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Truncated Triapeirogonal Tiling
In geometry, the truncated triapeirogonal tiling is a uniform tiling of the hyperbolic plane with a Schläfli symbol of tr. Symmetry The dual of this tiling represents the fundamental domains of ˆž,3 *∞32 symmetry. There are 3 small index subgroup constructed from ˆž,3by mirror removal and alternation. In these images fundamental domains are alternately colored black and white, and mirrors exist on the boundaries between colors. A special index 4 reflective subgroup, is ∞,∞,3) (*∞∞3), and its direct subgroup ∞,∞,3)sup>+, (∞∞3), and semidirect subgroup ∞,∞,3+) (3*∞).Norman W. Johnson and Asia Ivic Weiss, ''Quadratic Integers and Coxeter Groups'', Can. J. Math. Vol. 51 (6), 1999 pp. 1307–133/ref> Given ˆž,3with generating mirrors , then its index 4 subgroup has generators . An index 6 subgroup constructed as ˆž,3* becomes ∞,∞,∞) (*∞∞∞). Related polyhedra and tiling This tiling can be considered a member of a sequence of uniform ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is called a ''geometer''. Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry, which includes the notions of point, line, plane, distance, angle, surface, and curve, as fundamental concepts. During the 19th century several discoveries enlarged dramatically the scope of geometry. One of the oldest such discoveries is Carl Friedrich Gauss' ("remarkable theorem") that asserts roughly that the Gaussian curvature of a surface is independent from any specific embedding in a Euclidean space. This implies that surfaces can be studied ''intrinsically'', that is, as stand-alone spaces, and has been expanded into the theory of manifolds and Riemannian geometry. Later in the 19th century, it appeared that geometries ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orbifold Notation
In geometry, orbifold notation (or orbifold signature) is a system, invented by the mathematician William Thurston and promoted by John Conway, for representing types of symmetry groups in two-dimensional spaces of constant curvature. The advantage of the notation is that it describes these groups in a way which indicates many of the groups' properties: in particular, it follows William Thurston in describing the orbifold obtained by taking the quotient of Euclidean space by the group under consideration. Groups representable in this notation include the point groups on the sphere (S^2), the frieze groups and wallpaper groups of the Euclidean plane (E^2), and their analogues on the hyperbolic plane (H^2). Definition of the notation The following types of Euclidean transformation can occur in a group described by orbifold notation: * reflection through a line (or plane) * translation by a vector * rotation of finite order around a point * infinite rotation around a line in 3- ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
I32 Symmetry Azz
I3, I-3 or i3 may refer to: Science and technology * Core i3, a microprocessor line from Intel * i3 (window manager), a tiling window manager * Triiodide (I3−), an anion composed of 3 iodide atoms * Intermud-3, an InterMUD network method between MUDs * Axiom I3, a large cardinal axiom in rank-into-rank set theory Transport * Interstate 3, a proposed Interstate Highway in the U.S. states of Georgia and Tennessee * BMW i3, an electric car by BMW * Inline-triple engine, an internal combustion engine with three cylinders in a straight line * LB&SCR I3 class, a British LB&SCR locomotive Military * , a Type J1 submarine of the Imperial Japanese Navy * i3 fighter, a conceptual Japanese jet fighter proposed in 2010 * Mikoyan-Gurevich I-3, a heavy interceptor prototype fighter of the 1950s * Polikarpov I-3 The Polikarpov I-3 (russian: Поликарпов И-3) was a Soviet fighter designed during the late 1920s. It entered service in 1929, but was retired in 1935 with the advent o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
I32 Symmetry Mirrors-index4
I3, I-3 or i3 may refer to: Science and technology * Core i3, a microprocessor line from Intel * i3 (window manager), a tiling window manager * Triiodide (I3−), an anion composed of 3 iodide atoms * Intermud-3, an InterMUD network method between MUDs * Axiom I3, a large cardinal axiom in rank-into-rank set theory Transport * Interstate 3, a proposed Interstate Highway in the U.S. states of Georgia and Tennessee * BMW i3, an electric car by BMW * Inline-triple engine, an internal combustion engine with three cylinders in a straight line * LB&SCR I3 class, a British LB&SCR locomotive Military * , a Type J1 submarine of the Imperial Japanese Navy * i3 fighter, a conceptual Japanese jet fighter proposed in 2010 * Mikoyan-Gurevich I-3, a heavy interceptor prototype fighter of the 1950s * Polikarpov I-3 The Polikarpov I-3 (russian: Поликарпов И-3) was a Soviet fighter designed during the late 1920s. It entered service in 1929, but was retired in 1935 with the advent o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ii2 Symmetry Aaa
II is the Roman numeral for 2. II may also refer to: Biology and medicine * Image intensifier, medical imaging equipment *Invariant chain, a polypeptide involved in the formation and transport of MHC class II protein *Optic nerve, the second cranial nerve Economics * Income inequality, or the wealth gap, in economics * Internationalization Index, used by the UN to rank nations and companies in evaluating their degree of integration with the world economy * ''Institutional Investor'' (magazine), an American finance magazine Music * Supertonic, in music * ''ii'', a 2018 song by CHVRCHES Albums * ''II'' (2 Unlimited album), 1998 * ''II'' (Aquilo album), 2018 * ''II'' (Bad Books album), 2012 * ''II'' (Boyz II Men album), 1994 * ''II'' (Capital Kings album), 2015 * ''II'' (Charade album), 2004 * ''II'' (The Common Linnets album), 2015 * ''II'' (Compact Disco album), 2011 * ''II'' (Cursed album), 2005 * ''II'' (Darna album), 2003 * ''II'' (Espers album), 2006 * ''II'' (Fuzz al ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
I32 Symmetry Abb
I3, I-3 or i3 may refer to: Science and technology * Core i3, a microprocessor line from Intel * i3 (window manager), a tiling window manager * Triiodide (I3−), an anion composed of 3 iodide atoms * Intermud-3, an InterMUD network method between MUDs * Axiom I3, a large cardinal axiom in rank-into-rank set theory Transport * Interstate 3, a proposed Interstate Highway in the U.S. states of Georgia and Tennessee * BMW i3, an electric car by BMW * Inline-triple engine, an internal combustion engine with three cylinders in a straight line * LB&SCR I3 class, a British LB&SCR locomotive Military * , a Type J1 submarine of the Imperial Japanese Navy * i3 fighter, a conceptual Japanese jet fighter proposed in 2010 * Mikoyan-Gurevich I-3, a heavy interceptor prototype fighter of the 1950s * Polikarpov I-3 The Polikarpov I-3 (russian: Поликарпов И-3) was a Soviet fighter designed during the late 1920s. It entered service in 1929, but was retired in 1935 with the advent o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
I32 Symmetry Aaa
I3, I-3 or i3 may refer to: Science and technology * Core i3, a microprocessor line from Intel * i3 (window manager), a tiling window manager * Triiodide (I3−), an anion composed of 3 iodide atoms * Intermud-3, an InterMUD network method between MUDs * Axiom I3, a large cardinal axiom in rank-into-rank set theory Transport * Interstate 3, a proposed Interstate Highway in the U.S. states of Georgia and Tennessee * BMW i3, an electric car by BMW * Inline-triple engine, an internal combustion engine with three cylinders in a straight line * LB&SCR I3 class, a British LB&SCR locomotive Military * , a Type J1 submarine of the Imperial Japanese Navy * i3 fighter, a conceptual Japanese jet fighter proposed in 2010 * Mikoyan-Gurevich I-3, a heavy interceptor prototype fighter of the 1950s * Polikarpov I-3 The Polikarpov I-3 (russian: Поликарпов И-3) was a Soviet fighter designed during the late 1920s. It entered service in 1929, but was retired in 1935 with the advent o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Iiii Symmetry
In geometry, the order-4 apeirogonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of . Symmetry This tiling represents the mirror lines of *2∞ symmetry. It dual to this tiling represents the fundamental domains of orbifold notation *∞∞∞∞ symmetry, a square domain with four ideal vertices. : Uniform colorings Like the Euclidean square tiling there are 9 uniform colorings for this tiling, with 3 uniform colorings generated by triangle reflective domains. A fourth can be constructed from an infinite square symmetry (*∞∞∞∞) with 4 colors around a vertex. The checker board, r, coloring defines the fundamental domains of ∞,4,4) (*∞44) symmetry, usually shown as black and white domains of reflective orientations. Related polyhedra and tiling This tiling is also topologically related as a part of sequence of regular polyhedra and tilings with four faces per vertex, starting with the octahedron, with Schläfli symbol , and C ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
I2i2 Symmetry
In geometry, the tetraapeirogonal tiling is a uniform tilings in hyperbolic plane, uniform tiling of the hyperbolic geometry, hyperbolic plane with a Schläfli symbol of r. Uniform constructions There are 3 lower symmetry uniform construction, one with two colors of apeirogons, one with two colors of squares, and one with two colors of each: Symmetry The dual to this tiling represents the fundamental domains of *∞2∞2 symmetry group. The symmetry can be doubled by adding mirrors on either diagonal of the rhombic domains, creating ii2 symmetry, *∞∞2 and i44 symmetry, *∞44 symmetry. Related polyhedra and tiling See also * List of uniform planar tilings * Tilings of regular polygons * Uniform tilings in hyperbolic plane References * John Horton Conway, John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, ''The Symmetries of Things'' 2008, (Chapter 19, "The Hyperbolic Archimedean Tessellations") * External links * * {{Tessellation Apeirogonal tilings Hyp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Iii Symmetry
In geometry, the order-3 apeirogonal tiling is a regular tiling of the hyperbolic plane. It is represented by the Schläfli symbol , having three regular apeirogons around each vertex. Each apeirogon is inscribed in a horocycle. The order-2 apeirogonal tiling represents an infinite dihedron in the Euclidean plane as . Images Each apeirogon face is circumscribed by a horocycle, which looks like a circle in a Poincaré disk model, internally tangent to the projective circle boundary. : Uniform colorings Like the Euclidean hexagonal tiling, there are 3 uniform colorings of the ''order-3 apeirogonal tiling'', each from different reflective triangle group domains: Symmetry The dual to this tiling represents the fundamental domains of ∞,∞,∞)(*∞∞∞) symmetry. There are 15 small index subgroups (7 unique) constructed from ∞,∞,∞)by mirror removal and alternation. Mirrors can be removed if its branch orders are all even, and cuts neighboring branch orders in half. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
