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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.

 Tesseracts - Update 2013

A new page !     March 22, 2013

Recent developments - Contributors to Magic Tesseract theory - The 5-D Challenge

Introduction
What is a magic tesseract?
Coordinates of an order-3 magic tesseract
Directory to the rest of this section

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.

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?

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!

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.
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

1. Magic Hypercubes - Overview A general overview of magic squares, cubes, tesseract, etc. Their interrelationships and summations.
2. Hypercube Representations A review of how magic squares and cubes have been presented in publications, etc. through the ages.
3. Hypercube Representations - 2 A review of how the forth dimension and magic tesseracts have been illustrated (over a shorter time period).
4. Order-3 Magic Tesseracts Features of order 3 hypercubes and illustrations and listing for all 58 basic order-3 tesseracts.
5. Hypercube Aspects Illustrates the 8 aspects of the magic square and 48 aspects of the magic cube. The tesseract has 384 aspects.
6. Hypercube Classes Reviews classes of magic squares and cubes. Minimum requirements for the 18 of magic tesseract classes.
7. Associated Hypercubes Discusses features of  associated magic hypercubes, and embedded hypercubes of lower dimensions.
8. Hypercube Math Equations and comparison tables. Demonstrates close relationship between hypercubes of different dimensions.
9. More Tesseracts Orders 5 and 6 tesseracts, Inlaid tesseract, perfect tesseracts, plus links to some previously posted on other sites.
10. Hypercube - Cross-stitch A picture and discussion of a cross-stitch project demonstrating features of square, cube, and tesseract.
11. Tesseract Knight Tour Awani Kumar has successfully constructed an order 4 tesseract containing a magic knight tour.
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.

This page was originally posted November 2007
It was last updated March 25, 2013
Harvey Heinz   harveyheinz@shaw.ca
Copyright 1998-2009 by Harvey D. Heinz