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=0∑^m [_ji,_kl; #k=1] = S

n-agonal sums sums in n-agonal directions
_l=0∑^m [_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=0∑^m [_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 #(2^j-1) : ~(2ⁿ-1-2^j)
hyperoctant #(2^j+2^k-1) : ~(2^j+2^k)
(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)

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
monagonal sum	_l=0∑^m [_ji,_kl; #k=1] = S
n-agonal sums	sums in n-agonal directions
n-agonal sums	_l=0∑^m [_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=0∑^m [_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 #(2^j-1) : ~(2ⁿ-1-2^j) hyperoctant #(2^j+2^k-1) : ~(2^j+2^k) (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)