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Infinite-order Triangular Tiling
In geometry, the infinite-order triangular tiling is a regular tiling of the hyperbolic plane with a Schläfli symbol of . All vertices are ''ideal'', located at "infinity" and seen on the boundary of the Poincaré hyperbolic disk projection. Symmetry A lower symmetry form has alternating colors, and represented by cyclic symbol , . The tiling also represents the fundamental domains of the *∞∞∞ symmetry, which can be seen with 3 colors of lines representing 3 mirrors of the construction. Related polyhedra and tiling This tiling is topologically related as part of a sequence of regular polyhedra with Schläfli symbol . Other infinite-order triangular tilings A nonregular infinite-order triangular tiling can be generated by a recursive process from a central triangle as shown here: : See also * Infinite-order tetrahedral honeycomb *List of regular polytopes *List of uniform planar tilings *Tilings of regular polygons *Triangular tiling *Uniform tilings in hyperboli ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
H3 33inf UHS Plane At Infinity
H3, H03 or H-3 may refer to: Entertainment * ''H3'' (film), a 2001 film about the 1981 Irish hunger strike * ''H3 – Halloween Horror Hostel'', a 2008 German horror-parody television film * ''Happy Hustle High'', a manga series by Rie Takada, originally titled "H3 School!" * h3h3Productions, styled " 3, a satirical YouTube channel Science * Triatomic hydrogen (H3), an unstable molecule * Trihydrogen cation (), one of the most abundant ions in the universe * Tritium (3H), or hydrogen-3, an isotope of hydrogen * ATC code H03 ''Thyroid therapy'', a subgroup of the Anatomical Therapeutic Chemical Classification System * British NVC community H3, a heath community of the British National Vegetation Classification system * Histamine H3 receptor, a human gene * Histone H3, a component of DNA higher structure in eukaryotic cells * Hekla 3 eruption, a huge volcanic eruption around 1000 BC Computing * HTTP/3, the third major version of the Hypertext Transfer Protocol * Socket H3 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Infinite-order Tetrahedral Honeycomb
In the geometry of hyperbolic 3-space, the order-7 tetrahedral honeycomb is a regular space-filling tessellation (or honeycomb) with Schläfli symbol . It has seven tetrahedra around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many tetrahedra existing around each vertex in an order-7 triangular tiling vertex arrangement. Images Related polytopes and honeycombs It is a part of a sequence of regular polychora and honeycombs with tetrahedral cells, . It is a part of a sequence of hyperbolic honeycombs with order-7 triangular tiling vertex figures, . It is a part of a sequence of hyperbolic honeycombs, . Order-8 tetrahedral honeycomb In the geometry of hyperbolic 3-space, the order-8 tetrahedral honeycomb is a regular space-filling tessellation (or honeycomb) with Schläfli symbol . It has eight tetrahedra around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many tetrahedra exis ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isohedral Tilings
In geometry, a tessellation of dimension (a plane tiling) or higher, or a polytope of dimension (a polyhedron) or higher, is isohedral or face-transitive if all its faces are the same. More specifically, all faces must be not merely congruent but must be ''transitive'', i.e. must lie within the same '' symmetry orbit''. In other words, for any two faces and , there must be a symmetry of the ''entire'' figure by translations, rotations, and/or reflections that maps onto . For this reason, convex isohedral polyhedra are the shapes that will make fair dice. Isohedral polyhedra are called isohedra. They can be described by their face configuration. An isohedron has an even number of faces. The dual of an isohedral polyhedron is vertex-transitive, i.e. isogonal. The Catalan solids, the bipyramids, and the trapezohedra are all isohedral. They are the duals of the (isogonal) Archimedean solids, prisms, and antiprisms, respectively. The Platonic solids, which are eith ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isogonal Tilings
Isogonal, a mathematical term meaning "having similar angles", may refer to: *Isogonal figure or polygon, polyhedron, polytope or tiling * Isogonal trajectory, in curve theory *Isogonal conjugate __NOTOC__ In geometry, the isogonal conjugate of a point with respect to a triangle is constructed by reflecting the lines about the angle bisectors of respectively. These three reflected lines concur at the isogonal conjugate of . (Th ..., in triangle geometry See also * Isogonic line, in the study of Earth's magnetic field, a line of constant magnetic declination {{disambig Geometry ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hyperbolic Tilings
In hyperbolic geometry, a uniform hyperbolic tiling (or regular, quasiregular or semiregular hyperbolic tiling) is an edge-to-edge filling of the hyperbolic plane which has regular polygons as Face (geometry), faces and is vertex-transitive (Transitive group action, transitive on its vertex (geometry), vertices, isogonal, i.e. there is an isometry mapping any vertex onto any other). It follows that all vertices are Congruence (geometry), congruent, and the tessellation, tiling has a high degree of rotational and translational symmetry. Uniform tilings can be identified by their vertex configuration, a sequence of numbers representing the number of sides of the polygons around each vertex. For example, 7.7.7 represents the heptagonal tiling which has 3 heptagons around each vertex. It is also regular since all the polygons are the same size, so it can also be given the Schläfli symbol . Uniform tilings may be Regular polyhedron, regular (if also face- and edge-transitive), quasi- ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
John Horton Conway
John Horton Conway (26 December 1937 – 11 April 2020) was an English mathematician. He was active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. He also made contributions to many branches of recreational mathematics, most notably the invention of the cellular automaton called the Game of Life. Born and raised in Liverpool, Conway spent the first half of his career at the University of Cambridge before moving to the United States, where he held the John von Neumann Professorship at Princeton University for the rest of his career. On 11 April 2020, at age 82, he died of complications from COVID-19. Early life and education Conway was born on 26 December 1937 in Liverpool, the son of Cyril Horton Conway and Agnes Boyce. He became interested in mathematics at a very early age. By the time he was 11, his ambition was to become a mathematician. After leaving sixth form, he studied mathematics at Gonville and Caius Coll ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Uniform Tilings In Hyperbolic Plane
In hyperbolic geometry, a uniform hyperbolic tiling (or regular, quasiregular or semiregular hyperbolic tiling) is an edge-to-edge filling of the hyperbolic plane which has regular polygons as Face (geometry), faces and is vertex-transitive (Transitive group action, transitive on its vertex (geometry), vertices, isogonal, i.e. there is an isometry mapping any vertex onto any other). It follows that all vertices are Congruence (geometry), congruent, and the tessellation, tiling has a high degree of rotational and translational symmetry. Uniform tilings can be identified by their vertex configuration, a sequence of numbers representing the number of sides of the polygons around each vertex. For example, 7.7.7 represents the heptagonal tiling which has 3 heptagons around each vertex. It is also regular since all the polygons are the same size, so it can also be given the Schläfli symbol . Uniform tilings may be Regular polyhedron, regular (if also face- and edge-transitive), quasi- ... [...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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tilings Of Regular Polygons
Euclidean plane tilings by convex regular polygons have been widely used since antiquity. The first systematic mathematical treatment was that of Kepler in his (Latin: ''The Harmony of the World'', 1619). Notation of Euclidean tilings Euclidean tilings are usually named after Cundy & Rollett’s notation. This notation represents (i) the number of vertices, (ii) the number of polygons around each vertex (arranged clockwise) and (iii) the number of sides to each of those polygons. For example: 36; 36; 34.6, tells us there are 3 vertices with 2 different vertex types, so this tiling would be classed as a ‘3-uniform (2-vertex types)’ tiling. Broken down, 36; 36 (both of different transitivity class), or (36)2, tells us that there are 2 vertices (denoted by the superscript 2), each with 6 equilateral 3-sided polygons (triangles). With a final vertex 34.6, 4 more contiguous equilateral triangles and a single regular hexagon. However, this notation has two main problems relat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
List Of Uniform Planar Tilings
This table shows the 11 convex uniform tilings (regular and semiregular) of the Euclidean plane, and their dual tilings. There are three regular and eight semiregular tilings in the plane. The semiregular tilings form new tilings from their duals, each made from one type of irregular face. John Conway called these uniform duals ''Catalan tilings'', in parallel to the Catalan solid polyhedra. Uniform tilings are listed by their vertex configuration, the sequence of faces that exist on each vertex. For example ''4.8.8'' means one square and two octagons on a vertex. These 11 uniform tilings have 32 different ''uniform colorings''. A uniform coloring allows identical sided polygons at a vertex to be colored differently, while still maintaining vertex-uniformity and transformational congruence between vertices. (Note: Some of the tiling images shown below are ''not'' color-uniform.) In addition to the 11 convex uniform tilings, there are also 14 known nonconvex tilings, using ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
List Of Regular Polytopes
This article lists the regular polytopes in Euclidean, spherical and hyperbolic spaces. Overview This table shows a summary of regular polytope counts by rank. There are no Euclidean regular star tessellations in any number of dimensions. 1-polytopes There is only one polytope of rank 1 (1-polytope), the closed line segment bounded by its two endpoints. Every realization of this 1-polytope is regular. It has the Schläfli symbol , or a Coxeter diagram with a single ringed node, . Norman Johnson calls it a ''dion'' and gives it the Schläfli symbol . Although trivial as a polytope, it appears as the edges of polygons and other higher dimensional polytopes. It is used in the definition of uniform prisms like Schläfli symbol ×, or Coxeter diagram as a Cartesian product of a line segment and a regular polygon. 2-polytopes (polygons) The polytopes of rank 2 (2-polytopes) are called polygons. Regular polygons are equilateral and cyclic. A -gonal regular polygon is repre ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
