This paper challenges some of the common assumptions underlying the mathematics used to describe the physical world. We start by reviewing many of the assumptions underlying the concepts of real, physical, rigid bodies and the translational and rotational properties of such rigid bodies. Nearly all elementary and advanced texts make physical assumptions that are subtly different from ours, and as a result we develop a mathematical description that is subtly different from the standard mathematical structure. Using the homogeneity and isotropy of space, we investigate the translational and rotational features of rigid bodies in two and three dimensions. We find that the concept of rigid bodies and the concept of the homogeneity of space are intrinsically linked. The geometric study of rotations of rigid objects leads to a geometric product relationship for lines and vectors. By requiring this product to be both associative and to satisfy Pythagoras' theorem, we obtain a choice of Clifford algebras. We extend our arguments from space to include time. By assuming that ct=l and rewriting this in Lorentz invariant form as c^2t^2-x^2-y^2-z^2=0 we obtain a generalization of Pythagoras to spacetime. This leads us directly to establishing that the Clifford algebra CL(1,3) is an appropriate mathematical structure to describe spacetime. Clifford algebras are not division algebras. We show that the existence of non-invertible elements in the algebra is not a limitation of the usefulness to physics of the algebra but rather that it reflects accurately the spacetime properties of physical systems.
Inverse Elastic Scattering for Multiscale Rigid Bodies with A Single Far-field Pattern
Weierstrass Institute（韦伊尔斯特拉斯研究所）；South University of Science and Technology of China（南方科技大学）；University of North Carolina, Charlotte（北卡罗来纳大学夏洛特分校）；University of Graz（格拉茨大学）
