Once the concepts are in place it is easy to see how higher dimensional figures can be created. Problems arise in execution because of the rapidly increasing size as each dimension is added. See Statistics to get an appreciation of the increasing complexity. It is possible to visually look at the cubes. With 65,536 numbers, the tesseracts must be viewed in 256 square pieces. With 33,554,432 numbers, it would still be possible to look at the 5-D hypercube in square pieces, but the 32,768 squares are huge and their numbers too big to fit the 32x32 matrix easily on a page. Mathematically, however, even higher dimensions can be described with confidence that they indeed are perfect magic figures.
The tesseract is a 16x16x16x16 figure. It is too big to fit on a single sheet of paper or a computer screen. It can best be observed as 256 16x16 squares, but those squares must be logically ordered. This figure can be placed on an Excel spreadsheet as there are 256 columns, just enough for sixteen 16x16 squares. There are plenty of rows below this set to place the other fifteen sets of squares. Presented this way each square has 16 rows and 16 columns, the sets of 16 squares going left to right on the Excel sheet go in the direction of the pillars and the sets going down represent the files of the figure. I built my first tesseract using an Excel sheet like this with lots of calculation linkages.
The tesseract is 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 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. Many 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 aspects for each tesseract. This leaves 1.51E63 different magic tesseracts that can be generated using the base line method. All can be generated using the tesseract generator.
Until recently I would have said that creation of a 5-D magic hypercube was unreasonable. I could see how to do it. The base line concept is easily extendable to any dimension. The mechanics of placing 33,554,432 numbers into a 5 dimensional array using 25 of the 8,053,063,680 5-D possible base hypercubes was not fathomable. Proving it to be magic would not have seemed possible especially using an Excel spread sheet, my weapon of choice. Then I read that John Hendricks3 had done it. I was working on the Cube Generator and Xcode at the time and the seed was planted that it could be done.
The 5-D hypercube is a 32x32x32x32x32 figure containing a 33,554,432 number array. It would require 32,768 squares containing 32x32 number arrays to write out this cube and the numbers in those squares would have to use such a small font that they would be hard to read. It cannot be shown in any reasonable way on an Excel spreadsheet. It can, however, be represented using an array of 25 base cubes containing 5 base lines each. It can also be fully represented using 5 master base lines containing 32 numbers each. Both of these representations readily allow the determination of the number at any coordinate in the figure. With that ability it is easy to pick out sets of 32 numbers with their coordinates that correspond to any of the groups that add to the magic constant. Groups of 32 can be shown dynamically on a computer screen.
The 32-bit A, B, C, and D base lines are made by doubling the corresponding tesseract base lines. There are 32,768 E base lines consisting of the 16-bit equivalent of the numbers from 0 to 32,767 followed by their inverse. It is not even possible to show all the possible base lines on a computer screen. There are 1x2x8x128x32768 = 67,108,864 different base hypercubes in 5! dimensional directions or 8,053,063,680 possible 5-D base hypercubes. This means that there are 8,053,063,68025 = 4.46E+247 potential perfect magic 5-D hypercubes. I have no way of determining how many actually are magic but looking at the trends from the smaller figures even a conservative estimate would put the number over E+150.
The hypercube generator is capable of generating any 5-D hypercube that can be made using the base line method. Unlike the cube generator and the tesseract generator the user can make any choice regardless of whether it will lead to a valid magic hypercube. The user must make choices and then check to determine if the choices are valid before proceeding. The generator gives only minimal guidance in choosing base hypercubes. The user must understand the method in order to be successful with this generator.
Beyond 5 dimensions even the representations of the hypercubes using base line codes or master base lines become large. Despite this a simple strategy to create hypercubes of any size has been formulated. The strategy ensures that the resulting figure will contain all numbers in the range required for that figure. Numbers at any position can be determined from the representations allowing evaluation of magic constants. A 6-D hypercube example is shown and higher dimensions can be made using obvious extensions.