Magic HyperStar

In The Magic Encyclopedia ™ I defined two variations of the HyperStar thus:

 The Regular Hyperstar onSsm(sum) 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 (o) 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 The Magic sum partakes in the hyperstars notation since thus far contineous sequences are seldom and the actual sum is dependent on their use (trick is also to minimize that sum) The Irregular Hyperstar {o1,..,o?}nS{s1,..,s?}{m1,..,m?} Defined as is the regular hyperstar but allows varying step "s", order "m" and/or suborder "o" (currently loosely defined, but allows for 2S{2,2,3}6) Depending on future development subclassification might be defined.

In order to define a magic hyperstar of either type a amount of numbers are placed upon each line. For this moment I only concider one amount of number on each line, certainly possible however to define a varying amount of numbers. In order to stay consistent with regular star terminolgy the amount of numbers per line is called the hyperstars suborder, in the above notation this suborder takes the place of the subscripted r in front of the notation.

<Harvey Heinz> investigated in dimension 2 the normal star which is defined with suborder 4, thus:

 ADWORKS The Magic Encyclopedia ™ notation Harvey Heinz: Normal Star: Nsm ≡ 42Ssm The Magic Encyclopedia ™ article

with all possible steps 's' and orders 'm'.
Together we investigated the possibility of two intertwined pyramids, the only possible solution we could come up with was to leave three gaps in the number range of the figure giving the order 8 pyramid star 33S18 embedded in:

The line 3-9-15 was added as a satelite, the entire figure is a perfect 3 dimensional order 8 magic linefigure with suborder 3

The tetrahedrons below are in convenient notation the first number the tetrahedrons top, the second three numbers are actually plased on the tetrahedrons first floor connecting with the 1st, 3rd, and 5th number of the third line which makes up the tetrahedrons base triangle.

The Magic Tetrahedron 33S13
33S13(15) 01
06 09 11
08 02 05 07 03 04
33S13(21) 11
06 03 01
04 10 07 05 09 08
triangle based pyramid star 33S18
33S18(27) 02
11 13 17
14 01 12 07 08 05
06
05 11 17
16 01 10 13 04 07

The following table show magic square based pyramids in convenient square notation, the central number of each square is actually the pyramids top, the four numbers towards the corners is the first floor of the pyramids, the surrounding numbers the pyramids floor.

The Magic Square based pyramid 33S15
33S15(18) 01 15 02
08 07
14 09 12
06 05
03 11 04
33S15(24) 08 09 07
01 02
10 15 12
03 04
06 13 05
33S15(24) 08 07 09
15 14
06 01 04
13 12
10 03 11
33S15(30) 15 01 14
08 09
02 07 04
10 11
13 05 12
square based pyramid star 33S110
33S110(36)
01 33 02
08 07
32 27 30
06 05
03 29 04
09 17 10
08 07
16 19 14
06 05
11 13 12
33S110(42)
01 39 02
15 14
38 26 36
13 12
03 35 04
16 09 17
15 14
08 11 06
13 12
18 05 19
33S110(66)
33 01 32
26 27
02 07 04
28 29
31 05 30
25 17 24
26 27
18 15 20
28 29
23 21 22
33S110(78)
39 01 38
25 26
02 14 04
27 28
37 05 36
24 31 23
25 26
32 29 34
27 28
22 35 21