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Ii2 Symmetry
In geometry, the truncated order-4 apeirogonal tiling is a uniform tiling of the Hyperbolic geometry, hyperbolic plane. It has Schläfli symbol of t. Uniform colorings A half symmetry coloring is tr, has two types of apeirogons, shown red and yellow here. If the apeirogonal curvature is too large, it doesn't converge to a single ideal point, like the right image, red apeirogons below. Coxeter diagram are shown with dotted lines for these divergent, ultraparallel mirrors. Symmetry From [∞,∞] symmetry, there are 15 small index subgroup by mirror removal and alternation. Mirrors can be removed if its branch orders are all even, and cuts neighboring branch orders in half. Removing two mirrors leaves a half-order gyration point where the removed mirrors met. In these images fundamental domains are alternately colored black and white, and mirrors exist on the boundaries between colors. The symmetry can be doubled as i42 symmetry, ∞42 symmetry by adding a mirror bisecting the f ...
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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 ...
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Coxeter Notation
In geometry, Coxeter notation (also Coxeter symbol) is a system of classifying symmetry groups, describing the angles between fundamental reflections of a Coxeter group in a bracketed notation expressing the structure of a Coxeter-Dynkin diagram, with modifiers to indicate certain subgroups. The notation is named after H. S. M. Coxeter, and has been more comprehensively defined by Norman Johnson. Reflectional groups For Coxeter groups, defined by pure reflections, there is a direct correspondence between the bracket notation and Coxeter-Dynkin diagram. The numbers in the bracket notation represent the mirror reflection orders in the branches of the Coxeter diagram. It uses the same simplification, suppressing 2s between orthogonal mirrors. The Coxeter notation is simplified with exponents to represent the number of branches in a row for linear diagram. So the ''A''''n'' group is represented by ''n''−1 to imply ''n'' nodes connected by ''n−1'' order-3 branches. Exampl ...
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Ii2 Symmetry Bab
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 ...
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Ii2 Symmetry Bba
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 ...
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Ii2 Symmetry Abb
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 ...
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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 ...
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Ii2 Symmetry Ab0
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 ...
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Ii2 Symmetry 0ab
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 ...
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Ii2 Symmetry A0a
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 ...
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Ii2 Symmetry Aa0
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 ...
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Ii2 Symmetry 0bb
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 a ...
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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 ...
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