Titlebook: Symmetries; D. L. Johnson Textbook 2001 Springer-Verlag London 2001 Abelian group.Group theory.Groups.Symmetries.Symmetry group.polytope

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Metric Spaces and their Groups,oints. In mathematics the idea of distance, as a function that assigns a real number to a given pair of points in some space, is formalised in terms of a few reasonablelooking properties, or axioms, and the result is called a metric on that space. Having defined a structure such as this on a set, it

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aspersion

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Plane Crystallographic Groups: OR Case, of the groups ., . = 1,2,3,4,6, derived in the previous chapter. Thus . is determined by the values of ., . and of the conjugates ., ., .. As usual the treatment is case by case, greatly simplified by the following useful dichotomy.

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Tessellations of the Plane,f) at least one of the polygons. By “non-overlapping” we mean that two distinct polygons intersect, if at all, in a part of the boundary of each. Our polygons thus have a boundary and an interior, and we think of them as tiles, or tessera. For the sake of convenience, we make the additional assumpti

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Tessellations of the Sphere, points are those of any set on which distance is defined, that is, of a metric space. In this chapter we take this to be Euclidean 3-space with the Pythagorean metric. Sensibly letting the centre be the origin of coordinate and the radius be the unit of length, we get the .

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Regular Polytopes,ll be discrete and playa fundamental role: they arise even in the definition of the objects we wish to describe, which are natural generalisations to arbitrary dimension . of regular polygons in dimension 2 and polyhedra in dimension 3.

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