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List Of Polygons, Polyhedra And Polytopes
A polytope is a geometric object with flat sides, which exists in any general number of dimensions. The following list of polygons, polyhedra and polytopes gives the names of various classes of polytopes and lists some specific examples. Polytope elements Polygon (2-polytope) * Vertex the ''ridge'' or ''(n−2)-face'' of the polygon * Edge the ''facet'' or ''(n−1)-face'' of the polygon Polyhedron (3-polytope) * Vertex the ''peak'' or ''(n−3)-face'' of the polyhedron * Edge the ''ridge'' or ''(n−2)-face'' of the polyhedron *Face the ''facet'' or ''(n−1)-face'' of the polyhedron Polychoron (4-polytope) * Vertex the ''(n−4)-face'' of the polychoron * Edge the ''peak'' or ''(n−3)-face'' of the polychoron *Face the ''ridge'' or ''(n−2)-face'' of the polychoron * Cell the ''facet'' or ''(n−1)-face'' of the polychoron 5-polytope * Vertex the ''(n−5)-face'' of the 5-polytope * Edge the ''(n−4)-face'' of the 5-polytope *Face the ''peak'' or ''(n−3)-face'' of the 5-po ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vertex (geometry)
In geometry, a vertex (: vertices or vertexes), also called a corner, is a point (geometry), point where two or more curves, line (geometry), lines, or line segments Tangency, meet or Intersection (geometry), intersect. For example, the point where two lines meet to form an angle and the point where edge (geometry), edges of polygons and polyhedron, polyhedra meet are vertices. Definition Of an angle The ''vertex'' of an angle is the point where two Line (mathematics)#Ray, rays begin or meet, where two line segments join or meet, where two lines intersect (cross), or any appropriate combination of rays, segments, and lines that result in two straight "sides" meeting at one place. :(3 vols.): (vol. 1), (vol. 2), (vol. 3). Of a polytope A vertex is a corner point of a polygon, polyhedron, or other higher-dimensional polytope, formed by the intersection (Euclidean geometry), intersection of Edge (geometry), edges, face (geometry), faces or facets of the object. In a polygon, a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Right Triangle
A right triangle or right-angled triangle, sometimes called an orthogonal triangle or rectangular triangle, is a triangle in which two sides are perpendicular, forming a right angle ( turn or 90 degrees). The side opposite to the right angle is called the '' hypotenuse'' (side c in the figure). The sides adjacent to the right angle are called ''legs'' (or ''catheti'', singular: '' cathetus''). Side a may be identified as the side ''adjacent'' to angle B and ''opposite'' (or ''opposed to'') angle A, while side b is the side adjacent to angle A and opposite angle B. Every right triangle is half of a rectangle which has been divided along its diagonal. When the rectangle is a square, its right-triangular half is isosceles, with two congruent sides and two congruent angles. When the rectangle is not a square, its right-triangular half is scalene. Every triangle whose base is the diameter of a circle and whose apex lies on the circle is a right triangle, with the right angle at ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Parallelogram
In Euclidean geometry, a parallelogram is a simple polygon, simple (non-list of self-intersecting polygons, self-intersecting) quadrilateral with two pairs of Parallel (geometry), parallel sides. The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of equal measure. The congruence (geometry), congruence of opposite sides and opposite angles is a direct consequence of the Euclidean parallel postulate and neither condition can be proven without appealing to the Euclidean parallel postulate or one of its equivalent formulations. By comparison, a quadrilateral with at least one pair of parallel sides is a trapezoid in American English or a trapezium in British English. The three-dimensional counterpart of a parallelogram is a parallelepiped. The word "parallelogram" comes from the Greek παραλληλό-γραμμον, ''parallēló-grammon'', which means "a shape of parallel lines". Special cases *Rectangle – A par ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Golden Rhombus
In geometry, a golden rhombus is a rhombus whose diagonals are in the golden ratio: : = \varphi = \approx 1.618~034 Equivalently, it is the Varignon parallelogram formed from the edge midpoints of a golden rectangle. Rhombi with this shape form the faces of several notable polyhedra. The golden rhombus should be distinguished from the two rhombi of the Penrose tiling, which are both related in other ways to the golden ratio but have different shapes than the golden rhombus. Angles (See the characterizations and the basic properties of the general rhombus for angle properties.) The internal supplementary angles of the golden rhombus are:. See in particular table 1, p. 188. *Acute angle: \alpha=2\arctan ; :by using the arctangent addition formula (see inverse trigonometric functions): :\alpha=\arctan=\arctan=\arctan2\approx63.43495^\circ. : *Obtuse angle: \beta=2\arctan\varphi=\pi-\arctan2\approx116.56505^\circ, :which is also the dihedral angle of the dodecahedron. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rhombus
In plane Euclidean geometry, a rhombus (: rhombi or rhombuses) is a 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), diamonds suit in playing cards which resembles the projection of an Octahedron#Orthogonal projections, octahedral diamond, or a lozenge (shape), lozenge, though the former sometimes refers specifically to a rhombus with a 60° angle (which some authors call a calisson after calisson, the French sweet—also see Polyiamond), and the latter sometimes refers specifically to a rhombus with a 45° angle. Every rhombus is simple polygon, simple (non-self-intersecting), and is a special case of a parallelogram and a Kite (geometry), kite. A rhombus with right angles is a square. Etymology The word "rhombus" comes from , meaning something that spins, which derives from the verb , roman ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ailles Rectangle
The Ailles rectangle is a rectangle constructed from four right-angled triangles which is commonly used in geometry classes to find the values of trigonometric functions of 15° and 75°. It is named after Douglas S. Ailles who was a high school teacher at Kipling Collegiate Institute in Toronto. Construction A Special right triangle#30°–60°–90° triangle, 30°–60°–90° triangle has sides of length 1, 2, and \sqrt. When two such triangles are placed in the positions shown in the illustration, the smallest rectangle that can enclose them has width 1+\sqrt and height \sqrt. Drawing a line connecting the original triangles' top corners creates a Special right triangle#45°–45°–90° triangle, 45°–45°–90° triangle between the two, with sides of lengths 2, 2, and (by the Pythagorean theorem) 2\sqrt. The remaining space at the top of the rectangle is a right triangle with acute angles of 15° and 75° and sides of \sqrt-1, \sqrt+1, and 2\sqrt. Derived trig ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Silver Rectangle
In mathematics, the silver ratio is a geometrical proportion with exact value the positive solution of the equation The name ''silver ratio'' results from analogy with the golden ratio, the positive solution of the equation Although its name is recent, the silver ratio (or silver mean) has been studied since ancient times because of its connections to the square root of 2, almost-isosceles Pythagorean triples, square triangular numbers, Pell numbers, the octagon, and six polyhedra with octahedral symmetry. Definition If the ratio of two quantities is proportionate to the sum of two and their reciprocal ratio, they are in the silver ratio: \frac =\frac The ratio \frac is here denoted Substituting a=\sigma b \, in the second fraction, \sigma =\frac. It follows that the silver ratio is the positive solution of quadratic equation \sigma^2 -2\sigma -1 =0. The quadratic formula gives the two solutions 1 \pm \sqrt, the decimal expansion of the positive root begins as . ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Golden Rectangle
In geometry, a golden rectangle is a rectangle with side lengths in golden ratio \tfrac :1, or with approximately equal to or Golden rectangles exhibit a special form of self-similarity: if a square is added to the long side, or removed from the short side, the result is a golden rectangle as well. Construction Owing to the Pythagorean theorem, the diagonal dividing one half of a square equals the radius of a circle whose outermost point is the corner of a golden rectangle added to the square. Thus, a golden rectangle can be Straightedge and compass construction, constructed with only a straightedge and compass in four steps: # Draw a square # Draw a line from the midpoint of one side of the square to an opposite corner # Use that line as the radius to draw an arc that defines the height of the rectangle # Complete the golden rectangle A distinctive feature of this shape is that when a square (geometry), square section is added—or removed—the product is another golden re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Oblong (description)
In Euclidean plane geometry, a rectangle is a rectilinear convex polygon or 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°); or a parallelogram containing a right angle. A rectangle with four sides of equal length is a ''square''. The term "oblong" is 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) and ''angulus'' (angle). A crossed rectangle is a crossed (self-intersecting) quadrilateral which consists of two opposite sides of a rectangle along with the two diagonals (therefore only two sides are parallel). It is a special case of an antiparallelogram, and its angles are not right angles and not all equal, though opposite angles are equal. Other geometries, such as spherical, ell ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unit Square
In mathematics, a unit square is a square whose sides have length . Often, ''the'' unit square refers specifically to the square in the Cartesian plane with corners at the four points ), , , and . Cartesian coordinates In a Cartesian coordinate system with coordinates , a unit square is defined as a square consisting of the points where both and lie in a closed unit interval from to . That is, a unit square is the Cartesian product , where denotes the closed unit interval. Complex coordinates The unit square can also be thought of as a subset of the complex plane, the topological space formed by the complex numbers. In this view, the four corners of the unit square are at the four complex numbers , , , and . Rational distance problem It is not known whether any point in the plane is a rational distance from all four vertices of the unit square. See also * Unit circle In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequentl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Square (geometry)
In geometry, a square is a regular quadrilateral. It has four straight sides of equal length and four equal angles. Squares are special cases of rectangles, which have four equal angles, and of rhombuses, which have four equal sides. As with all rectangles, a square's angles are right angles (90 degrees, or /2 radians), making adjacent sides perpendicular. The area of a square is the side length multiplied by itself, and so in algebra, multiplying a number by itself is called squaring. Equal squares can tile the plane edge-to-edge in the square tiling. Square tilings are ubiquitous in tiled floors and walls, graph paper, image pixels, and game boards. Square shapes are also often seen in building floor plans, origami paper, food servings, in graphic design and heraldry, and in instant photos and fine art. The formula for the area of a square forms the basis of the calculation of area and motivates the search for methods for squaring the circle by compass and straightedge ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rectangle
In Euclidean geometry, Euclidean plane geometry, a rectangle is a Rectilinear polygon, rectilinear convex polygon or 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°); or a parallelogram containing a right angle. A rectangle with four sides of equal length is a ''square''. The term "wikt:oblong, oblong" is used to refer to a non-square rectangle. A rectangle with Vertex (geometry), 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) and ''angulus'' (angle). A #Crossed rectangles, crossed rectangle is a crossed (self-intersecting) quadrilateral which consists of two opposite sides of a rectangle along with the two diagonals (therefore only two sides are parallel). It is a special case of an antiparallelogram, and its angles are not right angles an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
