Rotations in the Space of Split Octonions

The geometrical application of split octonions is considered. The new representation of products of the basis units of split octonionic having David's star shape (instead of the Fano triangle) is presented. It is shown that active and passive transformations of coordinates in octonionic “eight-space” are not equivalent. The group of passive transformations that leave invariant the pseudonorm of split octonions isSO(4,4), while active rotations are done by the direct product ofO(3,4)-boosts and real noncompact form of the exceptional groupG2. In classical limit, these transformations reduce to the standard Lorentz group.

Random nested tetrahedra

In a real n-1 dimensional affine space E, consider a tetrahedron T 0, i.e. the convex hull of n points α1, α2, …, α n of E. Choose n independent points β1, β2, …, β n randomly and uniformly in T 0, thus obtaining a new tetrahedron T 1 contained in T 0. Repeat the operation with T 1 instead of T 0, obtaining T 2, and so on. The sequence of the T k shrinks to a point Y of T 0 and this note computes the distribution of the barycentric coordinates of Y with respect to (α1, α2, …, α n ) (Corollary 2.3). We also obtain the explicit distribution of Y in more general cases. The technique used is to reduce the problem to the study of a random walk on the semigroup of stochastic (n,n) matrices, and this note is a geometrical application of a former result of Chamayou and Letac (1994).

Random nested tetrahedra

In a real n-1 dimensional affine space E, consider a tetrahedron T0, i.e. the convex hull of n points α1, α2, …, αn of E. Choose n independent points β1, β2, …, βn randomly and uniformly in T0, thus obtaining a new tetrahedron T1 contained in T0. Repeat the operation with T1 instead of T0, obtaining T2, and so on. The sequence of the Tk shrinks to a point Y of T0 and this note computes the distribution of the barycentric coordinates of Y with respect to (α1, α2, …, αn) (Corollary 2.3). We also obtain the explicit distribution of Y in more general cases. The technique used is to reduce the problem to the study of a random walk on the semigroup of stochastic (n,n) matrices, and this note is a geometrical application of a former result of Chamayou and Letac (1994).

Perspectives in Open Spaces: A Geometrical Application of the Thouless Index

A geometrical model for computing ‘general perspectives’ is discussed. It is based on the power function r = p(d/t)1- i, where i is the Thouless index for the phenomenal regression to the real object, r is the real size of the object, p is the apparent size, d is the distance between the subject and the object, and t is the distance between the subject and the projection plane. This model assumes that i is invariant for different distances and this was verified in seventy children and adults at distances of 15 to 120 m. A computer program draws families of curved perspectives which are well-fitted to the actual shape of large visual alleys produced by experiment in open fields.