The Magic Encyclopedia ™

The Pan n-agonal Transform
(by Aale de Winkel)

Some time ago I constructed the "pan n-agonal transform" as a means to obtain {pan n-agonal magic} hypercubes, these also turn out to be {compact complete}. At that time Kathleen Hollerenshaw book on "most-perfect" squares came out. (Other files hold this qualification for the hypercubes obtained by my method, but I'm changing this to {compact complete}). The Transform be refered to as Pan(ⁿH_4k) ⁿH_4k being te "seed" or "generating hypercube" which serves as a starting point where the transform acts upon The order 2 hypercube ⁿN₂^analitic place an important role

The simplest of these objects I now call "Pan" is the transform seeded by 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
More hypercubes of the quality {pan-n-agonal compact complete} from "Pan" by application of dynamic numbering giving hypercubes "{GS} Pan", the n=2 (square) investigation show the group, combined with yet another group show that all {pandiagonal compact complete} square form a single group of grouporder 48. Simular thing expected for the cubes which is under investigation, seen several samples however the complete set of all {pantriagonal compact complete} order 4 cube isn't known yet (at least to me)

the pan n-agonal transform
The pan n-agonal tranform Pan(ⁿH_4k) constructs order 4k hypercubes
the complete proces is defined by the following two steps procedure
For all dimensions and orders 4k this defines the hypercube:
Pan == ⁿPan_4k = OddSwap(ⁿN_{4k _[0,..,2k-1,4k-1,..,2k]})

Reflector
ⁿN₂^analitic the order 2 n dimensional hypercube in analitic numberrange
ⁿN₂[_ji] = _j=0∑¹_ji m ^j
{pan n-agonal blockwise}

the first step of the transform is to divide the order 4k hypercube into
hyperquadrants, and reflect each quadrants content in the manner which
corresponds to the reflectors number which maps its number to the hyperquadrants
R(ⁿH_4k) == ⁿH_4k^{~ⁿN₂^analitic} == ⁿH_4k _{_[0,..,2k-1,4k-1,..,2k]}
Note: this defines the notation
while the last formulation identifies it as an n-agonal permutation

m/2 (= 2k) swap swapping numbers m/2 cells apart
the second step of the transform is to swap on each 1 agonal every other number
with the number m/2 further on the diagonal, note here that only half the
1-agonal is traversed and only the odd cells are swapped. where "odd-cell"
might be defined as the cells whose coordinate sum to an odd number
OddSwap(ⁿH_4k) :
R(ⁿH_4k)[_ji] <-> R(ⁿH_4k)[_ji + 2k] iff (_j=0∑ⁿ_ji % 2 == 1)

Although the transform is described by 2 steps the second is best be done by n seperate
steps for each 1-agonal direction at a time, but this is merely for practical reasons
seeding the transform
Currently the transform is applied on generating hypercubes that are {pan n-agonal} by nature
further these seeds are merely {blockwise) ie the 1-agonal sums are off by the same amount as
the 1-agonal reflected into the hyper cubes center. (curently the following are known to work:)

The Normal Hypercube
ⁿN_m each number in sequence
ⁿN_m[_ji] = _j=0∑^n-1_ji m ^j
(Note: expression result in analitic number range [0..mⁿ-1])
{pan n-agonal blockwise}

The Normal hypercube is the hypercube where all numbers are listed in their
natural sequence

direct multiplications
ⁿN₂ ⁿH_2k{pan n-agonal} direct multiplication of order 2 hypercube with {pan n-agonal} order 2k hypercube
{pan n-agonal blockwise}
the direct multiplication of the order 2 hypercube with any order 2k hypercube
form a {blockwise} hypercube, when the order 2k is {pan n-agonal} the blockwise
is also {pan n-agonal} and thus a good seed for the transform
note: that this is a recursive statement allowing the order 2k hypercube to be simular defined

though the quality of the seeds known to work is {pan n-agonal blockwise}, experience shows that thus
quaifiable hypercubes like N₂ Pan(N_4k) does not result in {pan n-agonal} but in {s-pan n-agonal}
hypercubes. Uptil now it seems that direct multiplications of Normal hypercubes work in whatever combination,
future examination might pinpoint the exact seed quality in more detail.

Note: the table below present some order 4 samples, these will also be uploaded onto the database more listings of will be presented there, I might even try the order 4 tesseract

Pan diagonal transform
complete derivaton of order 4 {compact complete pantriagonal magic} square families
²N₄
01 02 03 04
05 06 07 08
09 10 11 12
13 14 15 16 [ ²N₄ ]^{~²N₂^analitic}
01 02 04 03
05 06 08 07
13 14 16 15
09 10 12 11 half alternate swap
01 03 04 02
08 06 05 07
13 15 16 14
12 10 09 11 Pan(N₄)
01 15 04 14
12 06 09 07
13 03 16 02
08 10 05 11
²N₂N₂
01 02 05 06
03 04 07 08
09 10 13 14
11 12 15 16 [ ²N₂²N₂ ]^{~²N₂^analitic}
01 02 06 05
03 04 08 07
11 12 16 15
09 10 14 13 half alternate swap
01 05 06 02
08 04 03 07
11 15 16 12
14 10 09 13 Pan(N₂N₂)
01 15 06 12
14 04 09 07
11 05 16 02
08 10 03 13
²N₂N₂^t
01 03 05 07
02 04 06 08
09 11 13 15
10 12 14 16 [ ²N₂²N₂^t ]^{~²N₂^analitic}
01 03 07 05
02 04 08 06
10 12 16 14
09 11 15 13 half alternate swap
01 05 07 03
08 04 02 06
10 14 16 12
15 11 09 13 Pan(N₂N₂^t)
01 14 07 12
15 04 09 06
10 05 16 03
08 11 02 13

Pan triagonal transform
complete derivaton of order 4 {compact complete pantriagonal magic} cube families
³N₂ Aspect(³N₂)
00+a 00+b 08+a 08+b
00+c 00+d 08+c 08+d
16+a 16+b 24+a 24+b
16+c 16+d 24+c 24+d 00+e 00+f 08+e 08+f
00+g 00+h 08+g 08+h
16+e 16+f 24+e 24+f
16+g 16+h 24+g 24+h 32+a 32+b 40+a 40+b
32+c 32+d 40+c 40+d
48+a 48+b 56+a 56+b
48+c 48+d 56+c 56+d 32+e 32+f 40+e 40+f
32+g 32+h 40+g 40+h
48+e 48+f 56+e 56+f
48+g 48+h 56+g 56+h
[ ³N₂ Aspect(³N₂) ]^{~³N₂^analitic}
00+a 00+b 08+b 08+a
00+c 00+d 08+d 08+c
16+c 16+d 24+d 24+c
16+a 16+b 24+b 24+a 00+e 00+f 08+f 08+e
00+g 00+h 08+h 08+g
16+g 16+h 24+h 24+g
16+e 16+f 24+f 24+e 32+e 32+f 40+f 40+e
32+g 32+h 40+h 40+g
48+g 48+h 56+h 56+g
48+e 48+f 56+f 56+e 32+a 32+b 40+b 40+a
32+c 32+d 40+d 40+c
48+c 48+d 56+d 56+c
48+a 48+b 56+b 56+a
alternate swap of m/2 aparts (done 3 times)
00+a 08+a 08+b 00+b
08+d 00+d 00+c 08+c
16+c 24+c 24+d 16+d
24+b 16+b 16+a 24+a 08+f 00+f 00+e 08+e
00+g 08+g 08+h 00+h
24+h 16+h 16+g 24+g
16+e 24+e 24+f 16+f 32+e 40+e 40+f 32+f
40+h 32+h 32+g 40+g
48+g 56+g 56+h 48+h
56+f 48+f 48+e 56+e 40+b 32+b 32+a 40+a
32+c 40+c 40+d 32+d
56+d 48+d 48+c 56+c
48+a 56+a 56+b 48+b
00+a 24+c 08+b 16+d
24+b 00+d 16+a 08+c
16+c 08+b 24+d 00+b
08+d 16+b 00+c 24+a 24+h 00+f 16+g 08+e
00+g 24+e 08+h 16+f
08+f 16+h 00+e 24+g
16+e 08+g 24+f 00+h 32+e 56+g 40+f 48+h
56+f 32+h 48+e 40+g
48+g 40+e 56+h 32+f
40+h 48+f 32+g 56+e 56+d 32+b 48+c 40+a
32+c 56+a 40+d 48+b
40+b 48+d 32+a 56+c
48+a 40+c 56+b 32+d
Pan(³N₂ Aspect(³N₂))
00+a 56+g 08+b 48+h
56+f 00+d 48+e 08+c
16+c 40+e 24+d 32+f
40+h 16+b 32+g 24+a 56+d 00+f 48+c 08+e
00+g 56+a 08+h 48+b
40+b 16+h 32+a 24+g
16+e 40+c 24+f 32+d 32+e 24+c 40+f 16+d
24+b 32+h 16+a 40+g
48+g 08+b 56+h 00+b
08+d 48+f 00+c 56+e 24+h 32+b 16+g 40+a
32+c 24+e 40+d 16+f
08+f 48+d 00+e 56+c
48+a 08+g 56+b 00+h
Pan(³N₂ ³N₂)
01 63 10 56
62 04 53 11
19 45 28 38
48 18 39 25 60 06 51 13
07 57 16 50
42 24 33 31
21 43 30 36 37 27 46 20
26 40 17 47
55 09 64 02
12 54 03 61 32 34 23 41
35 29 44 22
14 52 05 59
49 15 58 08
Note: This cube is a {compact complete pantriagonal} cube
each order 2 sub(hyper)cube has a total sum of 260 (= 2² (4³+1) )
each triagonally cells 2 apart sum 65 (= 4³+1 )
magic sum 130 (= 2 (4³+1) ) of course

the pan n-agonal transform
The pan n-agonal tranform Pan(ⁿH_4k) constructs order 4k hypercubes the complete proces is defined by the following two steps procedure For all dimensions and orders 4k this defines the hypercube: Pan == ⁿPan_4k = OddSwap(ⁿN_{4k _[0,..,2k-1,4k-1,..,2k]})
Reflector ⁿN₂^analitic	the order 2 n dimensional hypercube in analitic numberrange
	ⁿN₂[_ji] = _j=0∑¹_ji m ^j
	{pan n-agonal blockwise}
	the first step of the transform is to divide the order 4k hypercube into hyperquadrants, and reflect each quadrants content in the manner which corresponds to the reflectors number which maps its number to the hyperquadrants
	R(ⁿH_4k) == ⁿH_4k^{~ⁿN₂^analitic} == ⁿH_4k _{_[0,..,2k-1,4k-1,..,2k]} Note: this defines the notation while the last formulation identifies it as an n-agonal permutation
m/2 (= 2k) swap	swapping numbers m/2 cells apart
	the second step of the transform is to swap on each 1 agonal every other number with the number m/2 further on the diagonal, note here that only half the 1-agonal is traversed and only the odd cells are swapped. where "odd-cell" might be defined as the cells whose coordinate sum to an odd number
	OddSwap(ⁿH_4k) : R(ⁿH_4k)[_ji] <-> R(ⁿH_4k)[_ji + 2k] iff (_j=0∑ⁿ_ji % 2 == 1)
Although the transform is described by 2 steps the second is best be done by n seperate steps for each 1-agonal direction at a time, but this is merely for practical reasons
seeding the transform
Currently the transform is applied on generating hypercubes that are {pan n-agonal} by nature further these seeds are merely {blockwise) ie the 1-agonal sums are off by the same amount as the 1-agonal reflected into the hyper cubes center. (curently the following are known to work:)
The Normal Hypercube ⁿN_m	each number in sequence
	ⁿN_m[_ji] = _j=0∑^n-1_ji m ^j (Note: expression result in analitic number range [0..mⁿ-1])
	{pan n-agonal blockwise}
	The Normal hypercube is the hypercube where all numbers are listed in their natural sequence
direct multiplications ⁿN₂ ⁿH_2k{pan n-agonal}	direct multiplication of order 2 hypercube with {pan n-agonal} order 2k hypercube
	{pan n-agonal blockwise}
	the direct multiplication of the order 2 hypercube with any order 2k hypercube form a {blockwise} hypercube, when the order 2k is {pan n-agonal} the blockwise is also {pan n-agonal} and thus a good seed for the transform note: that this is a recursive statement allowing the order 2k hypercube to be simular defined
though the quality of the seeds known to work is {pan n-agonal blockwise}, experience shows that thus quaifiable hypercubes like N₂ Pan(N_4k) does not result in {pan n-agonal} but in {s-pan n-agonal} hypercubes. Uptil now it seems that direct multiplications of Normal hypercubes work in whatever combination, future examination might pinpoint the exact seed quality in more detail.

Pan diagonal transform
complete derivaton of order 4 {compact complete pantriagonal magic} square families
²N₄ 01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16	[ ²N₄ ]^{~²N₂^analitic} 01 02 04 03 05 06 08 07 13 14 16 15 09 10 12 11	half alternate swap 01 03 04 02 08 06 05 07 13 15 16 14 12 10 09 11	Pan(N₄) 01 15 04 14 12 06 09 07 13 03 16 02 08 10 05 11
²N₂N₂ 01 02 05 06 03 04 07 08 09 10 13 14 11 12 15 16	[ ²N₂²N₂ ]^{~²N₂^analitic} 01 02 06 05 03 04 08 07 11 12 16 15 09 10 14 13	half alternate swap 01 05 06 02 08 04 03 07 11 15 16 12 14 10 09 13	Pan(N₂N₂) 01 15 06 12 14 04 09 07 11 05 16 02 08 10 03 13
²N₂N₂^t 01 03 05 07 02 04 06 08 09 11 13 15 10 12 14 16	[ ²N₂²N₂^t ]^{~²N₂^analitic} 01 03 07 05 02 04 08 06 10 12 16 14 09 11 15 13	half alternate swap 01 05 07 03 08 04 02 06 10 14 16 12 15 11 09 13	Pan(N₂N₂^t) 01 14 07 12 15 04 09 06 10 05 16 03 08 11 02 13
Pan triagonal transform
complete derivaton of order 4 {compact complete pantriagonal magic} cube families
³N₂ Aspect(³N₂)
00+a 00+b 08+a 08+b 00+c 00+d 08+c 08+d 16+a 16+b 24+a 24+b 16+c 16+d 24+c 24+d	00+e 00+f 08+e 08+f 00+g 00+h 08+g 08+h 16+e 16+f 24+e 24+f 16+g 16+h 24+g 24+h	32+a 32+b 40+a 40+b 32+c 32+d 40+c 40+d 48+a 48+b 56+a 56+b 48+c 48+d 56+c 56+d	32+e 32+f 40+e 40+f 32+g 32+h 40+g 40+h 48+e 48+f 56+e 56+f 48+g 48+h 56+g 56+h
[ ³N₂ Aspect(³N₂) ]^{~³N₂^analitic}
00+a 00+b 08+b 08+a 00+c 00+d 08+d 08+c 16+c 16+d 24+d 24+c 16+a 16+b 24+b 24+a	00+e 00+f 08+f 08+e 00+g 00+h 08+h 08+g 16+g 16+h 24+h 24+g 16+e 16+f 24+f 24+e	32+e 32+f 40+f 40+e 32+g 32+h 40+h 40+g 48+g 48+h 56+h 56+g 48+e 48+f 56+f 56+e	32+a 32+b 40+b 40+a 32+c 32+d 40+d 40+c 48+c 48+d 56+d 56+c 48+a 48+b 56+b 56+a
alternate swap of m/2 aparts (done 3 times)
00+a 08+a 08+b 00+b 08+d 00+d 00+c 08+c 16+c 24+c 24+d 16+d 24+b 16+b 16+a 24+a	08+f 00+f 00+e 08+e 00+g 08+g 08+h 00+h 24+h 16+h 16+g 24+g 16+e 24+e 24+f 16+f	32+e 40+e 40+f 32+f 40+h 32+h 32+g 40+g 48+g 56+g 56+h 48+h 56+f 48+f 48+e 56+e	40+b 32+b 32+a 40+a 32+c 40+c 40+d 32+d 56+d 48+d 48+c 56+c 48+a 56+a 56+b 48+b
00+a 24+c 08+b 16+d 24+b 00+d 16+a 08+c 16+c 08+b 24+d 00+b 08+d 16+b 00+c 24+a	24+h 00+f 16+g 08+e 00+g 24+e 08+h 16+f 08+f 16+h 00+e 24+g 16+e 08+g 24+f 00+h	32+e 56+g 40+f 48+h 56+f 32+h 48+e 40+g 48+g 40+e 56+h 32+f 40+h 48+f 32+g 56+e	56+d 32+b 48+c 40+a 32+c 56+a 40+d 48+b 40+b 48+d 32+a 56+c 48+a 40+c 56+b 32+d
Pan(³N₂ Aspect(³N₂))
00+a 56+g 08+b 48+h 56+f 00+d 48+e 08+c 16+c 40+e 24+d 32+f 40+h 16+b 32+g 24+a	56+d 00+f 48+c 08+e 00+g 56+a 08+h 48+b 40+b 16+h 32+a 24+g 16+e 40+c 24+f 32+d	32+e 24+c 40+f 16+d 24+b 32+h 16+a 40+g 48+g 08+b 56+h 00+b 08+d 48+f 00+c 56+e	24+h 32+b 16+g 40+a 32+c 24+e 40+d 16+f 08+f 48+d 00+e 56+c 48+a 08+g 56+b 00+h
Pan(³N₂ ³N₂)
01 63 10 56 62 04 53 11 19 45 28 38 48 18 39 25	60 06 51 13 07 57 16 50 42 24 33 31 21 43 30 36	37 27 46 20 26 40 17 47 55 09 64 02 12 54 03 61	32 34 23 41 35 29 44 22 14 52 05 59 49 15 58 08
Note: This cube is a {compact complete pantriagonal} cube each order 2 sub(hyper)cube has a total sum of 260 (= 2² (4³+1) ) each triagonally cells 2 apart sum 65 (= 4³+1 ) magic sum 130 (= 2 (4³+1) ) of course