While I was diving the the Philippines I saw this blue basket starfish and its dimensions and patterns struck an inspirational cord with me. I had been reading Michio Kaku’s book on Parallel Worlds that is a journey into higher dimensions and the string theory seemed to describe what I saw in the geometry of the starfish. Maybe is is a portal into a parallel dimension, after all it is a Starfish. What do you think?
Six-dimensional space is any space that has six dimensions, that is six degrees of freedom, and that needs six pieces of data, or coordinates, to specify a location in this space. There are an infinite number of these, but those of most interest are simpler ones that model some aspect of our environment. Of particular interest is six-dimensional Euclidean space, in which 6-polytopes and the 5-sphere are constructed. Six-dimensional elliptical space and hyperbolic spaces are also studied, with constant positive and negative curvature.
Formally six-dimensional Euclidean space, ?6, is generated by considering all real6-tuples as 6-vectors in this space. As such it has the properties of all Euclidian spaces, so it is linear, has a metric and a full set of vector operations. In particular the dot product between two 6-vectors is readily defined, and can be used to calculate the metric. 6 × 6 matrices can be used to describe transformations such as rotations that keep the origin fixed.
In three dimensional space a generalised transformation has six degrees of freedom, three translations along the three coordinate axes and three from the set of rotations, SO(3). Often these transformations are handled separately as they have very different geometrical structures, but there are ways of dealing with them that treat them as a single six-dimensional object.
A polytope in six dimensions is called a 6-polytope. The most studied are the regular polytopes, of which there are only three is six dimensions. A wider family are the uniform 6-polytopes, constructed from fundamental symmetry domains of reflection, each domain defined by a Coxeter-group. Each uniform polytope is defined by a ringed Coxeter-Dynkin diagram.
Six-dimensional space ?=?/E6 .0436111111111111111
The 221 has 27 vertices, and 99 facets: 27 5-orthoplexes and 72 5-simplices.
The 6-sphere, or hypersphere in seven dimensions, is the six-dimensional surface equidistant from a point. It has symbol S6. and the equation for the 6-sphere, radius r, centre the origin is
The volume of the space bounded by this 6-sphere is
which is 4.72477 × r7, or 0.0369 of the smalls 7-cube that contains the 6-sphere.
Note: Some of this is explained here: http://en.wikipedia.org/wiki/Six-dimensional_space
and here is a Nova program that investigates the String Theory: http://www.pbs.org/wgbh/nova/physics/imagining-other-dimensions.html
Description of art work: 2-21-polytope family polytopes in Coxeter plane symmetry provided by Tomruen 11/29/10