# MAGIC HYPERCUBE OVERVIEW

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 for the smallest possible nasik figures of n dimensions. These are all of order-2n. The order-4 square is small. It is possible to visually look at the 512 numbers of an order-8 cube on one page. With 65,536 numbers, the order-16 tesseracts are best viewed as 256 squares of 256 numbers. 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 nasik magic figures.

The order-16 tesseract is the smallest tesseract that can be nasik and these are easily made using binary base lines. It is possible to make smaller pan-magic tesseracts of odd orders by sacrificing some of the pan lines. Even the order-3 tesseracts with no pan features and only the main 4-D-agonals can be made using the base line construction method.

Larger tesseracts and other hypercubes can also be made by extension of the concepts developed for squares and cubes. A general method for construction of n-dimensional figures of order-2p is described in Überprüfen. Something similar appears attainable for prime orders but is currently without proof.

## TESSERACTS

The smallest nasik 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 sixteen 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 from base lines using an Excel sheet like this with lots of calculation linkages.

There is a tesseract generator available on the Downloads page. The Tesseract Guide describes the tesseract generator's features. There are 1.51E63 different order-16 tesseracts that can be generated using this generator.

The Order-2p Tesseracts page describes different order-8, order-16, and order-32 tesseracts that have been constructed using base lines. The HyperCubeLines spreadsheet available on the Downloads page contains the master base lines for these figures and a description of their features. Because of their size the larger tesseracts are not shown but excel spreadsheets are available from the Author. The compression, that the master base lines provides, is evident in these figures because the 64 numbers of the master base lines of the order-16 tesseracts contain all the information needed to place the 65536 numbers of an order-16 tesseract. A 1000 fold compression.

Tesseracts of prime order are discussed in Prime Order Tesseracts. The HyperCubeLines spreadsheet also contains the master base lines for these figures. The figures are also shown except for the largest due to its size. Again, I can provide the figures and the sums to anyone interested.

## 5-D HYPERCUBE

Until recently, I would have said that creation of a 5-D nasik 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 hypercube 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 hypercubes made from 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. However, these sums will not prove that the figure is a nasik hypercube since all combinations of 5-D base line type hypercubes will add correctly in all ways. It must be proved that all 33,554,432 numbers are present in the hypercube.

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 nasik 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. The 5-D Hypercube guide will help in making the choices and it gives an approach that should always work.

## n-D HYPERCUBE

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. The n-Dimensions page describes the construction process and the Überprüfen page gives a proof of its validity.