"term the areas of the two green triangles, the third term the areas of the four yellow triangles, and so on. Simplifying the fractions gives This is a geometric series with common ratio and the fractional part is equal to The sum is This computation uses the method of exhaustion, an early version of integration. Using calculus, the same area could be found by a definite integral. In the study of fractals, geometric series often arise as the perimeter, area, or volume of a self-similar figure. For example, the area inside the Koch snowflake can be described as the union"
"vertex and having consecutive sides of length 1, 2, ..., n units. The polygon may be self-crossing. Golygons are a special case of serial-sided isogons. The three-dimensional generalization of a golygon is called a golyhedron–a closed simply-connected solid figure confined to the faces of a cubical lattice and having face areas in the sequence 1, 2, ..., n, for some integer n. Golyhedrons have been found with values of n equal to 32, 15, 12, and 11 (the minimum possible). Golygon A golygon is any polygon with all right angles (a rectilinear polygon) whose sides are consecutive integer lengths. Golygons"
"Triptych A triptych ( ; from the Greek adjective ""τρίπτυχον"" """"triptukhon"""" (""three-fold""), from ""tri"", i.e., ""three"" and ""ptysso"", i.e., ""to fold"" or ""ptyx"", i.e., ""fold"") is a work of art (usually a panel painting) that is divided into three sections, or three carved panels that are hinged together and can be folded shut or displayed open. It is therefore a type of polyptych, the term for all multi-panel works. The middle panel is typically the largest and it is flanked by two smaller related works, although there are triptychs of equal-sized panels. The form can also be used for pendant"
"Trirectangular tetrahedron In geometry, a trirectangular tetrahedron is a tetrahedron where all three face angles at one vertex are right angles. That vertex is called the ""right angle"" of the trirectangular tetrahedron and the face opposite it is called the ""base"". The three edges that meet at the right angle are called the ""legs"" and the perpendicular from the right angle to the base is called the ""altitude"" of the tetrahedron. Only the bifurcating graph of the formula_1 Affine Coxeter group has a Trirectangular tetrahedron fundamental domain. If the legs have lengths ""a, b, c"", then the trirectangular tetrahedron has"
"Polytope In elementary geometry, a polytope is a geometric object with ""flat"" sides. It is a generalization in any number of dimensions of the three-dimensional polyhedron. Polytopes may exist in any general number of dimensions ""n"" as an ""n""-dimensional polytope or ""n""-polytope. Flat sides mean that the sides of a (""k""+1)-polytope consist of ""k""-polytopes that may have (""k""-1)-polytopes in common. For example, a two-dimensional polygon is a 2-polytope and a three-dimensional polyhedron is a 3-polytope. Some theories further generalize the idea to include such objects as unbounded apeirotopes and tessellations, decompositions or tilings of curved manifolds including spherical polyhedra, and"
"Sierpinski triangle The Sierpinski triangle (also with the original orthography ""Sierpiński""), also called the Sierpinski gasket or the Sierpinski sieve, is a fractal and attractive fixed set with the overall shape of an equilateral triangle, subdivided recursively into smaller equilateral triangles. Originally constructed as a curve, this is one of the basic examples of self-similar sets, i.e., it is a mathematically generated pattern that is reproducible at any magnification or reduction. It is named after the Polish mathematician Wacław Sierpiński, but appeared as a decorative pattern many centuries before the work of Sierpiński. There are many different ways of constructing"
"A triangle (""X"", ""Y"", ""Z"", ""u"", ""v"", ""w"") consists of 3 objects ""X"", ""Y"", and ""Z"", together with morphisms ""u"" : ""X"" → ""Y"", ""v"" : ""Y"" → ""Z"" and ""w"" : ""Z"" → ""X""[1]. Triangles are generally written in the unravelled form: Triangulated category In mathematics, a triangulated category is a category together with the additional structure of a ""translation functor"" and a class of ""distinguished triangles"". Prominent examples are the derived category of an abelian category and the stable homotopy category of spectra (more generally, the homotopy category of a stable ∞-category), both of which carry the structure"
"Trigonal trapezohedron In geometry, a trigonal trapezohedron or trigonal deltohedron is a three-dimensional figure formed by six congruent rhombi. Six identical rhombic faces can construct two configurations of trigonal trapezohedra. The ""acute"" or ""prolate"" form has three acute angles corners of the rhombic faces meeting at two polar axis vertices. The ""obtuse"" or ""oblate"" or ""flat"" form has three obtuse angle corners of the rhombic faces meeting at the two polar axis vertices. A trigonal trapezohedron is a special case of a rhombohedron. A general rhombohedron allows up to three types of rhombic faces. A trigonal trapezohedron is a special"
"or formula_16 with coprime formula_11. Trirectangular tetrahedron In geometry, a trirectangular tetrahedron is a tetrahedron where all three face angles at one vertex are right angles. That vertex is called the ""right angle"" of the trirectangular tetrahedron and the face opposite it is called the ""base"". The three edges that meet at the right angle are called the ""legs"" and the perpendicular from the right angle to the base is called the ""altitude"" of the tetrahedron. Only the bifurcating graph of the formula_1 Affine Coxeter group has a Trirectangular tetrahedron fundamental domain. If the legs have lengths ""a, b, c"","
"vertices and lines connecting the points in a closed chain, as well as the resulting interior points. Such shapes are called polygons and include triangles, squares, and pentagons. Other shapes may be bounded by curves such as the circle or the ellipse. Many three-dimensional geometric shapes can be defined by a set of vertices, lines connecting the vertices, and two-dimensional faces enclosed by those lines, as well as the resulting interior points. Such shapes are called polyhedrons and include cubes as well as pyramids such as tetrahedrons. Other three-dimensional shapes may be bounded by curved surfaces, such as the ellipsoid"
"Irregular polyhedra appear in nature as crystals. Notes Sources Polyhedron In geometry, a polyhedron (plural polyhedra or polyhedrons) is a solid in three dimensions with flat polygonal faces, straight edges and sharp corners or vertices. The word polyhedron comes from the Classical Greek πολύεδρον, as ""poly-"" (stem of πολύς, ""many"") + ""-hedron"" (form of ἕδρα, ""base"" or ""seat""). A convex polyhedron is the convex hull of finitely many points, not all on the same plane. Cubes and pyramids are examples of convex polyhedra. A polyhedron is a 3-dimensional example of the more general polytope in any number of dimensions. Convex"
"found is shown in the figure as a dark blue bar. Point P is called the pole of the neusis, line l the directrix, or guiding line, and line m the catch line. Length ""a"" is called the ""diastema"" (διάστημα; Greek for ""distance""). ""Neuseis"" have been important because they sometimes provide a means to solve geometric problems that are not solvable by means of compass and straightedge alone. Examples are the trisection of any angle in three equal parts, the doubling of the cube, and the construction of a regular heptagon, nonagon, or tridecagon (polygons with 7, 9, or 13"
"(*""n""32) reflectional symmetry. The triakis octahedron is also a part of a sequence of polyhedra and tilings, extending into the hyperbolic plane. These face-transitive figures have (*""n""42) reflectional symmetry. Triakis octahedron In geometry, a triakis octahedron (or trigonal trisoctahedron or kisoctahedron) 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"
"Face (geometry) In solid geometry, a face is a flat (planar) surface that forms part of the boundary of a solid object; a three-dimensional solid bounded exclusively by flat faces is a polyhedron. In more technical treatments of the geometry of polyhedra and higher-dimensional polytopes, the term is also used to mean an element of any dimension of a more general polytope (in any number of dimensions). In elementary geometry, a face is a polygon on the boundary of a polyhedron. Other names for a polygonal face include side of a polyhedron, and tile of a Euclidean plane tessellation. For"
"of the 2-circle figure is called a Vesica piscis, from Euclid. Two circles are also called Villarceau circles as a plane intersection of a torus. The areas inside one circle and outside the other circle is called a lune. The 3-circle figure resembles a depiction of borromean rings and is used in 3-set theory Venn diagrams. Its interior makes a unicursal path called a triquetra. The center of the 3-circle figure is called a reuleaux triangle. Some spherical polyhedra with edges along great circles can be stereographically projected onto the plane as overlapping circles. The 7-circle pattern has also been"
"a rectangle: The sides of a rhombus centered at the origin, with diagonals each falling on an axis, consist of all points (""x, y"") satisfying The vertices are at formula_11 and formula_12 This is a special case of the superellipse, with exponent 1. 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"
"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"
"Triakis octahedron In geometry, a triakis octahedron (or trigonal trisoctahedron or kisoctahedron) 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"
"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"
"Gosset–Elte figures In geometry, the Gosset–Elte figures, named by Coxeter after Thorold Gosset and E. L. Elte, are a group of uniform polytopes which are not regular, generated by a Wythoff construction with mirrors all related by order-2 and order-3 dihedral angles. They can be seen as ""one-end-ringed"" Coxeter–Dynkin diagrams. The Coxeter symbol for these figures has the form ""k"", where each letter represents a length of order-3 branches on a Coxeter–Dynkin diagram with a single ring on the end node of a ""k"" length sequence of branches. The vertex figure of ""k"" is (""k"" − 1), and each of"
"Dihedral symmetry in three dimensions In geometry, dihedral symmetry in three dimensions is one of three infinite sequences of point groups in three dimensions which have a symmetry group that as abstract group is a dihedral group Dih ( ""n"" ≥ 2 ). There are 3 types of dihedral symmetry in three dimensions, each shown below in 3 notation: Schönflies notation, Coxeter notation, and orbifold notation. For a given ""n"", all three have ""n""-fold rotational symmetry about one axis (rotation by an angle of 360°/""n"" does not change the object), and 2-fold about a perpendicular axis, hence about ""n"" of"
"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"
"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"
"3} has regular apeirogon faces. Hypercyclic apeirogons can also be isogonal or quasiregular, with truncated apeirogon faces, t{∞}, like the tiling tr{∞,3}, with two types of edges, alternately connecting to triangles or other apeirogons. Norman Johnson calls the general apeirogon (divergent mirror form) a pseudogon, circumscribed by a hypercycle, with and regular pseudogons as {iπ/λ}, where λ is the periodic distance between the divergent perpendicular mirrors. Apeirogon In geometry, an apeirogon (from the Greek word ἄπειρος ""apeiros"", ""infinite, boundless"" and γωνία ""gonia"", ""angle"") is a generalized polygon with a countably infinite number of sides. It can be considered as 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"
"Three-torus The three-dimensional torus, or triple torus, is defined as the Cartesian product of three circles, In contrast, the usual torus is the Cartesian product of two circles only. The triple torus is a three-dimensional compact manifold with no boundary. It can be obtained by ""gluing"" the three pairs of opposite faces of a cube, where being ""glued"" can be intuitively understood to mean that when a particle moving by inertia in the interior of the cube reaches a point on a face, it goes through it and appears to come forth from the corresponding point on the opposite face,"
"Polygon In elementary geometry, a polygon () is a plane figure that is bounded by a finite chain of straight line segments closing in a loop to form a closed polygonal chain or ""circuit"". These segments are called its ""edges"" or ""sides"", and the points where two edges meet are the polygon's ""vertices"" (singular: vertex) or ""corners"". The interior of the polygon is sometimes called its ""body"". An ""n""-gon is a polygon with ""n"" sides; for example, a triangle is a 3-gon. A polygon is a 2-dimensional example of the more general polytope in any number of dimensions. The basic"
"is a deltoid (broadly defined as a figure with three vertices connected by curves that are concave to the exterior of the deltoid, making the interior points a non-convex set). The vertices of the deltoid are at the midpoints of the medians; all points inside the deltoid are on three different area bisectors, while all points outside it are on just one. The sides of the deltoid are arcs of hyperbolas that are asymptotic to the extended sides of the triangle. The ratio of the area of the envelope of area bisectors to the area of the triangle is invariant"
"two equal sides, and for isosceles sets, sets of points every three of which form an isosceles triangle. In an isosceles triangle that has exactly two equal sides, the equal sides are called legs and the third side is called the base. The angle included by the legs is called the ""vertex angle"" and the angles that have the base as one of their sides are called the ""base angles"". The vertex opposite the base is called the apex. In the equilateral triangle case, since all sides are equal, any side can be called the base. Whether an isosceles triangle"
"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"
"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"
"Compound of three octahedra In mathematics, the compound of three octahedra or octahedron 3-compound is a polyhedral compound formed from three regular octahedra, all sharing a common center but rotated with respect to each other. Although appearing earlier in the mathematical literature, it was rediscovered and popularized by M. C. Escher, who used it in the central image of his 1948 woodcut ""Stars"". A regular octahedron can be circumscribed around a cube in such a way that the eight edges of two opposite squares of the cube lie on the eight faces of the octahedron. The three octahedra formed in"
"Charts on SO(3) In mathematics, the special orthogonal group in three dimensions, otherwise known as the rotation group SO(3), is a naturally occurring example of a manifold. The various charts on SO(3) set up rival coordinate systems: in this case there cannot be said to be a preferred set of parameters describing a rotation. There are three degrees of freedom, so that the dimension of SO(3) is three. In numerous applications one or other coordinate system is used, and the question arises how to convert from a given system to another. In geometry the rotation group is the group of"