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This is now an orphan site On October 30, 2010, I posted a consolidated version of my
three magic hypercube sites to
magic-squares.net. |
This page is the entry to my tesseract pages.
It will contain a
description of what a tesseract is, and the basic parts.
It will then include a list of all the pages in this section
(with links), and a brief description of each.
 |
Knight Tours | October, 2008 Awani Kumar's 4 and 5 dimensional Knight Tours |
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The Unfolded Tesseract | March, 2009 The 4-dimensional tesseract unfolded to show the 24 faces. |
Because of space constraints, I am forced
to put these pages on yet another site. My hypercube pages now include
My first site (Magic Squares and Stars) now at
http://www.magic-squares.net/
My Magic cubes at
http://members.shaw.ca/hdhcubes/
and, of course, this newest site.
This search engine returns results from all 3 of these sites.
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I also have a personal home page at
http://members.shaw.ca/harveyheinz/
I still maintain John Hendricks web site
at
http://members.shaw.ca/johnhendricksmath/
Introduction
Previous hypercube sections of these sites are 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.
John Hendricks passed away on July 7, 2007. I dedicate these pages to his memory.
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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, although several have not yet been constructed. More on this subject on my Classes page.
Figure 1 A. An order-3 magic tesseract B. The lines required to sum to 123 for simple magic tesseract
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! 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 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.
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| An alternate coordinate representation of the order-3
tesseract
3131 3231 3331 3132 3232 3332 3133 3233 3333 |
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
| Magic Hypercubes - Overview | A general overview of magic squares, cubes, tesseract, etc. Their interrelationships and summations. |
| Hypercube Representations | A review of how magic squares and cubes have been presented in publications, etc. through the ages. |
| Hypercube Representations - 2 | A review of how the forth dimension and magic tesseracts have been illustrated (over a shorter time period). |
| Order-3 Magic Tesseracts | Features of order 3 hypercubes and illustrations and listing for all 58 basic order-3 tesseracts. |
| Hypercube Aspects | Illustrates the 8 aspects of the magic square and 48 aspects of the magic cube. The tesseract has 384 aspects. |
| Hypercube Classes | Reviews classes of magic squares and cubes. Minimum requirements for the 18 of magic tesseract classes. |
| Associated Hypercubes | Discusses features of associated magic hypercubes, and embedded hypercubes of lower dimensions. |
| Hypercube Math | Equations and comparison tables. Demonstrates close relationship between hypercubes of different dimensions. |
| More Tesseracts | Orders 5 and 6 tesseracts, Inlaid tesseract, perfect tesseracts, plus links to some previously posted on other sites. |
| Hypercube - Cross-stitch | A picture and discussion of a cross-stitch project demonstrating features of square, cube, and tesseract. |
| Tesseract Knight Tour | Awani Kumar has successfully constructed an order 4 tesseract containing a magic knight tour. |
| The Unfolded Tesseract | Showing the 3-D illustration of a 4_D tesseract flattened out to 2 Dimensions to show all 24 faces. |
Please send me Feedback about my Web
site!
Harvey Heinz harveyheinz@shaw.ca
This page originated November, 2007
This page last updated
October 31, 2010
Copyright © 2007 by Harvey D. Heinz