Márton Módis, Flórián Kovács

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This paper covers aspects of establishing a relationship between the highest-order rotation-invariant moments of inertia and the order of symmetry of Platonic polyhedra. Moments of inertia about arbitrary, but centroidal axes are considered. After an introductory part which summarizes the possible applications of higher-order moments of area and inertia, the revision of the already solved two-dimensional version of this problem is presented, in other words the highest-order rotation-invariant moments of area about the origin of regular m-gons are studied. As a continuation, some aspects of the possible decomposition of spatial finite rotations into a sequence of rotations about given axes will be covered which are of great importance by the extension of the two-dimensional problem into three dimensions. Finally, the solution of the 3D problem will be presented, emphasizing the differences between the behavior of the even- and odd-order moments which is present in the 2D case, too, and also between the behaviour of the invariant moments of the tetrahedral and the octahedral or icosahedral solids. As a last part, possible applications will be presented.


moments of area and inertia, higher-order moments, rotation-invariant moments, rotation of inhomogeneous bodies, functionally graded materials

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