Self-similar Magic Cubes

This page is a natural extension into 3 dimensions of the subject on symmetrical magic squares as covered on my Self-similar 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 self-similar magic cubes.

Symmetrical (or selfcomplementary, or selfsimilar) Magic Cubes of Order 4
Walter Trump, 2003-03-02

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:

	Trump-1 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.
	Trump-2 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.
	Trump-3 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.
	Trump-4 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.

This is another way of looking at the Trump-3 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 'self-similar'.
Of course, if the cube is not symmetrical, exchanging each number with its complement will result in a completely different cube!

Self-similar 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 self-similar, 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

A set of m different integers from 1 to m³ 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.

I add some multimagic samples to your 4th-order symmetrical cubes, in using your terminology.

Both my 27th-order bimagic, 32nd-order bimagic and 256th-order trimagic cubes are center symmetrical cubes.
My 64th-order trimagic cube is a plane symmetrical cube.
But both my 16th-order bimagic and the Hendrick's 25th-order bimagic cubes are non-symmetrical cubes.