By Omer Cabrera
Desk of Contents
Chapter 1 - Symmetry
Chapter 2 - crew (Mathematics)
Chapter three - staff Action
Chapter four - general Polytope
Chapter five - Lie element Symmetry
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Additional info for Applications of Symmetry in Mathematics, Physics & Chemistry
This is analogous to the way a CAT scan reassembles two-dimensional images to form a 3-dimensional representation of the organs being scanned. The ideal would be an animated hologram of some sort, however, even a simple animation such as the one shown can already give some limited insight into the structure of the polytope. Another way a three-dimensional viewer can comprehend the structure of a fourdimensional polychoron is through being "immersed" in the object, perhaps via some form of virtual reality technology.
A cyclic group is a group all of whose elements are powers (when the group operation is written additively, the term 'multiple' can be used) of a particular element a. , where a2 means a • a, and a−3 stands for a−1 • a−1 • a−1=(a • a • a)−1 etc Such an element a is called a generator or a primitive element of the group. A typical example for this class of groups is the group of n-th complex roots of unity, given by complex numbers z satisfying zn = 1 (and whose operation is multiplication). Any cyclic group with n elements is isomorphic to this group.
There are exactly three in each higher dimension, which correspond to the tetrahedron, cube and octahedron: these are the regular simplices, measure polytopes and cross polytopes. Descriptions of these may be found in the List of regular polytopes. Also of interest are the nonconvex regular 4-polytopes, partially discovered by Schläfli. By the end of the 19th century, mathematicians such as Arthur Cayley and Ludwig Schläfli had developed the theory of regular polytopes in four and higher dimensions, such as the tesseract and the 24-cell.