This page is a natural extension into 3 dimensions of
the subject on symmetrical magic squares as covered on my
Selfsimilar Magic Squares page.
The subject as it pertains to magic squares, is extensively covered there with
the help of many examples.
The nucleus of this page is material supplied to me by
Walter Trump (Germany), as an Excel attachment to an email of March 3, 2003. He
has graciously permitted me to use this material and the 5 images associated
with it. I provide it here with minor editing and some additional comments.
Thanks Walter.
Following that presentation is material written by myself
to further expand on the subject of selfsimilar magic cubes.
Walter Trump has a Web page on SelfComplementary PanMagic
7x7Squares at
http://www.nefkom.net/trump/magicsquares/
Symmetrical (or selfcomplementary,
or selfsimilar) Magic Cubes of Order 4
Walter Trump, 20030302
All shown cubes have the following property:
If you replace each number i by its complement ( 65  i ) you will
get another aspect of the same magic cube
This new aspect can be transformed to the original one by a reflection with
respect to a point, an axis or a plane.
Thus the complementary cube is symmetrical to the original
cube.
There are four different possible types of symmetry:
 The center, the axis or the plane
of symmetry are coloured red.
 The two green cells show a
pair of symmetrical cells. They contain complementary numbers.
 The two blue cells show a
second example.

Trump1 Center symmetrical
cubes  associated
1 63 62 4 48 18 19 45 32 34 35 29 49 15 14 52
60 6 7 57 21 43 42 24 37 27 26 40 12 54 55 9
56 10 11 53 25 39 38 28 41 23 22 44 8 58 59 5
13 51 50 16 36 30 31 33 20 46 47 17 61 3 2 64
This cube is symmetric around a point in the center
of the cube.
It is the only type of symmetry that appears in odd order cubes (or
squares). This type of cube is called associated, or quite often it is
called symmetric. However, there is one other type of symmetry that can
appear in even order magic squares. There are three other types of even
order symmetrical cubes.
This is a disguised version of the Schubert cube of
1898
Why only center symmetry for odd order cubes (and
squares). A later comment from Walter.
Things are really different with odd orders.
Only the center symmetry is possible in that case.
All axis would contain more then one cell,
and there is only ONE number that is complementary to itself.


Trump2 Orthogonal axis
symmetrical cubes  not associated 1 24 62 43 16 25 51 38 61 44 7 18 52 37 10 31
48 57 19 6 33 56 30 11 20 5 42 63 29 12 39 50
59 46 8 17 54 35 9 32 2 23 60 45 15 26 53 36
22 3 41 64 27 14 40 49 47 58 21 4 34 55 28 13
This cube is symmetrical around an axis line
parallel to four of the cube faces.
The four horizontal squares are associative.
This type of symmetry works only on even order magic cubes (or squares).
This particular simple magic cube has a bonus
feature (not associated with symmetry).
The four horizontal planes and two of the six oblique planes are simple
magic squares.


Trump3 Diagonal axis
symmetrical cubes  not associated 1 4 62 63 23 31 44 32 42 40 18 30 64 55 6 5
10 58 45 17 25 53 36 16 34 12 38 46 61 7 11 51
59 54 8 9 47 27 13 43 21 29 52 28 3 20 57 50
60 14 15 41 35 19 37 39 33 49 22 26 2 48 56 24
This cube is symmetrical around a diagonal axis.
See further comments and illustration in the next section. 

Trump4 Plane symmetrical
cubes  not associated 1 5 61 63 40 15 44 31 25 50 21 34 64 60 4 2
14 58 52 6 18 55 27 30 47 10 38 35 51 7 13 59
62 43 9 16 46 37 11 36 19 28 54 29 3 22 56 49
53 24 8 45 26 23 48 33 39 42 17 32 12 41 57 20
There are billions of cubes with this symmetry.
This is an example of a cube that is symmetrical across an orthogonal
plane. 

Note: There
exists another geometrical symmetry of cubes. But 16 cells are positioned
directly in such a symmetry plane.
The numbers in those cells would be their own
complements.
Cubes of odd order have got only one number with
this property, cubes of even order none.
Thus there are no magic cubes that are symmetrical
with respect to such planes. 
These examples are all order 4 cubes, but the same
principles apply to all higher even orders.
Trump3 More on diagonal axis
symmetrical cubes

This is another way of looking at the
Trump3 cube that was shown above.
Here I show 10 complementary pairs of numbers. I have greyed out the
others to avoid confusion.For example:
62 and 3
is a pair, 63 and
2 is another pair. The axis is the dividing
line midway between all the pairs.
If the two numbers of each complementary pair in the
cube are exchanged, a different aspect of the same cube will be obtained.
This is true for any cube that has one of these four types of symmetry.
That is why Mutsumi Suzuki coined the term
'selfsimilar'.
Of course, if the cube is not symmetrical, exchanging each number with its
complement will result in a completely different cube! 
A complementary cube pair
Trump3 Diagonally axis symmetrical cube (reproduced for convenience)
1 4 62 63 23 31 44 32 42 40 18 30 64 55 6 5
10 58 45 17 25 53 36 16 34 12 38 46 61 7 11 51
59 54 8 9 47 27 13 43 21 29 52 28 3 20 57 50
60 14 15 41 35 19 37 39 33 49 22 26 2 48 56 24
This is the complement cube to the above
64 61 3 2 42 34 21 33 23 25 47 35 1 10 59 60
55 7 20 48 40 12 29 49 31 53 27 19 4 58 54 14
6 11 57 56 18 38 52 22 44 36 13 37 62 45 8 15
5 51 50 24 30 46 28 26 32 16 43 39 63 17 9 41
Selfsimilar complementary cube pairs always have exactly
the same characteristics. After all, they are the same cube, only differing by a
rotation or reflection.
However, if the cube is NOT selfsimilar, the
complementary pair are different cubes. But still have the
same characteristics?
Type of Symmetry 
Even or Odd Order 
# of coordinates that change between
members of the pair 
Magic Squares 


Center symmetrical 
Even or odd 
Two 
Orthogonal axis 
Even 
One 
Magic Cubes 


Center symmetrical 
Even or Odd 
Three 
Orthogonal axis 
Even 
Two 
Diagonal axis 
Even 
One or Three 
Plane symmetrical 
Even 
One 
Cube series
A set of m different integers from 1 to m^{3}
which sum up to the magic constant S.
Walter has also computed how many cube series there are
for each order of cube. The number increases very fast with increasing order.
Order of cube

Numbers used 
Number of different
series 
2 
1  8 
4 
3 
1  27 
85 
4 
1  64 
6,786 
5 
1  125 
1,142,341 
6 
1  216 
338,832,214 
7 
1  343 
156,623,626,331 
8 
1  512 
104,510,988,949,316 
From an email
– Christian Boyer to Walter Trump with cc to myself.
Dear Walter,
Thanks for this interesting document, nicely
presented.
I add some multimagic samples to your 4thorder
symmetrical cubes, in using your terminology.
Both my 27thorder bimagic, 32ndorder bimagic and
256thorder trimagic cubes are center symmetrical cubes.
My 64thorder trimagic cube is a plane symmetrical cube.
But both my 16thorder bimagic and the Hendrick's 25thorder bimagic cubes are
nonsymmetrical cubes.
(See the multimagic page
in this series.)
