PRIME ORDER TESSERACTS

Creation of prime order tesseracts is just an extension of the concepts developed for the creation of Prime Order Squares and Prime Order Cubes. The greater difficulty is the size of the figure and in visualization. Sums for the figures described below were checked for accuracy because I have not determined a general proof for these figures.

ORDER-3 TESSERACTS (ternary base lines)

The order-3 magic tesseract generator at right contains four sets of four base lines. Each set generates a base tesseract but the base tesseracts are not shown. Only their sum, after multiplication and addition of 1, is shown as the Order-3 Magic Tesseract. The sums section at the right side of the table determines the validity of the base lines entered. All six including the uniform integral distribution must be correct for a valid tesseract. Each base line must have a 0, 1, and 2 in any order. For a valid base tesseract, the sum of the middle numbers of the four base lines of each set must be 1, 4, or 7 (1 mod 3). Not all combinations of valid base tesseracts will yield a valid magic tesseract.

There are 58 order-3 magic tesseracts. Harvey Heinz has a listing of them and much more information about them on his Order-3 Magic Tesseracts page. Shown at right is an aspect of the tesseract designated as Index # 5 on that page. I am sure that all 58 can be generated by modifying the base lines in the figure although I have not done so. I am also sure that all 384 aspects of each of the 58 tesseracts can be generated with different arrangements of the base lines. There are 6^16 possible arrangements.

An aspect of each of the first four tesseracts on Heinz's Order-3 Magic Tesseracts page can be generated by changing all of the base lines in one of the base line sets to 012. Separate modification of each set will produce a different index numbered tesseract. The tesseract initially shown and these four are all the tesseracts that have a 1 as their lowest corner number and thus a zero for the first number of all base lines.

To obtain additional order-3 tesseracts some of the base lines must start with 1 or 2. If all of the base lines except one of the base lines in the last set start with zero then the order-3 tesseract will have as its lowest corner number 2 or 3 depending on what number the exception base line starts on. To make 4-9 the smallest corner number, the next to last base tesseract must have one of its base lines start with 1 or 2.

ORDER-5 TESSERACTS (quinary base lines)

There are only two quinary base lines that can be used to generate order-5 base figures. These are 01234 and 02413. They are generated from the equations x mod 5 and 2x mod 5 for 0 ≤ x ≤ 4. These lines can be shifted (12340, 23401, etc) and/or reversed (04321, 31420, etc.). But for purposes of building pan-magic figures, shifted and reversed lines are the same. An order-5 base pan tesseract must therefore be composed of all the same base line or some combination of the two different base lines.

Only the base tesseracts composed of the same base line produce an interesting result. These base tesseracts are pan-3-agonal. Combining base tesseracts with the same base line will yield an intermediate that does not have uniform integral distribution. Shifting base lines on one of the base tesseracts does not help, but reversing some lines will give tesseract combinations with uniform integral distribution. By systematically reversing lines an order-5 pan-3-agonal tesseract can be built.

The 5 tess worksheet of the TesseractLines Excel Spreadsheet available on the Downloads page gives the master base lines of two order-5 pan-3-agonal tesseracts. The first just has the pan feature while the second is also associated. The main quadragonals in the second tesseract also add to S. Sums for the tesseract are given in the spreadsheet.

ORDER-7 TESSERACTS

There are three base 7 base lines that can be used to generate order-7 base figures. These are 0123456, 0246135, and 0362514. They are generated from the equations x mod 7, 2x mod 7, and 3x mod 7 for 0 ≤ x ≤ 6. An order-7 base tesseract can therefore be composed of all the same base line, some combination of two different base lines, or a combination of all three base lines.

When base tesseracts are made from four sets of four base seven base lines, both order-7 pan-3-agonal and pan-4-agonal magic tesseracts can be made. The pan-3-agonal base tesseracts as above can be made using four identical base lines, but they can also be made from three of one base line and one of another. Other combinations of three and one will yield a pan-4-agonal tesseract. Combinations of all three base lines or of two of one and two of another do not yield order-7 pan-x-agonal base tesseracts of any type. Combining four base tesseracts, 343(tesseract 1) + 49(tesseract 2) + 7(tesseract 3) + tesseract 4, can yield a pan-x-agonal tesseract if all four base tesseracts are pan-x-agonal and x is always the same. Uniform integral distribution must be ensured.

The 7 tess worksheet of the TesseractLines Excel Spreadsheet available on the Downloads page gives the master base lines of four order-7 pan-x-agonal tesseracts. Two are pan-3-agonal and two are pan-4-agonal. In each pair, one is just pan-x-agonal while the other is also associated. The main quadragonals of the associated pan-3-agonal tesseract also add to S. The sums for the figures are included in the spreadsheet.

ORDER-11 TESSERACTS

There are five base 11 base lines that can be used to generate order-11 base figures. They are generated from the equations x mod 11, 2x mod 11, 3x mod 11, 4x mod 11, and 5x mod 11 for 0 ≤ x ≤ 10. As more base lines become available, more base line combinations become possible and more pan-x-agonal tesseract types become possible.

With order-11 base lines it is possible to make pan-3-agonal, pan-4-agonal, and pan-3,4-agonal magic tesseracts. The base line combinations that will give the various types of pan-x-agonal base tesseracts become more varied as the order gets larger. I haven't determined a pattern except that base tesseracts made from four of the same base line always give pan-3-tesseracts.

The 11 tess worksheet of the TesseractLines Excel Spreadsheet available on the Downloads page gives the master base lines of six order-11 pan-x-agonal tesseracts. There are two pan-3-agonal, two pan-4-agonal, and two pan-3,4-agonal. In each pair, one is just pan-x-agonal while the other is also associated. The main quadragonals of the associated pan-3-agonal tesseract also add to S. A rudimentary sum checker for the figures is included in the spreadsheet. I can provide a better checker for anyone interested.

ORDER-13 TESSERACTS

There are six base 13 base lines for generating order-13 base figures. In addition to the three tesseract types that can be generated for order-11 tesseracts, order-13 pan-2-agonal and pan-2,4-agonal tesseracts can be built.

The 13 tess worksheet of the TesseractLines Excel Spreadsheet available on the Downloads page gives the master base lines of ten order-13 pan-x-agonal tesseracts. There are two pan-2-agonal, two pan-3-agonal, two pan-4-agonal, two pan-2,4-agonal, and two pan-3,4-agonal tesseracts. In each pair, one is just pan-x-agonal while the other is also associated. The main quadragonals of the associated pan-2-agonal and pan-3-agonal tesseracts also add to S. A rudimentary sum checker for the figures is included in the spreadsheet. I can provide a better checker for anyone interested.

ORDER-17 TESSERACTS

Nasik tesseracts for prime orders are possible when o ≥ 17. These are constructed from base tesseracts made from four different base lines. If the four different base lines are picked correctly they will yield a nasik base tesseract otherwise they will generate pan-2-agonal, pan-2,3-agonal, or pan-2,4-agonal tesseracts. Two order-17 nasik tesseracts are given in the 17 tess worksheet of the TesseractLines Excel Spreadsheet available on the Downloads page.

For order-17 and larger tesseracts, all seven pan-x-agonal types are possible. The 17 tess worksheet has two of each of the seven pan-x-agonal types. One of each pair is just the pan-x-agonal tesseract and one is also associated. For the pan-2-agonal, pan-3-agonal, and pan-2,3-agonal tesseracts, the associated tesseract also has all main quadragonals equal to S.

LARGER PRIME ORDER NASIK TESSERACTS

It should be possible to make a nasik tesseract for any prime order ≥ 17. I offer the following procedure for an order-o tesseract without proof. I believe it to be correct based primarily on a lack of failure to date.

The 4 base lines x mod o, 2x mod o, 4x mod o, and 8x mod o with 0 ≤ x < o should always give valid base tesseracts. By systematically moving base lines in this, set a valid nasik tesseract can be built. The process is as follows.

  1. Let the four base lines be: a = x mod o, b = 2x mod o, c = 4x mod o, and d = 8x mod o for 0 ≤ x < o.
  2. For the first base tesseract let the row base line be a, the column b, the pillar c, and the file d.
  3. For the second base tesseract let the row base line be b, the column c, the pillar d, and the file a.
  4. For the third base tesseract let the row base line be c, the column d, the pillar a, and the file b.
  5. For the last base tesseract let the row base line be d, the column a, the pillar b, and the file c.

The four base tesseracts can be combined as o3(base tesseract 1) + o2(base tesseract 2) + o(base tesseract 3) + base tesseract 4 to make the master base lines. The value at position nijkl in the magic tesseract is: nijkl = o3((row1i + column1j + pillar1k + file1l) mod o) + o2((row2i + column2j + pillar2k + file2l) mod o) + o((row3i + column3j + pillar3k + file3l) mod o) + ((row4i + column4j + pillar4k + file4l) mod o).

addendum 6-2-2009

It is possible to determine which base line combinations do not produce nasik tesseracts and thus by exclusion determine which combinations are nasik. The procedure is not as straightforward as that for the cubes as some tial and error is involved. The procedure for a tesseract of order o and wxyz axes is given below:

  1. Select three base lines of order o: qx mod o, rx mod o, and sx mod o with x from 0 to o-1, s > r > q, and q, r, and s < o/2.
  2. Construct the lines (s+r)x mod o, (s-r)x mod o, (s+q)x mod o, (s-q)x mod o, (r+q)x mod o, (r-q)x mod o, (s+r+q)x mod o, (s+r-q)x mod o, (s-r+q)x mod o, and (s-r-q)x mod o. These will be the main diagonals of the first wx, wy, xy squares and the main triagonals of the first wxy cube respectively. These constructed base lines cannot be the forth base line of the tesseract because resonance between the diagonal or triagonal and the base line would result in repetition in one of the triagonals or quadragonals of the tesseract.
  3. Select the forth tesseract base line, t, such that t ≠ q, r, s, (s+r), (s-r), (s+q), (s-q), (r+q), (r-q), (s+r+q), (s+r-q), (s-r+q) or (s-r-q) with t < o/2. If there are no possible values for t return to step 1 and select three different base lines.
  4. Construct a base tesseract using the base lines qx mod o, rx mod o, sx mod o, and tx mod o. All such tesseracts will be nasik.

By swapping numbers in the base tesseracts, i.e. by making all the threes ones and all the ones threes, new base tesseracts are created. Multiple exchanges are permitted. Any two base tesseracts can be combined to to make pan-magic tesseracts. The new base tesseracts and magic tesseracts cannot be constructed directly from base lines, however.