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Elongated Pyramid
In geometry, the elongated pyramids are an infinite set of polyhedra, constructed by adjoining an pyramid to an prism. Along with the set of pyramids, these figures are topologically self-dual. There are three ''elongated pyramids'' that are Johnson solids: * Elongated triangular pyramid (), *Elongated square pyramid (), and * Elongated pentagonal pyramid (). Higher forms can be constructed with isosceles triangles. Forms See also * Gyroelongated bipyramid * Elongated bipyramid * Gyroelongated pyramid * Diminished trapezohedron In geometry, a diminished trapezohedron is a polyhedron in an infinite set of polyhedra, constructed by removing one of the polar vertices of a trapezohedron and replacing it by a new face (diminishment). It has one regular base face, triangle ... References * Norman W. Johnson, "Convex Solids with Regular Faces", Canadian Journal of Mathematics, 18, 1966, pages 169–200. Contains the original enumeration of the 92 solids and the c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 \triangle ABC. In 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. This article is about triangles in Euclidean geometry, and in particular, the Euclidean plane, except where otherwise noted. Types of triangle The terminology for categorizing triangles is more than two thousand years old, having been defined on the very first page of Euclid's Elements. The names used for modern classification are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elongated Pentagonal Pyramid
In geometry, the elongated pentagonal pyramid is one of the Johnson solids (). As the name suggests, it can be constructed by elongating a pentagonal pyramid () by attaching a pentagonal prism to its base. Formulae The following formulae for the height (H), surface area (A) and volume (V) can be used if all faces are regular, with edge length L: :H = L\cdot \left( 1 + \sqrt\right) \approx L\cdot 1.525731112 :A = L^2 \cdot \frac \approx L^2\cdot 8.88554091 :V = L^3 \cdot \left( \frac \right) \approx L^3\cdot 2.021980233 Dual polyhedron The dual of the elongated pentagonal pyramid has 11 faces: 5 triangular, 1 pentagonal and 5 trapezoidal. It is topologically identical to the Johnson solid. See also * Elongated pentagonal bipyramid In geometry, the elongated pentagonal bipyramid or pentakis pentagonal prism is one of the Johnson solids (). As the name suggests, it can be constructed by elongating a pentagonal bipyramid () by inserting a pentagonal prism betwe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Diminished Trapezohedron
In geometry, a diminished trapezohedron is a polyhedron in an infinite set of polyhedra, constructed by removing one of the polar vertices of a trapezohedron and replacing it by a new face (diminishment). It has one regular base face, triangle faces around the base, and kites meeting on top. The kites can also be replaced by rhombi with specific proportions. Along with the set of pyramids and elongated pyramids, these figures are topologically self-dual. It can also be seen as an augmented antiprism, with a pyramid augmented onto one of the faces, and whose height is adjusted so the upper antiprism triangle faces can be made coparallel to the pyramid faces and merged into kite-shaped faces. They're also related to the gyroelongated pyramids, as augmented antiprisms and which are Johnson solids for . This sequence has sets of two triangles instead of kite faces. Examples Special cases There are three special case geometries of the ''diminished trigonal trapezohedron' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gyroelongated Pyramid
In geometry, the gyroelongated pyramids (also called ''augmented antiprisms'') are an infinite set of polyhedra, constructed by adjoining an pyramid to an antiprism. There are two ''gyroelongated pyramids'' that are Johnson solids made from regular triangles and square, and pentagons. A triangular and hexagonal form can be constructed with coplanar In geometry, a set of points in space are coplanar if there exists a geometric plane that contains them all. For example, three points are always coplanar, and if the points are distinct and non-collinear, the plane they determine is unique. How ... faces. Others can be constructed allowing for isosceles triangles. Forms See also * Gyroelongated bipyramid * Elongated bipyramid * Elongated pyramid * Diminished trapezohedron References * Norman W. Johnson, "Convex Solids with Regular Faces", Canadian Journal of Mathematics, 18, 1966, pages 169–200. Contains the original enumeration of the 92 solids and the conjec ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elongated Bipyramid
In geometry, the elongated bipyramids are an infinite set of polyhedra, constructed by elongating an bipyramid (by inserting an prism between its congruent halves). There are three ''elongated bipyramids'' that are Johnson solids: * Elongated triangular bipyramid (), * Elongated square bipyramid (), and * Elongated pentagonal bipyramid (). Higher forms can be constructed with isosceles triangles. Forms See also * Gyroelongated bipyramid * Gyroelongated pyramid * Elongated pyramid * Diminished trapezohedron In geometry, a diminished trapezohedron is a polyhedron in an infinite set of polyhedra, constructed by removing one of the polar vertices of a trapezohedron and replacing it by a new face (diminishment). It has one regular base face, triangle ... References * Norman W. Johnson, "Convex Solids with Regular Faces", Canadian Journal of Mathematics, 18, 1966, pages 169–200. Contains the original enumeration of the 92 solids and the conjecture that there a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gyroelongated Bipyramid
In geometry, a Johnson solid is a strictly convex polyhedron each face of which is a regular polygon. There is no requirement that each face must be the same polygon, or that the same polygons join around each vertex. An example of a Johnson solid is the square-based pyramid with equilateral sides ( ); it has 1 square face and 4 triangular faces. Some authors require that the solid not be uniform (i.e., not Platonic solid, Archimedean solid, uniform prism, or uniform antiprism) before they refer to it as a “Johnson solid”. As in any strictly convex solid, at least three faces meet at every vertex, and the total of their angles is less than 360 degrees. Since a regular polygon has angles at least 60 degrees, it follows that at most five faces meet at any vertex. The pentagonal pyramid () is an example that has a degree-5 vertex. Although there is no obvious restriction that any given regular polygon cannot be a face of a Johnson solid, it turns out that the faces of John ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elongated Pentagonal Pyramid
In geometry, the elongated pentagonal pyramid is one of the Johnson solids (). As the name suggests, it can be constructed by elongating a pentagonal pyramid () by attaching a pentagonal prism to its base. Formulae The following formulae for the height (H), surface area (A) and volume (V) can be used if all faces are regular, with edge length L: :H = L\cdot \left( 1 + \sqrt\right) \approx L\cdot 1.525731112 :A = L^2 \cdot \frac \approx L^2\cdot 8.88554091 :V = L^3 \cdot \left( \frac \right) \approx L^3\cdot 2.021980233 Dual polyhedron The dual of the elongated pentagonal pyramid has 11 faces: 5 triangular, 1 pentagonal and 5 trapezoidal. It is topologically identical to the Johnson solid. See also * Elongated pentagonal bipyramid In geometry, the elongated pentagonal bipyramid or pentakis pentagonal prism is one of the Johnson solids (). As the name suggests, it can be constructed by elongating a pentagonal bipyramid () by inserting a pentagonal prism betwe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elongated Square Pyramid
In geometry, the elongated square pyramid is one of the Johnson solids (). As the name suggests, it can be constructed by elongating a square pyramid () by attaching a cube to its square base. Like any elongated pyramid, it is topologically (but not geometrically) self- dual. Formulae The following formulae for the height (H), surface area (A) and volume (V) can be used if all faces are regular, with edge length L: :H = L\cdot \left( 1 + \frac\right) \approx L\cdot 1.707106781 :A = L^2 \cdot \left( 5 + \sqrt \right) \approx L^2\cdot 6.732050808 :V = L^3 \left( 1 + \frac\right)\approx L^3\cdot 1.23570226 Dual polyhedron The dual of the elongated square pyramid has 9 faces: 4 triangular, 1 square and 4 trapezoidal. Related polyhedra and honeycombs The elongated square pyramid can form a tessellation of space with tetrahedra, similar to a modified tetrahedral-octahedral honeycomb The tetrahedral-octahedral honeycomb, alternated cubic honeycomb is a quasiregular ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elongated Triangular Pyramid
In geometry, the elongated triangular pyramid is one of the Johnson solids (). As the name suggests, it can be constructed by elongating a tetrahedron by attaching a triangular prism to its base. Like any elongated pyramid, the resulting solid is topologically (but not geometrically) self- dual. Formulae The following formulae for volume and surface area can be used if all faces are regular, with edge length ''a'': :V=\left(\frac\left(\sqrt+3\sqrt\right)\right)a^3\approx0.550864...a^3 :A=\left(3+\sqrt\right)a^2\approx4.73205...a^2 The height is given by :H = a\cdot \left( 1 + \frac\right) \approx a\cdot 1.816496581 If the edges are not the same length, use the individual formulae for the tetrahedron and triangular prism separately, and add the results together. Dual polyhedron Topologically, the elongated triangular pyramid is its own dual. Geometrically, the dual has seven irregular faces: one equilateral triangle, three isosceles triangles and three isosceles trap ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isosceles Triangle
In geometry, an isosceles triangle () is a triangle that has two sides of equal length. Sometimes it is specified as having ''exactly'' two sides of equal length, and sometimes as having ''at least'' two sides of equal length, the latter version thus including the equilateral triangle as a special case. Examples of isosceles triangles include the isosceles right triangle, the golden triangle, and the faces of bipyramids and certain Catalan solids. The mathematical study of isosceles triangles dates back to ancient Egyptian mathematics and Babylonian mathematics. Isosceles triangles have been used as decoration from even earlier times, and appear frequently in architecture and design, for instance in the pediments and gables of buildings. The two equal sides are called the legs and the third side is called the base of the triangle. The other dimensions of the triangle, such as its height, area, and perimeter, can be calculated by simple formulas from the lengths of the legs ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elongated Square Pyramid
In geometry, the elongated square pyramid is one of the Johnson solids (). As the name suggests, it can be constructed by elongating a square pyramid () by attaching a cube to its square base. Like any elongated pyramid, it is topologically (but not geometrically) self- dual. Formulae The following formulae for the height (H), surface area (A) and volume (V) can be used if all faces are regular, with edge length L: :H = L\cdot \left( 1 + \frac\right) \approx L\cdot 1.707106781 :A = L^2 \cdot \left( 5 + \sqrt \right) \approx L^2\cdot 6.732050808 :V = L^3 \left( 1 + \frac\right)\approx L^3\cdot 1.23570226 Dual polyhedron The dual of the elongated square pyramid has 9 faces: 4 triangular, 1 square and 4 trapezoidal. Related polyhedra and honeycombs The elongated square pyramid can form a tessellation of space with tetrahedra, similar to a modified tetrahedral-octahedral honeycomb The tetrahedral-octahedral honeycomb, alternated cubic honeycomb is a quasiregular ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Square (geometry)
In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90- degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length adjacent sides. It is the only regular polygon whose internal angle, central angle, and external angle are all equal (90°), and whose diagonals are all equal in length. A square with vertices ''ABCD'' would be denoted . Characterizations A convex quadrilateral is a square if and only if it is any one of the following: * A rectangle with two adjacent equal sides * A rhombus with a right vertex angle * A rhombus with all angles equal * A parallelogram with one right vertex angle and two adjacent equal sides * A quadrilateral with four equal sides and four right angles * A quadrilateral where the diagonals are equal, and are the perpendicular bisectors of each other (i.e., a rhombus with equal diagonals) * A convex quadrilateral wi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
