The Magic Encyclopedia ™

The {Pan n-agonal Associated} hypercubes of order 4
(investigative article)
(by Aale de Winkel)

The interesting investigation into the {pantriagonal associated} cubes of order 4 made me wonder about things for the {pan n-agonal associated} hypercubes of this order. The first part of the table is a little general recaption, most things here focusses on m=4 however.

the {pan n-agonal associated} hypercubes of order 4
For this investigation I use the hypercube notation, further we have the constants:
s = mn-1
S = m(mn-1)/2 = 2s (for m=4)
associated complementary value reside in associated position
[ji] = s - [j(m-1-i)]
[j(i+m/2)] = s - [j(m-1-(i+m/2))] = s - [j(m/2-1-i)]
which means {associated} invariant under @[j0,k(m/2)]
monagonal sum sums in monagonal direction
l=0m [ji,kl; #k=1] = S
n-agonal sums sums in n-agonal directions
l=0m [n-10,ji]+l<n-11,k1,q-1> = S
hyperoctant equalities
hyperoctant
sumpairs
equal sums of pairs of numbers in every hyperoctant
[j0,k0]+[j0,k1]= [j1,k1]+[j1,k0]
proof: (#k=1)
two n-agonal sums:
[n-10,ji,k1]+[n-11,j(i+1),k2]+ [n-12,j(i+2),k3]+[n-13,j(i+3),k0]=S
[n-10,ji,k0]+[n-11,j(i+1),k3]+ [n-12,j(i+2),k2]+[n-13,j(i+3),k1]=S

[n-10,ji,k1]+[n-11,j(i+1),k2]+ (s-[n-11,j(3-(i+2)),k0])+(s-[n-10,j(3-(i+3)),k3])=S
[n-10,ji,k0]+[n-11,j(i+1),k3]+ (s-[n-11,j(3-(i+2)),k1])+(s-[n-10,j(3-(i+3)),k2])=S

[n-10,ji,k1]+[n-11,j(i+1),k2] = [n-11,j(3-(i+2)),k0]+[n-10,j(3-(i+3)),k3]
[n-10,ji,k0]+[n-11,j(i+1),k3] = [n-11,j(3-(i+2)),k1]+[n-10,j(3-(i+3)),k2]

i=0: (focussing on 1st hyperoctant)
[n-10,j0,k1]+[n-11,j1,k2] = [n-11,j1,k0]+[n-10,j0,k3]
[n-10,j0,k0]+[n-11,j1,k3] = [n-11,j1,k1]+[n-10,j0,k2]

monagonal sum: [j0] connected
[j0,k0]+[j0,k1]+ [j0,k2]+[j0,k3]=S
[j0,k0]+[j0,k1]+ ([j0,k0]+[j1,k3]-[j1,k1])+ ([j0,k1]+[j1,k2]-[j1,k0])=S
2([j0,k0]+[j0,k1])+ ([j1,k3]+[j1,k2])- ([j1,k1]+[j1,k0])=
2([j0,k0]+[j0,k1])+ (S-2([j1,k1]+[j1,k0]))=S

[j0,k0]+[j0,k1]= [j1,k1]+[j1,k0]

other hyperoctants follow from the generality of this argument
and put onto other aspects of the thypercubes as well as panrelocations
@[j2,l0] on any of these aspects.
Q.E.D.
n=3
k=0: [0,0,0]+[0,0,1]=[1,1,1]+[1,1,0]
k=1: [0,0,0]+[0,1,0]=[1,1,1]+[1,0,1]
k=2: [0,0,0]+[1,0,0]=[1,1,1]+[0,1,1]
n=4
k=0: [0,0,0,0]+[0,0,0,1]=[1,1,1,1]+[1,1,1,0]
k=1: [0,0,0,0]+[0,0,1,0]=[1,1,1,1]+[1,1,0,1]
k=2: [0,0,0,0]+[0,1,0,0]=[1,1,1,1]+[1,0,1,1]
k=3: [0,0,0,0]+[1,0,0,0]=[1,1,1,1]+[0,1,1,1]
hyperoctant
differences
differences between hyperoctants
[ji,k0]-[ji,k2] = [j(1-i),k1]-[j(1-i),k3]
proof: (#k=1)
from the above:
[n-10,j0,k1]+[n-11,j1,k2] = [n-11,j1,k0]+[n-10,j0,k3]
[n-10,j0,k0]+[n-11,j1,k3] = [n-11,j1,k1]+[n-10,j0,k2]

[n-10,j0,k1]-[n-10,j0,k3] = [n-11,j1,k0]-[n-11,j1,k2]
[n-10,j0,k0]-[n-10,j0,k2] = [n-11,j1,k1]-[n-11,j1,k3]

which simplifies to ji ε [0,1]
[ji,k0]-[ji,k2] = [j(1-i),k1]-[j(1-i),k3]
Q.E.D.
these difference equations propagates through the entire hypercube.
NOTE: PRELIMINAIRY STATEMENTS NEED TO BE SCRUTINIZED
a parametric point of view analoguous to <Francis Gaspalou>'s cube might look like:
concider the folowing hypercube cells: [j0],[j0,k1],[j0,k2] and [j0,k1,l1] #k=#l=1 then:
hyperoctant sumpairs:
[l0,j0,k0]+[l0,j0,k1]= [l1,j1,k1]+[l1,j1,k0] => [l1,j1,k1]=[l1,j0,k0]+ [l1,j0,k1]-[l0,j1,k0]
[l0,j0,k1]+[l0,j1,k1]= [l1,j1,k0]+[l1,j0,k0] => [l0,j1,k1]=[l1,j1,k0]+ [l1,j0,k0]-[l0,j0,k1]
(which gives the entire first hyperoctant)
monagonal sum:
l=0m [ji,kl;#k=1]=S => [j0,k3]=S-[j0,k2]- [j0,k1]-[j0,k0]
(which makes all the axes known)
hyperoctant differences:
[l0,j0,k2]-[l0,j0,k0] = [l0,j1,k3]-[l0,j1,k1] => [l0,j1,k3]=[l0,j0,k2]+ [l0,j1,k1]-[l0,j0,k0]
[l0,j0,k3]-[l0,j0,k1] = [l0,j1,k2]-[l0,j1,k0] => [l0,j1,k2]=[l0,j0,k3]+ [l0,j1,k0]-[l0,j0,k1]
(need to check whether this defines the entire axial hyperoctant)
The in-hyperoctant equal sumpairs makes it possible to reflect the hyperoctants without disturbing quality
leaving the first hyperoctant as is (0 in [j0]) a closer inspection of the cube (n=3) situation taught
me that the n-agonals are not summing on various (if not most) possibilities save the one <Francis Gaspalou>
defined for the cube, generalising this onto the hypercubes it looks like the following description
hyperoctant #(2j-1) : ~(2n-1-2j)
hyperoctant #(2j+2k-1) : ~(2j+2k)
(YET TO BE VERIFIED (or proven) ON HYPERCUBES n > 3)
the other bijective transformation <Francis Gaspalou> defined for the cube
generalises into the following description
[ji,k0,l1 ; #k=#l=1] <=> [ji,k2,l3; #k=#l=1]
[ji,k0,l2 ; #k=#l=1] <=> [ji,k2,l0; #k=#l=1]
(YET TO BE VERIFIED (or proven) ON HYPERCUBES n > 3)