Which mathematical shape has four sides of equal length, but no right angles?
"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°). The dual polygon of a rectangle is a rhombus, as shown in the table below. 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"
"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"
"every square has four sides, even though not every four-sided shape is a square. The previous paragraph explained how a mathematician might weaken her conjecture in the face of counterexamples, but counterexamples can also be used to show that the assumptions and hypothesis are needed. Suppose that after a while the mathematician in question settled on the new conjecture ""All shapes that are rectangles and have four sides of equal length are squares"". This conjecture has two parts to the hypothesis: the shape must be 'a rectangle' and 'have four sides of equal length' and the mathematician would like to"
"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)"
"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"
"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"
"its sides be in a geometric progression. The 3–4–5 triangle is the unique right triangle (up to scaling) whose sides are in an arithmetic progression. Let be the side length of a regular decagon inscribed in the unit circle, where ""φ"" is the golden ratio. Let be the side length of a regular hexagon in the unit circle, and let {2}}</math>}} be the side length of a regular pentagon in the unit circle. Then , so these three lengths form the sides of a right triangle. The same triangle forms half of a golden rectangle. It may also be found"
"Császár polyhedron In geometry, the Császár polyhedron () is a nonconvex polyhedron, topologically a toroid, with 14 triangular faces. This polyhedron has no diagonals; every pair of vertices is connected by an edge. The seven vertices and 21 edges of the Császár polyhedron form an embedding of the complete graph formula_1 onto a topological torus. Of the 35 possible triangles from vertices of the polyhedron, only 14 are faces. The tetrahedron and the Császár polyhedron are the only two known polyhedra (having a manifold boundary) without any diagonals: every two vertices of the polygon are connected by an edge, so"
"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"
"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"
"(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"
"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"
"product of the two shorter sides of the triangle, at least one of which must be even. An example of a Heronian triangle which is not right-angled is the isosceles triangle with sidelengths 5, 5, and 6, whose area is 12. This triangle is obtained by joining two copies of the right-angled triangle with sides 3, 4, and 5 along the sides of length 4. This approach works in general, as illustrated in the adjacent picture. One takes a Pythagorean triple (""a"", ""b"", ""c""), with ""c"" being largest, then another one (""a"", ""d"", ""e""), with ""e"" being largest, constructs the"
"and no sides intersecting others. A common example of this type of pentagon is the regular pentagon. Equilateral pentagon In geometry an equilateral pentagon is a polygon with five sides of equal length. Its five internal angles, in turn, can take a range of sets of values, thus permitting it to form a family of pentagons. The requirement is that all angles must add up to 540 degrees and must be between 0 and 360 degrees but not equal to 180 degrees. In contrast, the regular pentagon is unique, because it is equilateral and moreover it is equiangular (its five"
"Yates in 1858. As W. A. Whitworth and D. Biddle proved in 1904, there are exactly three solutions, beyond the right triangles already listed, with sides (6,25,29), (7,15,20), and (9,10,17). The only equable rectangles with integer sides are the 4 × 4 square and the 3 × 6 rectangle. An integer rectangle is a special type of polyomino, and more generally there exist polyominoes with equal area and perimeter for any even integer area greater than or equal to 16. For smaller areas, the perimeter of a polyomino must exceed its area. In three dimensions, a shape is equable when"
"is the only triangle with integer sides and three rational angles as measured in degrees. The equilateral triangle is the only acute triangle that is similar to its orthic triangle (with vertices at the feet of the altitudes) (the heptagonal triangle being the only obtuse one). Equilateral triangles are found in many other geometric constructs. The intersection of circles whose centers are a radius width apart is a pair of equilateral arches, each of which can be inscribed with an equilateral triangle. They form faces of regular and uniform polyhedra. Three of the five Platonic solids are composed of equilateral"
"a sequence of expanded hypercubes: Octagon In geometry, an octagon (from the Greek ὀκτάγωνον ""oktágōnon"", ""eight angles"") is an eight-sided polygon or 8-gon. A ""regular octagon"" has Schläfli symbol {8} and can also be constructed as a quasiregular truncated square, t{4}, which alternates two types of edges. A truncated octagon, t{8} is a hexadecagon, t{16}. The sum of all the internal angles of any octagon is 1080°. As with all polygons, the external angles total 360°. If squares are constructed all internally or all externally on the sides of an octagon, then the midpoints of the segments connecting the centers"
"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"
"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. The octahemioctacron has four vertices at infinity. Octahemioctahedron In geometry, the octahemioctahedron or allelotetratetrahedron is a nonconvex uniform polyhedron, indexed as U. Its vertex figure is a crossed quadrilateral. It is one of nine hemipolyhedra, with 4 hexagonal faces passing through the model center. It is the only hemipolyhedron that is orientable, and the"
"4-manifold In mathematics, a 4-manifold is a 4-dimensional topological manifold. A smooth 4-manifold is a 4-manifold with a smooth structure. In dimension four, in marked contrast with lower dimensions, topological and smooth manifolds are quite different. There exist some topological 4-manifolds which admit no smooth structure and even if there exists a smooth structure it need not be unique (i.e. there are smooth 4-manifolds which are homeomorphic but not diffeomorphic). 4-manifolds are of importance in physics because, in General Relativity, spacetime is modeled as a pseudo-Riemannian 4-manifold. The homotopy type of a simply connected compact 4-manifold only depends on the"
"Equidiagonal quadrilateral In Euclidean geometry, an equidiagonal quadrilateral is a convex quadrilateral whose two diagonals have equal length. Equidiagonal quadrilaterals were important in ancient Indian mathematics, where quadrilaterals were classified first according to whether they were equidiagonal and then into more specialized types. Examples of equidiagonal quadrilaterals include the isosceles trapezoids, rectangles and squares. Among all quadrilaterals, the shape that has the greatest ratio of its perimeter to its diameter is an equidiagonal kite with angles π/3, 5π/12, 5π/6, and 5π/12. A convex quadrilateral is equidiagonal if and only if its Varignon parallelogram, the parallelogram formed by the midpoints of"
"Equilateral pentagon In geometry an equilateral pentagon is a polygon with five sides of equal length. Its five internal angles, in turn, can take a range of sets of values, thus permitting it to form a family of pentagons. The requirement is that all angles must add up to 540 degrees and must be between 0 and 360 degrees but not equal to 180 degrees. In contrast, the regular pentagon is unique, because it is equilateral and moreover it is equiangular (its five angles are equal; the measure is 108 degrees). Four intersecting equal circles arranged in a closed chain"
"Octagon In geometry, an octagon (from the Greek ὀκτάγωνον ""oktágōnon"", ""eight angles"") is an eight-sided polygon or 8-gon. A ""regular octagon"" has Schläfli symbol {8} and can also be constructed as a quasiregular truncated square, t{4}, which alternates two types of edges. A truncated octagon, t{8} is a hexadecagon, t{16}. The sum of all the internal angles of any octagon is 1080°. As with all polygons, the external angles total 360°. If squares are constructed all internally or all externally on the sides of an octagon, then the midpoints of the segments connecting the centers of opposite squares form 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"
"across any one of its edges produces another tile; one of them is a right kite, with 60°, 90°, and 120° angles. The tiling that it produces by its reflections is the deltoidal trihexagonal tiling. Among all quadrilaterals, the shape that has the greatest ratio of its perimeter to its diameter is an equidiagonal kite with angles π/3, 5π/12, 5π/6, 5π/12. Its four vertices lie at the three corners and one of the side midpoints of the Reuleaux triangle (above to the right). In non-Euclidean geometry, a Lambert quadrilateral is a right kite with three right angles. A quadrilateral is"
"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"
"Equable shape A two-dimensional equable shape (or perfect shape) is one whose area is numerically equal to its perimeter. For example, a right angled triangle with sides 5, 12 and 13 has area and perimeter both have a unitless numerical value of 30. An area cannot be equal to a length except relative to a particular unit of measurement. For example, if shape has an area of 5 square yards and a perimeter of 5 yards, then it has an area of and a perimeter of 15 feet (since 3 feet = 1 yard and hence 9 square feet ="
"Truncated cuboctahedron In geometry, the truncated cuboctahedron is an Archimedean solid, named by Kepler as a truncation of a cuboctahedron. It has 12 square faces, 8 regular hexagonal faces, 6 regular octagonal faces, 48 vertices and 72 edges. Since each of its faces has point symmetry (equivalently, 180° rotational symmetry), the truncated cuboctahedron is a zonohedron. The truncated cuboctahedron can tessellate with the octagonal prism. There is a nonconvex uniform polyhedron with a similar name, the nonconvex great rhombicuboctahedron. The Cartesian coordinates for the vertices of a truncated cuboctahedron having edge length 2 and centered at the origin are all"
"of of a perfect tetrahedron. Its integral edge lengths are 25, 39, 56, 120, 153 and 160. Heronian tetrahedron A Heronian tetrahedron or perfect tetrahedron is a tetrahedron whose edge lengths, face areas and volume are all rational numbers. The faces must therefore all be Heronian triangles. A regular tetrahedron (one with all faces being equilateral) with all sides rational is not a Heronian tetrahedron because its face areas and volume are not rational numbers. 117 is the smallest possible length of the longest edge of a perfect tetrahedron with integral edge lengths. Its other edge lengths are 51, 52,"
"digonal gyrobianticupola. It has the same topology and symmetry, but without equilateral triangles. It has 4 vertices in a square on a center plane as two anticupolae attached with rotational symmetry. Its dual has right-angled pentagons and can self-tessellate space. Snub disphenoid In geometry, the snub disphenoid, Siamese dodecahedron, triangular dodecahedron or dodecadeltahedron is a three-dimensional convex polyhedron with twelve equilateral triangles as its faces. It is not a regular polyhedron because some vertices have four faces and others have five. It is a dodecahedron, one of the eight deltahedra (convex polyhedra with equilateral triangle faces) and one of the"
"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"
"Octahemioctahedron In geometry, the octahemioctahedron or allelotetratetrahedron is a nonconvex uniform polyhedron, indexed as U. Its vertex figure is a crossed quadrilateral. It is one of nine hemipolyhedra, with 4 hexagonal faces passing through the model center. It is the only hemipolyhedron that is orientable, and the only uniform polyhedron with an Euler characteristic of zero (a topological torus). It shares the vertex arrangement and edge arrangement with the cuboctahedron (having the triangular faces in common), and with the cubohemioctahedron (having the hexagonal faces in common). By Wythoff construction it has tetrahedral symmetry (T), like the ""rhombitetratetrahedron"" construction for the"