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This page includes tesseracts not included on other pages in this section of my site.
It also includes links to tesseracts previously included on other pages.
Many tesseracts shown on these pages are NOT shown in normalized position.

A 5-D Magic Hypercube   A Panquadragonal Tesseract   An Inlaid Magic Tesseract  Other tesseracts  Addendum (1/1/2013)

 (21/1/2013

A 5-D Magic Hypercube

In a paper published in 1962, John Hendricks  introduced the modern way of illustrating a tesseract. This diagram permits the placement of numbers in a magic tesseract in a meaningful manner (similar to that used for magic squares and cubes). [1][2]

He first illustrated an order-3 magic tesseract (4-D). Then showed an 5-dimensional magic hypercube (illustrated below) consisting of 3 interconnected magic tesseracts. He concluded with a 6-dimensional magic hypercube (9 interconnected magic tesseracts).

I have shown the numbers in each tesseract in two colors to clarify the 3 horizontal cube layers.
Numbers in red illustrate the directions of the 1-agonals (orthogonal lines).

Example orthogonal lines are:

x = 1, 212,153;   
y = 1, 206, 159;   
z = 1, 132, 233;   
w = 1, 150, 215;
v = 1, 152, 213 (this is one of the lines connecting the
3 tesseracts.
The 16 5-agonals (which also must sum correctly) are:
1, 122, 243; 215, 122, 29; 10, 122, 234; 159, 122, 85; 153, 122, 91; 4, 122, 240; 162, 122, 82; 191, 122, 53; 99, 122, 145; 49, 122, 195; 227, 122, 17; 31, 122, 213; 233, 122, 11; 87, 122, 157; 242, 122, 2; 28, 122, 216.
 

Not shown are the 2, 3, and 4-agonals. These are not required to sum correctly. However, because all order-3 hypercubes are associated,
all main (1 segment) r-agonals in the central hypercubes sum correctly.
Several examples are:
a 2-agonal is 131, 122, 113;
a 3-agonal is 78, 122, 166;
a 4-agonal is 152, 122, 92.

A 5-D magic hypercube contains 5 central magic tesseracts. Each of these contain 4 central magic cubes. And each of these contain 3 central magic squares (because of the associated feature).

Required to sum correctly (to be magic) are
405 … 1-agonals (the orthogonal lines) and
16 … 5-agonals.
The magic constant is 366.

 Text listing of this 5-D order-3 magic hypercube

159  197  010     016  156  194     191  013  162
206  019  141     138  203  025     022  144  200
001  150  215     212  007  147     153  209  004

179  073  188     111  176  079     076  117  173
055  123  188     185  061  120     126  182  058
132  170  064     070  129  167     164  067  135

028  096  242     239  034  093     099  236  031
105  224  037     043  402  221     218  040  108
233  046  087     084  230  052     049  090  227


090  015  161     158  196  012     018  155  193
024  143  199     205  021  140     137  202  027
152  208  006     003  149  214     211  009  146

078  116  172     178  075  113     110  175  081
125  181  060     057  122  187     184  063  119
163  069  134     131  169  066     072  128  166

098  235  033     030  095  241     238  036  092
217  042  107     104  233  039     045  101  220
051  089  226     232  048  086     083  229  054


017  154  195     192  014  160     157  198  011
136  204  026     023  142  201     207  020  139
213  008  142     151  210  005     002  148  216

109  177  080     077  115  174     180  074  112
186  062  118     124  183  059     056  121  189
071  127  168     165  068  133     130  171  065

240  035  091     097  237  032     029  094  243
044  100  222     219  041  106     103  225  038
082  231  053     050  088  228     234  047  085

(The 6-D magic hypercube presented in the Bulletin consisted of 9 interconnected magic tesseracts!)

Others in recent years who have designed higher dimension hypercubes are: [3]

  • David M. Collison (1937-1991) Hypercubes from 2 to 8 dimensions, using matrices and modular equations.

  • Meredith Houlton (San Diego) Odd dimensions from 5 to 9

  • J. Hendricks later published a different 5-D magic hypercube [4]

[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] See excerpts from the above paper here.    
[3] John R. Hendricks, Magic Squares to Tesseracts by Computer, 1998, 0-9684700-9-0, preface.
[4] John R. Hendricks, A 5-Dimensional magic Hypercube of Order 3, Journal of Recreational Mathematics 21:4, 1989, pp245-248

A Panquadragonal Magic Tesseract

Almost all examples of tesseracts shown on these pages are order-3. Illustrated here is the first publication of an order-4 tesseract with the new type of diagram. This is also the introduction of the term pan-4-agonal, later renamed by Mr. Hendricks to panquadragonal. [1][2][3]

In the diagram, the I-rows (1-agonals) are easy to spot. Examples are:
14, 117, 195, 188 (x direction)
14, 161, 255, 84 (y)
14, 236, 51, 213 (z)
14, 87, 242, 171 (w)
All are required to sum to 514.

The 8 quadragonals (4-agonals) are not quite so easy to trace. They are:
14, 101, 243, 156;
171, 256, 86, 1;
53, 146, 204, 111;
84, 203, 173, 54;
68, 26, 189, 231;
213, 179, 44, 78;
239, 69, 18, 188;
186, 48, 71, 209.
All of these must also sum correctly for the tesseract to be magic.

Because this is a panquadragonal magic tesseract, all ‘broken’ quadragonals must also sum correctly. Following are 4-agonals that are parallel to the first main one mentioned above.
A 2 segment one is
200, 146, 57, 111.
A 3 segment one is
144, 235, 113, 22.
And a 4 segment quadragonals is 190, 232, 67, 25.

For this order-4 tesseract, there are 256 correct 1-agonals, and 512 correct 4-agonals. No 3 or 4-agonals are required to be correct, although some may be. Therefore there are probably no magic squares or cubes in this tesseract.

The red and green lines in the illustration are simply to differentiate between the 4 layers of cubes.

A panquadragonal magic tesseract is similar to a pandiagonal magic square and a pantriagonal magic cube. In a pandiagonal magic square, if you move a row or column (line) from one side to the other, you obtain a new pandiagonal magic square. With a pantriagonal magic cube, if you move a square (plane) from one side to the opposite side, you obtain a new pantriagonal magic cube. With a panquadragonal magic tesseract, if you move a cube from one side to the opposite side, you obtain a new panquadragonal magic tesseract.

 

Following is the text listing for this tesseract.

053  176  201  084     131  214  127  042     252  097  008  157     078  027  178  231
090  003  166  255     237  124  017  136     151  206  107  050     036  181  224  073
200  093  060  161     114  039  142  219     009  148  245  112     191  234  067  022
171  242  087  014     032  137  228  117     102  063  154  195     209  072  045  188

210  123  046  135     108  013  152  241     031  182  227  074     165  196  089  064
144  229  116  025     055  146  203  110     065  044  189  216     250  095  006  163
035  138  223  118     153  256  101  004     238  071  018  187     088  049  172  205
125  024  129  236     198  099  058  159     180  217  080  037     011  174  247  082

012  145  248  109     190  235  066  023     197  096  057  164     115  038  143  218
103  062  155  194     212  069  048  185     170  243  086  015     029  140  225  120
249  100  005  160     079  026  179  230     056  173  204  081     130  215  126  043
150  207  106  051     033  184  221  076     091  002  167  254     240  121  020  133

239  070  019  186     085  052  169  208     034  139  222  119     156  253  104  001
177  220  077  040     010  175  246  083     128  021  132  233     199  098  059  158
030  183  226  075     168  193  092  061     211  122  047  134     105  016  149  244
068  041  192  213     251  094  007  162     141  232  113  028     054  147  202  111

[1] John R. Hendricks, The Pan-4-agonal Magic Tesseract, American Math Monthly, 75:4:1968:384
[2]
John R. Hendricks, Magic Square Course, 1991, pp 497-499
[3] John R. Hendricks, Magic Squares to Tesseracts by Computer, 1998, 0-9684700-9-0, page 130.

An Inlaid Magic Tesseract

Over the years John Hendricks did a lot of work with inlaid magic squares and cubes. On October 15,1999, he announced the construction of the first inlaid magic tesseract [1][2]. It is an order-6, with an order-3 magic tesseract occupying one hexadecimant. A hexadecimant is the 4-D equivalent of the 2-D quadrant, and 3-D octant.

The order-6 uses the consecutive numbers from 1 to 1296. and sums to 3891 in 872 different ways from.
The order-3 uses the consecutive numbers from 568 to 848 and sums to 1824 in 116 different ways. This tesseract is associated (central symmetric) as are all order 3 hypercubes. Therefore all central cubes and squares are also associated magic.

See the 9 page portable document format (PDF) report on this tesseract here.   

[1] hendricks.htm#Inlaid Magic Tesseract  the first brief announcement to the world
[2] cube_inlaid.htm#magic tesseract         a brief description and diagrams

Other tesseracts

Before the posting of these tesseract pages (November 2007), I had posted the following tesseracts on other pages. 
(The first 4 are on the same page.)

A basic order-3 tesseract This is MT#9 (9th in Hendricks discovery order) and Index # 54 (normalized sort order)
Order-4 quadragonal tesseract A different tesseract from the one shown above.
A perfect (Nasik) order-16 The first magic tesseract constructed, where all possible lines sum correctly.
A perfect (Nasik) order-32 The first magic 5-dimensional hypercube where all possible lines sum correctly.
A basic order-3 tesseract This is MT#1 (1st in discovery order) and Index # 5 (normalized sort order)
Multimagic tesseracts Bimagic orders 32 and 64. Trimagic order 243.

Addendum January 21, 2013

Work continues to be done on magic tesseracts. Most notably by Mitsutoshi Nakamura.
Recently he added 2,3,4-agonal magic tesseracts of orders 10, 14, 15, and 18. to his website.
Go to http://homepage2.nifty.com/googol/magcube/en/ for links to the updated pages.

This page was originally posted November 2007
It was last updated January 21, 2013
Harvey Heinz   harveyheinz@shaw.ca
Copyright © 1998-2009 by Harvey D. Heinz