"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"
"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 ="
"5-orthoplex In five-dimensional geometry, a 5-orthoplex, or 5-cross polytope, is a five-dimensional polytope with 10 vertices, 40 edges, 80 triangle faces, 80 tetrahedron cells, 32 5-cell 4-faces. It has two constructed forms, the first being regular with Schläfli symbol {3,4}, and the second with alternately labeled (checkerboarded) facets, with Schläfli symbol {3,3,3} or Coxeter symbol 2. It is a part of an infinite family of polytopes, called cross-polytopes or ""orthoplexes"". The dual polytope is the 5-hypercube or 5-cube. The elements of the regular polytopes can be expressed in a configuration matrix. Rows and columns reference vertices, edges, faces, and cells,"
"5-simplex In five-dimensional geometry, a 5-simplex is a self-dual regular 5-polytope. It has six vertices, 15 edges, 20 triangle faces, 15 tetrahedral cells, and 6 5-cell facets. It has a dihedral angle of cos(), or approximately 78.46°. The 5-simplex is a solution to the problem: ""Make 20 equilateral triangles using 15 matchsticks, where each side of every triangle is exactly one matchstick."" It can also be called a hexateron, or hexa-5-tope, as a 6-facetted polytope in 5-dimensions. The name ""hexateron"" is derived from ""hexa-"" for having six facets and ""teron"" (with ""ter-"" being a corruption of ""tetra-"") for having four-dimensional"
"Pentagon In geometry, a pentagon (from the Greek πέντε ""pente"" and γωνία ""gonia"", meaning ""five"" and ""angle"") is any five-sided polygon or 5-gon. The sum of the internal angles in a simple pentagon is 540°. A pentagon may be simple or self-intersecting. A self-intersecting ""regular pentagon"" (or ""star pentagon"") is called a pentagram. A ""regular pentagon"" has Schläfli symbol {5} and interior angles are 108°. A ""regular pentagon"" has five lines of reflectional symmetry, and rotational symmetry of order 5 (through 72°, 144°, 216° and 288°). The diagonals of a convex regular pentagon are in the golden ratio to its"
"5-5 duoprism In geometry of 4 dimensions, a 5-5 duoprism or pentagonal duoprism is a polygonal duoprism, a 4-polytope resulting from the Cartesian product of two pentagons. It has 25 vertices, 50 edges, 35 faces (25 squares, and 10 pentagons), in 10 pentagonal prism cells. It has Coxeter diagram , and symmetry , order 200. Seen in a skew 2D orthogonal projection, 20 of the vertices are in two decagonal rings, while 5 project into the center. The 5-5 duoprism here has an identical 2D projective appearance to the 3D rhombic triacontahedron. In this projection, the square faces project into"
"connected to every vertex on the other. 5-5 duoprism In geometry of 4 dimensions, a 5-5 duoprism or pentagonal duoprism is a polygonal duoprism, a 4-polytope resulting from the Cartesian product of two pentagons. It has 25 vertices, 50 edges, 35 faces (25 squares, and 10 pentagons), in 10 pentagonal prism cells. It has Coxeter diagram , and symmetry , order 200. Seen in a skew 2D orthogonal projection, 20 of the vertices are in two decagonal rings, while 5 project into the center. The 5-5 duoprism here has an identical 2D projective appearance to the 3D rhombic triacontahedron. In"
"Pentahedron In geometry, a pentahedron (plural: pentahedra) is a polyhedron with five faces or sides. There are no face-transitive polyhedra with five sides and there are two distinct topological types. With regular polygon faces, the two topological forms are the square pyramid and triangular prism. The ""square pyramid"" can be seen as a ""triangular prism"" where one edge of its side edges is collapsed into a point, losing one edge and one vertex, and changing two squares into triangles. Geometric variations with irregular faces can also be constructed. Some irregular pentahedra with six vertices may be called wedges. An irregular"
"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"
"5-cube In five-dimensional geometry, a 5-cube is a name for a five-dimensional hypercube with 32 vertices, 80 edges, 80 square faces, 40 cubic cells, and 10 tesseract 4-faces. It is represented by Schläfli symbol {4,3,3,3} or {4,3}, constructed as 3 tesseracts, {4,3,3}, around each cubic ridge. It can be called a penteract, a portmanteau of tesseract (the ""4-cube"") and ""pente"" for five (dimensions) in Greek. It can also be called a regular deca-5-tope or decateron, being a 5-dimensional polytope constructed from 10 regular facets. It is a part of an infinite hypercube family. The dual of a 5-cube is 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"
"its surface area is numerically equal to its volume. As with equable shapes in two dimensions, you may find an equable solid, in which the volume is numerically equal to the surface area, by scaling any solid by an appropriate factor. For example a cube with side length six. 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"
"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"
"wide and narrow rhombi seen in penrose tiling. The regular complex polytope {4}, , in formula_1 has a real representation as a 5-5 duoprism in 4-dimensional space. {4} has 25 vertices, and 10 5-edges. Its symmetry is [4], order 50. It also has a lower symmetry construction, , or {}×{}, with symmetry [2], order 25. This is the symmetry if the red and blue 5-edges are considered distinct. The birectified order-5 120-cell, , constructed by all rectified 600-cells, a 5-5 duoprism vertex figure. The dual of a ""5-5 duoprism"" is called a 5-5 duopyramid or pentagonal duopyramid. It has 25"
"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"
"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)"
"with a 4-simplex base. 5-polytope In five-dimensional geometry, a five-dimensional polytope or 5-polytope is a 5-dimensional polytope, bounded by (4-polytope) facets. Each polyhedral cell being shared by exactly two 4-polytope facets. A 5-polytope is a closed five-dimensional figure with vertices, edges, faces, and cells, and 4-faces. A vertex is a point where five or more edges meet. An edge is a line segment where four or more faces meet, and a face is a polygon where three or more cells meet. A cell is a polyhedron, and a 4-face is a 4-polytope. Furthermore, the following requirements must be met: The"
"plane is called mirror symmetric (see mirror image). The axis of symmetry of a two-dimensional figure is a line such that, if a perpendicular is constructed, any two points lying on the perpendicular at equal distances from the axis of symmetry are identical. Another way to think about it is that if the shape were to be folded in half over the axis, the two halves would be identical: the two halves are each other's mirror image. Thus a square has four axes of symmetry, because there are four different ways to fold it and have the edges all match."
"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"
"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"
"a 5-fusil which can have a variety of subsymmetries. This polytope is one of 31 uniform 5-polytopes generated from the B Coxeter plane, including the regular 5-cube and 5-orthoplex. 5-orthoplex In five-dimensional geometry, a 5-orthoplex, or 5-cross polytope, is a five-dimensional polytope with 10 vertices, 40 edges, 80 triangle faces, 80 tetrahedron cells, 32 5-cell 4-faces. It has two constructed forms, the first being regular with Schläfli symbol {3,4}, and the second with alternately labeled (checkerboarded) facets, with Schläfli symbol {3,3,3} or Coxeter symbol 2. It is a part of an infinite family of polytopes, called cross-polytopes or ""orthoplexes"". The"
"Regular icosahedron In geometry, a regular icosahedron ( or ) is a convex polyhedron with 20 faces, 30 edges and 12 vertices. It is one of the five Platonic solids, and the one with the most sides. It has five equilateral triangular faces meeting at each vertex. It is represented by its Schläfli symbol {3,5}, or sometimes by its vertex figure as 3.3.3.3.3 or 3. It is the dual of the dodecahedron, which is represented by {5,3}, having three pentagonal faces around each vertex. A regular icosahedron is a gyroelongated pentagonal bipyramid and a biaugmented pentagonal antiprism in any of"
"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 numbers of"
"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"
"5-polytope In five-dimensional geometry, a five-dimensional polytope or 5-polytope is a 5-dimensional polytope, bounded by (4-polytope) facets. Each polyhedral cell being shared by exactly two 4-polytope facets. A 5-polytope is a closed five-dimensional figure with vertices, edges, faces, and cells, and 4-faces. A vertex is a point where five or more edges meet. An edge is a line segment where four or more faces meet, and a face is a polygon where three or more cells meet. A cell is a polyhedron, and a 4-face is a 4-polytope. Furthermore, the following requirements must be met: The topology of any given"
"Rhomboid Traditionally, 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 rhombohedron or a more general 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"
"Rectified 5-simplexes In five-dimensional geometry, a rectified 5-simplex is a convex uniform 5-polytope, being a rectification of the regular 5-simplex. There are three unique degrees of rectifications, including the zeroth, the 5-simplex itself. Vertices of the ""rectified 5-simplex"" are located at the edge-centers of the ""5-simplex"". Vertices of the ""birectified 5-simplex"" are located in the triangular face centers of the ""5-simplex"". In five dimensional geometry, a rectified 5-simplex, is a uniform 5-polytope with 15 vertices, 60 edges, 80 triangular faces, 45 cells (15 tetrahedral, and 30 octahedral), and 12 4-faces (6 5-cell and 6 rectified 5-cells). It is also called"
"Rectified 5-simplexes In five-dimensional geometry, a rectified 5-simplex is a convex uniform 5-polytope, being a rectification of the regular 5-simplex. There are three unique degrees of rectifications, including the zeroth, the 5-simplex itself. Vertices of the ""rectified 5-simplex"" are located at the edge-centers of the ""5-simplex"". Vertices of the ""birectified 5-simplex"" are located in the triangular face centers of the ""5-simplex"". In five dimensional geometry, a rectified 5-simplex, is a uniform 5-polytope with 15 vertices, 60 edges, 80 triangular faces, 45 cells (15 tetrahedral, and 30 octahedral), and 12 4-faces (6 5-cell and 6 rectified 5-cells). It is also called"
"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"
"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"
"mathematics, quasi-isometry can be used as a criterion to state that two shapes are approximately the same. Simple shapes can often be classified into basic geometric objects such as a point, a line, a curve, a plane, a plane figure (e.g. square or circle), or a solid figure (e.g. cube or sphere). However, most shapes occurring in the physical world are complex. Some, such as plant structures and coastlines, may be so complicated as to defy traditional mathematical description – in which case they may be analyzed by differential geometry, or as fractals. In geometry, two subsets of a Euclidean"
"Hexagram A hexagram (Greek) or sexagram (Latin) is a six-pointed geometric star figure with the Schläfli symbol {6/2}, 2{3}, or <nowiki></nowiki>. Since there are no true regular continuous hexagrams, the term is instead used to refer to the compound figure of two equilateral triangles shown to the right. The intersection is a regular hexagon. The hexagram is part of an infinite series of shapes which are compounds of two n-dimensional simplices. In three dimensions, the analogous compound is the stellated octahedron, and in four dimensions the compound of two 5-cells is obtained. It has been historically used in religious and"
"Cube In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. The cube is the only regular hexahedron and is one of the five Platonic solids. It has 6 faces, 12 edges, and 8 vertices. The cube is also a square parallelepiped, an equilateral cuboid and a right rhombohedron. It is a regular square prism in three orientations, and a trigonal trapezohedron in four orientations. The cube is dual to the octahedron. It has cubical or octahedral symmetry. The cube is the only convex polyhedron whose faces"