The Hyperspace IR^{n}  

The here defined concepts are most general, but closely related to the investigated by only a few extra restriction and notational addonns. The given notations are suggestions by the author, and will be used by him in future uploads the notational "r" for "real" can in actual cases be integer 

nPoint  a set of n numbers [x_{i} ; i = 0 .. n1] denoting a point in hyperspace  
nVector 
a set of n numbers <v_{i} ; i = 0 .. n1> denoting a vector in hyperspace ie the vector (arrow) between two nPoints 

the Hyperbeam ^{n}B_{{r1,..rn}} 
rectangular shaped object in hyperspace all sides free in length 

the Hypercube ^{n}B_{r} 
rectangular shaped object in hyperspace hyperbeam with equal sides 

in figures such as the hypercube the notational "r" (subscript in back) will in practical cases the hypercube order "o" which is the amount of numbers per line, thus the hypercube ^{n}B_{m} has: dimension "n" and order "m" ("B" was used to indicate the fact that a hypercube is a hyperbeam) 

the Hypersphere ^{n}S(r) 
All points equidistant from the spheres center point (say [0]), ie all nPoints (x_{i}) satisfying _{i=0}∑^{n1} x_{i}^{2} = r^{2} 

Hyperstar _{r}^{n}S^{s}_{m} 
The Hyperstar of dimension (n) can be defined with help of the n dimensional Hyperspere by placing a number of circles on the hypersphere, the intersecting points of these circles are the points of the star. The order of the star (m) is defined as the number of starpoint the step (s) is defined on a defining circle as the point that follows the previous, ie on a circle with s=1 one gets a polygon (or magon), with s=2 to m/2 the starlines intersect and the figure (per circle) shows a regular pointed star, (higher step numbers merely for a reflection of already defined figures. The stars suborder (r) is defined as the amount of numbers per line (not necessarily intersection points) (in case of even m (s = m/2) connects all points through the cicles center to the opposed point) The Hyperstar is "Regular" when all numbers (order, step and suborder) are the same on each circle. When these numbers change per circle, or on a circle the star is irregular 

in linefigures such as the hyperstar the notational "r" (subscript in front) will in practical cases the hyperstars suborder "o" which is the amount of numbers per line, thus the hyperstar _{o}^{n}S^{s}_{m} has: dimension "n", step "s", order "m" and suborder "o" 