"Square In geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree angles, or (100-gradian angles or right angles). It can also be defined as a rectangle in which two adjacent sides have equal length. A square with vertices ""ABCD"" would be denoted . A convex quadrilateral is a square if and only if it is any one of the following: A square is a special case of a rhombus (equal sides, opposite equal angles), a kite (two pairs of adjacent equal sides), a trapezoid (one pair of opposite sides"
"Quadrilateral In Euclidean plane geometry, a quadrilateral is a polygon with four edges (or sides) and four vertices or corners. Sometimes, the term quadrangle is used, by analogy with triangle, and sometimes tetragon for consistency with pentagon (5-sided), hexagon (6-sided) and so on. The origin of the word ""quadrilateral"" is the two Latin words ""quadri"", a variant of four, and ""latus"", meaning ""side"". Quadrilaterals are simple (not self-intersecting) or complex (self-intersecting), also called crossed. Simple quadrilaterals are either convex or concave. The interior angles of a simple (and planar) quadrilateral ""ABCD"" add up to 360 degrees of arc, that is"
"has 10 faces on the polar axis with 10 faces following the equator. Rhombus In plane 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"
"Rhombus In plane 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"
"Rectangle In Euclidean plane geometry, a rectangle is a 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. A rectangle with vertices ""ABCD"" would be denoted as . The word rectangle comes from the Latin ""rectangulus"", which is a combination of ""rectus"" (as an adjective, right, proper)"
"Disphenoid In geometry, a disphenoid (from Greek sphenoeides, ""wedgelike"") is a tetrahedron whose four faces are congruent acute-angled triangles. It can also be described as a tetrahedron in which every two edges that are opposite each other have equal lengths. Other names for the same shape are sphenoid, bisphenoid, isosceles tetrahedron, equifacial tetrahedron, almost regular tetrahedron, and tetramonohedron. All the solid angles and vertex figures of a disphenoid are the same, and the sum of the face angles at each vertex is equal to two right angles. However, a disphenoid is not a regular polyhedron, because, in general, its faces"
"Trapezoid In Euclidean geometry, a convex quadrilateral with at least one pair of parallel sides is referred to as a trapezoid () in American and Canadian English but as a trapezium () in English outside North America. The parallel sides are called the ""bases"" of the trapezoid and the other two sides are called the ""legs"" or the lateral sides (if they are not parallel; otherwise there are two pairs of bases). A ""scalene trapezoid"" is a trapezoid with no sides of equal measure, in contrast to the special cases below. The term ""trapezium"" has been in use in English"
"pattern in Knuth's up-arrow notation, , and so on, for any number of up arrows. (That is, for every positive integer ""n"", where is the hyperoperation.) A four-sided plane figure is a quadrilateral (quadrangle) which include kites, rhombi, rectangles and squares, sometimes also called a ""tetragon"". A circle divided by 4 makes right angles and four quadrants. Because of it, four (4) is the base number of the plane (mathematics). Four cardinal directions, four seasons, the duodecimal system, and the vigesimal system are based on four. A solid figure with four faces as well as four vertices is a tetrahedron,"
"Kite (geometry) In Euclidean geometry, a kite is a quadrilateral whose four sides can be grouped into two pairs of equal-length sides that are adjacent to each other. In contrast, a parallelogram also has two pairs of equal-length sides, but they are opposite to each other rather than adjacent. Kite quadrilaterals are named for the wind-blown, flying kites, which often have this shape and which are in turn named for a bird. Kites are also known as deltoids, but the word ""deltoid"" may also refer to a deltoid curve, an unrelated geometric object. A kite, as defined above, may be"
"Parallelogon A parallelogon is a polygon such that images of the polygon under translations only tile the plane when fitted together along entire sides. A parallelogon must have an even number of sides and opposite sides must be equal in length and parallel (hence the name). A less obvious restriction is that a parallelogon can only have four or six sides; a four-sided parallelogon is a parallelogram. In general a parallelogon has 180-degree rotational symmetry around its center. Quadrilateral and hexagonal parallelogons each have varied geometric symmetric forms. In general they all have central inversion symmetry, order 2. Every convex"
"Nonagon In geometry, a nonagon () or enneagon () is a nine-sided polygon or 9-gon. The name ""nonagon"" is a prefix hybrid formation, from Latin (""nonus"", ""ninth"" + ""gonon""), used equivalently, attested already in the 16th century in French ""nonogone"" and in English from the 17th century. The name ""enneagon"" comes from Greek ""enneagonon"" (εννεα, ""nine"" + γωνον (from γωνία = ""corner"")), and is arguably more correct, though less common than ""nonagon"". A ""regular nonagon"" is represented by Schläfli symbol {9} and has internal angles of 140°. The area of a regular nonagon of side length ""a"" is given by"
"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. Rectangle In Euclidean plane geometry, a rectangle is a 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"
"a ""puckered"" ring of four atoms. Historically the term gauche quadrilateral was also used to mean a skew quadrilateral. A skew quadrilateral together with its diagonals form a (possibly non-regular) tetrahedron, and conversely every skew quadrilateral comes from a tetrahedron where a pair of opposite edges is removed. Quadrilateral In Euclidean plane geometry, a quadrilateral is a polygon with four edges (or sides) and four vertices or corners. Sometimes, the term quadrangle is used, by analogy with triangle, and sometimes tetragon for consistency with pentagon (5-sided), hexagon (6-sided) and so on. The origin of the word ""quadrilateral"" is the two"
"divides this segment in the ratio (when taken from the short to the long side) If the angle bisectors to angles ""A"" and ""B"" intersect at ""P"", and the angle bisectors to angles ""C"" and ""D"" intersect at ""Q"", then The term ""trapezoid"" was once defined as a quadrilateral without any parallel sides in Britain and elsewhere. (The Oxford English Dictionary says ""Often called by English writers in the 19th century"".) According to the ""Oxford English Dictionary"", the sense of a figure with no sides parallel is the meaning for which Proclus introduced the term ""trapezoid"". This is retained in"
"pentagonal faces, with three meeting in each of the 20 vertices. However, the pentagons are not regular and the figure has no fivefold symmetry axes. Although regular dodecahedra do not exist in crystals, the tetartoid form does. The name tetartoid comes from the Greek root for one-fourth because it has one fourth of full octahedral symmetry, and half of pyritohedral symmetry. The mineral cobaltite can have this symmetry form. Its topology can be as a cube with square faces bisected into 2 rectangles like the pyritohedron, and then the bisection lines are slanted retaining 3-fold rotation at the 8 corners."
"Orthodiagonal quadrilateral In Euclidean geometry, an orthodiagonal quadrilateral is a quadrilateral in which the diagonals cross at right angles. In other words, it is a four-sided figure in which the line segments between non-adjacent vertices are orthogonal (perpendicular) to each other. 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. A rhombus is an orthodiagonal quadrilateral with two pairs of parallel sides (that is, an orthodiagonal quadrilateral"
"Pseudo-polyomino A pseudo-polyomino, also called a polyking, polyplet or hinged polyomino, is a plane geometric figure formed by joining one or more equal squares edge to edge or corner to corner at 90°. It is a polyform with square cells. The polyominoes are a subset of the polykings. The name ""polyking"" refers to the king in chess. The ""n""-kings are the ""n""-square shapes which could be occupied by a king on an infinite chessboard in the course of legal moves. Golomb uses the term ""pseudo-polyomino"" referring to kingwise-connected sets of squares. There are three common ways of distinguishing polyominoes and"
"Gömböc A gömböc or gomboc () is a convex three-dimensional homogeneous body which, when resting on a flat surface, has just one stable and one unstable point of equilibrium. Its existence was conjectured by Russian mathematician Vladimir Arnold in 1995 and proven in 2006 by Hungarian scientists Gábor Domokos and Péter Várkonyi. The gömböc shape is not unique; it has countless varieties, most of which are very close to a sphere and all have very strict shape tolerance (about 0.1 mm per 100 mm). The most famous solution has a sharpened top, as shown in the photo. Its shape helped"
"In Euclidean geometry, the six quadrilaterals illustrated are all different. Yet they have a common structure in the alternating chain of four vertices and four sides which gives them their name. They are said to be isomorphic or “structure preserving”. This common structure may be represented in an underlying abstract polytope, a purely algebraic partially-ordered set which captures the pattern of connections or ""incidences"" between the various structural elements. The measurable properties of traditional polytopes such as angles, edge-lengths, skewness, straightness and convexity have no meaning for an abstract polytope. What is true for traditional polytopes (also called classical or"
"Hexagon In geometry, a hexagon (from Greek ἕξ ""hex"", ""six"" and γωνία, ""gonía"", ""corner, angle"") is a six-sided polygon or 6-gon. The total of the internal angles of any simple (non-self-intersecting) hexagon is 720°. A ""regular hexagon"" has Schläfli symbol {1} and can also be constructed as a truncated equilateral triangle, t{3}, which alternates two types of edges. A regular hexagon is defined as a hexagon that is both equilateral and equiangular. It is bicentric, meaning that it is both cyclic (has a circumscribed circle) and tangential (has an inscribed circle). The common length of the sides equals the radius"
"Parallelogram In Euclidean geometry, a parallelogram is a simple (non-self-intersecting) quadrilateral with two pairs of parallel sides. The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of equal measure. The congruence of opposite sides and opposite angles is a direct consequence of the Euclidean parallel postulate and neither condition can be proven without appealing to the Euclidean parallel postulate or one of its equivalent formulations. By comparison, a quadrilateral with just one pair of parallel sides is a trapezoid in American English or a trapezium in British English. The"
"Complete quadrangle In mathematics, specifically projective geometry, a complete quadrangle is a system of geometric objects consisting of any four points in a plane, no three of which are on a common line, and of the six lines connecting the six pairs of points. Dually, a complete quadrilateral is a system of four lines, no three of which pass through the same point, and the six points of intersection of these lines. The complete quadrangle was called a tetrastigm by , and the complete quadrilateral was called a tetragram; those terms are occasionally still used. The six lines of a"
"Nonconvex great rhombicuboctahedron In geometry, the nonconvex great rhombicuboctahedron is a nonconvex uniform polyhedron, indexed as U. It is represented by Schläfli symbol t{4,3/2} and Coxeter-Dynkin diagram of . Its vertex figure is a crossed quadrilateral. This model shares the name with the convex ""great rhombicuboctahedron"", also called the truncated cuboctahedron. An alternate name for this figure is quasirhombicuboctahedron. From that derives its Bowers acronym: querco. It shares the vertex arrangement with the convex truncated cube. It additionally shares its edge arrangement with the great cubicuboctahedron (having the triangular faces and 6 square faces in common), and with the great"
"vertices. Regular apeirogons p{q}r are constrained by: 1/""p"" + 2/""q"" + 1/""r"" = 1. Edges have ""p"" vertices, and vertex figures are ""r""-gonal. Square tiling In geometry, the square tiling, square tessellation or square grid is a regular tiling of the Euclidean plane. It has Schläfli symbol of {4,4}, meaning it has ""4"" squares around every vertex. Conway calls it a quadrille. The internal angle of the square is 90 degrees so four squares at a point make a full 360 degrees. It is one of three regular tilings of the plane. The other two are the triangular tiling and"
"an axis of symmetry. In three dimensions, every figure that possesses a plane of symmetry or a center of symmetry is achiral. There are, however, achiral figures lacking both plane and center of symmetry. In terms of point groups, all chiral figures lack an improper axis of rotation (S). This means that they cannot contain a center of inversion (i) or a mirror plane (σ). Only figures with a point group designation of C, C, D, T, O, or I can be chiral. A knot is called achiral if it can be continuously deformed into its mirror image, otherwise it"
"(adding up to 180° or π radians). For a cyclic polygon with an odd number of sides, all angles are equal if and only if the polygon is regular. A cyclic polygon with an even number of sides has all angles equal if and only if the alternate sides are equal (that is, sides 1, 3, 5, ... are equal, and sides 2, 4, 6, ... are equal). A cyclic pentagon with rational sides and area is known as a Robbins pentagon; in all known cases, its diagonals also have rational lengths. In any cyclic ""n""-gon with even ""n"", the"
"different lengths. Cuboid In geometry, a cuboid is a convex polyhedron bounded by six quadrilateral faces, whose polyhedral graph is the same as that of a cube. While mathematical literature refers to any such polyhedron as a cuboid, other sources use ""cuboid"" to refer to a shape of this type in which each of the faces is a rectangle (and so each pair of adjacent faces meets in a right angle); this more restrictive type of cuboid is also known as a rectangular cuboid, right cuboid, rectangular box, rectangular hexahedron, right rectangular prism, or rectangular parallelepiped. By Euler's formula the"
"of an orthocentric tetrahedron can be expressed in terms of four edges ""a"", ""b"", ""c"", ""d"". The formula is where ""c"" and ""d"" are opposite edges, and formula_3. Orthocentric tetrahedron In geometry, an orthocentric tetrahedron is a tetrahedron where all three pairs of opposite edges are perpendicular. It is also known as an orthogonal tetrahedron since orthogonal means perpendicular. It was first studied by Simon Lhuilier in 1782, and got the name orthocentric tetrahedron by G. de Longchamps in 1890. In an orthocentric tetrahedron the four altitudes are concurrent. This common point is called the orthocenter, and it has the"
"Right triangle A right triangle (American English) or right-angled triangle (British English) is a triangle in which one angle is a right angle (that is, a 90-degree angle). The relation between the sides and angles of a right triangle is the basis for trigonometry. The side opposite the right angle is called the ""hypotenuse"" (side ""c"" in the figure). The sides adjacent to the right angle are called ""legs"" (or ""catheti"", singular: ""cathetus""). Side ""a"" may be identified as the side ""adjacent to angle B"" and ""opposed to"" (or ""opposite"") ""angle A"", while side ""b"" is the side ""adjacent to"
"Right triangle A right triangle (American English) or right-angled triangle (British English) is a triangle in which one angle is a right angle (that is, a 90-degree angle). The relation between the sides and angles of a right triangle is the basis for trigonometry. The side opposite the right angle is called the ""hypotenuse"" (side ""c"" in the figure). The sides adjacent to the right angle are called ""legs"" (or ""catheti"", singular: ""cathetus""). Side ""a"" may be identified as the side ""adjacent to angle B"" and ""opposed to"" (or ""opposite"") ""angle A"", while side ""b"" is the side ""adjacent to"
"Nonomino A nonomino (or 9-omino) is a polyomino of order 9, that is, a polygon in the plane made of 9 equal-sized squares connected edge-to-edge. The name of this type of figure is formed with the prefix non(a)-. When rotations and reflections are not considered to be distinct shapes, there are 1,285 different ""free"" nonominoes. When reflections are considered distinct, there are 2,500 ""one-sided"" nonominoes. When rotations are also considered distinct, there are 9,910 ""fixed"" nonominoes. The 1,285 free nonominoes can be classified according to their symmetry groups: Unlike octominoes, there are no nonominoes with rotational symmetry of order 4"
"Golygon A golygon is any polygon with all right angles (a rectilinear polygon) whose sides are consecutive integer lengths. Golygons were invented and named by Lee Sallows, and popularized by A.K. Dewdney in a 1990 ""Scientific American"" column (Smith). Variations on the definition of golygons involve allowing edges to cross, using sequences of edge lengths other than the consecutive integers, and considering turn angles other than 90°. In any golygon, all horizontal edges have the same parity as each other, as do all vertical edges. Therefore, the number ""n"" of sides must allow the solution of the system of equations"
"model center, cut off at a certain point that is convenient for the maker. Cubohemioctahedron In geometry, the cubohemioctahedron is a nonconvex uniform polyhedron, indexed as U. Its vertex figure is a crossed quadrilateral. It is given Wythoff symbol 4/3 4 | 3, although that is a double-covering of this figure. A nonconvex polyhedron has intersecting faces which do not represent new edges or faces. In the picture vertices are marked by golden spheres, and edges by silver cylinders. It is a hemipolyhedron with 4 hexagonal faces passing through the model center. The hexagons intersect each other and so only"