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List Of Tessellations
See also *Uniform tiling **Convex uniform honeycombs **List of k-uniform tilings **List of convex uniform tilings **Uniform tilings in hyperbolic plane {{DEFAULTSORT:Tessellations Mathematics-related lists Tessellation, * ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spherical Polyhedron
In geometry, a spherical polyhedron or spherical tiling is a tessellation, tiling of the sphere in which the surface is divided or partitioned by great arcs into bounded regions called spherical polygons. Much of the theory of symmetrical polyhedron, polyhedra is most conveniently derived in this way. The most familiar spherical polyhedron is the Ball (association football), soccer ball, thought of as a spherical truncated icosahedron. The next most popular spherical polyhedron is the beach ball, thought of as a hosohedron. Some #Improper_cases, "improper" polyhedra, such as hosohedron, hosohedra and their dual polyhedron, duals, dihedron, dihedra, exist as spherical polyhedra, but their flat-faced analogs are Degeneracy (mathematics), degenerate. The example hexagonal beach ball, is a hosohedron, and is its dual dihedron. History The first known man-made polyhedra are spherical polyhedra stone carving, carved in stone. Many have been found in Scotland, and appear to date fr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prism (geometry)
In geometry, a prism is a polyhedron comprising an polygon Base (geometry), base, a second base which is a Translation (geometry), translated copy (rigidly moved without rotation) of the first, and other Face (geometry), faces, necessarily all parallelograms, joining corresponding sides of the two bases. All Cross section (geometry), cross-sections parallel to the bases are translations of the bases. Prisms are named after their bases, e.g. a prism with a pentagonal base is called a pentagonal prism. Prisms are a subclass of prismatoids. Like many basic geometric terms, the word ''prism'' () was first used in Euclid's Elements. Euclid defined the term in Book XI as “a solid figure contained by two opposite, equal and parallel planes, while the rest are parallelograms”. However, this definition has been criticized for not being specific enough in relation to the nature of the bases, which caused confusion among later geometry writers. Oblique prism An oblique prism is a pr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Triangular Tiling
In geometry, the triangular tiling or triangular tessellation is one of the three regular tilings of the Euclidean plane, and is the only such tiling where the constituent shapes are not parallelogons. Because the internal angle of the equilateral triangle is 60 degrees, six triangles at a point occupy a full 360 degrees. The triangular tiling has Schläfli symbol of English mathematician John Conway called it a deltille, named from the triangular shape of the Greek letter delta (Δ). The triangular tiling can also be called a kishextille by a kis operation that adds a center point and triangles to replace the faces of a hextille. It is one of three regular tilings of the plane. The other two are the square tiling and the hexagonal tiling. Uniform colorings There are 9 distinct uniform colorings of a triangular tiling. (Naming the colors by indices on the 6 triangles around a vertex: 111111, 111112, 111212, 111213, 111222, 112122, 121212, 121213, 121314) Three of them can ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Square Tiling Uniform Coloring 7
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 with successiv ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Square Tiling
In geometry, the square tiling, square tessellation or square grid is a regular tiling of the Euclidean plane. It has Schläfli symbol of meaning it has 4 squares around every vertex. Conway called it a quadrille. The internal angle of the square is 90 degrees so four squares at a point make a full 360 degrees. It is one of three regular tilings of the plane. The other two are the triangular tiling and the hexagonal tiling. Uniform colorings There are 9 distinct uniform colorings of a square tiling. Naming the colors by indices on the 4 squares around a vertex: 1111, 1112(i), 1112(ii), 1122, 1123(i), 1123(ii), 1212, 1213, 1234. (i) cases have simple reflection symmetry, and (ii) glide reflection symmetry. Three can be seen in the same symmetry domain as reduced colorings: 1112i from 1213, 1123i from 1234, and 1112ii reduced from 1123ii. Related polyhedra and tilings This tiling is topologically related as a part of sequence of regular polyhedra and tilings, extending ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Apeirogonal Tiling
In geometry, an apeirogonal tiling is a tessellation of the Euclidean plane, hyperbolic plane, or some other two-dimensional space by apeirogons. Tilings of this type include: *Order-2 apeirogonal tiling, Euclidean tiling of two half-spaces *Order-3 apeirogonal tiling, hyperbolic tiling with 3 apeirogons around a vertex *Order-4 apeirogonal tiling, hyperbolic tiling with 4 apeirogons around a vertex *Order-5 apeirogonal tiling, hyperbolic tiling with 5 apeirogons around a vertex *Infinite-order apeirogonal tiling, hyperbolic tiling with an infinite number of apeirogons around a vertex See also *Apeirogonal antiprism *Apeirogonal prism *Apeirohedron In geometry, a skew apeirohedron is an infinite skew polyhedron consisting of nonplanar Face (geometry), faces or nonplanar vertex figures, allowing the figure to extend indefinitely without folding round to form a Surface (topology)#Closed_surfac ... {{set index article, mathematics Apeirogonal tilings ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Order-2 Apeirogonal Tiling
In geometry, an order-2 apeirogonal tiling, apeirogonal dihedron, or infinite dihedronConway (2008), p. 263 is a tiling of the plane consisting of two apeirogons. It may be considered an improper regular tiling of the Euclidean plane, with Schläfli symbol Two apeirogons, joined along all their edges, can completely fill the entire plane as an apeirogon is infinite in size and has an interior angle of 180°, which is half of a full 360°. Related tilings and polyhedra The apeirogonal tiling is the arithmetic limit of the family of dihedra , as ''p'' tends to infinity, thereby turning the dihedron into a Euclidean tiling. Similarly to the uniform polyhedra and the uniform tilings, eight uniform tilings may be based from the regular apeirogonal tiling. The rectified and cantellated forms are duplicated, and as two times infinity is also infinity, the truncated and omnitruncated forms are also duplicated, therefore reducing the number of unique forms to four: the apeirogonal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Apeirogonal Hosohedron
In geometry, an apeirogonal hosohedron or infinite hosohedronConway (2008), p. 263 is a tiling of the plane consisting of two vertices at infinity. It may be considered an improper regular tiling of the Euclidean plane, with Schläfli symbol Related tilings and polyhedra The apeirogonal hosohedron is the arithmetic limit of the family of hosohedra , as ''p'' tends to infinity, thereby turning the hosohedron into a Euclidean tiling. All the vertices have then receded to infinity and the digonal faces are no longer defined by closed circuits of finite edges. Similarly to the uniform polyhedra and the uniform tilings, eight uniform tilings may be based from the regular apeirogonal tiling. The rectified and cantellated forms are duplicated, and as two times infinity is also infinity, the truncated and omnitruncated forms are also duplicated, therefore reducing the number of unique forms to four: the apeirogonal tiling, the apeirogonal hosohedron, the apeirogonal prism, and the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spherical Hexagonal Trapezohedron
A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the centre of the sphere, and is the sphere's radius. The earliest known mentions of spheres appear in the work of the ancient Greek mathematicians. The sphere is a fundamental object in many fields of mathematics. Spheres and nearly-spherical shapes also appear in nature and industry. Bubbles such as soap bubbles take a spherical shape in equilibrium. The Earth is often approximated as a sphere in geography, and the celestial sphere is an important concept in astronomy. Manufactured items including pressure vessels and most curved mirrors and lenses are based on spheres. Spheres roll smoothly in any direction, so most balls used in sports and toys are spherical, as are ball bearings. Basic terminology As mentioned earlier is the sphere's r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Trapezohedron
In geometry, an trapezohedron, -trapezohedron, -antidipyramid, -antibipyramid, or -deltohedron is the dual polyhedron of an antiprism. The faces of an are congruent and symmetrically staggered; they are called ''twisted kites''. With a higher symmetry, its faces are ''kites'' (also called ''trapezoids'', or ''deltoids''). The "" part of the name does not refer to faces here, but to two arrangements of each vertices around an axis of symmetry. The dual antiprism has two actual faces. An trapezohedron can be dissected into two equal pyramids and an antiprism. Terminology These figures, sometimes called deltohedra, must not be confused with deltahedra, whose faces are equilateral triangles. ''Twisted'' ''trigonal'', ''tetragonal'', and ''hexagonal trapezohedra'' (with six, eight, and twelve ''twisted'' congruent kite faces) exist as crystals; in crystallography (describing the crystal habits of minerals), they are just called ''trigonal'', ''tetragonal'', and ''he ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spherical Hexagonal Bipyramid2
A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the centre of the sphere, and is the sphere's radius. The earliest known mentions of spheres appear in the work of the ancient Greek mathematicians. The sphere is a fundamental object in many fields of mathematics. Spheres and nearly-spherical shapes also appear in nature and industry. Bubbles such as soap bubbles take a spherical shape in equilibrium. The Earth is often approximated as a sphere in geography, and the celestial sphere is an important concept in astronomy. Manufactured items including pressure vessels and most curved mirrors and lenses are based on spheres. Spheres roll smoothly in any direction, so most balls used in sports and toys are spherical, as are ball bearings. Basic terminology As mentioned earlier is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bipyramid
A (symmetric) -gonal bipyramid or dipyramid is a polyhedron formed by joining an -gonal pyramid and its mirror image base-to-base. An -gonal bipyramid has triangle faces, edges, and vertices. The "-gonal" in the name of a bipyramid does not refer to a face but to the internal polygon base, lying in the mirror plane that connects the two pyramid halves. (If it were a face, then each of its edges would connect three faces instead of two.) "Regular", right bipyramids A ''"regular"'' bipyramid has a ''regular'' polygon base. It is usually implied to be also a ''right'' bipyramid. A ''right'' bipyramid has its two apices ''right'' above and ''right'' below the center or the ''centroid'' of its polygon base. A "regular" right (symmetric) -gonal bipyramid has Schläfli symbol . A right (symmetric) bipyramid has Schläfli symbol , for polygon base . The "regular" right (thus face-transitive) -gonal bipyramid with regular vertices is the dual of the -gonal uniform (thus right) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
