The Unfolded Tesseract

In the study of magic cubes, a variation is to find a solution for a cube with a magic square on each of the 6 faces. [1]
A variation in the study of magic knight tours is also to find such a tour on the surface of the cube. [2]

In both cases this is accomplished on the six surfaces of this 3-dimensional hypercube.
This is analogous to performing these tasks on the one surface of the 2-dimensional hypercube (the square).
The question thus naturally comes to mind ‘Is it possible to do these tasks on the 24 surfaces of a 4-dimensional hypercube?’

But the first requirement is to lay out these 24 surfaces on the 2-dimensional plane!

Figure 1 is a Schlegel diagram of a hypercube of 4 dimensions. [3]
This is the traditional method of illustrating this 4-dimensional object with a two dimensional drawing.
I have arbitrarily labelled the eight corners for the purpose of the following discussion.

Three of the 8 bounding cubes are shown in separate colors to make the object easier to understand.
They are ABONFGJK, EFGHIJKL, and CDMPEHIL.
The other 5 cubes are: ABCDEFGH,ABCDMNOP, ADEFKLMN, BCHGIJOP, and IJKLMNOP

When studying and displaying magic tesseracts, the schlegel diagram [3] is not very suitable. Because all lines and planes do not show as orthogonals, displaying the numbers in the magic hypercube is hopelessly confusing.

Figure 2.
In 1962, John R. Hendricks published a grid type representation of the 4-dimensional magic tesseract that was similar to that being used to illustrate the 3-dimensional magic cube. [4]

To allow comparison of the schlegel and Hendricks representations, the diagram shown here has corners and edges labeled to correspond with the schlegel illustration. The line colors also correspond.

This (figure 2) form is the one now commonly used to illustrate magic tesseracts. However, for simplicity, only the outline is shown here.
To illustrate an actual magic tesseract, including all the required integers, interior planes would be required. The number of these would depend on the order of the tesseract.

Figure 1--A schlegel diagram Flat of a tesseract with the 16 corners and 32 edges labeled.

Figure 2--Hendricks tesseract representation with corners and edges labeled.

The 8 cubes in this tesseract have already been identified. This table list the 24 faces and the 32 edges in alphabetical order.

The 24 faces with identification numbers The 32 EDGES with identification numbers

1	ABCD	9	BGJO	17	FGJK	1	AB	9	CH	17	FK	25	JO
2	ABFG	10	CDEH	18	GHIJ	2	AD	10	CP	18	GH	26	KL
3	ABNO	11	CDMP	19	IJKL	3	AF	11	DE	19	GJ	27	KN
4	ADEF	12	CHIP	20	IJOP	4	AN	12	DM	20	HI	28	LM
5	ADMN	13	DELM	21	ILMP	5	BC	13	EF	21	IJ	29	MN
6	AFKN	14	EFGH	22	JKNO	6	BG	14	EH	22	IL	30	MP
7	BCGH	15	EFKL	23	KLMN	7	BO	15	EL	23	IP	31	NO
8	BCPO	16	EHIL	24	MNOP	8	CD	16	FG	24	JK	32	OP

Figure 3 illustrates the faces of the above tesseract diagram laid out in 2-dimensional form.

This is only one of many possible arrangements of the 24 faces.
In this diagram, you will notice the corner letter designations are repeated several times. Because this represents a 4-dimensional object, there are 3 surfaces (planes) connected at each edge!
(The 3-D cube has 2 planes adjoining, the 2-D square has only one.)

Who will be the first to find a solution where all orthogonal planes (including the 24 faces) are magic squares?

Who will be the first to find a solution where the cells on all 24 faces are connected by a magic knight tour?
Awani Kumar May, 2009 See Figure 7.

Below I show examples of these two accomplishments on the six surfaces of a cube.

Figure 3. The unfolded tesseract

Figure 4 This cube was announced jointly by Walter Trump and Christian Boyer in November 2003. All six surface squares are magic. In fact all 15 orthogonal planes are magic. And as a bonus, all 6 oblique planes are also magic. Because all four triagonals sum to the constant also, this is a magic cube.

Notice that rows or columns adjacent to the edges are duplicated on the joining plane. This is because two planes are joined at each edge. The tesseract is 4-dimensional so three planes will have identical numbers along their adjacent edge!

In the Hendricks hypercube classification system this is a
Diagonal magic cube, and order 5 is the smallest order possible for this class!. Under one of the older classification systems this is referred to as perfect. More information and links are here.

NOTE: This diagram shows the surface layers of a 3-dimensional hypercube. This is the reason for duplicate rows or columns on adjacent edges.
The magic knight tours shown in Figures 5 and 7 (and the magic squares in figure 6) are on the surfaces (not as a layer of a 3-D figure).

Figure 4 The six faces of the Trump-Boyer Diagonal magic cube of order-5.

Figure 5 This solution (Awani Kumar 2007) is a magic knight tour. That is, If each step is numbered, all rows and columns sum to the constant of 1540. Note that these are NOT magic squares though, because the diagonals do not sum to the constant. More information is at my Cube Update-5 page.

Figure 6 It is simple, but tedious to create 24 magic squares with an identical constant to put on the surfaces of the tesseract. I went for the tedious with Figure 6. In the example, I have simply distributed the numbers from 1 to 384 evenly among the 24 squares, using the Dürer square as the placement pattern.

The simplest solution to all planes being magic squares would be the diagonal magic tesseract. Mitsutoshi Nakamura shows a tesseract of this type here.
It will be a more difficult matter (then magic squares), to cover the 24 surfaces with a knight tour and still more difficult to obtain a solution where all rows and columns sum to a constant (i.e. a magic knight tour!).
One solution, by Awani Kumar, is shown in Figure 7.
I show Awani Kumar solutions for 4 and 5 dimensional knight tours (not just the surfaces) here.

Figure 5. A magic knight tour on the six faces of an order-8 cube.

Figure 6 24 order-4 magic squares placed on the faces of a tesseract.

Figure 7. A Magic Knight Tour of the 24 tesseract faces.

Here is shown one solution of a magic knight tour on the 24 faces of a tesseract. Each face contains an order-8 semi-magic square. All rows and columns sum to 6148, but no main diagonals sum correctly.
There is a chess knight jump between the consecutive numbers 1 to 1536. Because most adjacent faces cannot be shown adjacent in this 2-dimensional representation, the connection points of the tour are indicated with the red or green numbers.
As with the cube faces knight tour of Fig. 5, each face consists of two tours of 32 numbers, and so appear in two colors.

This magic knight tour was discovered by Awani Kumar in May 2009. It was subsequently checked by Aale de Winkel and Harvey Heinz. A modified version of Awani's Excel spreadsheet showing the placement of the numbers may be downloaded here.

Figure 8. Another Unfolded Arrangement

As mentioned above, there are many possible arrangements of the 24 faces of the tesseract.

This arrangement, by Aale de Winkel, shows two of the eight bounding cubes. The two cubes are highlighted with face numbers shown in blue and violet.

The edge and face labels are arbitrary, but again are chosen to correspond to figure 2 at the top of this page. A careful comparison between these two figures will help in understanding the relationship between the 4-D hypercube and the 2-D representation.

Unfortunately, this arrangement requires that some seemingly adjacent edges are, in fact, NOT adjacent in the four dimensional hypercube. These edges are shown in red.

Footnotes

1. The order-5 cube (Walter Trump 2007) shown at this link consists of an order 3 magic cube surrounded by 6 order 5 magic squares http://members.shaw.ca/hdhcubes/cube_modulo.htm#Trump%20mod%202,%203,%2010,%20and%2062%20cubes

2. The Knight tour on the 6 surfaces of an order 8 cube at this link was published by H. E. Dudeney in 1917 and is shown at http://members.shaw.ca/hdhcubes/cube_unusual.htm#Knight Tour

3. The Schlegel diagram was introduced about 1883. See http://en.wikipedia.org/wiki/Schlegel_diagram

4. The Hendricks diagram was developed in 1950 but not published until 1962. See http://members.shaw.ca/tesseracts/t_represent-2.htm#Hendricks