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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.
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| 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. |
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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. |
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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 |
3.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
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...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.
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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.
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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).
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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?
Perfect?
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.
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