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Triangle
A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices A, B, and C is denoted △ A B C displaystyle triangle ABC . In Euclidean geometry
Euclidean geometry
any three points, when non-collinear, determine a unique triangle and simultaneously, a unique plane (i.e. a two-dimensional Euclidean space). In other words, there is only one plane that contains that triangle, and every triangle is contained in some plane. If the entire geometry is only the Euclidean plane, there is only one plane and all triangles are contained in it; however, in higher dimensional Euclidean spaces this is no longer true
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Tally Marks
Tally marks, also called hash marks, are a unary numeral system. They are a form of numeral used for counting. They are most useful in counting or tallying ongoing results, such as the score in a game or sport, as no intermediate results need to be erased or discarded. However, because of the length of large numbers, tallies are not commonly used for static text
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Degree (angle)
A degree (in full, a degree of arc, arc degree, or arcdegree), usually denoted by ° (the degree symbol), is a measurement of a plane angle, defined so that a full rotation is 360 degrees. It is not an SI unit, as the SI unit
SI unit
of angular measure is the radian, but it is mentioned in the SI brochure as an accepted unit.[4] Because a full rotation equals 2π radians, one degree is equivalent to π/180 radians.Contents1 History 2 Subdivisions 3 Alternative units 4 See also 5 Notes 6 References 7 External linksHistory[edit] See also: DecansA circle with an equilateral chord (red). One sixtieth of this arc is a degree. Six such chords complete the circle.The original motivation for choosing the degree as a unit of rotations and angles is unknown
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Euclid's Elements
The Elements (Ancient Greek: Στοιχεῖα Stoicheia) is a mathematical treatise consisting of 13 books attributed to the ancient Greek mathematician Euclid
Euclid
in Alexandria, Ptolemaic Egypt
Ptolemaic Egypt
c. 300 BC. It is a collection of definitions, postulates, propositions (theorems and constructions), and mathematical proofs of the propositions. The books cover plane and solid Euclidean geometry, elementary number theory, and incommensurable lines. Elements is the oldest extant large-scale deductive treatment of mathematics. It has proven instrumental in the development of logic and modern science, and its logical rigor was not surpassed until the 19th century. Euclid's Elements
Euclid's Elements
has been referred to as the most successful[1][2] and influential[3] textbook ever written
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Isotoxal Figure
In geometry, a polytope (for example, a polygon or a polyhedron), or a tiling, is isotoxal or edge-transitive if its symmetries act transitively on its edges. Informally, this means that there is only one type of edge to the object: given two edges, there is a translation, rotation and/or reflection that will move one edge to the other, while leaving the region occupied by the object unchanged. The term isotoxal is derived from the Greek τοξον meaning arc.Contents1 Isotoxal polygons 2 Isotoxal polyhedra and tilings 3 See also 4 ReferencesIsotoxal polygons[edit] An isotoxal polygon is an equilateral polygon, but not all equilateral polygons are isotoxal. The duals of isotoxal polygons are isogonal polygons. In general, an isotoxal 2n-gon will have Dn (*nn) dihedral symmetry
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Isogonal Figure
In geometry, a polytope (a polygon, polyhedron or tiling, for example) is isogonal or vertex-transitive if all its vertices are equivalent under the symmetries of the figure. This implies that each vertex is surrounded by the same kinds of face in the same or reverse order, and with the same angles between corresponding faces. Technically, we say that for any two vertices there exists a symmetry of the polytope mapping the first isometrically onto the second. Other ways of saying this are that the group of automorphisms of the polytope is transitive on its vertices, or that the vertices lie within a single symmetry orbit. All vertices of a finite n-dimensional isogonal figure exist on an (n-1)-sphere.[citation needed] The term isogonal has long been used for polyhedra
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Plane (mathematics)
In mathematics, a plane is a flat, two-dimensional surface that extends infinitely far. A plane is the two-dimensional analogue of a point (zero dimensions), a line (one dimension) and three-dimensional space
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Vertex (geometry)
In geometry, a vertex (plural: vertices or vertexes) is a point where two or more curves, lines, or edges meet
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Euclid
Euclid
Euclid
(/ˈjuːklɪd/; Greek: Εὐκλείδης Eukleidēs [eu̯.klěː.dɛːs]; fl. 300 BC), sometimes given the name Euclid
Euclid
of Alexandria[1] to distinguish him from Euclides of Megara, was a Greek mathematician, often referred to as the "founder of geometry"[1] or the "father of geometry". He was active in Alexandria
Alexandria
during the reign of Ptolemy I (323–283 BC). His Elements is one of the most influential works in the history of mathematics, serving as the main textbook for teaching mathematics (especially geometry) from the time of its publication until the late 19th or early 20th century.[2][3][4] In the Elements, Euclid
Euclid
deduced the theorems of what is now called Euclidean geometry
Euclidean geometry
from a small set of axioms
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Internal Angle
2D anglesRight Interior Exterior2D angle pairsAdjacent Vertical Complementary Supplementary Transversal3D anglesDihedralInternal and External anglesIn geometry, an angle of a polygon is formed by two sides of the polygon that share an endpoint. For a simple (non-self-intersecting) polygon, regardless of whether it is convex or non-convex, this angle is called an interior angle (or internal angle) if a point within the angle is in the interior of the polygon. A polygon has exactly one internal angle per vertex. If every internal angle of a simple polygon is less than 180°, the polygon is called convex. In contrast, an exterior angle (or external angle) is an angle formed by one side of a simple polygon and a line extended from an adjacent side.[1][2]:pp. 261-264The sum of the internal angle and the external angle on the same vertex is 180°. The sum of all the internal angles of a simple polygon is 180(n–2)° where n is the number of sides
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Exterior Angle
2D anglesRight Interior Exterior2D angle pairsAdjacent Vertical Complementary Supplementary Transversal3D anglesDihedralInternal and External anglesIn geometry, an angle of a polygon is formed by two sides of the polygon that share an endpoint. For a simple (non-self-intersecting) polygon, regardless of whether it is convex or non-convex, this angle is called an interior angle (or internal angle) if a point within the angle is in the interior of the polygon. A polygon has exactly one internal angle per vertex. If every internal angle of a simple polygon is less than 180°, the polygon is called convex. In contrast, an exterior angle (or external angle) is an angle formed by one side of a simple polygon and a line extended from an adjacent side.[1][2]:pp. 261-264The sum of the internal angle and the external angle on the same vertex is 180°. The sum of all the internal angles of a simple polygon is 180(n–2)° where n is the number of sides
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Euclidean Plane
Two-dimensional space
Two-dimensional space
or bi-dimensional space is a geometric setting in which two values (called parameters) are required to determine the position of an element (i.e., point). In Mathematics, it is commonly represented by the symbol ℝ2. For a generalization of the concept, see dimension. Two-dimensional space
Two-dimensional space
can be seen as a projection of the physical universe onto a plane
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Polytope
In elementary geometry, a polytope is a geometric object with "flat" sides. It is a generalisation 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 set-theoretic abstract polytopes. Polytopes in more than three dimensions were first discovered by Ludwig Schläfli
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Plane Figure
A geometric shape is the geometric information which remains when location, scale, orientation and reflection are removed from the description of a geometric object.[1] That is, the result of moving a shape around, enlarging it, rotating it, or reflecting it in a mirror is the same shape as the original, and not a distinct shape. Objects that have the same shape as each other are said to be similar. If they also have the same scale as each other, they are said to be congruent. Many two-dimensional geometric shapes can be defined by a set of points or 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
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Euler Diagram
An Euler diagram
Euler diagram
(/ˈɔɪlər/, OY-lər) is a diagrammatic means of representing sets and their relationships. Typically they involve overlapping shapes, and may be scaled, such that the area of the shape is proportional to the number of elements it contains. They are particularly useful for explaining complex hierarchies and overlapping definitions. They are often confused with Venn diagrams. Unlike Venn diagrams, which show all possible relations between different sets, the Euler diagram
Euler diagram
shows only relevant relationships. The first use of "Eulerian circles" is commonly attributed to Swiss mathematician Leonhard Euler
Leonhard Euler
(1707–1783). In the United States, both Venn and Euler diagrams were incorporated as part of instruction in set theory as part of the new math movement of the 1960s
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Tetrakis Square Tiling
In geometry, the tetrakis square tiling is a tiling of the Euclidean plane. It is a square tiling with each square divided into four isosceles right triangles from the center point, forming an infinite arrangement of lines. It can also be formed by subdividing each square of a grid into two triangles by a diagonal, with the diagonals alternating in direction, or by overlaying two square grids, one rotated by 45 degrees from the other and scaled by a factor of √2. Conway calls it a kisquadrille,[1] represented by a kis operation that adds a center point and triangles to replace the faces of a square tiling (quadrille)
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