Quadrant Model of Reality: Quadrant Model of Reality Book 23 Philosophy/ Math

Philosophy Chapter

QMRA primitive cell is a unit cell constructed so that it contains only one lattice point (each vertex of the cell sits on a lattice point which is shared with the surrounding cells, each lattice point is said to contribute 1/n to the total number of lattice points in the cell where n is the number of cells sharing the lattice point).[1] A primitive cell is built on the primitive basis of the direct lattice, namely a crystallographic basis of the vector lattice L such that every lattice vector t of L may be obtained as an integral linear combination of the basis vectors, a, b, c.

Used predominantly in geometry, solid state physics, and mineralogy, particularly in describing crystal structure, a primitive cell is a minimum volume cell corresponding to a single lattice point of a structure with translational symmetry in 2 dimensions, 3 dimensions, or other dimensions. A lattice can be characterized by the geometry of its primitive cell.

Primitive cells are four sided quadrants

2-dimensional primitive cell[edit]

A 2-dimensional primitive cell is a parallelogram, which in special cases may have orthogonal angles, or equal lengths, or both.

2-dimensional primitive cells

Wallpaper group diagram p1.svg

Parallelogram Wallpaper group diagram p1 rect.svg

Rectangle Wallpaper group diagram p1 rhombic.svg

Rhombus Wallpaper group diagram p1 square.svg

Square

The primitive cell is a primitive unit. A primitive unit is a section of the tiling (usually a parallelogram or a set of neighboring tiles) that generates the whole tiling using only translations, and is as small as possible.

The primitive cell is a fundamental domain with respect to translational symmetry only. In the case of additional symmetries a fundamental domain is smaller.

3-dimensional primitive cell[edit]

A crystal can be categorized by its lattice and the atoms that lie in a primitive cell (the basis). A cell will fill all the lattice space without leaving gaps by repetition of crystal translation operations.

A 3-dimensional primitive cell is a parallelepiped, which in special cases may have orthogonal angles, or equal lengths, or both.

3-dimensional primitive cells

Triclinic.svg

Triclinic Monoclinic2.svg

Monoclinic Rhombohedral.svg

Rhombohedral Orthorhombic.svg

Orthorhombic Tetragonal.svg

Tetragonal Cubic.svg

Cubic

Primitive translation vectors are used to define a crystal translation vector, \vec T , and also gives a lattice cell of smallest volume for a particular lattice. The lattice and translation vectors \vec a_1 , \vec a_2 , and \vec a_3 are primitive if the atoms look the same from any lattice points using integers u_1 , u_2 , and u_3 .

\vec T = u_1\vec a_1 + u_2\vec a_2 + u_3\vec a_3

The primitive cell is defined by the primitive axes (vectors) \vec a_1 , \vec a_2 , and \vec a_3 . The volume, V_p , of the primitive cell is given by the parallelepiped from the above axes as

V_p = | \vec a_1 \cdot ( \vec a_2 \times \vec a_3 ) |.

QMRIn 1787, Franklin served as a delegate to the Philadelphia Convention. He held an honorary position and seldom engaged in debate. He is the only Founding Father who is a signatory of all four of the major documents of the founding of the United States: the Declaration of Independence, the Treaty of Alliance with France, the Treaty of Paris and the United States Constitution.

QMR"The Big Four" was the name popularly given to the famous and influential businessmen, philanthropists and railroad tycoons who built the Central Pacific Railroad, (C.P.R.R.), which formed the western portion through the Sierra Nevada and the Rocky Mountains of the First Transcontinental Railroad in the United States, built from the mid-continent at the Mississippi River to the Pacific Ocean during the middle and late 1860s.[1] Composed of Leland Stanford, (1824–1893), Collis Potter Huntington, (1821–1900), Mark Hopkins, (1813–1878), and Charles Crocker, (1822–1888), the four themselves however, personally preferred to be known as "The Associates."[2]

The jungle gym, monkey bars, or climbing frame, is a piece of playground equipment made of many pieces of material, such as metal pipe or rope, on which children can climb, hang, or sit. The monkey bar designation refers to the rambunctious, climbing play of monkeys.

The pipe of a traditional set are in quadrant formation

QMRA traditional jungle gym. A traditional jungle gym is made up of lattices that look like quadrants

QMRA glider or platform rocker is a type of rocking chair that moves as a swing seat, where the entire frame consists of a seat attached to the base by means of a double-rocker four-bar linkage. The non-parallel suspension arms of the linkage cause the chair to simulate a rocking-chair motion as it swings back and forth.

Gliders are used as alternatives to porch swings, and are also popular as nursery furnishing for assisting parents in feeding newborn babies. Because pinch points are moved away from the floor, a glider is marginally safer for pets and toddlers.

History[edit]

Early patents described different mechanisms for glider chairs, such as rails[1] and four-bar linkages supported by springs.[2] Patents using a swinging seat suspended from a four-bar linkage appeared in 1939, and this is now the general configuration used by most glider chairs.[3]

In the southern United States, porch gliders were referred to as divans. Especially popular was the "basket weave" pattern in the hot non- air conditioned South of the 1950s and 1960s

QMRPantograph (four-bar, two degrees of freedom, i.e., only one pivot joint is fixed.)

Crank-slider, (four-bar, one degree of freedom)

Double wishbone suspension

Watt's linkage and Chebyshev linkage (linkages that approximate straight line motion)

Biological linkages

Bicycle suspension

QMRBennett's linkage[edit]

A Bennett's linkage

Another example of an overconstrained mechanism is Bennett's linkage, which consists of four links connected by four revolute joints.[2]

A general spatial linkage formed from four links and four hinged joints has mobility

M = 6(N - 1 - j) + \sum_{i=1}^j f_i = 6(4-1-4) + 4 = -2,

which is a highly constrained system.

As in the case of the Sarrus linkage, it is a particular set of dimensions that makes the Bennett linkage movable.[3][4]

The dimensional constraints that makes Bennett's linkage movable are the following. Let us number the links in order that links with consecutive index are joined (first and fourth links are also joined). For the i-th link, let us denote by di and ai respectively the distance and the oriented angle of the axes of the revolute joints of the link. Bennett's linkage must satisfies the following constraints:

\begin{align}

&d_1 = d_3, \quad a_1 = a_3\\

&d_2 = d_4, \quad a_2 = a_4\\

&\frac{d_1^2}{sin^2a_1} = \frac{d_2^2}{sin^2a_2}.

\end{align}

Moreover, the links are assembled in such a way that, for two links that are joined together, the common perpendicular to the joint axes of the first link intersects the common perpendicular of the joint axes of the second link.

Below is an external link to an animation of a Bennett's linkage.

QMRA four-bar linkage, also called a four-bar, is the simplest movable closed chain linkage. It consists of four bodies, called bars or links, connected in a loop by four joints. Generally, the joints are configured so the links move in parallel planes, and the assembly is called a planar four-bar linkage.[1]

If the linkage has four hinged joints with axes angled to intersect in a single point, then the links move on concentric spheres and the assembly is called a spherical four-bar linkage. Bennett's linkage is a spatial four-bar linkage with hinged joints that have their axes angled in a particular way that makes the system movable.[2]

QMRThe Peaucellier–Lipkin linkage (or Peaucellier–Lipkin cell, or Peaucellier–Lipkin inversor), invented in 1864, was the first true planar straight line mechanism – the first planar linkage capable of transforming rotary motion into perfect straight-line motion, and vice versa. It is named after Charles-Nicolas Peaucellier (1832–1913), a French army officer, and Yom Tov Lipman Lipkin (1846–1876), a Lithuanian Jew and son of the famed Rabbi Israel Salanter.[1][2]

It has a quadrilateral in it

Until this invention, no planar method existed of producing exact straight-line motion without reference guideways, making the linkage especially important as a machine component and for manufacturing. In particular, a piston head needs to keep a good seal with the shaft in order to retain the driving (or driven) medium. The Peaucellier linkage was important in the development of the steam engine.[dubious – discuss]

The mathematics of the Peaucellier–Lipkin linkage is directly related to the inversion of a circle.

QMRSlider-rocker four-bar

acts as the driver of the Peaucellier–Lipkin linkage

QMR Hart's inversor is a mechanism that provides a perfect straight line motion without sliding guides.[1]

It was invented and published by Harry Hart in 1874–5.[1][2]

It can be used to convert rotary motion to a perfect straight line by fixing a point on one short link and driving a point on another link in a circular arc.[1][3] (The fixed points and driving arm make it a 6-bar linkage.) It is an anti parallelogram (it has a cross)

QMRIn geometry, an antiparallelogram is a quadrilateral having, like a parallelogram, two opposite pairs of equal-length sides, but in which the sides of one pair cross each other. The longer of the two pairs will always be the one that crosses. Antiparallelograms are also called contraparallelograms[1] or crossed parallelograms.[2]

A crossed parallelograms is a special case of a crossed quadrilateral with unequal edges.[3] A special form of the crossed parallelogram is a crossed rectangle where the short edges are parallel.

It has a crossing like a quadrant

Four-bar linkages[edit]

The antiparallelogram has been used as a form of four-bar linkage, in which four rigid beams of fixed length (the four sides of the antiparallelogram) may rotate with respect to each other at joints placed at the four vertices of the antiparallelogram. In this context it is also called a butterfly or bow-tie linkage. As a linkage, it has a point of instability in which it can be converted into a parallelogram and vice versa.

Fixing the short edge of an antiparallelogram linkage causes the crossing point to trace out an ellipse.

If one of the short (uncrossed) edges of an antiparallelogram linkage is fixed in place, and the remaining linkage moves freely, then the crossing point of the antiparallelogram traces out an ellipse that has the fixed edge's endpoints as its foci. The other moving short edge of the antiparallelogram has as its endpoints the foci of another moving ellipse, formed from the first one by reflection across a tangent line through the crossing point.[2][6]

An antiparallelogram linkage with one of its long edges ΦΛ fixed as a base. The two points F and L move in circles of equal radius but opposite direction.

For both the parallelogram and antiparallelogram linkages, if one of the long (crossed) edges of the linkage is fixed as a base, the free joints move on equal circles, but in a parallelogram they move in the same direction with equal velocities while in the antiparallelogram they move in opposite directions with unequal velocities.[7] As James Watt discovered, if an antiparallelogram has its long side fixed in this way it forms a variant of Watt's linkage, and the midpoint of the unfixed long edge will trace out a lemniscate or figure eight curve. For the antiparallelogram formed by the sides and diagonals of a square, it is the lemniscate of Bernoulli.[8]

The antiparallelogram is an important feature in the design of Hart's inversor, a linkage that (like the Peaucellier–Lipkin linkage) can convert rotary motion to straight-line motion.[9] An antiparallelogram-shaped linkage can also be used to connect the two axles of a four-wheeled vehicle, decreasing the turning radius of the vehicle relative to a suspension that only allows one axle to turn.[2] A pair of nested antiparallelograms was used in a linkage defined by Alfred Kempe as part of his universality theorem stating that any algebraic curve may be traced out by the joints of a suitably defined linkage. Kempe called the nested-antiparallelogram linkage a "multiplicator", as it could be used to multiply an angle by an integer.[1]

Celestial mechanics[edit]

In the n-body problem, the study of the motions of point masses under Newton's law of universal gravitation, an important role is played by central configurations, solutions to the n-body problem in which all of the bodies rotate around some central point as if they were rigidly connected to each other. For instance, for three bodies, there are five solutions of this type, given by the five Lagrangian points. For four bodies, with two pairs of the bodies having equal masses, numerical evidence indicates that there exists a continuous family of central configurations, related to each other by the motion of an antiparallelogram linkage.[

Uniform polyhedra and their duals[edit]

The small rhombihexahedron. Slicing off any vertex of this shape gives an antiparallelogram cross-section as the vertex figure.

The small rhombihexacron, a polyhedron with antiparallelograms (formed by pairs of coplanar triangles) as its faces

Several nonconvex uniform polyhedra, including the tetrahemihexahedron, cubohemioctahedron, octahemioctahedron, small rhombihexahedron, small icosihemidodecahedron, and small dodecahemidodecahedron, have antiparallelograms as their vertex figures, the cross-sections formed by slicing the polyhedron by a plane that passes near a vertex, perpendicularly to the axis between the vertex and the center.[5]

For uniform polyhedra of this type in which the faces do not pass through the center point of the polyhedron, the dual polyhedron has antiparallelograms as its faces; examples of dual uniform polyhedra with antiparallelogram faces include the small rhombihexacron, the great rhombihexacron, the small rhombidodecacron, the great rhombidodecacron, the small dodecicosacron, and the great dodecicosacron. The antiparallelograms that form the faces of these dual uniform polyhedra are the same antiparallelograms that form the vertex figure of the original uniform polyhedron.

QMR Hart's inversor is a mechanism that provides a perfect straight line motion without sliding guides.[1]

It was invented and published by Harry Hart in 1874–5.[1][2]

QMRSlider-rocker four-bar

acts as the driver of the Peaucellier–Lipkin linkage

It has a quadrilateral in it

QMRIn mathematics, a root of unity, occasionally called a de Moivre number, is any complex number that gives 1 when raised to some positive integer power n. Roots of unity are used in many branches of mathematics, and are especially important in number theory, the theory of group characters, and the discrete Fourier transform.

In field theory and ring theory the notion of root of unity also applies to any ring with a multiplicative identity element. Any algebraically closed field has exactly n nth roots of unity, if n is not divisible by the characteristic of the field.

A complex plane is a quadrant grid

QMRIn mathematics, an nth-order Argand system (named after French mathematician Jean-Robert Argand) is a coordinate system constructed around the nth roots of unity. From the origin, n axes extend such that the angle between each axis and the axes immediately before and after it is 360/n degrees. For example, the number line is the 2nd-order Argand system because the two axes extending from the origin represent 1 and −1, the 2nd roots of unity. The complex plane (sometimes called the Argand plane, also named after Argand) is the 4th-order Argand system because the 4 axes extending from the origin represent 1, i, −1, and −i, the 4th roots of unity.

QMRAlgebraic expression of roots[edit]

All roots of polynomials with rational coefficients are algebraic numbers, by definition of the latter. If the degree n of the polynomial is no greater than 4, all the roots of the polynomial can be written as algebraic expressions in terms of the coefficients—that is, applying only the four basic arithmetic operations and the extraction of n-th roots. But by the Abel-Ruffini theorem, this cannot be done in general for higher-degree equations.

QMRA rectilinear polygon is a polygon all of whose edges meet at right angles. Thus the interior angle at each vertex is either 90° or 270°. Rectilinear polygons are a special case of isothetic polygons.

QMRThere are four models commonly used for hyperbolic geometry: the Klein model, the Poincaré disk model, the Poincaré half-plane model, and the Lorentz or hyperboloid model. These models define a hyperbolic plane which satisfies the axioms of a hyperbolic geometry. Despite their names, the first three mentioned above were introduced as models of hyperbolic space by Beltrami, not by Poincaré or Klein. All these models are extendable to more dimensions.

QMRComparison with Euclidean geometry[edit]

In Euclidean geometry, a figure can be scaled up or scaled down indefinitely, and the resulting figures are similar, i.e., they have the same angles and the same internal proportions. In elliptic geometry this is not the case. For example, in the spherical model we can see that the distance between any two points must be strictly less than half the circumference of the sphere (because antipodal points are identified). A line segment therefore cannot be scaled up indefinitely. A geometer measuring the geometrical properties of the space he or she inhabits can detect, via measurements, that there is a certain distance scale that is a property of the space. On scales much smaller than this one, the space is approximately flat, geometry is approximately Euclidean, and figures can be scaled up and down while remaining approximately similar.

A great deal of Euclidean geometry carries over directly to elliptic geometry. For example, the first and fourth of Euclid's postulates, that there is a unique line between any two points and that all right angles are equal, hold in elliptic geometry. Postulate 3, that one can construct a circle with any given center and radius, fails if "any radius" is taken to mean "any real number", but holds if it is taken to mean "the length of any given line segment". Therefore any result in Euclidean geometry that follows from these three postulates will hold in elliptic geometry, such as proposition 1 from book I of the Elements, which states that given any line segment, an equilateral triangle can be constructed with the segment as its base.

Elliptic geometry is also like Euclidean geometry in that space is continuous, homogeneous, isotropic, and without boundaries. Isotropy is guaranteed by the fourth postulate, that all right angles are equal. For an example of homogeneity, note that Euclid's proposition I.1 implies that the same equilateral triangle can be constructed at any location, not just in locations that are special in some way. The lack of boundaries follows from the second postulate, extensibility of a line segment.

One way in which elliptic geometry differs from Euclidean geometry is that the sum of the interior angles of a triangle is greater than 180 degrees. In the spherical model, for example, a triangle can be constructed with vertices at the locations where the three positive Cartesian coordinate axes intersect the sphere, and all three of its internal angles are 90 degrees, summing to 270 degrees. For sufficiently small triangles, the excess over 180 degrees can be made arbitrarily small.

The Pythagorean theorem fails in elliptic geometry. In the 90°–90°–90° triangle described above, all three sides have the same length, and consequently do not satisfy a^2+b^2=c^2. The Pythagorean result is recovered in the limit of small triangles.

The ratio of a circle's circumference to its area is smaller than in Euclidean geometry. In general, area and volume do not scale as the second and third powers of linear dimensions.

QMRRelation to Euclid's postulates[edit]

Spherical geometry obeys two of Euclid's postulates: the second postulate ("to produce [extend] a finite straight line continuously in a straight line") and the fourth postulate ("that all right angles are equal to one another"). However, it violates the other three: contrary to the first postulate, there is not a unique shortest route between any two points (antipodal points such as the north and south poles on a spherical globe are counterexamples); contrary to the third postulate, a sphere does not contain circles of arbitrarily great radius; and contrary to the fifth (parallel) postulate, there is no point through which a line can be drawn that never intersects a given line.[8]

A statement that is equivalent to the parallel postulate is that there exists a triangle whose angles add up to 180°. Since spherical geometry violates the parallel postulate, there exists no such triangle on the surface of a sphere. The sum of the angles of a triangle on a sphere is 180°(1 + 4f), where f is the fraction of the sphere's surface that is enclosed by the triangle. For any positive value of f, this exceeds 180°.

QMRTiling rectangles with polyaboloes.

Tiling rectangles with copies of a single polyabolo[edit]

Tiling rectangles with polyaboloes.

In 1968, David A. Klarner defined the order of a polyomino. Similarly, the order of a polyabolo P can be defined as the minimum number of congruent copies of P that can be assembled (allowing translation, rotation, and reflection) to form a rectangle.

A polyabolo has order 1 if and only if it is itself a rectangle. Polyaboloes of order 2 are also easily recognisable. Solomon W. Golomb found polyaboloes, including a triabolo, of order 8.[2] Michael Reid found a heptabolo of order 6.[3] Higher orders are possible.

QMRTiling rectangles with polyaboloes.

qMRA polyomino is a plane geometric figure formed by joining one or more equal squares edge to edge. It is a polyform whose cells are squares. It may be regarded as a finite subset of the regular square tiling with a connected interior.

Polyominoes are classified according to how many cells they have:

QMRThe United Kingdom (UK) comprises four countries: England, Scotland, Wales and Northern Ireland.[1][2]

Within the United Kingdom, a unitary sovereign state, Northern Ireland, Scotland and Wales have gained a degree of autonomy through the process of devolution. The UK Parliament and British Government deal with all reserved matters for Northern Ireland and Scotland and all non-transferred matters for Wales, but not in general matters that have been devolved to the Northern Ireland Assembly, Scottish Parliament and National Assembly for Wales. Additionally, devolution in Northern Ireland is conditional on co-operation between the Northern Ireland Executive and the Government of Ireland (see North/South Ministerial Council) and the British Government consults with the Government of Ireland to reach agreement on some non-devolved matters for Northern Ireland (see British–Irish Intergovernmental Conference). England, comprising the majority of the population and area of the United Kingdom,[3][4] remains fully the responsibility of the UK Parliament centralised in London.

England, Northern Ireland, Scotland and Wales are not themselves listed in the International Organization for Standardization (ISO) list of countries. However the ISO list of the subdivisions of the UK, compiled by British Standards and the UK's Office for National Statistics, uses "country" to describe England, Scotland and Wales.[5] Northern Ireland, in contrast, is described as a "province" in the same lists.[5] Each have separate national governing bodies for sports and compete separately in many international sporting competitions, including the Commonwealth Games. Northern Ireland also forms joint All-Island sporting bodies with the Republic of Ireland for most sports, including rugby union.[6]

The Channel Islands and the Isle of Man are dependencies of the Crown and are not part of the UK. Similarly, the British overseas territories, remnants of the British Empire, are not part of the UK.

Historically, from 1801, following the Acts of Union, until 1921 the whole island of Ireland was a country within the UK. Ireland was split into two separate jurisdictions in 1921: Southern Ireland and Northern Ireland. Southern Ireland left the United Kingdom under the Irish Free State Constitution Act 1922.

The four countries of the United Kingdom.

QMRAfter the end of World War II in 1945, the UK was one of the Big Four powers (the Soviet Union, the United Kingdom, the U.S. and China) who met to plan the post-war world;[100][101] it was an original signatory to the Declaration of the United Nations.

QMRSIL Ethnologue lists six "living" Celtic languages, of which four have retained a substantial number of native speakers. These are the Gaelic languages (i.e. the Irish language and Scottish Gaelic - both descended from Old Irish), and the Brittonic languages (i.e. Welsh and the Breton language - both descended from Old Brittonic).

QMRThe The United Kingdom consists of four countries: England, Scotland, Wales, and Northern Ireland.[13] The latter three have devolved administrations,[14] each with varying powers,[15][16] based in their capitals, Edinburgh, Cardiff, and Belfast, respectively. The nearby Isle of Man, Bailiwick of Guernsey and Bailiwick of Jersey are not part of the United Kingdom, being Crown dependencies with the British Government responsible for defence and international representation.

QMRThe four unities is a concept in the common law of real property describing conditions that must exist in order for certain kinds of property interests to be created. Specifically, these four unities must be met in order for two or more people to own property as joint tenants with right of survivorship, or for a married couple to own property as tenants by the entirety. Some jurisdictions may require additional unities.

The four unities[edit]

Unity of time

Interest must be acquired by both tenants at the same time.

In common law, the "time" requirement could be satisfied only by using a "straw man" to create a joint tenancy. The party creating the joint tenancy would have to convey title to a straw man, who would then transfer title to the two parties as joint tenants.

Unity of title

The interests held by the co-owners must arise out of the same instrument.[1]

Unity of interest

Both tenants must have the same interest in the property.

This means that the joint tenants must have the same type of interest, and the interest must run for the same duration. For example, if X and Y create a joint tenancy, both X and Y's interests must be in fee simple absolute. If, for example, X has a fee simple absolute and Y has a life estate, there is no unity of interest.

Unity of possession

Both tenants must have the right to possess the whole property.

If any of the four unities is broken and it is not a joint tenancy, the ownership reverts to a tenancy in common.

QMREnglish property law refers to the law of acquisition, sharing and protection of valuable assets in England and Wales. While part of the United Kingdom, many elements of Scots property law are different. In England, property law encompasses four main topics:

English land law, or the law of "real property"

English trusts law

English personal property law

United Kingdom intellectual property law

Property in land is the domain of the law of real property. The law of personal property is particularly important for commercial law and insolvency. Trusts affects everything in English property law. Intellectual property is also an important branch of the law of property. For unregistered land see Unregistered land in English law.

QMRThe bronze medal is 1 3/8 inches in width. The obverse is a figure of Liberation standing full length with head turned to dexter looking to the dawn of a new day, right foot resting on a war god’s helmet with the hilt of a broken sword in the right hand and the broken blade in the left hand, the inscription WORLD WAR II placed immediately below the center. On the reverse are inscriptions for the Four Freedoms: FREEDOM FROM FEAR AND WANT and FREEDOM OF SPEECH AND RELIGION separated by a palm branch, all within a circle composed of the words UNITED STATES OF AMERICA 1941 1945.[2]

The suspension and service ribbon of the medal is 1 3/8 inches wide and consists of the following stripes: 3/8 inch double rainbow in juxtaposition (blues, greens, yellows, reds (center), yellows greens and blues); 1/32 inch White 67101; center 9/16 inch Old Glory Red 67156; 1/32 inch White; and 3/8 inch double rainbow in juxtaposition. The rainbow on each side of the ribbon is a miniature of the pattern used in the WWI Victory Medal.[2]

QMRThe subdivisions of England consist of up to four levels of subnational division controlled through a variety of types of administrative entities created for the purposes of local government. The flag of England is a cross

QMRAdministrative divisions

Main articles: Administrative divisions of China, Districts of Hong Kong and Municipalities of Macau

The People's Republic of China has administrative control over 22 provinces and considers Taiwan to be its 23rd province, although Taiwan is currently and independently governed by the Republic of China, which disputes the PRC's claim.[175] China also has five subdivisions officially termed autonomous regions, each with a designated minority group; four municipalities; and two Special Administrative Regions (SARs), which enjoy a degree of political autonomy. These 22 provinces, five autonomous regions, and four municipalities can be collectively referred to as "mainland China", a term which usually excludes the SARs of Hong Kong and Macau. None of these divisions are recognized by the ROC government, which claims the entirety of the PRC's territory

QMRIn Euclidean plane geometry, a rectangle is any quadrilateral with four right angles. It can also be defined as an equiangular quadrilateral, since equiangular means that all of its angles are equal (360°/4 = 90°). It can also be defined as a parallelogram containing a right angle. A rectangle with four sides of equal length is a square. The term oblong is occasionally used to refer to a non-square rectangle.[1][2][3] A rectangle with vertices ABCD would be denoted as Rectanglen.PNG ABCD.

The word rectangle comes from the Latin rectangulus, which is a combination of rectus (right) and angulus (angle).

A so-called crossed rectangle is a crossed (self-intersecting) quadrilateral which consists of two opposite sides of a rectangle along with the two diagonals.[4] It is a special case of an antiparallelogram, and its angles are not right angles. Other geometries, such as spherical, elliptic, and hyperbolic, have so-called rectangles with opposite sides equal in length and equal angles that are not right angles.

Rectangles are involved in many tiling problems, such as tiling the plane by rectangles or tiling a rectangle by polygons.

A convex quadrilateral is a rectangle if and only if it is any one of the following:[5][6]

an equiangular quadrilateral

a quadrilateral with four right angles

a parallelogram with at least one right angle

a parallelogram with diagonals of equal length

a parallelogram ABCD where triangles ABD and DCA are congruent

a convex quadrilateral with successive sides a, b, c, d whose area is \tfrac{1}{4}(a+c)(b+d).[7]:fn.1

a convex quadrilateral with successive sides a, b, c, d whose area is \tfrac{1}{2} \sqrt{(a^2+c^2)(b^2+d^2)}.[7]

Traditional hierarchy[edit]

A rectangle is a special case of a parallelogram in which each pair of adjacent sides is perpendicular.

A parallelogram is a special case of a trapezium (known as a trapezoid in North America) in which both pairs of opposite sides are parallel and equal in length.

A trapezium is a convex quadrilateral which has at least one pair of parallel opposite sides.

A convex quadrilateral is

Simple: The boundary does not cross itself.

Star-shaped: The whole interior is visible from a single point, without crossing any edge.

Alternative hierarchy[edit]

De Villiers defines a rectangle more generally as any quadrilateral with axes of symmetry through each pair of opposite sides.[8] This definition includes both right-angled rectangles and crossed rectangles. Each has an axis of symmetry parallel to and equidistant from a pair of opposite sides, and another which is the perpendicular bisector of those sides, but, in the case of the crossed rectangle, the first axis is not an axis of symmetry for either side that it bisects.

Quadrilaterals with two axes of symmetry, each through a pair of opposite sides, belong to the larger class of quadrilaterals with at least one axis of symmetry through a pair of opposite sides. These quadrilaterals comprise isosceles trapezia and crossed isosceles trapezia (crossed quadrilaterals with the same vertex arrangement as isosceles trapezia).

Symmetry[edit]

A rectangle is cyclic: all corners lie on a single circle.

It is equiangular: all its corner angles are equal (each of 90 degrees).

It is isogonal or vertex-transitive: all corners lie within the same symmetry orbit.

It has two lines of reflectional symmetry and rotational symmetry of order 2 (through 180°).

Rectangle-rhombus duality[edit]

The dual polygon of a rectangle is a rhombus, as shown in the table below.[9]

Rectangle Rhombus

All angles are equal. All sides are equal.

Alternate sides are equal. Alternate angles are equal.

Its centre is equidistant from its vertices, hence it has a circumcircle. Its centre is equidistant from its sides, hence it has an incircle.

Its axes of symmetry bisect opposite sides. Its axes of symmetry bisect opposite angles.

Diagonals are equal in length. Diagonals intersect at equal angles.

The figure formed by joining, in order, the midpoints of the sides of a rectangle is a rhombus and vice versa.

Miscellaneous[edit]

The two diagonals are equal in length and bisect each other. Every quadrilateral with both these properties is a rectangle.

A rectangle is rectilinear: its sides meet at right angles.

A rectangle in the plane can be defined by five independent degrees of freedom consisting, for example, of three for position (comprising two of translation and one of rotation), one for shape (aspect ratio), and one for overall size (area).

Two rectangles, neither of which will fit inside the other, are said to be incomparable.

Crossed rectangles[edit]

A crossed (self-intersecting) quadrilateral consists of two opposite sides of a non-self-intersecting quadrilateral along with the two diagonals. Similarly, a crossed rectangle is a crossed quadrilateral which consists of two opposite sides of a rectangle along with the two diagonals. It has the same vertex arrangement as the rectangle. It appears as two identical triangles with a common vertex, but the geometric intersection is not considered a vertex.

A crossed quadrilateral is sometimes likened to a bow tie or butterfly. A three-dimensional rectangular wire frame that is twisted can take the shape of a bow tie. A crossed rectangle is sometimes called an "angular eight".

The interior of a crossed rectangle can have a polygon density of ±1 in each triangle, dependent upon the winding orientation as clockwise or counterclockwise.

A crossed rectangle is not equiangular. The sum of its interior angles (two acute and two reflex), as with any crossed quadrilateral, is 720°.[13]

A rectangle and a crossed rectangle are quadrilaterals with the following properties in common:

Opposite sides are equal in length.

The two diagonals are equal in length.

It has two lines of reflectional symmetry and rotational symmetry of order 2 (through 180°).

Crossed rectangles.png

Other rectangles

Other rectangles[edit]

A saddle rectangle has 4 nonplanar vertices, alternated from vertices of a cuboid, with a unique minimal surface interior defined as a linear combination of the four vertices, creating a saddle surface. This example shows 4 blue edges of the rectangle, and two green diagonals, all being diagonal of the cuboid rectangular faces.

In spherical geometry, a spherical rectangle is a figure whose four edges are great circle arcs which meet at equal angles greater than 90°. Opposite arcs are equal in length. The surface of a sphere in Euclidean solid geometry is a non-Euclidean surface in the sense of elliptic geometry. Spherical geometry is the simplest form of elliptic geometry.

In elliptic geometry, an elliptic rectangle is a figure in the elliptic plane whose four edges are elliptic arcs which meet at equal angles greater than 90°. Opposite arcs are equal in length.

In hyperbolic geometry, a hyperbolic rectangle is a figure in the hyperbolic plane whose four edges are hyperbolic arcs which meet at equal angles less than 90°. Opposite arcs are equal in length.

Tessellations[edit]

The rectangle is used in many periodic tessellation patterns, in brickwork, for example, these tilings:

Stacked bond.png

Stacked bond Wallpaper group-cmm-1.jpg

Running bond Wallpaper group-p4g-1.jpg

Basket weave Basketweave bond.svg

Basket weave Herringbone bond.svg

Herringbone pattern

Squared, perfect, and other tiled rectangles[edit]

A rectangle tiled by squares, rectangles, or triangles is said to be a "squared", "rectangled", or "triangulated" (or "triangled") rectangle respectively. The tiled rectangle is perfect[14][15] if the tiles are similar and finite in number and no two tiles are the same size. If two such tiles are the same size, the tiling is imperfect. In a perfect (or imperfect) triangled rectangle the triangles must be right triangles.

A rectangle has commensurable sides if and only if it is tileable by a finite number of unequal squares.[14][16] The same is true if the tiles are unequal isosceles right triangles.

The tilings of rectangles by other tiles which have attracted the most attention are those by congruent non-rectangular polyominoes, allowing all rotations and reflections. There are also tilings by congruent polyaboloes.

Polyominoes are classified according to how many cells they have:

QMRIn geometry, a golden rectangle is a rectangle whose side lengths are in the golden ratio, 1 : \tfrac{1 + \sqrt{5}}{2}, which is 1:\varphi (the Greek letter phi), where \varphi is approximately 1.618.

Construction[edit]

A method to construct a golden rectangle. The square is outlined in red. The resulting dimensions are in the golden ratio.

A golden rectangle can be constructed with only straightedge and compass by four simple steps:

Construct a simple square.

Draw a line from the midpoint of one side of the square to an opposite corner.

Use that line as the radius to draw an arc that defines the height of the rectangle.

Complete the golden rectangle.

silver ratios squares quadrants square root of 2 is quadrant

The word rectangle comes from the Latin rectangulus, which is a combination of rectus (right) and angulus (angle).

Rectangles are involved in many tiling problems, such as tiling the plane by rectangles or tiling a rectangle by polygons.

QMRIn geometry, a golden rhombus is a rhombus whose diagonals are in the ratio \frac{p}{q}=\varphi\!, where \varphi\! is the golden ratio.

Golden rhombohedra[edit]

There are two distinct convex golden rhombohedra constructed from six golden rhombi as a trigonal trapezohedron. These are sometimes called acute or prolate and the obtuse or oblate golden rhombohedron.

The rhombic triacontahedron is constructed with 30 golden rhombic faces, alternating 3 and 5 around every vertex. The dihedral angle between adjacent rhombi of the rhombic triacontahedron is 144°, which can be constructed by placing the short sides of two golden triangles back-to-back. The rhombic icosahedron is also constructed with golden rhombi.

The nonconvex rhombic hexecontahedron can be constructed by 20 acute golden rhombohedra. It also represents a stellation of the rhombic triacontahedron.

QMRGolden triangle, rhombus, and rhombic triacontahedron

One of the rhombic triacontahedron's rhombi

All of the faces of the rhombic triacontahedron are golden rhombi

A golden rhombus is a rhombus whose diagonals are in the golden ratio. The rhombic triacontahedron is a convex polytope that has a very special property: all of its faces are golden rhombi. In the rhombic triacontahedron the dihedral angle between any two adjacent rhombi is 144°, which is twice the isosceles angle of a golden triangle and four times its most acute angle

QMRA pentagram colored to distinguish its line segments of different lengths. The four lengths are in golden ratio to one another.

Pentagram

A pentagram colored to distinguish its line segments of different lengths. The four lengths are in golden ratio to one another.

The golden ratio plays an important role in the geometry of pentagrams. Each intersection of edges sections other edges in the golden ratio. Also, the ratio of the length of the shorter segment to the segment bounded by the two intersecting edges (a side of the pentagon in the pentagram's center) is φ, as the four-color illustration shows.

QMRIn Euclidean geometry, Ptolemy's theorem is a relation between the four sides and two diagonals of a cyclic quadrilateral (a quadrilateral whose vertices lie on a common circle). The theorem is named after the Greek astronomer and mathematician Ptolemy (Claudius Ptolemaeus).[1] Ptolemy used the theorem as an aid to creating his table of chords, a trigonometric table that he applied to astronomy.

If the quadrilateral is given with its four vertices A, B, C, and D in order, then the theorem states that:

|\overline{AC}|\cdot |\overline{BD}|=|\overline{AB}|\cdot |\overline{CD}|+|\overline{BC}|\cdot |\overline{AD}|

where the vertical lines denote the lengths of the line segments between the named vertices. In the context of geometry, the above equality is often simply written as

AC·BD=AB·CD+BC·AD.

This relation may be verbally expressed as follows:

If a quadrilateral is inscribable in a circle then the product of the measures of its diagonals is equal to the sum of the products of the measures of the pairs of opposite sides.

Moreover, the converse of Ptolemy's theorem is also true:

In a quadrilateral, if the sum of the products of its two pairs of opposite sides is equal to the product of its diagonals, then the quadrilateral can be inscribed in a circle.

Formulation of the theorem[edit]

t_{12} \cdot t_{34}+t_{41}\cdot t_{23}-t_{13}\cdot t_{24}=0

Let \,O be a circle of radius \,R. Let \,O_1, O_2, O_3, O_4 be (in that order) four non-intersecting circles that lie inside \,O and tangent to it. Denote by \,t_{ij} the length of the exterior common bitangent of the circles \,O_i, O_j. Then:[1]

\,t_{12} \cdot t_{34}+t_{41} \cdot t_{23}=t_{13}\cdot t_{24}.

Note that in the degenerate case, where all four circles reduce to points, this is exactly Ptolemy's theorem.

QMRIn mathematics, Casey's theorem, also known as the generalized Ptolemy's theorem, is a theorem in Euclidean geometry named after the Irish mathematician John Casey.

QMRThébault's theorem is the name given variously to one of the geometry problems proposed by the French mathematician Victor Thébault, individually known as Thébault's problem I, II, and III.

Thébault's problem I[edit]

Given any parallelogram, construct on its sides four squares external to the parallelogram. The quadrilateral formed by joining the centers of those four squares is a square.

It is a special case of van Aubel's theorem and a square version of the Napoleon's theorem.

Tiling pattern based on Thébault's problem I

Thébault's problem II[edit]

Given a square, construct equilateral triangles on two adjacent edges, either both inside or both outside the square. Then the triangle formed by joining the vertex of the square distant from both triangles and the vertices of the triangles distant from the square is equilateral.

Thébault's problem III[edit]

Given any triangle ABC, and any point M on BC, construct the incircle and circumcircle of the triangle. Then construct two additional circles, each tangent to AM, BC, and to the circumcircle. Then their centers and the center of the incircle are colinear.

Until 2003, academia thought this third problem of Thébault the most difficult to prove. It was published in the American Mathematical Monthly in 1938, and proved by Dutch mathematician H. Streefkerk in 1973. However, in 2003, Jean-Louis Ayme discovered that Y. Sawayama, an instructor at The Central Military School of Tokyo, independently proposed and solved this problem in 1905.[1]

An "external" version of this theorem, where the incircle is replaced by an excircle and the two additional circles are external to the circumcircle, is found in Shay Gueron (2002). [2] A proof based on Casey's theorem is in the paper.

QMRIn geometry, the Petr–Douglas–Neumann theorem (or the PDN-theorem) is a result concerning arbitrary planar polygons. The theorem asserts that a certain procedure when applied to an arbitrary polygon always yields a regular polygon having the same number of sides as the initial polygon. The theorem was first published by Karel Petr (1868–1950) of Prague in 1908.[1][2] The theorem was independently rediscovered by Jesse Douglas (1897–1965) in 1940[3] and also by B H Neumann (1909–2002) in 1941.[2][4] The naming of the theorem as Petr–Douglas–Neumann theorem, or as the PDN-theorem for short, is due to Stephen B Gray.[2] This theorem has also been called Douglas’s theorem, the Douglas–Neumann theorem, the Napoleon–Douglas–Neumann theorem and Petr’s theorem.[2]

The PDN-theorem is a generalisation of the Napoleon's theorem which is concerned about arbitrary triangles and of the van Aubel's theorem which is related to arbitrary quadrilaterals.

Specialisation to quadrilaterals[edit]

In the case of quadrilaterals, the value of n is 4 and that of n − 2 is 2. There are two possible values for k, namely 1 and 2, and so two possible apex angles, namely:

(2×1×π)/4 = π/2 = 90° ( corresponding to k = 1 )

(2×2×π)/4 = π = 180° ( corresponding to k = 2 ).

According to the PDN-theorem the quadrilateral A2 is a regular 4-gon, that is, a square. The two-stage process yielding the square A2 can be carried out in two different ways. (The apex Z of an isosceles triangle with apex angle π erected over a line segment XY is the midpoint of the line segment XY.)

Construct A1 using apex angle π/2 and then A2 with apex angle π.[edit]

In this case the vertices of A1 are the free apices of isosceles triangles with apex angles π/2 erected over the sides of the quadrilaetral A0. The vertices of the quadrilateral A2 are the midpoints of the sides of the quadrilateral A1. By the PDN theorem, A2 is a square.

The vertices of the quadrilateral A1 are the centers of squares erected over the sides of the quadrilateral A0. The assertion that quadrilateral A2 is a square is equivalent to the assertion that the diagonals of A1 are equal and perpendicular to each other. The latter assertion is the content of van Aubel's theorem.

Thus van Aubel's theorem is a special case of the PDN-theorem.

Construct A1 using apex angle π and then A2 with apex angle π/2.[edit]

In this case the vertices of A1 are the midpoints of the sides of the quadrilateral A0 and those of A2 are the apices of the triangles with apex angles π/2 erected over the sides of A1. The PDN-theorem asserts that A2 is a square in this case also.

Images illustrating application of the theorem to quadrilaterals

Petr–Douglas–Neumann theorem as

applied to a quadrilateral A0 = ABCD.

A1 = EFGH is constructed using

apex angle π/2 and A2 = PQRS

with apex angle π. Petr–Douglas–Neumann theorem as

applied to a quadrilateral A0 = ABCD.

A1 = EFGH is constructed using

apex angle π and A2 = PQRS

with apex angle π/2.

PDN Theorem For Self Intersection Quadrilateral Case1.svg PDN Theorem For Self Intersection Quadrilateral Case2.svg

Petr–Douglas–Neumann theorem as

applied to a self-intersecting

quadrilateral A0 = ABCD.

A1 = EFGH is constructed using

apex angle π/2 and A2 = PQRS

with apex angle π. Petr–Douglas–Neumann theorem as

applied to a self-intersecting

quadrilateral A0 = ABCD.

A1 = EFGH is constructed using

apex angle π and A2 = PQRS

with apex angle π/2.

QMRIn plane geometry, Van Aubel's theorem describes a relationship between squares constructed on the sides of a quadrilateral. Starting with a given quadrilateral (a polygon having four sides), construct a square on each side. Van Aubel's theorem states that the two line segments between the centers of opposite squares are of equal lengths and are at right angles to one another. Another way of saying the same thing is that the center points of the four squares form the vertices of an equidiagonal orthodiagonal quadrilateral. The theorem is named after H. H. van Aubel, who published it in 1878.[1]

QMRIn Euclidean geometry, an orthodiagonal quadrilateral is a quadrilateral in which the diagonals cross at right angles (making a quadrant). In other words, it is a four-sided figure in which the line segments between non-adjacent vertices are orthogonal (perpendicular) to each other.

Special cases[edit]

A kite is an orthodiagonal quadrilateral in which one diagonal is a line of symmetry. The kites are exactly the orthodiagonal quadrilaterals that contain a circle tangent to all four of their sides; that is, the kites are the tangential orthodiagonal quadrilaterals.[1]

A rhombus is an orthodiagonal quadrilateral with two pairs of parallel sides (that is, an orthodiagonal quadrilateral that is also a parallelogram).

A square is a limiting case of both a kite and a rhombus.

Orthodiagonal equidiagonal quadrilaterals in which the diagonals are at least as long as all of the quadrilateral's sides have the maximum area for their diameter among all quadrilaterals, solving the n = 4 case of the biggest little polygon problem. The square is one such quadrilateral, but there are infinitely many others.

QMRRhombus

From Wikipedia, the free encyclopedia

For other uses, see Rhombus (disambiguation).

rhombus

Rhombus.svg

Two rhombi.

Type quadrilateral

Edges and vertices 4

Schläfli symbol { } + { }

Coxeter diagram CDel node f1.pngCDel 2x.pngCDel node f1.png

Symmetry group Dih2, [2], (*22), order 4

Area \tfrac{pq}{2} (half the product of the diagonals)

Dual polygon rectangle

Properties convex, isotoxal

The rhombus has a square as a special case, and is a special case of a kite and parallelogram.

In Euclidean geometry, a rhombus(◊), plural rhombi or rhombuses, is a simple (non-self-intersecting) quadrilateral whose four sides all have the same length. Another name is equilateral quadrilateral, since equilateral means that all of its sides are equal in length. The rhombus is often called a diamond, after the diamonds suit in playing cards which resembles the projection of an octahedral diamond, or a lozenge, though the former sometimes refers specifically to a rhombus with a 60° angle (see Polyiamond), and the latter sometimes refers specifically to a rhombus with a 45° angle.

Every rhombus is a parallelogram and a kite. A rhombus with right angles is a square.[1][2]

Etymology[edit]

The word "rhombus" comes from Greek ῥόμβος(rhombos), meaning something that spins,[3] which derives from the verb ρέμβω (rhembō), meaning "to turn round and round".[4] The word was used both by Euclid and Archimedes, who used the term "solid rhombus" for two right circular cones sharing a common base.[5]

Characterizations[edit]

A simple (non self-intersecting) quadrilateral is a rhombus if and only if it is any one of the following:[6][7]

a quadrilateral with four sides of equal length (by definition)

a quadrilateral in which the diagonals are perpendicular and bisect each other

a quadrilateral in which each diagonal bisects two opposite interior angles

a parallelogram in which a diagonal bisects an interior angle

a parallelogram in which at least two consecutive sides are equal in length

a parallelogram in which the diagonals are perpendicular (an orthodiagonal parallelogram)

Basic properties[edit]

Every rhombus has two diagonals connecting pairs of opposite vertices, and two pairs of parallel sides. Using congruent triangles, one can prove that the rhombus is symmetric across each of these diagonals. It follows that any rhombus has the following properties:

Opposite angles of a rhombus have equal measure

The two diagonals of a rhombus are perpendicular; that is, a rhombus is an orthodiagonal quadrilateral

Its diagonals bisect opposite angles

The first property implies that every rhombus is a parallelogram. A rhombus therefore has all of the properties of a parallelogram: for example, opposite sides are parallel; adjacent angles are supplementary; the two diagonals bisect one another; any line through the midpoint bisects the area; and the sum of the squares of the sides equals the sum of the squares of the diagonals (the parallelogram law). Thus denoting the common side as a and the diagonals as p and q, in every rhombus

\displaystyle 4a^2=p^2+q^2.

Not every parallelogram is a rhombus, though any parallelogram with perpendicular diagonals (the second property) is a rhombus. In general, any quadrilateral with perpendicular diagonals, one of which is a line of symmetry, is a kite. Every rhombus is a kite, and any quadrilateral that is both a kite and parallelogram is a rhombus.

A rhombus is a tangential quadrilateral.[8] That is, it has an inscribed circle that is tangent to all four sides.

As a Varignon parallelogram[edit]

The Varignon parallelogram of an equidiagonal quadrilateral is a rhombus.[11]

As the faces of a polyhedron[edit]

A rhombohedron is a three-dimensional figure like a cube, except that its six faces are rhombi instead of squares.

The rhombic dodecahedron is a convex polyhedron with 12 congruent rhombi as its faces.

The rhombic triacontahedron is a convex polyhedron with 30 golden rhombi (rhombi whose diagonals are in the golden ratio) as its faces.

The great rhombic triacontahedron is a nonconvex isohedral, isotoxal polyhedron with 30 intersecting rhombic faces.

The rhombic hexecontahedron is a stellation of the rhombic triacontahedron. It is nonconvex with 60 golden rhombic faces with icosahedral symmetry.

The rhombic enneacontahedron is a polyhedron composed of 90 rhombic faces, with three, five, or six rhombi meeting at each vertex. It has 60 broad rhombi and 30 slim ones.

The trapezo-rhombic dodecahedron is a convex polyhedron with 6 rhombic and 6 trapezoidal faces.

The rhombic icosahedron is a polyhedron composed of 20 rhombic faces, of which three, four, or five meet at each vertex. It has 10 faces on the polar axis with 10 faces following the equator.

QMRRhombic Chess is a chess variant for two players created by Tony Paletta in 1980.[1][2] The gameboard has an overall hexagonal shape and comprises 72 rhombi in three alternating colors. Each player commands a full set of standard chess pieces.

The game was first published in Chess Spectrum Newsletter 2 by the inventor. It was included in World Game Review No. 10 edited by Michael Keller.[3]

QMRThysanoteuthis rhombus, also known as the diamond squid or diamondback squid, is a large species of squid growing to 100 cm in mantle length[1] and a maximum weight of 30 kg,[2] although it averages around 20 kg.[3] The species occurs worldwide, throughout tropical and subtropical waters. Arms have two series of suckers, whereas the tentacular clubs have four. It lacks photophores. T. rhombus is named for its fins, which run in equal length along the mantle, giving the appearance of a rhombus. The species is commercially fished in Japan, specifically in the Sea of Japan and Okinawa.[4]

QMRThe diamondback squid, Thysanoteuthis rhombus, has full-length rhomboid fins

QMRTraditionally, in two-dimensional geometry, a rhomboid is a parallelogram in which adjacent sides are of unequal lengths and angles are non-right angled.

A parallelogram with sides of equal length (equilateral) is a rhombus but not a rhomboid.

A parallelogram with right angled corners is a rectangle but not a rhomboid.

The term rhomboid is now more often used for a parallelepiped, a solid figure with six faces in which each face is a parallelogram and pairs of opposite faces lie in parallel planes. Some crystals are formed in three-dimensional rhomboids. This solid is also sometimes called a rhombic prism. The term occurs frequently in science terminology referring to both its two- and three-dimensional meaning.

Euclid introduced the term in his Elements in Book I, Definition 22,

Of quadrilateral figures, a square is that which is both equilateral and right-angled; an oblong that which is right-angled but not equilateral; a rhombus that which is equilateral but not right-angled; and a rhomboid that which has its opposite sides and angles equal to one another but is neither equilateral nor right-angled. And let quadrilaterals other than these be called trapezia.

— Translation from the page of D.E. Joyce, Dept. Math. & Comp. Sci., Clark University [1]

Euclid never used the definition of rhomboid again and introduced the word parallelogram in Proposition 31 of Book I; "In parallelogrammic areas the opposite sides and angles are equal to one another, and the diameter bisects the areas." Heath suggests that rhomboid was an older term already in use.

Symmetries[edit]

The rhomboid has no line of symmetry, but it has rotational symmetry of order 2.

In biology[edit]

In biology, rhomboid may describe a geometric rhomboid (e.g. the rhomboid muscles) or a bilaterally-symmetrical kite-shaped or diamond-shaped outline, as in leaves[1] or cephalopod fins.[2]

In medicine[edit]

In a type of arthritis called pseudogout, crystals of calcium pyrophosphate dihydrate accumulate in the joint, causing inflammation. Aspiration of the joint fluid reveals rhomboid-shaped crystals under a microscope.

QMRConstruction of equilateral triangle with compass and straightedge

equilateral triangle wiki

QMRA regular tetrahedron is made of four equilateral triangles.

carbon is the miracle element with four valence electrons

Methane and carbon are greenhouse gases and they both have carbon

QMRStrengthening of the greenhouse effect through human activities is known as the enhanced (or anthropogenic) greenhouse effect.[23] This increase in radiative forcing from human activity is attributable mainly to increased atmospheric carbon dioxide levels.[24] According to the latest Assessment Report from the Intergovernmental Panel on Climate Change, "atmospheric concentrations of carbon dioxide, methane and nitrous oxide are unprecedented in at least the last 800,000 years. Their effects, together with those of other anthropogenic drivers, have been detected throughout the climate system and are extremely likely to have been the dominant cause of the observed warming since the mid-20th century".[25]

CO2 is produced by fossil fuel burning and other activities such as cement production and tropical deforestation.[26] Measurements of CO2 from the Mauna Loa observatory show that concentrations have increased from about 313 ppm[27] in 1960 to about 389 ppm in 2010. It reached the 400ppm milestone on May 9, 2013.[28] The current observed amount of CO2 exceeds the geological record maxima (~300 ppm) from ice core data.[29] The effect of combustion-produced carbon dioxide on the global climate, a special case of the greenhouse effect first described in 1896 by Svante Arrhenius, has also been called the Callendar effect.

Over the past 800,000 years,[30] ice core data shows that carbon dioxide has varied from values as low as 180 parts per million (ppm) to the pre-industrial level of 270ppm.[31] Paleoclimatologists consider variations in carbon dioxide concentration to be a fundamental factor influencing climate variations over this time scale.[

QMRGreenhouse gases

Main article: Greenhouse gas

Atmospheric gases only absorb some wavelengths of energy but are transparent to others. The absorption patterns of water vapor (blue peaks) and carbon dioxide (pink peaks) overlap in some wavelengths. Carbon dioxide is not as strong a greenhouse gas as water vapor, but it absorbs energy in wavelengths (12-15 micrometers) that water vapor does not, partially closing the "window" through which heat radiated by the surface would normally escape to space. (Illustration NASA, Robert Rohde)[20]

By their percentage contribution to the greenhouse effect on Earth the four major gases are:[21][22]

water vapor, 36–70%

carbon dioxide, 9–26%

methane, 4–9%

ozone, 3–7%

It is not physically realistic to assign a specific percentage to each gas because the absorption and emission bands of the gases overlap (hence the ranges given above). The major non-gas contributor to Earth's greenhouse effect, clouds, also absorb and emit infrared radiation and thus have an effect on the radiative properties of the atmosphere.

QMRIn geometry, demihypercubes (also called n-demicubes, n-hemicubes, and half measure polytopes) are a class of n-polytopes constructed from alternation of an n-hypercube, labeled as hγn for being half of the hypercube family, γn. Half of the vertices are deleted and new facets are formed. The 2n facets become 2n (n-1)-demicubes, and 2n (n-1)-simplex facets are formed in place of the deleted vertices.[1]

They have been named with a demi- prefix to each hypercube name: demicube, demitesseract, etc. The demicube is identical to the regular tetrahedron, and the demitesseract is identical to the regular 16-cell. The demipenteract is considered semiregular for having only regular facets. Higher forms don't have all regular facets but are all uniform polytopes.

The vertices and edges of a demihypercube form two copies of the halved cube graph.

QMRTetrahemihexacron[edit]

Tetrahemihexacron

Tetrahemihexacron.png

Type Star polyhedron

Face —

Elements F = 6, E = 12

V = 7 (χ = 1)

Symmetry group Td, [3,3], *332

Index references DU04

dual polyhedron Tetrahemihexahedron

The tetrahemihexacron is the dual of the tetrahemihexahedron, and is one of nine dual hemipolyhedra.

Since the hemipolyhedra have faces passing through the center, the dual figures have corresponding vertices at infinity; properly, on the real projective plane at infinity.[2] In Magnus Wenninger's Dual Models, they are represented with intersecting prisms, each extending in both directions to the same vertex at infinity, in order to maintain symmetry. In practice the model prisms are cut off at a certain point that is convenient for the maker. Wenninger suggested these figures are members of a new class of stellation figures, called stellation to infinity. However, he also suggested that strictly speaking they are not polyhedra because their construction does not conform to the usual definitions.

Topologically it is considered to contain seven vertices. The three vertices considered at infinity (the real projective plane at infinity) correspond directionally to the three vertices of the hemi-octahedron, an abstract polyhedron. The other four vertices exist at alternate corners of a central cube (a demicube, in this case a tetrahedron).

It looks like two quadrants

QMRTetrahedral symmetry

From Wikipedia, the free encyclopedia

Point groups in three dimensions

Sphere symmetry group cs.png

Involutional symmetry

Cs, (*)

[ ] = CDel node c2.png Sphere symmetry group c3v.png

Cyclic symmetry

Cnv, (*nn)

[n] = CDel node c1.pngCDel n.pngCDel node c1.png Sphere symmetry group d3h.png

Dihedral symmetry

Dnh, (*n22)

[n,2] = CDel node c1.pngCDel n.pngCDel node c1.pngCDel 2.pngCDel node c1.png

Polyhedral group, [n,3], (*n32)

Sphere symmetry group td.png

Tetrahedral symmetry

Td, (*332)

[3,3] = CDel node c1.pngCDel 3.pngCDel node c1.pngCDel 3.pngCDel node c1.png Sphere symmetry group oh.png

Octahedral symmetry

Oh, (*432)

[4,3] = CDel node c2.pngCDel 4.pngCDel node c1.pngCDel 3.pngCDel node c1.png Sphere symmetry group ih.png

Icosahedral symmetry

Ih, (*532)

[5,3] = CDel node c2.pngCDel 5.pngCDel node c2.pngCDel 3.pngCDel node c2.png

A regular tetrahedron, an example of a solid with full tetrahedral symmetry

A regular tetrahedron has 12 rotational (or orientation-preserving) symmetries, and a symmetry order of 24 including transformations that combine a reflection and a rotation.

The group of all symmetries is isomorphic to the group S4, the symmetric group of permutations of four objects, since there is exactly one such symmetry for each permutation of the vertices of the tetrahedron. The set of orientation-preserving symmetries forms a group referred to as the alternating subgroup A4 of S4

Details[edit]

Chiral and full (or achiral tetrahedral symmetry and pyritohedral symmetry) are discrete point symmetries (or equivalently, symmetries on the sphere). They are among the crystallographic point groups of the cubic crystal system.

Seen in stereographic projection the edges of the tetrakis hexahedron form 6 circles (or centrally radial lines) in the plane. Each of these 6 circles represent a mirror line in tetrahedral symmetry. The intersection of these circles meet at order 2 and 3 gyration points

Chiral tetrahedral symmetry[edit]

Sphere symmetry group t.png

The tetrahedral rotation group T with fundamental domain; for the triakis tetrahedron, see below, the latter is one full face Tetrahedral group 2.svg

A tetrahedron can be placed in 12 distinct positions by rotation alone. These are illustrated above in the cycle graph format, along with the 180° edge (blue arrows) and 120° vertex (reddish arrows) rotations that permute the tetrahedron through those positions. Tetrakishexahedron.jpg

In the tetrakis hexahedron one full face is a fundamental domain; other solids with the same symmetry can be obtained by adjusting the orientation of the faces, e.g. flattening selected subsets of faces to combine each subset into one face, or replacing each face by multiple faces, or a curved surface.

T, 332, [3,3]+, or 23, of order 12 – chiral or rotational tetrahedral symmetry. There are three orthogonal 2-fold rotation axes, like chiral dihedral symmetry D2 or 222, with in addition four 3-fold axes, centered between the three orthogonal directions. This group is isomorphic to A4, the alternating group on 4 elements; in fact it is the group of even permutations of the four 3-fold axes: e, (123), (132), (124), (142), (134), (143), (234), (243), (12)(34), (13)(24), (14)(23).

The conjugacy classes of T are:

identity

4 × rotation by 120° clockwise (seen from a vertex): (234), (143), (412), (321)

4 × rotation by 120° counterclockwise (ditto)

3 × rotation by 180°

The rotations by 180°, together with the identity, form a normal subgroup of type Dih2, with quotient group of type Z3. The three elements of the latter are the identity, "clockwise rotation", and "anti-clockwise rotation", corresponding to permutations of the three orthogonal 2-fold axes, preserving orientation.

A4 is the smallest group demonstrating that the converse of Lagrange's theorem is not true in general: given a finite group G and a divisor d of |G|, there does not necessarily exist a subgroup of G with order d: the group G = A4 has no subgroup of order 6. Although it is a property for the abstract group in general, it is clear from the isometry group of chiral tetrahedral symmetry: because of the chirality the subgroup would have to be C6 or D3, but neither applies.

Subgroups of chiral tetrahedral symmetry

Achiral tetrahedral symmetry[edit]

The full tetrahedral group Td with fundamental domain

Td, *332, [3,3] or 43m, of order 24 – achiral or full tetrahedral symmetry, also known as the (2,3,3) triangle group. This group has the same rotation axes as T, but with six mirror planes, each through two 3-fold axes. The 2-fold axes are now S4 (4) axes. Td and O are isomorphic as abstract groups: they both correspond to S4, the symmetric group on 4 objects. Td is the union of T and the set obtained by combining each element of O \ T with inversion. See also the isometries of the regular tetrahedron.

The conjugacy classes of Td are:

identity

8 × rotation by 120°

3 × rotation by 180°

6 × reflection in a plane through two rotation axes

6 × rotoreflection by 90°

Subgroups of achiral tetrahedral symmetry[edit]

Pyritohedral symmetry[edit]

The pyritohedral group Th with fundamental domain

The seams of a volleyball have pyritohedral symmetry

Th, 3*2, [4,3+] or m3, of order 24 – pyritohedral symmetry. This group has the same rotation axes as T, with mirror planes through two of the orthogonal directions. The 3-fold axes are now S6 (3) axes, and there is a central inversion symmetry. Th is isomorphic to T × Z2: every element of Th is either an element of T, or one combined with inversion. Apart from these two normal subgroups, there is also a normal subgroup D2h (that of a cuboid), of type Dih2 × Z2 = Z2 × Z2 × Z2. It is the direct product of the normal subgroup of T (see above) with Ci. The quotient group is the same as above: of type Z3. The three elements of the latter are the identity, "clockwise rotation", and "anti-clockwise rotation", corresponding to permutations of the three orthogonal 2-fold axes, preserving orientation.

It is the symmetry of a cube with on each face a line segment dividing the face into two equal rectangles, such that the line segments of adjacent faces do not meet at the edge. The symmetries correspond to the even permutations of the body diagonals and the same combined with inversion. It is also the symmetry of a pyritohedron, which is extremely similar to the cube described, with each rectangle replaced by a pentagon with one symmetry axis and 4 equal sides and 1 different side (the one corresponding to the line segment dividing the cube's face); i.e., the cube's faces bulge out at the dividing line and become narrower there. It is a subgroup of the full icosahedral symmetry group (as isometry group, not just as abstract group), with 4 of the 10 3-fold axes.

The conjugacy classes of Th include those of T, with the two classes of 4 combined, and each with inversion:

identity

8 × rotation by 120°

3 × rotation by 180°

inversion

8 × rotoreflection by 60°

3 × reflection in a plane

Solids with chiral tetrahedral symmetry[edit]

Snub tetrahedron.png The Icosahedron colored as a snub tetrahedron has chiral symmetry.

Solids with full tetrahedral symmetry[edit]

Class Name Picture Faces Edges Vertices

Platonic solid tetrahedron Tetrahedron 4 6 4

Archimedean solid truncated tetrahedron Truncated tetrahedron 8 18 12

Catalan solid triakis tetrahedron Triakis tetrahedron 12 18 8

Near-miss Johnson solid Truncated triakis tetrahedron Truncated triakis tetrahedron.png 16 42 28

Tetrated dodecahedron Tetrated Dodecahedron.gif 28 54 28

Uniform star polyhedron Tetrahemihexahedron Tetrahemihexahedron.png 7 12 6

Tetrahemihexahedron

From Wikipedia, the free encyclopedia

Tetrahemihexahedron

Tetrahemihexahedron.png

Type Uniform star polyhedron

Elements F = 7, E = 12

V = 6 (χ = 1)

Faces by sides 4{3}+3{4}

Wythoff symbol 3/2 3 | 2 (double-covering)

Symmetry group Td, [3,3], *332

Index references U04, C36, W67

Dual polyhedron Tetrahemihexacron

Vertex figure Tetrahemihexahedron vertfig.svg

3.4.3/2.4

Bowers acronym Thah

In geometry, the tetrahemihexahedron or hemicuboctahedron is a uniform star polyhedron, indexed as U4. It has 6 vertices and 12 edges, and 7 faces: 4 triangular and 3 square. Its vertex figure is a crossed quadrilateral. Its Coxeter-Dynkin diagram is CDel node 1.pngCDel 3x.pngCDel rat.pngCDel d2.pngCDel node.pngCDel 3.pngCDel node 1.png (although this is a double covering of the tetrahemihexahedron).

It is the only non-prismatic uniform polyhedron with an odd number of faces. Its Wythoff symbol is 3/2 3 | 2, but actually that represents a double covering of the tetrahemihexahedron with 8 triangles and 6 squares, paired and coinciding in space. (It can more intuitively be seen as two coinciding tetrahemihexahedra.)

It is a hemipolyhedron. The "hemi" part of the name means some of the faces form a group with half as many members as some regular polyhedron—here, three square faces form a group with half as many faces as the regular hexahedron, better known as the cube—hence hemihexahedron. Hemi faces are also oriented in the same direction as the regular polyhedron's faces. The three square faces of the tetrahemihexahedron are, like the three facial orientations of the cube, mutually perpendicular.

The "half-as-many" characteristic also means that hemi faces must pass through the center of the polyhedron, where they all intersect each other. Visually, each square is divided into four right triangles, with two visible from each side.

It is the three-dimensional demicross polytope.

Related surfaces[edit]

It is a non-orientable surface. It is unique as the only uniform polyhedron with an Euler characteristic of 1 and is hence a projective polyhedron, yielding a representation of the real projective plane[1] very similar to the Roman surface.

Related polyhedra[edit]

It has the same vertices and edges as the regular octahedron. It also shares 4 of the 8 triangular faces of the octahedron, but has three additional square faces passing through the centre of the polyhedron.

The dual figure is the tetrahemihexacron.

It is 2-covered by the cuboctahedron,[1] which accordingly has the same abstract vertex figure (2 triangles and two squares: 3.4.3.4) and twice the vertices, edges, and faces. It has the same topology as the abstract polyhedron hemi-cuboctahedron.

It may also be constructed as a crossed triangular cuploid, being a reduced version of the {3/2}-cupola (retrograde triangular cupola, ratricu) by its {6/2}-gonal base.

QMRIn geometry, a tetrahedrally diminished[1] dodecahedron (also tetrahedrally stellated icosahedron) is a topologically self-dual polyhedron made of 16 vertices, 30 edges, and 16 faces (4 equilateral triangles and 12 identical quadrilaterals).[2]

It has chiral tetrahedral symmetry, and so its geometry can be constructed from pyritohedral symmetry of the pseudoicosahedron with 4 faces stellated, or from the pyritohedron, with 4 vertices diminished. Within its tetrahedral symmetry, it has geometric varied proportions. By Dorman Luke dual construction, a unique geometric proportion can be defined. The kite faces have edges of length ratio ~ 1:0.6325.

As a self-dual hexadecahedron, it is one of 302404 forms, 1476 with at least order 2 symmetry, and the only one with tetrahedral symmetry.[3]

As a diminished regular dodecahedron, with 4 vertices removed, the quadrilaterals faces are trapezoids.

As a stellation of the regular icosahedron it is one of 32 stellations defined with tetrahedral symmetry. It has kite faces.[4]

In Conway polyhedron notation, it can represented as pT, applying George W. Hart's propeller operator to a regular tetrahedron.[5]

Related polytopes and honeycombs[edit]

This polyhedron represents the vertex figure of a hyperbolic uniform honeycomb, the partially diminished icosahedral honeycomb, pd{3,5,3}, with 12 pentagonal antiprisms and 4 dodecahedron cells meeting at every vertex.

Partial truncation order-3 icosahedral honeycomb verf.png

Vertex figure projected as Schlegel diagram

QMRTetrated dodecahedron

From Wikipedia, the free encyclopedia

Tetrated dodecahedron

Type near-miss Johnson solid

Faces 4+12 triangles

12 pentagons

Edges 54

Vertices 28

Vertex configuration 4 (5.5.5)

12 (3.5.3.5)

12 (3.3.5.5)

Symmetry group Td

Properties convex

The tetrated dodecahedron is a near-miss Johnson solid. It was first discovered in 2002 by Alex Doskey. It was then independently rediscovered in 2003 and named by Robert Austin.[1]

It has 28 faces: twelve regular pentagons arranged in four panels of three pentagons each, four equilateral triangles (shown in blue), and six pairs of isosceles triangles (shown in yellow). All edges of the tetrated dodecahedron have the same length, except for the shared bases of these isosceles triangles, which are approximately 1.07 times as long as the other edges. This polyhedron has tetrahedral symmetry.

Net[edit]

The 12 pentagons and 16 triangles are colored in this net by their locations within the tetrahedral symmetry.

QMRTruncated triakis tetrahedron

From Wikipedia, the free encyclopedia

Not to be confused with triakis truncated tetrahedron.

This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. (September 2014)

Truncated triakis tetrahedron

Conway notation t6kT

Faces 4 hexagons

12 pentagons

Edges 42

Vertices 28

Dual Hexakis truncated tetrahedron

Vertex configuration 4 (5.5.5)

24 (5.5.6)

Symmetry group Td

Properties convex

The truncated triakis tetrahedron is a convex polyhedron with 16 faces: 4 sets of 3 pentagons arranged in a tetrahedral arrangement, with 4 hexagons in the gaps. It is constructed from taking a triakis tetrahedron by truncating the order-6 vertices. This creates 4 regular hexagon faces, and leaves 12 irregular pentagons.

A topologically similar equilateral polyhedron can be constructed by using 12 regular pentagons with 4 equilateral but nonplanar hexagons, each vertex with internal angles alternating between 108 and 132 degrees.

Full truncation[edit]

If all of a triakis tetrahedron's vertices, of both kinds, are truncated, the resulting solid is an irregular icosahedron, whose dual is a trihexakis truncated tetrahedron.

Truncation of only the simpler vertices yields what looks like a tetrahedron with each face raised by a low triangular frustum. The dual to that truncation will be the triakis truncated tetrahedron.

qMRIn geometry, a triakis tetrahedron (or kistetrahedron[1]) is an Archimedean dual solid, or a Catalan solid. Its dual is the truncated tetrahedron.

It can be seen as a tetrahedron with triangular pyramids added to each face; that is, it is the Kleetope of the tetrahedron. It is very similar to the net for the 5-cell, as the net for a tetrahedron is a triangle with other triangles added to each edge, the net for the 5-cell a tetrahedron with pyramids attached to each face. This interpretation is expressed in the name.

If the triakis tetrahedron has shorter edge lengths 1, it has area \tfrac{5}{3} \scriptstyle{\sqrt{11}} and volume \tfrac{25}{36} \scriptstyle{\sqrt{2}}.

A triakis tetrahedron with equilateral triangle faces represents a net of the four-dimensional regular polytope known as the 5-cell.

QMRTetrakis hexahedron

From Wikipedia, the free encyclopedia

(Redirected from Tetrakis cube)

Tetrakis hexahedron

Tetrakishexahedron.jpg

(Click here for rotating model)

Type Catalan solid

Coxeter diagram CDel node f1.pngCDel 3.pngCDel node f1.pngCDel 4.pngCDel node.png

Conway notation kC

Face type V4.6.6 DU08 facets.png

isosceles triangle

Faces 24

Edges 36

Vertices 14

Vertices by type 6{4}+8{6}

Symmetry group Oh, B3, [4,3], (*432)

Rotation group O, [4,3]+, (432)

Dihedral angle 143° 7' 48"

\arccos ( -\frac{4}{5} )

Properties convex, face-transitive

Truncated octahedron.png

Truncated octahedron

(dual polyhedron) Tetrakis hexahedron Net

Net

In geometry, a tetrakis hexahedron (also known as a tetrahexahedron, hextetrahedron and kiscube[1]) is a Catalan solid. Its dual is the truncated octahedron, an Archimedean solid.

It also can be called a disdyakis hexahedron as the dual of an omnitruncated tetrahedron.

Orthogonal projections[edit]

The tetrakis hexahedron, dual of the truncated octahedron has 3 symmetry positions, two located on vertices and one mid-edge.

Orthogonal projections[edit]

The tetrakis hexahedron, dual of the truncated octahedron has 3 symmetry positions, two located on vertices and one mid-edge.

Uses[edit]

Naturally occurring (crystal) formations of tetrahexahedra are observed in copper and fluorite systems.

Polyhedral dice shaped like the tetrakis hexahedron are occasionally used by gamers.

A 24-cell viewed under a vertex-first perspective projection has a surface topology of a tetrakis hexahedron and the geometric proportions of the rhombic dodecahedron, with the rhombic faces divided into two triangles.

The tetrakis hexahedron appears as one of the simplest examples in building theory. Consider the Riemannian symmetric space associated to the group SL4(R). Its Tits boundary has the structure of a spherical building whose apartments are 2-dimensional spheres. The partition of this sphere into spherical simplices (chambers) can be obtained by taking the radial projection of a tetrakis hexahedron.

Symmetry[edit]

With Td, [3,3] (*332) tetrahedral symmetry, the triangular faces represent the 24 fundamental domains of tetrahdral symmetry. This polyhedron can be constructed from 6 great circles on a sphere. It can also be seen by a cube with its square faces triangulated by their vertices and face centers and a tetrahedron with its faces divided by vertices, mid-edges, and a central point.

The edges of the tetrakis hexahedron form 6 circles (or centrally radial lines) in the plane. Each of these 6 circles represent a mirror line in tetrahedral symmetry. The 6 circles can be grouped into 3 sets of 2 pairs of orthogonal circles. These edges can also be seen as a compound of 3 othogonal square hosohedrons.

QMRIn geometry, the truncated tetrahedron is an Archimedean solid. It has 4 regular hexagonal faces, 4 equilateral triangle faces, 12 vertices and 18 edges (of two types). It can be constructed by truncating all 4 vertices of a regular tetrahedron at one third of the original edge length.

A deeper truncation, removing a tetrahedron of half the original edge length from each vertex, is called rectification. The rectification of a tetrahedron produces an octahedron.[1]

A truncated tetrahedron is the Goldberg polyhedron GIII(1,1), containing triangular and hexagonal faces.

A truncated tetrahedron can be called a cantic cube, with Coxeter diagram, CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.png, having half of the vertices of the cantellated cube (rhombicuboctahedron), CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.png. There are two dual positions of this construction, and combining them creates the uniform compound of two truncated tetrahedra.

QMRIn geometry, a triakis octahedron (or kisoctahedron[1]) is an Archimedean dual solid, or a Catalan solid. Its dual is the truncated cube.

It can be seen as an octahedron with triangular pyramids added to each face; that is, it is the Kleetope of the octahedron. It is also sometimes called a trisoctahedron, or, more fully, trigonal trisoctahedron. Both names reflect the fact that it has three triangular faces for every face of an octahedron. The tetragonal trisoctahedron is another name for the deltoidal icositetrahedron, a different polyhedron with three quadrilateral faces for every face of an octahedron.

This convex polyhedron is topologically similar to the concave stellated octahedron. They have the same face connectivity, but the vertices are in different relative distances from the center.

If its shorter edges have length 1, its surface area and volume are:

A=3\sqrt{7+4\sqrt{2}}

V=\frac{1}{2}(3+2\sqrt{2}).

QMRIn geometry, an octahedron (plural: octahedra) is a polyhedron with eight faces. A regular octahedron is 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 also a triangular antiprism in any of four orientations.

Orthogonal projections[edit]

The octahedron has four special orthogonal projections, centered, on an edge, vertex, face, and normal to a face. The second and third correspond to the B2 and A2 Coxeter planes.

Geometric relations[edit]

The octahedron represents the central intersection of two tetrahedra

The interior of the compound of two dual tetrahedra is an octahedron, and this compound, called the stella octangula, is its first and only stellation. Correspondingly, a regular octahedron is the result of cutting off from a regular tetrahedron, four regular tetrahedra of half the linear size (i.e. rectifying the tetrahedron). The vertices of the octahedron lie at the midpoints of the edges of the tetrahedron, and in this sense it relates to the tetrahedron in the same way that the cuboctahedron and icosidodecahedron relate to the other Platonic solids. One can also divide the edges of an octahedron in the ratio of the golden mean to define the vertices of an icosahedron. This is done by first placing vectors along the octahedron's edges such that each face is bounded by a cycle, then similarly partitioning each edge into the golden mean along the direction of its vector. There are five octahedra that define any given icosahedron in this fashion, and together they define a regular compound.

Octahedra and tetrahedra can be alternated to form a vertex, edge, and face-uniform tessellation of space, called the octet truss by Buckminster Fuller. This is the only such tiling save the regular tessellation of cubes, and is one of the 28 convex uniform honeycombs. Another is a tessellation of octahedra and cuboctahedra.

The octahedron is unique among the Platonic solids in having an even number of faces meeting at each vertex. Consequently, it is the only member of that group to possess mirror planes that do not pass through any of the faces.

Using the standard nomenclature for Johnson solids, an octahedron would be called a square bipyramid. Truncation of two opposite vertices results in a square bifrustum.

The octahedron is 4-connected, meaning that it takes the removal of four vertices to disconnect the remaining vertices. It is one of only four 4-connected simplicial well-covered polyhedra, meaning that all of the maximal independent sets of its vertices have the same size. The other three polyhedra with this property are the pentagonal dipyramid, the snub disphenoid, and an irregular polyhedron with 12 vertices and 20 triangular faces.[1]

Faceting[edit]

The uniform tetrahemihexahedron is a tetrahedral symmetry faceting of the regular octahedron, sharing edge and vertex arrangement. It has four of the triangular faces, and 3 central squares.

QMRThe great icosahedron is one of the four regular star Kepler-Poinsot polyhedra. Its Schläfli symbol is {3,

}. Like the convex form, it also has 20 equilateral triangle faces, but its vertex figure is a pentagram rather than a pentagon, leading to geometrically intersecting faces. The intersections of the triangles do not represent new edges

Four views of an icosahedron with tetrahedral symmetry, with eight equilateral triangles (red and yellow), and 12 blue isosceles triangles. Yellow and red triangles are the same color in pyritohedral symmetry.

Pyritohedral symmetry[edit]

Pyritohedral and tetrahedral symmetries

Coxeter diagrams CDel node h.pngCDel 3.pngCDel node h.pngCDel 4.pngCDel node.png (pyritohedral) Uniform polyhedron-43-h01.svg

CDel node h.pngCDel 3.pngCDel node h.pngCDel 3.pngCDel node h.png (tetrahedral) Uniform polyhedron-33-s012.svg

Schläfli symbol s{3,4}

sr{3,3} or s\begin{Bmatrix} 3 \\ 3 \end{Bmatrix}

Faces 20 triangles:

8 equilateral

12 isosceles

Edges 30 (6 short + 24 long)

Vertices 12

Symmetry group Th, [4,3+], (3*2), order 24

Rotation group Td, [3,3]+, (332), order 12

Dual polyhedron Pyritohedron

Properties convex

Pseudoicosahedron flat.png

Net

Construction from the vertices of a truncated octahedron, showing internal rectangles.

A regular icosahedron can be constructed with pyritohedral symmetry, and is called a snub octahedron or snub tetratetrahedron or snub tetrahedron. this can be seen as an alternated truncated octahedron. If all the triangles are equilateral, the symmetry can also be distinguished by colouring the 8 and 12 triangle sets differently.

Pyritohedral symmetry has the symbol (3*2), [3+,4], with order 24. Tetrahedral symmetry has the symbol (332), [3,3]+, with order 12. These lower symmetries allow geometric distortions from 20 equilateral triangular faces, instead having 8 equilateral triangles and 12 congruent isosceles triangles.

These symmetries offer Coxeter diagrams: CDel node.pngCDel 4.pngCDel node h.pngCDel 3.pngCDel node h.png and CDel node h.pngCDel 3.pngCDel node h.pngCDel 3.pngCDel node h.png respectively, each representing the lower symmetry to the regular icosahedron CDel node 1.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png, (*532), [5,3] icosahedral symmetry of order 120.

Cartesian coordinates[edit]

The coordinates of the 12 vertices can be defined by the vectors defined by all the possible cyclic permutations and sign-flips of coordinates of the form (2, 1, 0). These coordinates represent the truncated octahedron with alternated vertices deleted.

This construction is called a snub tetrahedron in its regular icosahedron form, generated by the same operations carried out starting with the vector (φ, 1, 0), where φ is the golden ratio.

QMRTetrad (index notation)

From Wikipedia, the free encyclopedia

For preliminary discussion, see Cartan connection applications.

In Riemannian geometry, we can introduce a coordinate system over the Riemannian manifold (at least, over a chart), giving n coordinates

x_{i}\;\text{,}\qquad i = 1, \dots, n

for an n-dimensional manifold. Locally, at least, this gives a basis for the 1-forms, dxi where d is the exterior derivative. The dual basis for the tangent space T is ei.

Now, let's choose an orthonormal basis for the fibers of T. The rest is index manipulation.

QMRA gridshell is a structure which derives its strength from its double curvature (in a similar way that a fabric structure derives strength from double curvature), but is constructed of a grid or lattice.

The grid can be made of any material, but is most often wood (similar to garden trellis) or steel.

Gridshells were pioneered in the 1896 by Russian engineer Vladimir Shukhov in constructions of exhibition pavilions of the All-Russia industrial and art exhibition 1896 in Nizhny Novgorod.[1]

Large span timber gridshells are commonly constructed by initially laying out the main lath members flat in a regular square or rectangular lattice, and subsequently deforming this into the desired doubly curved form. This can be achieved by pushing the members up from the ground, as in the Mannheim Multihalle.[2] More recent projects such as the Savill Garden gridshell were constructed by laying the laths on top of a sizeable temporary scaffolding structure which is removed in phases to let the laths settle into the desired curvature.- The grids look like quadrants

QMRIn mapmaking, a quadrilateralized spherical cube, or quad sphere for short, is an equal-area mapping and binning scheme for data collected on a spherical surface (either that of the Earth or the celestial sphere). It was first proposed in 1975 by Chan and O'Neill for the Naval Environmental Prediction Research Facility.[1]

This scheme is also often called the COBE sky cube,[2] because it was designed to hold data from the Cosmic Background Explorer (COBE) project.[3]

QMRAnother approach gaining favour uses geodesic sphere grids generated by the subdivision of a platonic solid into cells or by iteratively bisecting the edges of the polyhedron and projecting the new cells onto a sphere. In this geodesic grid, each of the vertices in the resulting geodesic sphere corresponds to a cell. One implementation uses an icosahedron as the base polyhedron, hexagonal cells, and the Snyder equal area projection is known as the Icosahedron Snyder Equal Area (ISEA) grid.[3] Another method, using the intersection of a tetrahedron into triangular quadtrees, is known as the Quaternary Triangular Mesh (QTM). A triangular mesh conforms well to representation in a graphics pipeline, and its dual cells are hexagons, convenient for encoding data. The hexagonal geodesic grid inherits many of the virtues of 2D hexagonal grids, and specifically avoids problems with singularities and oversampling near the poles. Along the same line, different Platonic solids could also be used as a starting point instead of an icosahedron or tetrahedron — e.g. cubes that are common in video games can be used to represent the Earth with a small aperture and an efficient equal area projection.[4]

The quadrilateralized spherical cube is a kind of geodesic grid based on subdividing a cube into equal-area cells that are approximately square.

QMRBarycentric coordinates on tetrahedra[edit]

Barycentric coordinates may be easily extended to three dimensions. The 3D simplex is a tetrahedron, a polyhedron having four triangular faces and four vertices. Once again, the barycentric coordinates are defined so that the first vertex \mathbf{r}_1 maps to barycentric coordinates \lambda = (1,0,0,0), \mathbf{r}_2 \to (0,1,0,0), etc.

This is again a linear transformation, and we may extend the above procedure for triangles to find the barycentric coordinates of a point \mathbf{r} with respect to a tetrahedron:

\left(\begin{matrix}\lambda_1 \\ \lambda_2 \\ \lambda_3\end{matrix}\right) = \mathbf{T}^{-1} ( \mathbf{r}-\mathbf{r}_4 )

where \mathbf{T} is now a 3×3 matrix:

\mathbf{T} = \left(\begin{matrix}

x_1-x_4 & x_2-x_4 & x_3-x_4\\

y_1-y_4 & y_2-y_4 & y_3-y_4\\

z_1-z_4 & z_2-z_4 & z_3-z_4

\end{matrix}\right)

Once again, the problem of finding the barycentric coordinates has been reduced to inverting a 3×3 matrix. 3D barycentric coordinates may be used to decide if a point lies inside a tetrahedral volume, and to interpolate a function within a tetrahedral mesh, in an analogous manner to the 2D procedure. Tetrahedral meshes are often used in finite element analysis because the use of barycentric coordinates can greatly simplify 3D interpolation.

QMRSpecial cases[edit]

Quadrilaterals by symmetry

Rhomboid – A quadrilateral whose opposite sides are parallel and adjacent sides are unequal, and whose angles are not right angles[1]

Rectangle – A parallelogram with four angles of equal size

Rhombus – A parallelogram with four sides of equal length.

Square – A parallelogram with four sides of equal length and angles of equal size (right angles).

QMRThe corner of Euclid and Imperial in Southeast San Diego is known as the 4 (Four) Corners of Death, so we decided to take a trip to this notorious intersection. Upon our arrival, we witnessed a young man being chased down Imperial avenue by the San Diego Police Department. He was caught and placed under arrest right before our cameras were rolling.

- See more at: http://www.streetgangs.com/features/031010_south_east#sthash.IqQPvUnF.dpuf

QMRVideo gaming has traditionally been a social experience. Multiplayer video games are those that can be played either competitively, sometimes in Electronic Sports, or cooperatively by using either multiple input devices, or by hotseating. Tennis for Two, arguably the first video game, was a two player game, as was its successor Pong. The first commercially available game console, the Magnavox Odyssey, had two controller inputs.

Since then, most consoles have been shipped with two or four controller inputs. Some have had the ability to expand to four, eight or as many as 12 inputs with additional adapters, such as the Multitap. Multiplayer arcade games typically feature play for two to four players, sometimes tilting the monitor on its back for a top-down viewing experience allowing players to sit opposite one another.

QMRThe VRIO framework, in a wider scope, is part of a much larger strategic scheme of a firm. The basic strategic process that any firm goes through begins with a vision statement, and continues on through objectives, internal & external analysis, strategic choices (both business-level and corporate-level), and strategic implementation. The firm will hope that this process results in a competitive advantage in the marketplace they operate in.

VRIO falls into the internal analysis step of these procedures, but is used as a framework in evaluating just about all resources and capabilities of a firm, regardless of what phase of the strategic model it falls under.

VRIO is an abbreviation for the four question framework you ask about a resource or capability to determine its competitive potential: the question of Value, the question of Rarity, the question of Imitability (Ease/Difficulty to Imitate), and the question of Organization (ability to exploit the resource or capability).

The Question of Value: "Is the firm able to exploit an opportunity or neutralize an external threat with the resource/capability?"

The Question of Rarity: "Is control of the resource/capability in the hands of a relative few?"

The Question of Imitability: "Is it difficult to imitate, and will there be significant cost disadvantage to a firm trying to obtain, develop, or duplicate the resource/capability?"

The Question of Organization: "Is the firm organized, ready, and able to exploit the resource/capability?" "Is the firm organized to capture value?"[1]

QMRThe Four Corners Office/Retail Complex is an office complex in Uptown Houston, Texas.[1][2] The complex, with a total of 395,473 square feet (36,740.6 m2) of office space and 28,290 square feet (2,628 m2) of retail space,[2] sits on 7.5 acres (3.0 ha) of land.[3] The buildings include the 11 story TeleCheck Plaza, the 10 story 5333 Westheimer Road, the 21,120 square feet (1,962 m2) retail tenant building Sage Plaza, the single story 8,800 square feet (820 m2) Savoy Salons, and a 3,892 square feet (361.6 m2) single story Region's Bank with a drive-through.[2] The complex is located at the southwest corner of Westheimer Road and Sage, taking one full square block across the street from The Galleria.[3]

QMRGap analysis is a formal study of what a business is doing currently and where it wants to go in the future. It can be conducted, in different perspectives, as follows:

Organization (e.g., Human Resources)

Business direction

Business processes

Information technology

Quadrant Model of Reality

Tuesday, February 23, 2016

Quadrant Model of Reality Book 23 Philosophy/ Math

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