Features of the four dimensional magic hypercube

 

A hyper cube is a cube which has more than three dimensions. What?

A magic square is flat and that is two dimensional. A magic cube is three dimensional. But what is four dimensional? See below a four dimensional (hyper) cube (source: http://mathworld.wolfram.com/ MagicTesseract.html).

 



 

 

A four dimensional 3x3x3x3 cube (n.b.: a 4D cube is also called a tesseract) consists of three 3x3x3 cubes (see above red, blew and green) which are put over each other. That gives besides rows and columns (= two dimensional) and pillars (= three dimensional), posts (= four dimensional). There are also space diagonals, but these are four (in stead of three) dimensional. It is easier to put the three 3x3x3 cubes next to (in stead of over) each other, so I can make all magic features clear:

 

 

T1, level I

 

 

T2, level I

 

 

T3, level I

a1

b1

c1

 

 

a1

b1

c1

 

 

a1

b1

c1

a2

b2

c2

 

 

a2

b2

c2

 

 

a2

b2

c2

a3

b3

c3

 

 

a3

b3

c3

 

 

a3

b3

c3

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

T1, level II

 

 

T2, level II

 

 

T3, level II

a1

b1

c1

 

 

a1

b1

c1

 

 

a1

b1

c1

a2

b2

c2

 

 

a2

b2

c2

 

 

a2

b2

c2

a3

b3

c3

 

 

a3

b3

c3

 

 

a3

b3

c3

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

T1, level III

 

 

T2, level III

 

 

T3, level III

a1

b1

c1

 

 

a1

b1

c1

 

 

a1

b1

c1

a2

b2

c2

 

 

a2

b2

c2

 

 

a2

b2

c2

a3

b3

c3

 

 

a3

b3

c3

 

 

a3

b3

c3

 

 

Simple magic cube (boundary magic features):

  • all rows give magic sum (27x): a1 + b1 + c1 = a2 + b2 + c2 = a3 + b3 + c3
  • all columns give magic sum (27x): a1 + a2 + a3 = b1 + b2 + b3 = c1 + c2 + c3
  • all pillars give magic sum (81x): I a1+IIa1+III a1=Ia2+IIa2+IIIa2=...=Ic3+IIc3+IIIc3
  • all posts give magic sum (81x): T1Ia1+T2Ia1+T3Ia1= ... = T1III c3+T2III c3+T3 III c3
  • all 4 dimensional diagonals (8x) is 2 cells with same colour + middle cell (b2) give magic sum.

  

Magic cube, pandiagonal in levels (additional magic feature):

  • all (pan)diagonals in all levels give magic sum: a1+b2+c3 = c1+b2+a3 = b1+c2+a3 = c1+a2+a3 = c2+b3+a1 = c3+b1+a2

 

Diagonal magic cube (additional magic features):

  • All diagonals in levels give magic sum: a1 + b2 + c3 = c1 + b2 + a3
  • All diagonals through levels give magic sum: Ia/b/c1 + IIa/b/c2 + IIa/b/c3 =        Ia1/2/3 + IIb1/2/3 + IIIc1/2/3

 

Pandiagonal magic cube (additional magic features):

  • All (pan)diagonals in all levels give magic sum: a1+b2+c3 = c1+b2+a3 = b1+c2+a3 = c1+a2+a3 = c2+b3+a1 = c3+b1+a2
  • All diagonals through levels give magic sum: Ia/b/c1+IIa/b/c2+IIa/b/c3 = Ia1/2/3+IIb1/2/3+IIIc1/2/3
  • All pandiagonals through levels give magic sum: Ia/b/c2+IIa/b/c3+IIa/b/c1 = Ib1/2/3+IIc1/2/3+IIIa1/2/3

 

Pantriagonal magic cube (additional magic feature):

  • All (pan)triagonals through the levels give magic sum: e.g. Ia1+IIb2+IIIc3=Ic2+IIb3+IIIa1

 

Panquadragonal magic cube (additional magic feature):

  • All (pan)triagonals through the cubes give magic sum: e.g. T1 Ia1+T2 IIb2+T3 IIIc3 = T2 Ic2+T3 IIb3+T1 IIIa1

 

A four dimensional magic cube (= tesseract) can also have a combination of the above mentioned magic features. A Nasik magic tesseract is pandiagonal, pantriagonal and panquadragonal magic. The smallest possible Nasik tesseract is a 16x16x16x16 tesseract.