scholarly journals On dimensions of visible parts of self-similar sets with finite rotation groups

2021 ◽  
Author(s):  
Esa Järvenpää ◽  
Maarit Järvenpää ◽  
Ville Suomala ◽  
Meng Wu
Keyword(s):  
Finite Rotation ◽  
Self Similar ◽  
Rotation Groups

Finite Rotation Groups in Low Dimensions

1968 ◽  
Vol 20 ◽  
pp. 711-719 ◽  
Author(s):  
Donald K. Friesen
Keyword(s):  
Vector Space ◽  
Finite Groups ◽  
Rotation Group ◽  
Finite Rotation ◽  
Low Dimensions ◽  
Rotation Groups

Let F be a vector space of dimension two, three, or four over a field of characteristic not two, and let V have a non-singular orthogonal metric. The problem discussed in this paper is the determination of all finite groups that can occur as subgroups of the rotation group of V.

Finite rotation groups and crystal classes in four dimensions

1951 ◽  
Vol 47 (4) ◽  
pp. 650-661 ◽  
Author(s):  
A. C. Hurley
Keyword(s):  
Euclidean Space ◽  
Three Dimensional ◽  
Three Dimensions ◽  
Explicit Calculation ◽  
Dimensional Euclidean Space ◽  
Finite Rotation ◽  
Four Dimensions ◽  
Higher Dimensional ◽  
Motion Groups ◽  
Rotation Groups

The groups of symmetries of three-dimensional lattices have been known for some time. They consist of finite rotation groups, the crystal classes, and infinite discrete motion groups, which include both rotations and translations. The general theory of the corresponding groups in higher dimensional Euclidean spaces has also been developed. This theory includes a demonstration that in Euclidean space of n dimensions the number of motion groups is finite, and leads to a method† for calculating the motion groups, the first step being to determine the crystal classes. The explicit calculation of the various groups by the general method is not simple, and has so far been confined to the case of two and three dimensions. In the special case of the crystal classes in four dimensions, however, we may make use of the results of a paper by Goursat‡. In this paper Goursat sets up a correspondence between the finite rotation groups in four-dimensional Euclidean space, and a set of groups each of which is formed by associating two of Klein's groups of linear non-homogeneous substitutions in one variable. Using this result he is able to evaluate explicitly all the proper and improper finite four-dimensional rotation groups which include the element −I, where I is the four-rowed unit matrix.

Self-similar collapse with non-radial motions

2006 ◽  
Vol 20 ◽  
pp. 1-4
Author(s):  
A. Nusser
Keyword(s):  
Self Similar
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