The Magic Encyclopedia ™

The Hypercube called "Pan"
(by Aale de Winkel)

The hypercube I call "Pan" is obtained by using the pan-n-agonal transform on the order 4k normal hypercube ⁿN_4k and thus can be formulized as:
Pan == ⁿPan_4k = Pan(ⁿN_4k) = OddSwap(ⁿN_{4k _[0,..,2k-1,4k-1,..,2k]})

which as indicated defines the hypercube "Pan" for all order 4k in any dimension
A mere peek at some of these hypercubes through various dimensions revealed to me the the hypercubes are of the curious quality
{pan-odd-agonal pan-n-agonal}
The nature of the transform I merely expected {monagonal pan-n-agonal} however it seems that any "odd" pathfinder meets up with and alternating "even" and "odd" positions and the transform places the numbers thus that all "odd" pathfinder digits sum up correctly. Investigating this further gave me new insights in "Pan", and this article intents to describe these insights and might hold other things starting from "Pan".

The spreadsheet Pan Hypercube.xls implements the formula along a path and thus far confirms the postulated for n=2..10, and doubly even orders up to 20

The Hypercube ⁿPan_4k
ⁿPan_4k = OddSwap(ⁿN_{4k _[0,..,2k-1,4k-1,..,2k]})

{pan-(2l+1 ; l=0..n\2)-agonal pan-n-agonal compact complete}

ⁿPan_4k([x_i]) = ⁿN_4k({perm[(x + ((Odd([x_i]) ? 2k : 0)) % 4k]}_i) =
_i=0∑^n-1 {perm[(x + ((Odd([x_i]) ? 2k : 0)) % 4k]}_i (4k)ⁱ
with perm = [0,..,2k-1,4k-1,..,2k]

"even" position
"odd" position an n-Point with values summing either too an odd or even number
Even([x_i]) are: [x_i] with: (_i=0∑^n-1 x_i) % 2 = 0
Odd([x_i]) are: [x_i] with: (_i=0∑^n-1 x_i) % 2 = 1

"even" vector
"odd" vector an n-Vector with values summing either too an odd or even number
Even(<x_i>) are: <x_i> with: (_i=0∑^n-1 x_i) % 2 = 0
Odd(<x_i>) are: <x_i> with: (_i=0∑^n-1 x_i) % 2 = 1

OddSwap(n-Point) half n-agonal swapping of odd n-Points
OddSwap([x_i]) = [x_i] when Even([x_i])
OddSwap([x_i]) = [((x + 2k) % 4k)_i] when Odd([x_i])

The Panmagic squares order 4 starting with ²Pan₄^{^[1,0]} (=square #178)
group of (group-)order 48 combining two groups:
GS-group of order 6 generated by {N4} {2N1 2N4} and {N2 N2} (tables vertical)
group of order 8 generated by @[2,1]_1[0,3,2,1] and _3[0,3,2,1] (tables horizontal)

square #178 @[2,1]_1[0,3,2,1] (@[2,1]_1[0,3,2,1])² (@[2,1]_1[0,3,2,1])³

_3[0,3,2,1] _3[0,3,2,1] _3[0,3,2,1]

_3[0,3,2,1] @[2,1]_1[0,3,2,1] @[2,1]_2[0,3,2,1] @[0,2] @[0,2]_3[0,3,2,1] @[2,3]_1[0,3,2,1] @[2,3]_2[0,3,2,1]

{N4} 01 12 13 08
15 06 03 10
04 09 16 05
14 07 02 11 01 08 13 12
14 11 02 07
04 05 16 09
15 10 03 06 03 06 15 10
16 09 04 05
02 07 14 11
13 12 01 08 03 10 15 06
13 08 01 12
02 11 14 07
16 05 04 09 04 09 16 05
14 07 02 11
01 12 13 08
15 06 03 10 04 05 16 09
15 10 03 06
01 08 13 12
14 11 02 07 02 07 14 11
13 12 01 08
03 06 15 10
16 09 04 05 02 11 14 07
16 05 04 09
03 10 15 06
13 08 01 12

{2N1 2N4} 01 14 07 12
15 04 09 06
10 05 16 03
08 11 02 13 01 12 07 14
08 13 02 11
10 03 16 05
15 06 09 04 09 04 15 06
16 05 10 03
02 11 08 13
07 14 01 12 09 06 15 04
07 12 01 14
02 13 08 11
16 03 10 05 10 05 16 03
08 11 02 13
01 14 07 12
15 04 09 06 10 03 16 05
15 06 09 04
01 12 07 14
08 13 02 11 02 11 08 13
07 14 01 12
09 04 15 06
16 05 10 03 02 13 08 11
16 03 10 05
09 06 15 04
07 12 01 14

{N2 N2} 01 14 11 08
15 04 05 10
06 09 16 03
12 07 02 13 01 08 11 14
12 13 02 07
06 03 16 09
15 10 05 04 05 04 15 10
16 09 06 03
02 07 12 13
11 14 01 08 05 10 15 04
11 08 01 14
02 13 12 07
16 03 06 09 06 09 16 03
12 07 02 13
01 14 11 08
15 04 05 10 06 03 16 09
15 10 05 04
01 08 11 14
12 13 02 07 02 07 12 13
11 14 01 08
05 04 15 10
16 09 06 03 02 13 12 07
16 03 06 09
05 10 15 04
11 08 01 14

{{2N1 2N4} (2N1 2N4)} 01 08 11 14
15 10 05 04
06 03 16 09
12 13 02 07 01 14 11 08
12 07 02 13
06 09 16 03
15 04 05 10 05 10 15 04
16 03 06 09
02 13 12 07
11 08 01 14 05 04 15 10
11 14 01 08
02 07 12 13
16 09 06 03 06 03 16 09
12 13 02 07
01 08 11 14
15 10 05 04 06 09 16 03
15 04 05 10
01 14 11 08
12 07 02 13 02 13 12 07
11 08 01 14
05 10 15 04
16 03 06 09 02 07 12 13
16 09 06 03
05 04 15 10
11 14 01 08

{{N2 N2} (2N1 2N4)} 01 12 07 14
15 06 09 04
10 03 16 05
08 13 02 11 01 14 07 12
08 11 02 13
10 05 16 03
15 04 09 06 09 06 15 04
16 03 10 05
02 13 08 11
07 12 01 14 09 04 15 06
07 14 01 12
02 11 08 13
16 05 10 03 10 03 16 05
08 13 02 11
01 12 07 14
15 06 09 04 10 05 16 03
15 04 09 06
01 14 07 12
08 11 02 13 02 13 08 11
07 12 01 14
09 06 15 04
16 03 10 05 02 11 08 13
16 05 10 03
09 04 15 06
07 14 01 12

{{2N1 2N4} (N2 N2)} 01 08 13 12
15 10 03 06
04 05 16 09
14 11 02 07 01 12 13 08
14 07 02 11
04 09 16 05
15 06 03 10 03 10 15 06
16 05 04 09
02 11 14 07
13 08 01 12 03 06 15 10
13 12 01 08
02 07 14 11
16 09 04 05 04 05 16 09
14 11 02 07
01 08 13 12
15 10 03 06 04 09 16 05
15 06 03 10
01 12 13 08
14 07 02 11 02 11 14 07
13 08 01 12
03 10 15 06
16 05 04 09 02 07 14 11
16 09 04 05
03 06 15 10
13 12 01 08

The Hypercube ⁿPan_4k
ⁿPan_4k = OddSwap(ⁿN_{4k _[0,..,2k-1,4k-1,..,2k]})
{pan-(2l+1 ; l=0..n\2)-agonal pan-n-agonal compact complete}
ⁿPan_4k([x_i]) = ⁿN_4k({perm[(x + ((Odd([x_i]) ? 2k : 0)) % 4k]}_i) = _i=0∑^n-1 {perm[(x + ((Odd([x_i]) ? 2k : 0)) % 4k]}_i (4k)ⁱ with perm = [0,..,2k-1,4k-1,..,2k]
"even" position "odd" position	an n-Point with values summing either too an odd or even number
"even" position "odd" position	Even([x_i]) are: [x_i] with: (_i=0∑^n-1 x_i) % 2 = 0 Odd([x_i]) are: [x_i] with: (_i=0∑^n-1 x_i) % 2 = 1
"even" vector "odd" vector	an n-Vector with values summing either too an odd or even number
"even" vector "odd" vector	Even(<x_i>) are: <x_i> with: (_i=0∑^n-1 x_i) % 2 = 0 Odd(<x_i>) are: <x_i> with: (_i=0∑^n-1 x_i) % 2 = 1
OddSwap(n-Point)	half n-agonal swapping of odd n-Points
OddSwap(n-Point)	OddSwap([x_i]) = [x_i] when Even([x_i]) OddSwap([x_i]) = [((x + 2k) % 4k)_i] when Odd([x_i])

The Panmagic squares order 4 starting with ²Pan₄^{^[1,0]} (=square #178)
group of (group-)order 48 combining two groups: GS-group of order 6 generated by {N4} {2N1 2N4} and {N2 N2} (tables vertical) group of order 8 generated by @[2,1]_1[0,3,2,1] and _3[0,3,2,1] (tables horizontal)
	square #178		@[2,1]_1[0,3,2,1]		(@[2,1]_1[0,3,2,1])²		(@[2,1]_1[0,3,2,1])³
	square #178			_3[0,3,2,1]		_3[0,3,2,1]		_3[0,3,2,1]
		_3[0,3,2,1]	@[2,1]_1[0,3,2,1]	@[2,1]_2[0,3,2,1]	@[0,2]	@[0,2]_3[0,3,2,1]	@[2,3]_1[0,3,2,1]	@[2,3]_2[0,3,2,1]
{N4}	01 12 13 08 15 06 03 10 04 09 16 05 14 07 02 11	01 08 13 12 14 11 02 07 04 05 16 09 15 10 03 06	03 06 15 10 16 09 04 05 02 07 14 11 13 12 01 08	03 10 15 06 13 08 01 12 02 11 14 07 16 05 04 09	04 09 16 05 14 07 02 11 01 12 13 08 15 06 03 10	04 05 16 09 15 10 03 06 01 08 13 12 14 11 02 07	02 07 14 11 13 12 01 08 03 06 15 10 16 09 04 05	02 11 14 07 16 05 04 09 03 10 15 06 13 08 01 12
{2N1 2N4}	01 14 07 12 15 04 09 06 10 05 16 03 08 11 02 13	01 12 07 14 08 13 02 11 10 03 16 05 15 06 09 04	09 04 15 06 16 05 10 03 02 11 08 13 07 14 01 12	09 06 15 04 07 12 01 14 02 13 08 11 16 03 10 05	10 05 16 03 08 11 02 13 01 14 07 12 15 04 09 06	10 03 16 05 15 06 09 04 01 12 07 14 08 13 02 11	02 11 08 13 07 14 01 12 09 04 15 06 16 05 10 03	02 13 08 11 16 03 10 05 09 06 15 04 07 12 01 14
{N2 N2}	01 14 11 08 15 04 05 10 06 09 16 03 12 07 02 13	01 08 11 14 12 13 02 07 06 03 16 09 15 10 05 04	05 04 15 10 16 09 06 03 02 07 12 13 11 14 01 08	05 10 15 04 11 08 01 14 02 13 12 07 16 03 06 09	06 09 16 03 12 07 02 13 01 14 11 08 15 04 05 10	06 03 16 09 15 10 05 04 01 08 11 14 12 13 02 07	02 07 12 13 11 14 01 08 05 04 15 10 16 09 06 03	02 13 12 07 16 03 06 09 05 10 15 04 11 08 01 14
{{2N1 2N4} (2N1 2N4)}	01 08 11 14 15 10 05 04 06 03 16 09 12 13 02 07	01 14 11 08 12 07 02 13 06 09 16 03 15 04 05 10	05 10 15 04 16 03 06 09 02 13 12 07 11 08 01 14	05 04 15 10 11 14 01 08 02 07 12 13 16 09 06 03	06 03 16 09 12 13 02 07 01 08 11 14 15 10 05 04	06 09 16 03 15 04 05 10 01 14 11 08 12 07 02 13	02 13 12 07 11 08 01 14 05 10 15 04 16 03 06 09	02 07 12 13 16 09 06 03 05 04 15 10 11 14 01 08
{{N2 N2} (2N1 2N4)}	01 12 07 14 15 06 09 04 10 03 16 05 08 13 02 11	01 14 07 12 08 11 02 13 10 05 16 03 15 04 09 06	09 06 15 04 16 03 10 05 02 13 08 11 07 12 01 14	09 04 15 06 07 14 01 12 02 11 08 13 16 05 10 03	10 03 16 05 08 13 02 11 01 12 07 14 15 06 09 04	10 05 16 03 15 04 09 06 01 14 07 12 08 11 02 13	02 13 08 11 07 12 01 14 09 06 15 04 16 03 10 05	02 11 08 13 16 05 10 03 09 04 15 06 07 14 01 12
{{2N1 2N4} (N2 N2)}	01 08 13 12 15 10 03 06 04 05 16 09 14 11 02 07	01 12 13 08 14 07 02 11 04 09 16 05 15 06 03 10	03 10 15 06 16 05 04 09 02 11 14 07 13 08 01 12	03 06 15 10 13 12 01 08 02 07 14 11 16 09 04 05	04 05 16 09 14 11 02 07 01 08 13 12 15 10 03 06	04 09 16 05 15 06 03 10 01 12 13 08 14 07 02 11	02 11 14 07 13 08 01 12 03 10 15 06 16 05 04 09	02 07 14 11 16 09 04 05 03 06 15 10 13 12 01 08