Hypercube Representations - 2



The previous page considered ways to present magic squares and cubes (2-D and 3-D objects).

This page considers magic tesseracts (4-dimensional objects).

Historical representations             Hendricks modern form             The first tesseract

Historical representations

  • Magic squares are 2-dimensional objects consisting of 4 edges that meet 4 corners at right angles. They are easy to illustrate on a 2-dimensional surface.
  • Magic cubes are 3-dimensional objects consisting of  12 edges meeting 8 corners at right angles . They are easy to see in 3-dimensional space, but can be illustrated on a surface (2-D)  only by introducing distortion.
  • A magic tesseract is a 4-dimensional object consisting of  32 edges meeting 16 corners at right angles. A model of a tesseract could be built in 3-dimensional space, but only by introducing distortion (the corners would not be right angles). A drawing of a tesseract obviously requires still more distortion!
In trying to visualize a tesseract, 2 types of drawings have been used for many years. This schlegel diagram shows a small cube 'suspended' inside a large cube and 'supported' by six distorted cubes.

This is the best illustration to show how each tesseract is 'bounded' by 8 cubes.


This drawing illustrates the second visualization (the figure to the right).

The drawing as a whole attempts to show how a hypercube may be 'dragged' through the dimensions to produce a hypercube illustration for each. i.e here we go from 0-dimension to 1, 2, 3, and finally arrive at the forth dimension.

The numbers help to identify the corners. As an 'extra', I have arranged the numbers so that each square (and rhomboid) is perimeter magic. Remember that in these two cases, the rhomboids are actually distorted squares!

In both examples, the drawing is obviously not suitable to display the numbers in the magic tesseract. Even for order-3, the lowest, there are 81 numbers to display.

Hendricks modern form

In 1962, John R. Hendricks published a new method of drawing the magic tesseract [1][2].  He did it in grand style by describing a 6-dimension magic hypercube of order-3. This hypercube used the numbers from 1 to 729 (729 = 36) and required 9 order-3 dimension 4 (tesseract) figures to display (a 6-D figure was just too complicated to comprehend). Each tesseract sums on its own in 4 ways. The fifth direction is found by jumping from tesseract to tesseract horizontally, and the sixth direction by jumping vertically. The resulting normal 6-dimensional magic hypercube sums to 1095 in at least 1490 ways (6m5 + 32).
He also showed a 5-dimension order-3 magic hypercube. It required three order 3 tesseract diagrams. See this hypercube here.

He introduced the subject with figures 1, 2, and 3; showing an order 3 magic square, magic cube, and magic tesseract.

Notice that none of the three magic figures are normalized.

 He came up with that idea (for cubes and tesseracts) at a later date, when he realized that a system was required for listing solutions in order.

This tesseract is an aspect of index # 5.

The magic square has traditionally been illustrated with each number occupying a 2-dimensional 'cell', not as intersections of a 'grid' as suggested in Figure 1 (above). The magic cube was also shown that way at the start, but now is normally shown with the numbers placed at grid intersections.

This tesseract diagram is now the one in universal use for displaying small orders of  4-dimension magic hypercubes.
For the larger orders, a simple text listing is still the most practical.

This is one way of listing the above tesseract

09   76   38      64   35   24      50   12   61
46   17   60      05   75   43      72   31   20
68   30   25      54   13   56      01   80   42

74   45   04      33   19   71      16   59   48
15   55   53      79   41   03      29   27   67
34   23   66      11   63   49      78   37   08

40   02   81      26   69   27      57   52   14
62   51   10      39   07   77      22   65   36
21   70   32      58   47   18      44   06   73

The purpose of this  page is to show methods of illustrating a magic tesseract. Other pages on this site will discuss features and characteristics of the magic tesseract, and the relationships between this 4-d magic figure and it's 2-D and 3-D cousins.)

[1]John R. Hendricks, The Five and Six Dimensional Magic Hypercubes of Order 3, Canadian Mathematical Bulletin, vol. 5, no. 2, 1962, pp 171-189
[2]John R. Hendricks, Magic Squares to Tesseracts by Computer, Self-published 1998,  0-9684700-0-9  the preface contains some history of this discovery

The first magic tesseract?

In 1905, Dr. C. Planck published a paper [1] [2] for private circulation. Called The Theory of Paths Nasik, It dealt with perfect magic squares and cubes.

Dr. A. H. Frost defined the term Nasik in 1878 [3] as requiring that all paths sum correctly. He stated that the smallest order Nasik magic square is order-4 ( we call it pandiagonal), and the smallest order Nasik magic cube is order-8.

In explaining his theory, Dr. Planck constructed the order-9 pandiagonal magic square shown in 2. He then took the nine order-3 sub-squares and arranged them as shown in illustration 1. He called this a 'crude-magic octahedron.

1. 2.

As an order-9 pandiagonal magic square


Another aspect of the same magic tesseract

hh-1 (to the right) is the modern method of displaying the 4-dimension magic hypercube illustrated above.
All lines parallel to the edges, and also the 8 quadragonals, sum correctly to the constant 123.

I quote from his paper

...we shall obtain the three cubes of order 3 shown in figure 11, which form a perfect crude-magic octahedroid of order 3. These three cubes are sections of the four-fold made by three parallel equidistant spaces, laid out in perspective in one space of three dimensions.

Some points

  • The 3x3 squares in figure 11 are not magic, nor are the 3x3x3 cubes that may be formed from them. This is because the diagonals and triagonals do not sum correctly. This is not a requirement of a magic tesseract. The eight Quadragonals of the tesseract (such as 47+41+35) is a requirement and does sum correctly.

  • Figure 11a is shown in green in the modern drawing with the other two cubes parallel to it. The three cubes of fig. 12 are parallel to the front of the drawing.

  • This tesseract is 1 of 384 aspects of the 58 basic magic tesseracts of order 3. It is a non-normalized aspect of index number 57 (of the 58).

  • Usually, to make the drawing simpler, only the outline lines are shown (see previous section).

  • hh-2 is another aspect of the same 'octahedroid'. Do you see where Planck,s cubes fit in this drawing?


Planck's paper expanded on Frost original definition of nasik, applying it to hypercubes where all lines sum correctly i.e. perfect hypercubes.

I must emphasize, though, that this tesseract is not perfect (and his paper did not suggest it was). Like all order-3 magic hypercubes, it is classed as a simple order-3 magic hypercube.

hh-1 Modern day presentation of planck's 'octahedroid'

hh-2  This is another aspect of the same tesseract

The smallest nasik (perfect) magic tesseract possible is order-16. John Hendricks constructed the first one in 1999, and Clifford Pickover confirmed that it summed correctly to 524,296 in the required 163,840 ways (straight line paths only). [4] [5]

[1] Dr. C. Planck, The Theory of Paths Nasik, self-published in Haywards Heath, (England) in November 1905.  18 pages self-cover.
[2] W. S. Andrews, Magic Squares and Cubes,2nd Edition, Open Court, 1917, pages 363-375 written by C. Planck.
     This book republished by Dover Publ. in 1960. The above fig. 10 and 11 (from the 1905 paper appear in Andrews as fig. 687 and 688.
[3] A.H. Frost, On the General Properties of Nasik Cubes, Quarterly Journal of Mathematics, 15, 1878, pp 34-49.
[4] John R. Hendricks, Magic Squares to Tesseracts by Computer, Self-published 1998,  0-9684700-0-9  pp 126-127 (and private correspondence)
[5] Clifford A. Pickover, The Zen of Magic Squares, Circles, and Stars, Princeton University Press, 2002, 0-691-07041-5,  page 121.

This page was originally posted November 2007
It was last updated November 28, 2009
Harvey Heinz   harveyheinz@shaw.ca
Copyright 1998-2009 by Harvey D. Heinz