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If a point has coordinates, P(x, y, z, w), then x 2 + y 2 + z 2 + w 2 = 1 characterizes those points on the unit 3-sphere centered at the origin. This 3-sphere is an example of a 3-manifold: a space which is 'looks locally' like 3-D space.
where u = (u 1, u 2, u 3) is a vector in R 3 and ‖ u ‖ 2 = u 1 2 + u 2 2 + u 3 2. In the second equality above, we have identified p with a unit quaternion and u = u 1 i + u 2 j + u 3 k with a pure quaternion. (Note that the numerator and denominator commute here even though quaternionic multiplication is generally noncommutative).
Star graphs with m equal to 1 or 2 need only dimension 1. The dimension of a complete bipartite graph K m , 2 {\displaystyle K_{m,2}} , for m ≥ 3 {\displaystyle m\geq 3} , can be drawn as in the figure to the right, by placing m vertices on a circle whose radius is less than a unit, and the other two vertices one each side of the plane of the ...
The dimension of a vector space is the number of vectors in any basis for the space, i.e. the number of coordinates necessary to specify any vector. This notion of dimension (the cardinality of a basis) is often referred to as the Hamel dimension or algebraic dimension to distinguish it from other notions of dimension.
These can be interpreted as the basis of a hypercomplex number system. Unlike the basis {e 1, ..., e k}, the remaining basis elements need not anti-commute, depending on how many simple exchanges must be carried out to swap the two factors. So e 1 e 2 = −e 2 e 1, but e 1 (e 2 e 3) = +(e 2 e 3)e 1.
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The seven lattice systems and their Bravais lattices in three dimensions. In geometry and crystallography, a Bravais lattice, named after Auguste Bravais (), [1] is an infinite array of discrete points generated by a set of discrete translation operations described in three dimensional space by
The number 1 has only a single factor, itself; each prime number has two factors, itself and 1; composite numbers are divisible by at least three different factors. Using the size of the dot representing an integer to indicate the number of factors and coloring prime numbers red and composite numbers blue produces the figure shown.