ORDER-2p TESSERACTS

Creation of order-2p tesseracts is just an extension of the concepts developed for the creation of order-2p Magic Squares and order-2p Magic Cubes. The difficulty comes in the size of the figure and in visualizing the four dimensions. The validity of their creation using base lines is discussed in Überprüfen.

ORDER-8 MAGIC TESSERACTS

It is possible to make an order-8 pan-magic tesseract using base lines but it cannot be nasik. An order-8 pan-4-agonal base tesseract can be created from three different C type base lines and either an A or B type base line. Combining twelve of these can give an order-8 pan-4-agonal magic tesseract. An example is given in the 8 tess worksheet of the HyperCubeLines Excel Spreadsheet available on the Downloads page. That tesseract is also {4compact5}

ORDER-16 MAGIC TESSERACTS

There is an order-16 tesseract generator available on the Downloads page. It is capable of generating all the nasik tesseracts that can be made from base lines. The generator will also allow many permutations of the tesseracts to create different tesseracts or different aspects of the same tesseract. Sums for agonals and corners of small tesseracts (compact type features) can also be viewed interactively. The tesseract generator and its functions are described in more detail in the Tesseract Guide.

Like the order-8 pan-2,3-agonal cubes, the 4-dimensional properties of order-16 pan-2,3,4-agonal tesseracts are in general homogeneous, but it is possible to create features within the embedded cubes and squares. Also, like the cubes it is possible to create order-16 tesseracts with some of the x-agonals missing. The master base lines for six tesseracts are shown on the 16 tess worksheet of the HyperCubeLines Excel Spreadsheet available on the Downloads page. The tesseract is quite large and requires many calculations to create on an Excel sheet. The sheet with sum calculations is available by request from the author.

The tesseracts are easier to describe mathematically than visually. The tesseract has 16-bit base lines. The A, B, and C base lines are made by doubling the base lines used in the order-8 cube. There are 128 D base lines. The D base lines consist of the 8-bit equivalent of the numbers from 0 to 127 followed by the inverse of the first 8 bits. Most of these base lines do not have any apparent symmetries or periodicities.

There are twenty-four, 4!, possible arrangements of the A, B, C, and D base lines into the four dimensions of the tesseract. For any one of these arrangements there are 1x2x8x128 = 2048 base tesseracts. There are thus 24x2048 or 49,152 possible base tesseracts. Since a 16x16x16x16 tesseract will require 65,536 or 216 numbers, 16 base tesseracts will be required to construct the magic tesseract. This means that there are 4915216 or 1.16E75 possible magic tesseracts. Only about 500 ppt or 5.80E65 of these are actual magic tesseracts. There are 4 dimensions that can be reversed or 24 possible reversal combinations. With the 24 dimensional arrangements there are 384, 16x24, aspects for each tesseract. This leaves 1.51E63, 5.80E65/384, different magic tesseracts that can be generated using the base line method.

Tesseract 1 was made in the late 90's using an Excel spreadsheet. Like all order-16 nasik tesseracts, it is {4compact2,3,5,9}. It has no other special features.

Tesseract 2 is based on the structured construction described in n-Dimensions and Überprüfen. It is an order-16 pan-2,3,4-agonal, or nasik, tesseract and thus is {4compact2,3,5,9}. It has special features in some lower dimensions. Let the rows, columns, pillars, and files be the w, x, y, and z axes. Then, all wxy cubes are {3compact2,3}. All wxz cubes are {3compact2}. All wyz cubes are {3compact5}. All xyz cubes are {3compact3,5}. All wx squares are {2compact2}. All xy squares are {2compact3}. And all yz squares are {2compact5}.

The next three tesseracts are just modifications of the last base tesseract of tesseract 2. They illustrate removal of some x-agonals. The master base lines of the three differ by at most ±1 from tesseract 2 for any given number. If instead of having different lettered base lines for each dimension of the base tesseract some of the base lines are the same letter type, then the x-agonals are affected.

In Tesseract 3 one D and three C base lines are used in the last base tesseract. As a result, not all of the diagonals add to S but the other agonals are all OK. The result is a pan-3,4-agonal tesseract.

Tesseract 4 has three D and one A base line in the last base tesseract. Some diagonals and some triagonals do not add to S in Tesseract 4. Three C base lines and either an A or B base line or three D base lines and either a B or C type base line will cause the same effect, the creation of a pan-4-agonal tesseract.

Tesseract 5 contains 4 D type base lines in the last base tesseract resulting in some diagonals and some quadragonals not adding to S. This gives a pan-3-agonal tesseract.

Tesseract 6 is an order-16 nasik tesseract with all that entails. In addition all 256 squares in the wx plane contain order-8 pan-magic squares in their quadrants. These order-8 squares are all {2compact2} and contain Franklin V and W bent diagonals. The order-16 squares in the wx plane are {2compact2} and contain Franklin V, W, and WW bent diagonals. The 256 order-16 squares in the yz plane contain Franklin V, W, and WW bent diagonals and {zigzag2} lines and they are {2compact3}. The order-16 wxy and wxz cubes are {3compact2}. And the order-16 wyz and xyz cubes are {3compact3}.

ORDER-32 MAGIC TESSERACTS

Order-32 pan-2,3,4-agonal tesseracts should allow the freedom to make many interesting figures, however confirmation of uniform integral distribution becomes a problem, at least for me. On the 32 tess worksheet of the HyperCubeLines Excel Spreadsheet available on the Downloads page are the master base lines for three order-32 tesseracts. The tesseracts would be quite large and are not shown on that sheet. I have not made the tesseracts themselves. They exist and have the properties stated based solely on the proof in Überprüfen. It is not necessary to build the figure and confirm the sums if the correct base lines and base figures are chosen. All three of the described tesseracts are nasik.

The first order-32 tesseract described is complete and {4compact2,3,5,17}. All wxy cubes within the tesseract are {3compact2,3}. All wxz cubes are {3compact2}. All wyz cubes are {3compact5}. All xyz cubes are {3compact3,5}. All wx squares are {2compact2}. All xy squares are {2compact3}. And all yz squares are {2compact5}.

The second order-32 tesseract is complete and {4compact3,5,9,17}. All wxy cubes within the tesseract are {3compact3,5}. All wxz cubes are {3compact3}. All wyz cubes are {3compact9}. All xyz cubes are {3compact5,9}. All wx squares are {2compact3}. All xy squares are {2compact5}. And all yz squares are {2compact9}.

The third tesseract is {4compact2,3,5}. All wxy cubes within the tesseract are {3compact2,3}. All wxz cubes are {3compact2}. All wyz cubes are {3compact5}. All xyz cubes are {3compact3,5}. All wx squares are {2compact2}. All xy squares are {2compact3}. All yz squares are {2compact5}. There are order-16 nasik tesseracts in the hexadecants. The cubes and squares within these order-16 tesseracts have the same compact properties in the same dimensions as the order-32 tesseract.