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This
page is the entry to my tesseract pages.
It contains a description of what a tesseract is, and the basic parts.
It then includes a list of all the pages in this section (with links), and
a brief description of each.
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Tesseracts
- Update 2013 |
A new page ! March 22,
2013
Recent developments -
Contributors to Magic Tesseract theory - The 5-D Challenge |
Introduction
Previous
hypercube sections of this site is concerned with Magic Squares (2-D), and
Magic Cubes (3-D). Only cursory mention is made of higher dimension magic
hypercubes.
This
section will feature Tesseracts, the 4-dimension magic hypercube. It will
also discuss the connections between hypercubes of the different
dimensions.
John R.
Hendricks, more then any other single person, advanced the knowledge of
the higher dimensions of hypercubes. A great deal of the material I am
presenting in this set of pages comes from his notes , articles, and
books.
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John Hendricks passed away on
July 7, 2007. I dedicate these pages to his memory.
I still
maintain John Hendricks web site at
http://members.shaw.ca/johnhendricksmath/
What is a magic
tesseract?
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A magic tesseract is a 4-dimensional array
of m4 consecutive numbers (normally from 1 to m4),
arranged so that the sum of the m numbers in each of the m3
rows, m3 columns, m3 pillars, and m3
posts, as well as the 8 quadragonals, sum to a constant sum S, and
m = the order of the magic tesseract. |
Features of a magic tesseract are the same as
those in a magic square or cube, only more extensive, due to the extra
dimension.
Above is the definition for a simple
magic tesseract. Additional lines summing correctly qualifies the tesseract to
be classified accordingly. There are 2 classes of magic square, and 6 classes of
magic cube. Mitsutoshi
Nakamura, a current prodigious investigator of magic tesseracts, established
that there are 18 classes of magic tesseract, and has constructed examples of
all of them. More on this subject on my Classes
page.
Figure 1A is a graphical presentation of an
order 3 magic tesseract. In these pages, order-3 will normally be used for
examples, because the higher orders are more complicated to illustrate or
list (an order-3 has 81 numbers, an order-4 has 256 numbers).
Figure 1B shows the minimum lines that are required to sum correctly in
order for the tesseract to be magic.
Point of Interest
The above diagram was invented by John Hendricks in 1950, although it was
not published until 1962. The older methods of illustrating the tesseract
were not suitable for showing the placement of the numbers in a magic
tesseract. Hendricks interesting account of this event is
here.
This is a listing the
tesseract in text form.
52 66
05 60 26 37 11 31
81
02 49 72 43 57 23 78 17 28
69 08 46
20 40 63 34 75
14
03 50 70 44 55 24 76 18 29
67 09 47 21 41 61 35 73 15
53 64 06 58 27 38 12 32 79
68 07 48
19 42 62 36 74
13
54 65 04 59 25 39 10 33 80
01 51 71
45 56 22 77 16
30 |
Notice
that the 3x3 arrays do not form magic squares. The rows and columns sum
correctly, but the diagonals do not. Likewise, the 3 groups of 3x3 arrays do
not form magic cubes because the triagonals do not sum correctly.
In the case of order-3
hypercubes, however, that statement is not quite correct. Because all
order-3 hypercubes are associated (center symmetric), the central hypercubes
within the main hypercube are magic.
In this example, the
center column of 3x3 arrays do form a magic cube, and the center 3x3
array of this cube is a magic square! |
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If we do a listing like
this for each of the other three orientations, the center column will be a
different order-3 magic cube with a magic square in the center.
This will be discussed in
more detail on my associated page. |
Coordinates of an order-3 magic
tesseract
Coordinates are useful for finding your way
around the tesseract diagram, or for specifying a particular location.
The x, y, z, and w coordinates may be
combined in one 4-digit word, without the need for delimiting commas, because
all coordinates are single digit positive integers. As is normal, the origin is
the lower left corner.
The number of coordinates that change as you
travel along a line indicates the type of line it is.
- Only 1 coordinate changes as you travel along
a 1-agonal (a row, column, pillar, or file).
Example: 1111, 1211,1311 or 1, 45, 77 in above tesseract example.
- 2 coordinates change when you move along a 2-agonal
(diagonal).
Example: 1111, 1221, 1331, or 1, 56, 30.
- 3 coordinates change when you move along a 3-agonal
(triagonal).
Example: 1111, 2122, 3133, or 1, 9, 5.
- 4 coordinates change when you move along a 4-agonal
(quadriagonal).
Example: 1111, 2222, 3333, or 1, 41, 81 in fig. 1.
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| An alternate coordinate representation of
the order-3 tesseract
3131 3231 3331 3132 3232 3332
3133 3233 3333
3121 3221 3321 3122 3222 3322 3123 3223 3323
3111 3211 3311 3112 3212 3312 3113 3213 3313
2131 2231 2331 2132 2232 2332 2133 2233 2333
2121 2221 2321 2122 2222 2322 2123 2223 2323
2111 2211 2311 2112 2211 2312 2113 2213 2313
1131 1231 1331 1132 1232 1332 1133 1233 1333
1121 1221 1321 1122 1222 1322 1123 1223 1323
1111 1211 1311 1112 1212 1312 1113 1213 1313 |
Each of the 4-digit coordinate numbers to
the left may be replaced with a number that would then appear in that
position of the tesseract. The system of
coordinates is required when working with modular equations, one of the
methods used to construct magic hypercubes of any dimension or order.
The colored coordinates in the above illustration show
one of the four central cubes. In an order-3 tesseract (or an associated
tesseract of any other order), the 4 central cubes are associated magic. |
Directory to the rest of this section
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1.
Magic Hypercubes - Overview |
A general overview
of magic squares, cubes, tesseract, etc. Their interrelationships and
summations. |
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2.
Hypercube Representations |
A review of how
magic squares and cubes have been presented in publications, etc. through
the ages. |
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Hypercube Representations - 2 |
A review of how the
forth dimension and magic tesseracts have been illustrated (over a shorter
time period). |
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4.
Order-3 Magic Tesseracts |
Features of order 3
hypercubes and illustrations and listing for all 58 basic order-3
tesseracts. |
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5.
Hypercube Aspects |
Illustrates the 8
aspects of the magic square and 48 aspects of the magic cube. The tesseract
has 384 aspects. |
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6.
Hypercube Classes |
Reviews classes of
magic squares and cubes. Minimum requirements for the 18 of magic tesseract
classes. |
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7.
Associated Hypercubes |
Discusses features
of associated magic hypercubes, and embedded hypercubes of lower
dimensions. |
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8.
Hypercube Math |
Equations and
comparison tables. Demonstrates close relationship between hypercubes of
different dimensions. |
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9.
More Tesseracts |
Orders 5 and 6
tesseracts, Inlaid tesseract, perfect tesseracts, plus links to some
previously posted on other sites. |
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10.
Hypercube - Cross-stitch |
A picture and
discussion of a cross-stitch project demonstrating features of square, cube,
and tesseract. |
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11.
Tesseract Knight Tour |
Awani Kumar has
successfully constructed an order 4 tesseract containing a magic knight
tour. |
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12.
The Unfolded Tesseract |
Showing the 3-D
illustration of a 4_D tesseract flattened out to 2 Dimensions to show all 24
faces. |
| 13.
Tesseracts - Update 2013 |
Some new developments over the
last few years. Contributors to Tesseract knowledge. 5-D challenge. |
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