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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.
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Addendum: Aug. 28/11 Today I discovered an
article in the Journal of Recreational Mathematics. Unfolding
the Tesseract, Peter Turney, 17:1:1984-85: 1-16. The author
demonstrates his method and shows that there are 261 different
unfolded tesseracts. There is only 1 unfolded square. There are 11
unfolded cubes. |
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Figure 3. The unfolded tesseract |
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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. |
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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.
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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. |
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Footnotes
1.
The order-5 cube (Walter Trump 2007) shown
here consists of an order 3
magic cube surrounded by 6 order 5 magic squares
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
here.
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 more on my
tesseract-represent-2 page.
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