# USING THE TESSERACT GENERATOR

The generators were designed for a resolution of 1024x640. A distorted screen will result at any resolution less than this.

Many of the text fields are buttons. Their use may not be intuitive. It is suggested that the following section be read prior to using the generator.

The generator opens in a build mode. Unlike the cube generator, there are no alternate build modes. Generation of a 16x16x16x16 tesseract involves manipulation of a 65,536 number array. This is two orders of magnitude greater than the 8x8x8 cube and limits what can be done on many fronts. There are 49,152 possible base tesseracts versus 96 base cubes. Because of the time required to manipulate such a large amount of data, the tesseract generator has fewer options than the cube generator even though the resulting figure has far more possibilities for manipulation.

Once the tesseract is built it can be manipulated to make a family of 16! different cubes or to make visually different modifications of any of the family's 16! valid tesseracts. Because of the size of the figure, only one of 256 squares of the tesseract can be viewed at any one time.

One can also scan some of the many patterns that add to the magic constant.

## TESSERACT CONSTRUCTION

Upon opening the tesseract generator the screen is split into two parts. On the left is a large square with a 16x16 matrix of zeros. This represents just one of the 256 squares used to illustrate the tesseract. Initially the screen is set to look at the wx square that is the first pillar of the first file.

On the right are the familiar 11 A, B, and C base lines used for the cube. The cube's base lines are doubled in order to make the 16-bit base lines of the tesseract from the 8-bit base lines of the cube. There are also 128 D base lines which must be used in one of the four dimensional directions. The D base lines consist of the 8-bit binary equivalents of the numbers from 0 to 127 followed by the inverse of the first 8-bits in the second 8-bits.

### Building the Tesseract

The tesseract is built in the same way as the base line method of the cube generator. The individual base lines are picked from the array in the order, row, column, pillar, and file. Each group of four base lines completes a base tesseract. Each base tesseract requires the A0, one B, one C, and one D base line in any order. Sixteen base tesseracts must be completed in order to make a magic tesseract. Each pick will be shown above the big square in 16 groups of four base lines.

As each pick is made, the array will be reset to allow only valid picks for the next step. Determining that base lines are valid is very fast at first but as more base tesseracts are completed a delay may be noticed. There are 1 x 2 x 8 x 128 x 24 or 49,152 possible base tesseracts. 12,288 of these have an A0 in the first dimension. If a valid result with an A0 in the first dimension is found, the program stops checking for A0's and goes to B0, etc. The time it takes to find a valid result for all the codes determines how long the delay is. The more complete the tesseract is, the more likely invalid results will be found resulting in longer delays. I have observed delays of 2-3 seconds on my computer. On other platforms, it could be longer. The delays are usually longest at the beginning of a new base tesseract of a nearly completed magic tesseract.

The tesseract generator is unable to complete a tesseract about 10% of the time in my experience. When this happens there are no selections possible for the next step. When looking for valid next base tesseracts, the assumption is made that if it is possible to add a next base tesseract it is possible to complete the tesseract. This appears to be a valid assumption for the cube but not for the tesseract. Normally the dead end occurs after completion of the 15th base tesseract. It may occur earlier. The dead end will only appear after completion of a base tesseract because of the way that the algorithm is set up. If it is possible to pick the first base line of the base tesseract it will always be possible to complete that base tesseract.

If a dead end is reached one can either EXIT and restart or the last base tesseract or partial tesseract can be deleted. The last base tesseract can be deleted back to the first. This is accomplished by selecting the BACK button above the right corner of the big square. The first base tesseract can only be removed by restarting. Often when a dead end is encountered, it will be necessary to remove two or three of the base tesseracts in order to find a path that will yield a completed tesseract.

### Viewing Other Parts of the Tesseract

When initially opened, the left side of the screen shows the rows and columns in the first pillar and first file as a 16 by 16 matrix of 256 numbers. The rows and columns are in the w and x dimensions of the figure. The pillars and files go in the y and z directions respectively. There are 255 additional pages representing the remainder of the tesseract that can be viewed.

Pressing the CHANGE button in the lower right corner of the screen accesses the second page of the tesseract generator. This can be done at any time. Clicking on RETURN restores the original screen. The new screen still shows the big square, but in place of the dimensional codes is a 16 by 16 matrix of small squares. This matrix is a representation of the 256 wx squares in the pillars and files of the tesseract. Any square can be selected by clicking on it. This will reveal the values present in that square. A description of which square is actually shown appears above the big square of this screen and in the lower right corner of the first screen.

Above the array of small squares is a grouping of the four dimensions as the w, x, y, and z axes. These correspond to the rows, columns, pillars, and files of the tesseract. The top two members of the group correspond to the codes that appear in the direction of the rows and columns of the displayed big square. The bottom two indicate the dimensional directions for the rows and columns of the 256 small squares. The order of the w, x, y, and z axes can be rearranged so that any of the dimensions can be shown in either direction of the big square and little squares. This is done by dragging the letter to a new position.

## BASE LINE EDITOR

When a tesseract is successfully completed, a message appears above the big square. The right side of the screen will also change to a tesseract version of the base line editor. The 256 square array is still accessible through the CHANGE button.

### Buttons

There are five new buttons. One allows access to the MAGIC CONSTANT checker. The second will RESET everything to the values they had when the tesseract was completed. This will not change the settings made using the CHANGE screen and it will not change the TRADITIONAL tesseract back to the BASE LINE tesseract.

The third button, the RESTART, will erase the current tesseract and allow a new tesseract to be built. The SAVE button will save the tesseract to a file called tesseractdrop. The last new button will convert the tesseract from the BASE LINE range of 0-65,535 to the TRADITIONAL magic tesseract range of 1-65,536.

### Translations

The upper left side of the base line editor contains four sliders that indicate the placement of the 0 of the tesseract at the w, x, y, and z coordinates. The same vector translates the remainder of the tesseract. This allows the zero (or one for the traditional tesseract) to be placed in any of the 65,536 positions of the tesseract.

Translations do not create different magic tesseracts. Because of the wraparound properties of these figures, all translations are equal. The translation just gives a visually different appearance.

### Aspects of the Tesseract

Below the sliders is a 16 x 4 array of the CURRENT TESSERACT CONFORMATION. From top to bottom, the rows represent the row, column, pillar, and file dimensions. These can be rearranged in any order by clicking and dragging the row when it is encircled by a rectangle. It needs to be noted that the similar action performed on the CHANGE page of the builder is different. Dragging those axes will determine which two dimensions are presented on the screen. The base line editor action is to change which groups of 16 base lines are in the row, column, pillar, and file dimension of the tesseract. This distinction also effects how the tesseract is ordered when it is saved. The former action does not affect the appearance of what is saved, the latter does. There are 24, 4!, different arrangements of the four axes.

In the upper left corner of the editor are four buttons that allow the corresponding base lines to be reversed. The reversal is coupled with a translation in order to maintain the zero (or one for the traditional tesseract) at the same position in the tesseract. There are 16, 24, different combinations of normal and reversed lines. Combined with the 24 different orders there are 384 different arrangements for each tesseract. In the vernacular of magic figures, they are just different aspects of the same tesseract.

### Reordering Base Tesseracts

Above each column in the array is a black oval. Selecting one of these allows the corresponding column to be dragged to any position. The columns may be arranged in any order. Shuffling columns creates 16! different tesseracts each of which can be shown in all the aspects and translations discussed above. Do not drag the columns too quickly or they will not change correctly.

### Unique Tesseracts

This set of 16! tesseracts, 384 aspects, and 65,536 translations can all be converted to the UNIQUE TESSERACT CONFORMATION shown at bottom. These Unique Tesseracts are derived in a manner similar to the Unique Cubes as described in Magic Cube Guide. There are about 7.22E49 Unique Tesseracts.

## MAGIC CONSTANT CHECKER

The tesseract magic constant checker screen is split into two parts. On the left is a 16x16 grid of squares. Each square represents one of the 256 wx squares of the tesseract. The square grid represents the y and z axes from left to right and from top to bottom respectively. Within the squares, the w axes goes from left to right and the x from top to bottom. At the top of the upper left square is a white line when the screen is first opened. This represents the 16 elements of the row associated with the 0,0,0,0 position of the tesseract. Each element of the row is given a white dot at its position. The white dots represent the elements that are being summed by the magic constant checker. It appears as a line because the dots are grouped. Other patterns will show the individual dots.

### Slider Controls for the Magic Constant Checker

On the right side of the screen are the controls that determine which groups are summed. At the top left are four sliders, one for each axis. The position of the sliders determines the position of the first element of the group being displayed and summed. The other elements of the group of 16 numbers are placed relative to the first number and appear as white dots on the screen. To the right of the sliders is a display of the values of the elements being summed and their coordinates. The sum at the bottom really does correspond to the sum of the 16 numbers. It is always 524,480 for the cube using the numbers 0-65,535 or 524,496 for the range 1-65,536.

### Magic Constant Types and Sub Types

There are 39 group types listed with short descriptions of the patterns they represent. Clicking on a type button displays the pattern on the display at left relative to the w,x,y,z slider coordinates selected. Each of the types have several sub types that can be different spacings or different orientations of the main type. There should be no duplication of types and sub types but there are redundancies within some sub types. For instance picking any of the numbers in a row as the first element will display the same 16 numbers, albeit in a different order in the display of values.

### Scanners

In the center of the right hand part of the screen are two buttons that will allow automatic scanning of the position coordinates and the types and sub types respectively. Both buttons have two speeds, a relatively slow speed that will allow observation of the patterns in the array, and a fast speed that I used to monitor the unwavering sum while troubleshooting. If there were problems, the sum would flicker indicating that there were incorrect calculations. In the final version there should be an unflickering sum indicating all sums are correct.

### Lines of a Nasik Tesseract

The first type in the magic constant checker is the 1-D lines of the cube. There are four sub types, one parallel to each of the four axes of the tesseract. By convention, they are the rows, columns, pillars, and files of the tesseract. There are 16,384 1-D lines altogether.

The tesseract has six square types inscribed within it corresponding to the six combinations of two of the four dimensions. All of the squares have two diagonals. There are thus 12 sub types of square or 2-D diagonals defined by the tesseract. There are 256 squares in each direction, each with a total of 32 diagonals and broken diagonals. There are thus 6 x 32 x 256 = 49,152 2-D diagonals plus broken diagonals in each tesseract.

The tesseract also has four cube types inscribed within it each of which corresponds to the combination of three of the dimensions. Each 3-D cube has four diagonals so that there are 12 sub types of 3-D cube diagonals in the tesseract. There are 64 different cubes in the tesseract, each with a total of 1024 diagonals and broken diagonals making a total of 65,536 different 3-D diagonals plus broken diagonals in each tesseract.

The tesseract itself has eight diagonals, thus eight sub types. There are 4096 diagonals and broken diagonals for each 4-D diagonal in the tesseract. Altogether there are 32,768 4-D diagonals and broken diagonals in the magic tesseract.

There are 4096 different examples of each line and diagonal sub type. Having all lines, all diagonals, and all possible broken diagonals of squares, cubes, and tesseracts equal the magic constant is the traditional definition of a nasik tesseract. The tesseracts generated here have many more ways to sum to the magic constant.

### Tesseract Corners and Tesseract Rectangular Prism Corners

The corners of tesseracts of all sizes sum to the magic constant. There are eight sub types of tesseract corners listed in the menu. Because of the wrap around property of the magic tesseract, the other seven potential sub types are redundant. That is corners of the 10x10x10x10 tesseract are equivalent to the corners of an 8x8x8x8 tesseract and also equivalent to a 10x8x10x8, 8x8x8x10, etc. tesseract. The 11's and 7's, 12's and 6's, 13's and 5's, 14's, and 4's, 15's and 3's, and 16's and2's are also interchangeable for both the tesseracts and prisms. With one exception all tesseract and prism sub types have 65,536 examples. The one exception is the 9x9x9x9 tesseract that only has 4096 different examples. For this tesseract, wraparound still gives a 9x9x9x9 tesseract.

There are 21 types of four dimensional prism corners described. Seven of these have 14 sub types and fourteen have 12 sub types. The 21 types represent all the prism corner types for which all 65,536 examples add to the magic constant. All other potential prism corner types have some examples that add to the magic constant and some that don't. All possible combinations of 2's, 4's, 6's, and 8's are included in the types as are all combinations of 3's and 7's. For an order-32 magic figure, 10's, 12's, 14's and 16s would be added to the former group and 11's and 15's to the latter. The new pair 5's and 13's would also be added.

### Shifted Lines and Shifted Diagonals

There are four types of shifted lines, each with seven sub types. The lines consist of eight numbers in a row, column, pillar, or file with a continuation of the line shifted a vector of 8 in one, two, or three of the other dimensions. The six 2-D shifted diagonals each have six sub types. The six types correspond to the six plane directions of the tesseract. The shifted diagonals consist of eight numbers in one of the two diagonals of a square with the continuation shifted a vector of 8 in one or both of the dimensions perpendicular to the square. There are twelve sub types of 3-D shifted diagonals. These correspond to the four diagonals of each of the four cube sets nested within the tesseract. The diagonals consist of eight numbers in one cube with the remaining eight continuation shifted a vector of 8 into the fourth dimension. There are 32,768 different example of each of the shifted figures because selection of the coordinates of the same relative points on either of the two line segments sums the same 16 numbers.

### Shifted Cubes and Beaded Squares

The last two types shown illustrate two 4-D analogs of the shifted squares described in the Magic Cube Guide. The shifted cubes consist of the corners of two 2x2x2 cubes in the same three dimensions that are separated either by a (8,8,8) vector in those three dimensions or by a (8,8,8,8) vector. Since there are four cube configurations in the tesseract there are eight sub types for this type. The cubes could be 3x3x3, 4x4x4, etc. as well but this is not shown. The last type is the beaded squares. These are a series of four 2x2 squares each shifted in the same plane a (4,4) vector from the previous square. As there are six square directions and two diagonals for each square there are twelve sub types. Many variations on this theme are also possible but again not shown. There are 32,768 examples of the shifted cubes and 16,384 of the beaded squares.

### Other Patterns

There are undoubtedly millions of patterns that could be translated throughout the tesseract and always sum to the magic constant. Those shown are primarily extrapolations from the square and cube. For example, in the square, there are two sub types and 20 examples of corners of squares, in the cube there are 4 types and 1600 examples of corners of cubes, and in the tesseract 8 types and 462,848 examples of corners of tesseracts. If the similar corners of prisms are included, the explosion is even more dramatic with over 18 million examples in the tesseracts. An examination of potential patterns in the cube suggests thousands for the cube all of which probably have 4-D analogues. The tesseract undoubtedly has many more.