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In geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles …

Crossed-square

3-simplex (3D)

Image: Tetragonal dihedron

Image: Square on hyperbolic plane

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1. Square – In geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal 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 ◻ ABCD. e. A rhombus with equal diagonals a convex quadrilateral with sides a, b, c, d whose area is A =12 =12. Opposite sides of a square are both parallel and equal in length, all four angles of a square are equal. All four sides of a square are equal, the diagonals of a square are equal. The square is the n=2 case of the families of n-hypercubes and n-orthoplexes, a truncated square, t, is an octagon. An alternated square, h, is a digon, the perimeter of a square whose four sides have length ℓ is P =4 ℓ and the area A is A = ℓ2. In classical times, the power was described in terms of the area of a square. This led to the use of the square to mean raising to the second power. The area can also be calculated using the diagonal d according to A = d 22. In terms of the circumradius R, the area of a square is A =2 R2, since the area of the circle is π R2, in terms of the inradius r, the area of the square is A =4 r 2. Because it is a polygon, a square is the quadrilateral of least perimeter enclosing a given area. Dually, a square is the quadrilateral containing the largest area within a given perimeter. Indeed, if A and P are the area and perimeter enclosed by a quadrilateral, then the isoperimetric inequality holds,16 A ≤ P2 with equality if. The diagonals of a square are 2 times the length of a side of the square and this value, known as the square root of 2 or Pythagoras constant, was the first number proven to be irrational. A square can also be defined as a parallelogram with equal diagonals that bisect the angles, if a figure is both a rectangle and a rhombus, then it is a square. If a circle is circumscribed around a square, the area of the circle is π /2 times the area of the square, if a circle is inscribed in the square, the area of the circle is π /4 times the area of the square. A square has an area than any other quadrilateral with the same perimeter

2. Geometry – Geometry is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. A mathematician who works in the field of geometry is called a geometer, Geometry arose independently in a number of early cultures as a practical way for dealing with lengths, areas, and volumes. Geometry began to see elements of mathematical science emerging in the West as early as the 6th century BC. By the 3rd century BC, geometry was put into a form by Euclid, whose treatment, Euclids Elements. Geometry arose independently in India, with texts providing rules for geometric constructions appearing as early as the 3rd century BC, islamic scientists preserved Greek ideas and expanded on them during the Middle Ages. By the early 17th century, geometry had been put on a solid footing by mathematicians such as René Descartes. Since then, and into modern times, geometry has expanded into non-Euclidean geometry and manifolds, while geometry has evolved significantly throughout the years, there are some general concepts that are more or less fundamental to geometry. These include the concepts of points, lines, planes, surfaces, angles, contemporary geometry has many subfields, Euclidean geometry is geometry in its classical sense. The mandatory educational curriculum of the majority of nations includes the study of points, lines, planes, angles, triangles, congruence, similarity, solid figures, circles, Euclidean geometry also has applications in computer science, crystallography, and various branches of modern mathematics. Differential geometry uses techniques of calculus and linear algebra to problems in geometry. It has applications in physics, including in general relativity, topology is the field concerned with the properties of geometric objects that are unchanged by continuous mappings. In practice, this often means dealing with large-scale properties of spaces, convex geometry investigates convex shapes in the Euclidean space and its more abstract analogues, often using techniques of real analysis. It has close connections to convex analysis, optimization and functional analysis, algebraic geometry studies geometry through the use of multivariate polynomials and other algebraic techniques. It has applications in areas, including cryptography and string theory. Discrete geometry is concerned mainly with questions of relative position of simple objects, such as points. It shares many methods and principles with combinatorics, Geometry has applications to many fields, including art, architecture, physics, as well as to other branches of mathematics. The earliest recorded beginnings of geometry can be traced to ancient Mesopotamia, the earliest known texts on geometry are the Egyptian Rhind Papyrus and Moscow Papyrus, the Babylonian clay tablets such as Plimpton 322. For example, the Moscow Papyrus gives a formula for calculating the volume of a truncated pyramid, later clay tablets demonstrate that Babylonian astronomers implemented trapezoid procedures for computing Jupiters position and motion within time-velocity space

3. Octagon – In geometry, an octagon is an eight-sided polygon or 8-gon. A regular octagon has Schläfli symbol and can also be constructed as a truncated square, t. A truncated octagon, t is a hexadecagon, t, the sum of all the internal angles of any octagon is 1080°. As with all polygons, the external angles total 360°, the midpoint octagon of a reference octagon has its eight vertices at the midpoints of the sides of the reference octagon. A regular octagon is a figure with sides of the same length. It has eight lines of symmetry and rotational symmetry of order 8. A regular octagon is represented by the Schläfli symbol, the internal angle at each vertex of a regular octagon is 135°. The area of an octagon of side length a is given by A =2 cot π8 a 2 =2 a 2 ≃4.828 a 2. In terms of the circumradius R, the area is A =4 sin π4 R2 =22 R2 ≃2.828 R2. In terms of the r, the area is A =8 tan π8 r 2 =8 r 2 ≃3.314 r 2. These last two coefficients bracket the value of pi, the area of the unit circle. The area can also be expressed as A = S2 − a 2, where S is the span of the octagon, or the second-shortest diagonal, and a is the length of one of the sides, or bases. This is easily proven if one takes an octagon, draws a square around the outside and then takes the corner triangles and places them with right angles pointed inward, the edges of this square are each the length of the base. Given the length of a side a, the span S is S = a 2 + a + a 2 = a ≈2.414 a. The area is then as above, A =2 − a 2 =2 a 2 ≈4.828 a 2, expressed in terms of the span, the area is A =2 S2 ≈0.828 S2. Another simple formula for the area is A =2 a S, more often the span S is known, and the length of the sides, a, is to be determined, as when cutting a square piece of material into a regular octagon. From the above, a ≈ S /2.414, the two end lengths e on each side, as well as being e = a /2, may be calculated as e = /2. The circumradius of the octagon in terms of the side length a is R = a

4. Alternation (geometry) – In geometry, an alternation or partial truncation, is an operation on a polygon, polyhedron, tiling, or higher dimensional polytope that removes alternate vertices. Coxeter labels an alternation by a prefixed by an h, standing for hemi or half, because alternation reduce all polygon faces to half as many sides, it can only be applied for polytopes with all even-sided faces. An alternated square face becomes a digon, and being degenerate, is reduced to a single edge. More generally any vertex-uniform polyhedron or tiling with a configuration consisting of all even-numbered elements can be alternated. For example, the alternation a vertex figure with 2a. 2b. 2c is a.3. b.3. c.3 where the three is the number of elements in this vertex figure. A special case is square faces whose order divide in half into degenerate digons, a snub can be seen as an alternation of a truncated regular or truncated quasiregular polyhedron. In general a polyhedron can be snubbed if its truncation has only even-sided faces, all truncated rectified polyhedra can be snubbed, not just from regular polyhedra. The snub square antiprism is an example of a general snub and this alternation operation applies to higher-dimensional polytopes and honeycombs as well, but in general most of the results of this operation will not be uniform. The voids created by the vertices will not in general create uniform facets. Examples, Honeycombs An alternated cubic honeycomb is the tetrahedral-octahedral honeycomb, an alternated hexagonal prismatic honeycomb is the gyrated alternated cubic honeycomb. 4-polytope An alternated truncated 24-cell is the snub 24-cell, 4-honeycombs, An alternated truncated 24-cell honeycomb is the snub 24-cell honeycomb. A hypercube can always be alternated into a uniform demihypercube, cube → Tetrahedron → Tesseract → 16-cell → Penteract → demipenteract Hexeract → demihexeract. Coxeter also used the operator a, which contains both halves, so retains the original symmetry, for even-sided regular polyhedra, a represents a compound polyhedron with two opposite copies of h. For odd-sided, greater than 3, regular polyhedra a, becomes a star polyhedron, Norman Johnson extended the use of the altered operator a, b for blended, and c for converted, as, and respectively. The compound polyhedron, stellated octahedron can be represented by a, the star-polyhedron, small ditrigonal icosidodecahedron, can be represented by a, and. Here all the pentagons have been alternated into pentagrams, and triangles have been inserted to take up the free edges. A similar operation can truncate alternate vertices, rather than just removing them, below is a set of polyhedra that can be generated from the Catalan solids. These have two types of vertices which can be alternately truncated, truncating the higher order vertices and both vertex types produce these forms, Conway polyhedral notation Wythoff construction Coxeter, H. S. M