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(nH4k) nH4k being te "seed" or "generating hypercube" which serves as a starting point where the transform acts upon The order 2 hypercube nN2analitic place an important role

The simplest of these objects I now call "Pan" is the transform seeded by the order 4k normal hypercube nN4k and thus can be formulized as:
Pan == nPan4k = Pan(nN4k) = OddSwap(nN4k _[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(nH4k) 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 == nPan4k = OddSwap(nN4k _[0,..,2k-1,4k-1,..,2k])
Reflector
nN2analitic
the order 2 n dimensional hypercube in analitic numberrange
nN2[ji] = j=01ji 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(nH4k) == nH4k~nN2analitic == nH4k _[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(nH4k) :
R(nH4k)[ji] <-> R(nH4k)[ji + 2k] iff (j=0nji % 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
nNm
each number in sequence
nNm[ji] = j=0n-1ji 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 nH2k{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(N4k) 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 ]~2N2analitic
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 ]~2N2analitic
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
2N2N2t
01 03 05 07
02 04 06 08
09 11 13 15
10 12 14 16
[ 2N22N2t ]~2N2analitic
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(N2N2t)
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) ]~3N2analitic
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