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Icosian Calculus
The icosian calculus is a non-commutative algebraic structure discovered by the Irish mathematician William Rowan Hamilton in 1856. In modern terms, he gave a group presentation of the icosahedral group, icosahedral rotation group by Generating set of a group, generators and relations. Hamilton's discovery derived from his attempts to find an algebra of tuple, "triplets" or 3-tuples that he believed would reflect the three Cartesian coordinate system#Cartesian coordinates in three dimensions, Cartesian axes. The symbols of the icosian calculus correspond to moves between vertices on a dodecahedron. (Hamilton originally thought in terms of moves between the faces of an icosahedron, which is equivalent by Platonic solid#Dual polyhedra, duality. This is the origin of the name "icosian".) Hamilton's work in this area resulted indirectly in the terms Hamiltonian circuit and Hamiltonian path in graph theory. He also invented the icosian game as a means of illustrating and popularising hi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Algebraic Structure
In mathematics, an algebraic structure or algebraic system consists of a nonempty set ''A'' (called the underlying set, carrier set or domain), a collection of operations on ''A'' (typically binary operations such as addition and multiplication), and a finite set of identities (known as ''axioms'') that these operations must satisfy. An algebraic structure may be based on other algebraic structures with operations and axioms involving several structures. For instance, a vector space involves a second structure called a field, and an operation called ''scalar multiplication'' between elements of the field (called '' scalars''), and elements of the vector space (called '' vectors''). Abstract algebra is the name that is commonly given to the study of algebraic structures. The general theory of algebraic structures has been formalized in universal algebra. Category theory is another formalization that includes also other mathematical structures and functions between structu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Icosahedron
In geometry, an icosahedron ( or ) is a polyhedron with 20 faces. The name comes . The plural can be either "icosahedra" () or "icosahedrons". There are infinitely many non- similar shapes of icosahedra, some of them being more symmetrical than others. The best known is the ( convex, non- stellated) regular icosahedron—one of the Platonic solids—whose faces are 20 equilateral triangles. Regular icosahedra There are two objects, one convex and one nonconvex, that can both be called regular icosahedra. Each has 30 edges and 20 equilateral triangle faces with five meeting at each of its twelve vertices. Both have icosahedral symmetry. The term "regular icosahedron" generally refers to the convex variety, while the nonconvex form is called a ''great icosahedron''. Convex regular icosahedron The convex regular icosahedron is usually referred to simply as the ''regular icosahedron'', one of the five regular Platonic solids, and is represented by its Schläfli symbol , contai ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Graph Theory
In mathematics and computer science, graph theory is the study of ''graph (discrete mathematics), graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of ''Vertex (graph theory), vertices'' (also called ''nodes'' or ''points'') which are connected by ''Glossary of graph theory terms#edge, edges'' (also called ''arcs'', ''links'' or ''lines''). A distinction is made between undirected graphs, where edges link two vertices symmetrically, and directed graphs, where edges link two vertices asymmetrically. Graphs are one of the principal objects of study in discrete mathematics. Definitions Definitions in graph theory vary. The following are some of the more basic ways of defining graphs and related mathematical structures. Graph In one restricted but very common sense of the term, a graph is an ordered pair G=(V,E) comprising: * V, a Set (mathematics), set of vertices (also called nodes or points); * ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Icosian
In mathematics, the icosians are a specific set of Hamiltonian quaternions with the same symmetry as the 600-cell. The term can be used to refer to two related, but distinct, concepts: * The icosian Group (mathematics), group: a multiplicative group of 120 quaternions, positioned at the vertices of a 600-cell of unit radius. This group is group isomorphism, isomorphic to the binary icosahedral group of order of a group, order 120. * The icosian Ring (mathematics), ring: all finite sums of the 120 unit icosians. Unit icosians The icosian group, consisting of the 120 unit icosians, comprises the distinct even permutations of * ½(±2, 0, 0, 0) (resulting in 8 icosians), * ½(±1, ±1, ±1, ±1) (resulting in 16 icosians), * ½(0, ±1, ±1''/φ'', ±''φ'') (resulting in 96 icosians). In this case, the vector (''a'', ''b'', ''c'', ''d'') refers to the quaternion ''a'' + ''b''i + ''c''j + ''d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
History
History is the systematic study of the past, focusing primarily on the human past. As an academic discipline, it analyses and interprets evidence to construct narratives about what happened and explain why it happened. Some theorists categorize history as a social science, while others see it as part of the humanities or consider it a hybrid discipline. Similar debates surround the purpose of history—for example, whether its main aim is theoretical, to uncover the truth, or practical, to learn lessons from the past. In a more general sense, the term ''history'' refers not to an academic field but to the past itself, times in the past, or to individual texts about the past. Historical research relies on primary and secondary sources to reconstruct past events and validate interpretations. Source criticism is used to evaluate these sources, assessing their authenticity, content, and reliability. Historians strive to integrate the perspectives of several sources to develop a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Séminaire Lotharingien De Combinatoire
The ''Séminaire Lotharingien de Combinatoire'' (English: ''Lotharingian Seminar of Combinatorics'') is a peer-reviewed academic journal specialising in combinatorial mathematics, named after Lotharingia. It has existed since 1980 as a regular joint seminar in Combinatorics for the Universities of Bayreuth, Erlangen and Strasbourg Strasbourg ( , ; ; ) is the Prefectures in France, prefecture and largest city of the Grand Est Regions of France, region of Geography of France, eastern France, in the historic region of Alsace. It is the prefecture of the Bas-Rhin Departmen .... In 1994, it was decided to create a journal under the same name. The regular meetings continue to this day. See also * M. Lothaire References External links Séminaire Lotharingien de Combinatoire Combinatorics journals Open access journals {{math-journal-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Dessin D'enfant
In mathematics, a dessin d'enfant is a type of graph embedding used to study Riemann surfaces and to provide combinatorial Invariant (mathematics), invariants for the action of the absolute Galois group of the rational numbers. The name of these embeddings is French language, French for a "child's drawing"; its plural is either ''dessins d'enfant'', "child's drawings", or ''dessins d'enfants'', "children's drawings". A dessin d'enfant is a undirected graph, graph, with its vertex (graph theory), vertices colored alternately black and white, graph embedding, embedded in an Orientability, oriented surface that, in many cases, is simply a Plane (mathematics), plane. For the coloring to exist, the graph must be bipartite graph, bipartite. The faces of the embedding are required to be topological disks. The surface and the embedding may be described combinatorially using a rotation system, a cyclic order of the edges surrounding each vertex of the graph that describes the order in which ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Geometric Group Theory
Geometric group theory is an area in mathematics devoted to the study of finitely generated groups via exploring the connections between algebraic properties of such groups and topological and geometric properties of spaces on which these groups can act non-trivially (that is, when the groups in question are realized as geometric symmetries or continuous transformations of some spaces). Another important idea in geometric group theory is to consider finitely generated groups themselves as geometric objects. This is usually done by studying the Cayley graphs of groups, which, in addition to the graph structure, are endowed with the structure of a metric space, given by the so-called word metric. Geometric group theory, as a distinct area, is relatively new, and became a clearly identifiable branch of mathematics in the late 1980s and early 1990s. Geometric group theory closely interacts with low-dimensional topology, hyperbolic geometry, algebraic topology, computational group ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Combinatorial Group Theory
In mathematics, combinatorial group theory is the theory of free groups, and the concept of a presentation of a group by generators and relations. It is much used in geometric topology, the fundamental group of a simplicial complex having in a natural and geometric way such a presentation. A very closely related topic is geometric group theory, which today largely subsumes combinatorial group theory, using techniques from outside combinatorics besides. It also comprises a number of algorithmically insoluble problems, most notably the word problem for groups; and the classical Burnside problem. History See the book by Chandler and Magnus for a detailed history of combinatorial group theory. A proto-form is found in the 1856 icosian calculus of William Rowan Hamilton, where he studied the icosahedral symmetry group via the edge graph of the dodecahedron. The foundations of combinatorial group theory were laid by Walther von Dyck, student of Felix Klein Felix Christ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Presentation Of A Group
In mathematics, a presentation is one method of specifying a group. A presentation of a group ''G'' comprises a set ''S'' of generators—so that every element of the group can be written as a product of powers of some of these generators—and a set ''R'' of relations among those generators. We then say ''G'' has presentation :\langle S \mid R\rangle. Informally, ''G'' has the above presentation if it is the "freest group" generated by ''S'' subject only to the relations ''R''. Formally, the group ''G'' is said to have the above presentation if it is isomorphic to the quotient of a free group on ''S'' by the normal subgroup generated by the relations ''R''. As a simple example, the cyclic group of order ''n'' has the presentation :\langle a \mid a^n = 1\rangle, where 1 is the group identity. This may be written equivalently as :\langle a \mid a^n\rangle, thanks to the convention that terms that do not include an equals sign are taken to be equal to the group identity ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Symmetric Graph
In the mathematical field of graph theory, a graph is symmetric or arc-transitive if, given any two ordered pairs of adjacent vertices (u_1,v_1) and (u_2,v_2) of , there is an automorphism :f : V(G) \rightarrow V(G) such that :f(u_1) = u_2 and f(v_1) = v_2. In other words, a graph is symmetric if its automorphism group acts transitively on ordered pairs of adjacent vertices (that is, upon edges considered as having a direction). Such a graph is sometimes also called -transitive or flag-transitive. By definition (ignoring and ), a symmetric graph without isolated vertices must also be vertex-transitive. Since the definition above maps one edge to another, a symmetric graph must also be edge-transitive. However, an edge-transitive graph need not be symmetric, since might map to , but not to . Star graphs are a simple example of being edge-transitive without being vertex-transitive or symmetric. As a further example, semi-symmetric graphs are edge-transitive and regular, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Icosian Calculus Iota2
In mathematics, the icosians are a specific set of Hamiltonian quaternions with the same symmetry as the 600-cell. The term can be used to refer to two related, but distinct, concepts: * The icosian Group (mathematics), group: a multiplicative group of 120 quaternions, positioned at the vertices of a 600-cell of unit radius. This group is group isomorphism, isomorphic to the binary icosahedral group of order of a group, order 120. * The icosian Ring (mathematics), ring: all finite sums of the 120 unit icosians. Unit icosians The icosian group, consisting of the 120 unit icosians, comprises the distinct even permutations of * ½(±2, 0, 0, 0) (resulting in 8 icosians), * ½(±1, ±1, ±1, ±1) (resulting in 16 icosians), * ½(0, ±1, ±1''/φ'', ±''φ'') (resulting in 96 icosians). In this case, the vector (''a'', ''b'', ''c'', ''d'') refers to the quaternion ''a'' + ''b''i + ''c''j + ''d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
