true for normal integer arithmetic. Even ± = even; the division of two whole numbers does not result in a whole number. For example, 1 divided by 4 equals 1/4, neither nor odd, since the concepts and odd apply only to integers.
But when the quotient is an integer, it will be if and only if the dividend has more factors of two than the divisor. The ancient Greeks considered 1, the monad, to be neither odd nor even; some of this sentiment survived into the 19th century: Friedrich Wilhelm August Fröbel's 1826 The Education of Man instructs the teacher to drill students with the claim that 1 is neither nor odd, to which Fröbel attaches the philosophical afterthought, It is well to direct the pupil's attention here at once to a great far-reaching law of nature and of thought. It is this, that between two different things or ideas there stands always a third, in a sort of balance, seeming to unite the two. Thus, there is here between odd and numbers one number, neither of the two. In form, the right angle stands between the acute and obtuse angles. A thoughtful teacher and a pupil taught to think for himself can scarcely help noticing this and other important laws. Integer coordinates of points in Euclidean spaces of two or more dimensions have a parity defined as the parity of the sum of the coordinates.
For instance, the face-centered cubic lattice and its higher-dimensional generalizations, the Dn lattices, consist of all of the integer points whose sum of coordinates is even. This feature manifests itself in chess, where the parity of a square is indicated by its color: bishops are constrained to squares of the same parity; this form of parity was famously used to solve the mutilated chessboard problem: if two opposite corner squares are removed from a chessboard the remaining board cannot be covered by dominoes, because each domino covers one square of each parity and there are two more squares of one parity than of the other. The parity of an ordinal number may be defined to be if the number is a limit ordinal, or a limit ordinal plus a finite number, odd otherwise. Let R be a commutative ring and let I be an ideal of R whose index is 2. Elements of the coset 0 + I may be called while elements of the coset 1 + I may be called odd; as an example, let R = Z be the localization of Z at the prime ideal.
An element of R is or odd if and only if its numerator is so in Z. The numbers form an ideal in the ring of integers, but the odd numbers do not — this is clear from the fact that the identity element for addition, zero, is an element of the numbers only. An integer is if it is congruent to 0 modulo this ideal, in other words if it is congruent to 0 modulo 2, odd if it is congruent to 1 modulo 2. All prime numbers are odd, with one exception: the prime number 2. All known perfect numbers are even. Goldbach's conjecture states that every integer greater than 2 can be represented as a sum of two prime numbers. Modern computer calculations have shown this conjecture to
Cairo pentagonal tiling
In geometry, the Cairo pentagonal tiling is a dual semiregular tiling of the Euclidean plane. It is given its name, it is one of 15 known monohedral pentagon tilings. It is called MacMahon's net after Percy Alexander MacMahon and his 1921 publication New Mathematical Pastimes. Conway calls it a 4-fold pentille; as a 2-dimensional crystal net, it shares a special feature with the honeycomb net. Both nets are examples of standard realization, the notion introduced by M. Kotani and T. Sunada for general crystal nets; these are not regular pentagons: their sides are not equal, their angles in sequence are 120°, 120°, 90°, 120°, 90°. It is represented by with face configuration V3.3.4.3.4. It is similar to the prismatic pentagonal tiling with face configuration V3.3.3.4.4, which has its right angles adjacent to each other. The Cairo pentagonal tiling has two lower symmetry forms given as monohedral pentagonal tilings types 4 and 8: It is the dual of the snub square tiling, made of two squares and three equilateral triangles around each vertex.
This tiling can be seen as the union of two perpendicular hexagonal tilings, flattened by a ratio of 3. Each hexagon is divided into four pentagons; the two hexagons can be distorted to be concave, leading to concave pentagons. Alternately one of the hexagonal tilings can remain regular, the second one stretched and flattened by 3 in each direction, intersecting into 2 forms of pentagons; as a dual to the snub square tiling the geometric proportions are fixed for this tiling. However it can be adjusted to other geometric forms with the same topological connectivity and different symmetry. For example, this rectangular tiling is topologically identical. Truncating the 4-valence nodes creates a form related to the Goldberg polyhedra, can be given the symbol 2,1; the pentagons are truncated into heptagons. The dual 2,1 has all triangle, related to the geodesic polyhedra, it can be seen as a snub square tiling with its squares replaced by 4 triangles. The Cairo pentagonal tiling is similar to the prismatic pentagonal tiling with face configuration V3.3.3.4.4, two 2-uniform dual tilings and 2 3-uniform duals which mix the two types of pentagons.
They are drawn here with k-isohedral pentagons. The Cairo pentagonal tiling is in a sequence of dual snub polyhedra and tilings with face configuration V3.3.4.3.n. It is in a sequence of dual snub polyhedra and tilings with face configuration V3.3.n.3.n. Tilings of regular polygons List of uniform tilings Grünbaum, Branko. C.. Tilings and Patterns. New York: W. H. Freeman. ISBN 0-7167-1193-1. CS1 maint: Multiple names: authors list Williams, Robert; the Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. p. 38. ISBN 0-486-23729-X. Wells, The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, p. 23, 1991. Keith Critchlow, Order in Space: A design source book, 1970, p. 77-76, pattern 3 Weisstein, Eric W. "Cairo Tessellation". MathWorld
Trapezohedron
The n-gonal trapezohedron, antibipyramid or deltohedron is the dual polyhedron of an n-gonal antiprism. With a highest symmetry, its 2n faces are congruent kites; the faces are symmetrically staggered. The n-gon part of the name does not reference the faces here but arrangement of vertices around an axis of symmetry; the dual n-gonal antiprism has two actual n-gon faces. An n-gonal trapezohedron can be dissected into an n-gonal antiprism; these figures, sometimes called deltohedra, must not be confused with deltahedra, whose faces are equilateral triangles. In texts describing the crystal habits of minerals, the word trapezohedron is used for the polyhedron properly known as a deltoidal icositetrahedron; the symmetry group of an n-gonal trapezohedron is Dnd of order 4n, except in the case of a cube, which has the larger symmetry group Od of order 48, which has four versions of D3d as subgroups. The rotation group is Dn of order 2n, except in the case of a cube, which has the larger rotation group O of order 24, which has four versions of D3 as subgroups.
One degree of freedom within Dn symmetry changes the kites into congruent quadrilaterals with 3 edges lengths. In the limit, one edge of each quadrilateral goes to zero length, these become bipyramids. If the kites surrounding the two peaks are of different shapes, it can only have Cnv symmetry, order 2n; these can be called asymmetric trapezohedra. The dual is an unequal antiprism, with the bottom polygons of different radii. If it twisted and unequal its symmetry is reduced to cyclic symmetry, Cn symmetry, order n. A n-trapezohedron has 2n quadrilateral faces, with 2n+2 vertices. Two vertices are on the polar axis, the others are in two regular n-gonal rings of vertices. Special cases: n=2: A degenerate form, form a geometric tetrahedron with 6 vertices, 8 edges, 4 degenerate kite faces that are degenerated into triangles, its dual is a degenerate form of antiprism a tetrahedron. N=3: In the case of the dual of a triangular antiprism the kites are rhombi, hence these trapezohedra are zonohedra.
They are called rhombohedra. They are cubes scaled in the direction of a body diagonal, they are the parallelepipeds with congruent rhombic faces. A special case of a rhombohedron is one in which the rhombi which form the faces have angles of 60° and 120°, it can be decomposed into a regular octahedron. Since parallelepipeds can fill space, so can a combination of regular tetrahedra and regular octahedra. Crystal arrangements of atoms can repeat in space with hexagonal trapezohedral cells; the pentagonal trapezohedron is the only polyhedron other than the Platonic solids used as a die in roleplaying games such as Dungeons & Dragons. Having 10 sides, it can be used in repetition to generate any decimal-based uniform probability desired. Two dice of different colors are used for the two digits to represent numbers from 00 to 99. Self-intersecting trapezohedron exist with a star polygon central figure, defined by kite faces connecting each polygon edge to these two points. A p/q-trapezohedron has Coxeter-Dynkin diagram.
Diminished trapezohedron Rhombic dodecahedron Rhombic triacontahedron Bipyramid Conway polyhedron notation Anthony Pugh. Polyhedra: A visual approach. California: University of California Press Berkeley. ISBN 0-520-03056-7. Chapter 4: Duals of the Archimedean polyhedra and antiprisms Weisstein, Eric W. "Trapezohedron". MathWorld. Weisstein, Eric W. "Isohedron". MathWorld. Virtual Reality Polyhedra The Encyclopedia of Polyhedra VRML models <3> <4> <5> <6> <7> <8> <9> <10> Conway Notation for Polyhedra Try: "dAn", where n=3,4,5... example "dA5" is a pentagonal trapezohedron. Paper model tetragonal trapezohedron
Bipyramid
An n-gonal bipyramid or dipyramid is a polyhedron formed by joining an n-gonal pyramid and its mirror image base-to-base. An n-gonal bipyramid has 2n triangle faces, 3n edges, 2 + n vertices; the referenced n-gon in the name of the bipyramids is not an external face but an internal one, existing on the primary symmetry plane which connects the two pyramid halves. A right bipyramid has two points below the centroid of its base. Nonright bipyramids are called oblique bipyramids. A regular bipyramid has a regular polygon internal face and is implied to be a right bipyramid. A right bipyramid can be represented as + P for internal polygon P, a regular n-bipyramid +. A concave bipyramid has a concave interior polygon; the face-transitive regular bipyramids are the dual polyhedra of the uniform prisms and will have isosceles triangle faces. A bipyramid can be projected on a sphere or globe as n spaced lines of longitude going from pole to pole, bisected by a line around the equator. Bipyramid faces, projected as spherical triangles, represent the fundamental domains in the dihedral symmetry Dnh.
Indeed, an n-tonal bipyramid can be seen as the Kleetope of the respective n-gonal dihedron. The volume of a bipyramid is V =2/3Bh where B is the area of the base and h the height from the base to the apex; this works for any location of the apex, provided that h is measured as the perpendicular distance from the plane which contains the base. The volume of a bipyramid whose base is a regular n-sided polygon with side length s and whose height is h is therefore: V = n 6 h s 2 cot π n. Only three kinds of bipyramids can have all edges of the same length: the triangular and pentagonal bipyramids; the tetragonal bipyramid with identical edges, or regular octahedron, counts among the Platonic solids, while the triangular and pentagonal bipyramids with identical edges count among the Johnson solids. If the base is regular and the line through the apexes intersects the base at its center, the symmetry group of the n-gonal bipyramid has dihedral symmetry Dnh of order 4n, except in the case of a regular octahedron, which has the larger octahedral symmetry group Oh of order 48, which has three versions of D4h as subgroups.
The rotation group is Dn of order 2n, except in the case of a regular octahedron, which has the larger symmetry group O of order 24, which has three versions of D4 as subgroups. The digonal faces of a spherical 2n-bipyramid represents the fundamental domains of dihedral symmetry in three dimensions: Dnh, order 4n; the reflection domains can be shown as alternately colored triangles as mirror images. An asymmetric right bipyramid joins two unequal height pyramids. An inverted form can have both pyramids on the same side. A regular n-gonal asymmetry right pyramid has order 2n; the dual polyhedron of an asymmetric bipyramid is a frustum. A scalenohedron is topologically identical to a 2n-bipyramid, but contains congruent scalene triangles. There are two types. In one type the 2n vertices around the center alternate in rings below the center. In the other type, the 2n vertices are on the same plane, but alternate in two radii; the first has 2-fold rotation axes mid-edge around the sides, reflection planes through the vertices, n-fold rotation symmetry on its axis, representing symmetry Dnd, order 2n.
In crystallography, 8-sided and 12-sided scalenohedra exist. All of these forms are isohedra; the second has order 2n. The smallest scalenohedron is topologically identical to the regular octahedron; the second type is a rhombic bipyramid. The first type has 6 vertices can be represented as, where z is a parameter between 0 and 1, creating a regular octahedron at z = 0, becoming a disphenoid with merged coplanar faces at z = 1. For z > 1, it becomes concave. Self-intersecting bipyramids exist with a star polygon central figure, defined by triangular faces connecting each polygon edge to these two points. A bipyramid has Coxeter diagram. Isohedral even-sided stars can be made with zig-zag offplane vertices, in-out isotoxal forms, or both, like this form: The dual of the rectification of each convex regular 4-polytopes is a cell-transitive 4-polytope with bipyramidal cells. In the following, the apex vertex of the bipyramid is A and an equator vertex is E; the distance between adjacent vertices on the equator EE = 1, the apex to equator edge is AE and the distance between the apices is AA.
The bipyramid 4-polytope will have VA vertices. It will have VE vertices. NAE bipyramids meet along each type AE edge. NEE bipyramids meet along each type EE edge. CAE is the cosine of the dihedral angle along an AE edge. CEE is the cosine of the dihedral angle along an EE edge; as cells must fit around an edge, NAA cos−1 ≤ 2π, NAE cos−1 ≤ 2π. * The rectified 16-cell is the regular 24-cell and vertices are all equivalent – octahedra are regular bipyramids. ** Given numerically due to more complex form. In general, a bipyramid can be seen as an n-polytope constructed with a -polytope in a hyperplane with two points in opposite directions, equal distance perpendicular from the hyperplane. If the -polytope is a regular polytope, it will have identical pyramids facets. An example is the 16-cell, an octahedral bipyramid, more an n-orthoplex is an -orth