Abstract
The role of coordinate systems is to associate to the points of space three numbers, its coordinates. Since we will consider not only points but also lines, surfaces and volumes, we need a reference structure analogous to coordinate systems: these are the cell complexes. They are composed of vertices, edges, faces and cells, i.e. of the four types of space elements. Since a space element can be endowed with one of the two types of orientation, an inner or an outer one, we obtain eight distinct oriented space elements. To represent space elements endowed with an outer orientation, it appears natural to introduce the dual of a cell complex. These eight space elements can be easily organized in a classification diagram.
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- 1.
Isaacson and Keller [99, p. 364].
- 2.
Coxeter [44].
- 3.
- 4.
- 5.
- 6.
Also called a Dirichlet complex. See Frey and Cavendish [76].
- 7.
Cavendish et al. [37].
- 8.
- 9.
Klein [114, p. 17].
- 10.
Franz [74, p. 31].
- 11.
Hocking and Young [94, p. 223].
- 12.
Veblen and Whitehead [241, p. 55].
- 13.
A flow is uniform in a region when the velocity of all particles is invariant under translation.
- 14.
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Tonti, E. (2013). Cell Complexes. In: The Mathematical Structure of Classical and Relativistic Physics. Modeling and Simulation in Science, Engineering and Technology. Birkhäuser, New York, NY. https://doi.org/10.1007/978-1-4614-7422-7_4
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