# More Tesseracts

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.

(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

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.)

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