The Magic Encyclopedia ™

The Hyperspace IRⁿ
(by Aale de Winkel)

In order to define several objects I resort to regular mathematical hyperspace IRⁿ
The is not a recaption of IRⁿ vector algebra, but just a definition of
the stuff needed in our invstigations of magic objects

The Hyperspace IRⁿ
The here defined concepts are most general, but closely related to the investigated
by only a few extra restriction and notational add-onns. 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

n-Point a set of n numbers [x_i ; i = 0 .. n-1] denoting a point in hyperspace

n-Vector a set of n numbers <v_i ; i = 0 .. n-1> denoting a vector in hyperspace
ie the vector (arrow) between two n-Points

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

the Hypercube ⁿB_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 ⁿB_m has: dimension "n" and order "m"
("B" was used to indicate the fact that a hypercube is a hyperbeam)

the Hypersphere ⁿS(r) All points equidistant from the spheres center point (say [0]),
ie all n-Points (x_i) satisfying _i=0∑^n-1 x_i² = r²

Hyperstar _rⁿS^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 m-agon),
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 _oⁿS^s_m has:
dimension "n", step "s", order "m" and suborder "o"

The Hyperspace IRⁿ
The here defined concepts are most general, but closely related to the investigated by only a few extra restriction and notational add-onns. 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
n-Point	a set of n numbers [x_i ; i = 0 .. n-1] denoting a point in hyperspace
n-Vector	a set of n numbers <v_i ; i = 0 .. n-1> denoting a vector in hyperspace ie the vector (arrow) between two n-Points
the Hyperbeam ⁿB_{r1,..rn}	rectangular shaped object in hyperspace all sides free in length
the Hypercube ⁿB_r	rectangular shaped object in hyperspace hyperbeam with equal sides
the Hypercube ⁿB_r	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 ⁿB_m has: dimension "n" and order "m" ("B" was used to indicate the fact that a hypercube is a hyperbeam)
the Hypersphere ⁿS(r)	All points equidistant from the spheres center point (say [0]), ie all n-Points (x_i) satisfying _i=0∑^n-1 x_i² = r²
Hyperstar _rⁿS^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 m-agon), 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
Hyperstar _rⁿS^s_m	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 _oⁿS^s_m has: dimension "n", step "s", order "m" and suborder "o"