the {pan n-agonal associated} hypercubes of order 4 | ||||
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For this investigation I use the hypercube notation, further we have the constants: s = m^{n}-1 S = m(m^{n}-1)/2 = 2s (for m=4) |
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associated | complementary value reside in associated position | |||
[_{j}i] = 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 @[_{j}0,_{k}(m/2)] |
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monagonal sum | sums in monagonal direction | |||
_{l=0}∑^{m} [_{j}i,_{k}l; #k=1] = S | ||||
n-agonal sums | sums in n-agonal directions | |||
_{l=0}∑^{m} [_{n-1}0,_{j}i]+l<_{n-1}1,_{k}1,_{q}-1> = S | ||||
hyperoctant equalities | ||||
hyperoctant sumpairs |
equal sums of pairs of numbers in every hyperoctant | |||
[_{j}0,_{k}0]+[_{j}0,_{k}1]= [_{j}1,_{k}1]+[_{j}1,_{k}0] | ||||
proof: (#k=1) two n-agonal sums: [_{n-1}0,_{j}i,_{k}1]+[_{n-1}1,_{j}(i+1),_{k}2]+ [_{n-1}2,_{j}(i+2),_{k}3]+[_{n-1}3,_{j}(i+3),_{k}0]=S [_{n-1}0,_{j}i,_{k}0]+[_{n-1}1,_{j}(i+1),_{k}3]+ [_{n-1}2,_{j}(i+2),_{k}2]+[_{n-1}3,_{j}(i+3),_{k}1]=S [_{n-1}0,_{j}i,_{k}1]+[_{n-1}1,_{j}(i+1),_{k}2]+ (s-[_{n-1}1,_{j}(3-(i+2)),_{k}0])+(s-[_{n-1}0,_{j}(3-(i+3)),_{k}3])=S [_{n-1}0,_{j}i,_{k}0]+[_{n-1}1,_{j}(i+1),_{k}3]+ (s-[_{n-1}1,_{j}(3-(i+2)),_{k}1])+(s-[_{n-1}0,_{j}(3-(i+3)),_{k}2])=S [_{n-1}0,_{j}i,_{k}1]+[_{n-1}1,_{j}(i+1),_{k}2] = [_{n-1}1,_{j}(3-(i+2)),_{k}0]+[_{n-1}0,_{j}(3-(i+3)),_{k}3] [_{n-1}0,_{j}i,_{k}0]+[_{n-1}1,_{j}(i+1),_{k}3] = [_{n-1}1,_{j}(3-(i+2)),_{k}1]+[_{n-1}0,_{j}(3-(i+3)),_{k}2] i=0: (focussing on 1st hyperoctant) [_{n-1}0,_{j}0,_{k}1]+[_{n-1}1,_{j}1,_{k}2] = [_{n-1}1,_{j}1,_{k}0]+[_{n-1}0,_{j}0,_{k}3] [_{n-1}0,_{j}0,_{k}0]+[_{n-1}1,_{j}1,_{k}3] = [_{n-1}1,_{j}1,_{k}1]+[_{n-1}0,_{j}0,_{k}2] monagonal sum: [_{j}0] connected [_{j}0,_{k}0]+[_{j}0,_{k}1]+ [_{j}0,_{k}2]+[_{j}0,_{k}3]=S [_{j}0,_{k}0]+[_{j}0,_{k}1]+ ([_{j}0,_{k}0]+[_{j}1,_{k}3]-[_{j}1,_{k}1])+ ([_{j}0,_{k}1]+[_{j}1,_{k}2]-[_{j}1,_{k}0])=S 2([_{j}0,_{k}0]+[_{j}0,_{k}1])+ ([_{j}1,_{k}3]+[_{j}1,_{k}2])- ([_{j}1,_{k}1]+[_{j}1,_{k}0])= 2([_{j}0,_{k}0]+[_{j}0,_{k}1])+ (S-2([_{j}1,_{k}1]+[_{j}1,_{k}0]))=S [_{j}0,_{k}0]+[_{j}0,_{k}1]= [_{j}1,_{k}1]+[_{j}1,_{k}0] other hyperoctants follow from the generality of this argument and put onto other aspects of the thypercubes as well as panrelocations @[_{j}2,_{l}0] on any of these aspects. Q.E.D. |
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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] |
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hyperoctant differences |
differences between hyperoctants | |||
[_{j}i,_{k}0]-[_{j}i,_{k}2] = [_{j}(1-i),_{k}1]-[_{j}(1-i),_{k}3] | ||||
proof: (#k=1) from the above: [_{n-1}0,_{j}0,_{k}1]+[_{n-1}1,_{j}1,_{k}2] = [_{n-1}1,_{j}1,_{k}0]+[_{n-1}0,_{j}0,_{k}3] [_{n-1}0,_{j}0,_{k}0]+[_{n-1}1,_{j}1,_{k}3] = [_{n-1}1,_{j}1,_{k}1]+[_{n-1}0,_{j}0,_{k}2] [_{n-1}0,_{j}0,_{k}1]-[_{n-1}0,_{j}0,_{k}3] = [_{n-1}1,_{j}1,_{k}0]-[_{n-1}1,_{j}1,_{k}2] [_{n-1}0,_{j}0,_{k}0]-[_{n-1}0,_{j}0,_{k}2] = [_{n-1}1,_{j}1,_{k}1]-[_{n-1}1,_{j}1,_{k}3] which simplifies to _{j}i ε [0,1] [_{j}i,_{k}0]-[_{j}i,_{k}2] = [_{j}(1-i),_{k}1]-[_{j}(1-i),_{k}3] Q.E.D. |
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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: [_{j}0],[_{j}0,_{k}1],[_{j}0,_{k}2] and [_{j}0,_{k}1,_{l}1] #k=#l=1 then: hyperoctant sumpairs: [_{l}0,_{j}0,_{k}0]+[_{l}0,_{j}0,_{k}1]= [_{l}1,_{j}1,_{k}1]+[_{l}1,_{j}1,_{k}0] => [_{l}1,_{j}1,_{k}1]=[_{l}1,_{j}0,_{k}0]+ [_{l}1,_{j}0,_{k}1]-[_{l}0,_{j}1,_{k}0] [_{l}0,_{j}0,_{k}1]+[_{l}0,_{j}1,_{k}1]= [_{l}1,_{j}1,_{k}0]+[_{l}1,_{j}0,_{k}0] => [_{l}0,_{j}1,_{k}1]=[_{l}1,_{j}1,_{k}0]+ [_{l}1,_{j}0,_{k}0]-[_{l}0,_{j}0,_{k}1] (which gives the entire first hyperoctant) monagonal sum: _{l=0}∑^{m} [_{j}i,_{k}l;#k=1]=S => [_{j}0,_{k}3]=S-[_{j}0,_{k}2]- [_{j}0,_{k}1]-[_{j}0,_{k}0] (which makes all the axes known) hyperoctant differences: [_{l}0,_{j}0,_{k}2]-[_{l}0,_{j}0,_{k}0] = [_{l}0,_{j}1,_{k}3]-[_{l}0,_{j}1,_{k}1] => [_{l}0,_{j}1,_{k}3]=[_{l}0,_{j}0,_{k}2]+ [_{l}0,_{j}1,_{k}1]-[_{l}0,_{j}0,_{k}0] [_{l}0,_{j}0,_{k}3]-[_{l}0,_{j}0,_{k}1] = [_{l}0,_{j}1,_{k}2]-[_{l}0,_{j}1,_{k}0] => [_{l}0,_{j}1,_{k}2]=[_{l}0,_{j}0,_{k}3]+ [_{l}0,_{j}1,_{k}0]-[_{l}0,_{j}0,_{k}1] (need to check whether this defines the entire axial hyperoctant) |
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The in-hyperoctant equal sumpairs makes it possible to reflect the hyperoctants without disturbing quality leaving the first hyperoctant as is (0 in [_{j}0]) 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) : ~(2^{n}-1-2^{j}) hyperoctant #(2^{j}+2^{k}-1) : ~(2^{j}+2^{k}) (YET TO BE VERIFIED (or proven) ON HYPERCUBES n > 3) |
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the other bijective transformation <Francis Gaspalou>
defined for the cube generalises into the following description [_{j}i,_{k}0,_{l}1 ; #k=#l=1] <=> [_{j}i,_{k}2,_{l}3; #k=#l=1] [_{j}i,_{k}0,_{l}2 ; #k=#l=1] <=> [_{j}i,_{k}2,_{l}0; #k=#l=1] (YET TO BE VERIFIED (or proven) ON HYPERCUBES n > 3) |