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5-D HYPERCUBES

The smallest possible nasik 5-D hypercube is of order-32. This requires the proper placement of 33,554,432 consecutive numbers. The smallest possible prime order nasik 5-D hypercube is of order-37. This requires the placement of 69,343,957 consecutive numbers. These figures cannot be displayed in any reasonable way, yet with master base lines they can be fully described in a relatively small space.

A 5-D hypercube generator is available on the Downloads page. A more detailed description of how to use the generator and its features is given below under USING THE 5-D HYPERCUBE GENERATOR. It is difficult to use even with the hints below, but it is capable of making 5-D nasik hypercubes of order-32. All such hypercubes are {⁵compact_2,3,5,9,17}

ORDER-32 5-D HYPERCUBES

On the 5-D 32 worksheet of the HyperCubeLines Excel Spreadsheet available on the Downloads page there are master base line sets for two order-32 nasik 5-D hypercubes. There is also a rudimentary sum checker. The first 5-D hypercube is the same as the hypercube described below using alphanumeric codes. The alphanumeric codes can be entered into the hypercube generator for verification if desired. It was created using the hypercube generator before I discovered the structured approach. It has no special features that I have determined.

The second 5D hypercube described on the 5-D 32 worksheet was created using the structured pattern described in n-Dimensions. This hypercube has special features in many of its lower dimensional magic figures. Let the rows, columns, pillars, files, and 5^th-D be the v, w, x, y, and z axes. Then, all vwxy tesseracts are {⁴compact_2,3,5}. All vwxz tesseracts are {⁴compact_2,3}. All vwyz tesseracts are {⁴compact_2,9}. All vxyz tesseracts are {⁴compact_5,9}. All wxyz tesseracts are {⁴compact_3,5,9}. All vwx cubes are {³compact_2,3}. All vwy and vwz cubes are {³compact₂}. All vxy cubes are {³compact₅}. All vyz and wyz cubes are {³compact₉}. All wxy cubes are {³compact_3,5}. All wxz cubes are {³compact₃}. All xyz cubes are {³compact_5,9}. All vw squares are {²compact₂}. All wx squares are {²compact₃}. All xy squares are {²compact₅}. And all yz squares are {²compact₉}.

ORDER-37 5-D HYPERCUBES

Based on extrapolation from lower order prime figures it should be possible to describe prime order 5-D figures using base lines. In order to be nasik the prime must be larger that 32, thus order-37 is the smallest possible prime nasik 5-D hypercube. Master base lines for an order-37 5-D nasik hypercube are given in the 5-D 37 worksheet of the HyperCubeLines Excel Spreadsheet available on the Downloads page along with a rudimentary sum checker. The master base lines are derived as described below.

1. Let the five base lines of the 5-D hypercubes be: a = x mod 37, b = 2x mod 37, c = 4x mod 37, d = 8x mod 37, and e = 16x mod 37 for 0 ≤ x < 37.
2. For the first base 5-D hypercube let the row base line be a, the column b, the pillar c, the file d, and the 5^th-D e.
3. For the second base 5-D hypercube let the row base line be b, the column c, the pillar d, the file e, and the 5^th-D a.
4. For the third base 5-D hypercube let the row base line be c, the column d, the pillar e, the file a, and the 5^th-D b.
5. For the fourth base 5-D hypercube let the row base line be d, the column e, the pillar a, the file b, and the 5^th-D c.
6. For the last base 5-D hypercube let the row base line be e, the column a, the pillar b, the file c, and the 5^th-D d.
7. The five base 5-D hypercubes are combined as 37⁴(base 5-D hypercube 1) +37³(base 5-D hypercube 2) + 37²(base 5-D hypercube 3) + 37(base 5-D hypercube 4) + base 5-D hypercube 5 to make the master base lines.

The value at position n_ijklm in the 5-D hypercube is: n_ijklm = 37⁴((row1_i + column1_j + pillar1_k + file1_l + 5^th-D1_m) mod 37) + 37³((row2_i + column2_j + pillar2_k + file2_l + 5^th-D2_m) mod 37) + 37²((row3_i + column3_j + pillar3_k + file3_l + 5^th-D3_m) mod 37) + 37((row4_i + column4_j + pillar4_k + file4_l + 5^th-D4_m) mod 37) + ((row5_i + column5_j + pillar5_k + file5_l + 5^th-D5_m) mod 37).

More generally any prime, o, greater than 37 can be substituted for 37 in the above procedure to make an order-o nasik 5-D hypercube. At this point I make the above statements for prime 5-D hypercubes without a formal proof.

USING THE 5-D HYPERCUBE GENERATOR

Even with XCode it is not easy to manipulate a database of the size required to make the order-32 5-D hypercube. The computer is capable but time becomes an issue. Validation of a set of 5-D hypercubes requires the examination of all 33,554,432 positions to determine uniform integral distribution. The evaluation of just one set takes a noticeable amount of time and nearly all of the memory in my computer. Help in picking from the huge number of possibilities is therefore limited. To successfully build a 5-D hypercube requires some skill. The generator just provides a framework.

I have given some hints on how I built the example below. The hints are based on my observations of what works with the tesseract and are not necessarily the best or even a valid approach. I built this hypercube prior to my discovery of the generalized n-dimensional approach. The hints should allow greater diversity of solutions than that given by the approach in n-Dimensions but following them will miss some solutions that a more random approach would find.

5-D HYPERCUBE GENERATOR

The generator opens to a split screen with buttons in the middle. The left half of the screen is where base lines will be entered and the right side will list potential base lines to enter. If the SEARCH button is pressed a randomly selected list of possible first base hypercubes will be presented. One of these base line sets can be entered into the first line of Dimensional Base Lines array. Selecting the SEARCH button again will present a new list of possible base lines that are compatible with the first set. The base line sets should be entered in the same order as they appear in the Potential Next Base Lines set. This approach can be continued, unfortunately one would have to be extremely lucky to actually complete a 5-D hypercube using just this approach.

Code Entry

One can enter base lines into the grid at any time with or without using the help tools. Each row must contain A, B, C, D, and E base lines in any order. Move the selector over the desired location. A rectangle should now surround the position. Type the letter code and then a number. The A code does not require a number. Only valid numbers will be allowed. If an error is made use the delete key to remove the entry.

The base lines must also be completed in order so that the intermediate figures can be checked for validity. Combinations of base cubes may be checked at any time by selecting the VERIFY button. If the matrix is not filled out properly an error message will describe the problem that must be rectified. The VERIFY button must be pressed after the last line is completed in order to access the MAGIC CONSTANT CHECKER. The checker will only be accessed if the completed hypercube is valid.

Scan Buttons

A less random method of building the hypercube uses the SCAN D'S and SCAN E'S buttons. The setup for these buttons is a little different from the setup for the SEARCH and VERIFY buttons. The D or E base line in the last row must be left blank before selection. A search for just potential D or E base lines to fill the vacant position will then be done. Scanning for D's is sequential but only valid entries are shown. Since there are 128 D base lines and only 125 positions in the grid not all possibilities can be shown. You can be sure however that if 125 are shown, the last three are valid also. Scanning for E's is done randomly among the 32,768 possible E base lines. It is possible to get duplications.

The E GROUP Button

There are groups of E base lines that are compatible as defined in Base Cube Rules under Compatible Base Line Rules. There are a maximum of five allowed in any dimensional direction. Compatible E's can be found using the E GROUP button. There will be duplicates, especially when searching for the fifth member of a dimensional direction. Selection of compatible base lines is fast relative to the other searches because it is done by an algorithm that randomly looks only at E base lines, it does not verify the entire hypercube. The down side is that just because the new E base line is compatible with the other E base lines in that column it will not necessarily create a valid intermediate hypercube. A compatible E group in a column is very useful but the remaining base lines in the E group's rows can negate the effect. It is meant to be used following the rules of the n-Dimensional hypercube. A stepwise approach for the 5-D hypercube is shown below.

5-D HYPERCUBE GENERATOR HINTS

As more base hypercubes are added, more incompatibilities with potential next base hypercubes occur. Eventually random selection of a next potential base hypercube using just the SEARCH tool finds valid results to display very slowly. They may be out there but verification speed is an issue. The STILL SEARCHING blink rate indicates the rate that new random sets are tested. There will eventually be many blinks until a valid next hypercube is found. I have gotten to the 19^th base hypercube but only by carefully selecting potential base hypercubes from the list.

Limit Dimensional Arrangements

There are 120 different dimensional arrangements of the five types of base lines. In general, it is best to limit the number of different dimensional arrangements used. This is true for the tesseract as well. It is very hard to build a valid tesseract that has 16 different dimensional arrangements of its four base line types. It is probably possible to make a hypercube with 25 different dimensional arrangements of the five base line types, but I believe that it will be very difficult. The first five base hypercubes in the example hypercube have the same dimensional pattern. There is also one other base hypercube with the same pattern. There are no other base hypercubes in the example that have the same dimensional pattern.

When the same dimensional arrangement is used more than once, the SCAN D'S and SCAN E'S buttons become more useful. There are limits to the number of base hypercubes of any one dimensional arrangement. An E base line must be present at least once in every dimension requiring at least five dimensional arrangements.

Limit Number of Base Lines

The D base lines in the example are mostly D₄'s, the C's mostly C₁'s, and the B's, B₀'s. Using random values for the base lines makes it more difficult to complete the figure. The E base lines are mostly E₁₇₂₆₉'s and the other E base lines are either E₀, E₅₈₈₀, E₄₀₃₈, or E₃₂₅₄₄. This is a set of compatible E base lines. If other C and/or D base lines are used in a base hypercube, it is useful that they be compatible with other C and D base lines in the figure as well. The B₀ and B₁ base lines are compatible.

An Approach That Always Works

The approach outlined in n-Dimensions will always give a valid hypercube. A simple way to generate many 5-D hypercubes follows:

1. Across the first row enter A, B_x, C_y, D_z, E_g, where x is 0 or 1, y is 0-7, z is 0-127, and g is 0-32767. The A, B, C, and D codes may be in any order in the first four boxes, but there must be one of each. The E code must be in the fifth.
2. Across the second, row enter the same four A, B, C, and D codes in the same order as in the first four boxes of the first row. Then click on the E GROUP button. Enter one of the displayed E base lines into the last box in the second row.
3. Repeat step two for rows 3, 4, and 5.
4. In row six enter any A, B, C and D codes as described above in boxes 1, 2, 3, and 5 in any order. There must be one A, one B, one C, and one D code on the line. Enter E_g as described above into the fourth box of the row.
5. In the seventh row, enter the same three codes as row six in the first three boxes. In the last box enter the same letter code as in line six and any legal number for that code. Click on the E GROUP button to get a compatible E code to enter in the fourth box.
6. Repeat step five for rows 8, 9, and 10.
7. In row 11 enter any A, B, C, and D codes in boxes 1, 2, 4, and 5. Enter E_g as described above in the third box. It doesn't matter what order the A, B, C, and D codes are entered but there must be one of each.
8. For row twelve enter the same two codes as row 11 in the first two boxes. In the fourth box and fifth box enter the same two letter codes as in row 11 in either order and with any of their legal number subscripts. Use the E GROUP button to get a compatible E code for the third box. Repeat for rows 13-15.
9. In row 16 enter any A, B, C, and D codes in boxes 1, 3, 4, and 5. Enter E_g in box 2. It doesn't matter what order the A, B, C, and D codes are entered but there must be one of each.
10. Rows 17-20 follow the same pattern as row 16 except the second box is determined using the E GROUP button to get a compatible E code. The code used in the first box in row 16 must be used for these rows as well. In the third, fourth, and fifth box enter the same three letter codes as in row 16 in any order and with any of their legal number subscripts.
11. For rows 21-25, the E codes are in the first box followed by any A, B, C, and D codes in the remaining boxes in any order as long as it is the same for all five rows. Any legal subscripts may be used for the A, B, C, and D codes.

The structured order of the above procedure is not required to obtain a nasik magic hypercube as the example shows. The row code sets of a valid magic hypercube can be reordered into any of the possible orders to make 25! different magic hypercubes.

MAGIC CONSTANT CHECKER

When the last base hypercube is verified, a 5-D hypercube magic constant checker appears in the middle and right of the display. The base line matrix at the left remains. The checker is similar to the tesseract's checker except there is no pattern display. In the center are five Position Sliders that indicate the position of the first member of the group being summed. To the right are the types and sub types of patterns that can be summed. Only the lines, diagonals, and corners of various sized cubes are given. Many more patterns are possible. On the far right of the display are the 32 numbers being summed and their v, w, x, y, and z positions. The position of the first number corresponds to the position of the sliders. The numbers and the sum represent the range from 0 to 33,554,432.

In the middle at the top are two scan buttons. Both have two scan speeds. The position scanner must go through 33,554,432 combinations before recycling. This takes a long time even at the fast scan rate. To evaluate all the sums would take a very long time. Fortunately, this is not necessary as discussed under base cube addition properties in Magic Cube Basics. Verification that all 33,554,432 numbers are present in the cube was done prior to granting access to the checker. Therefore, the cube is a nasik 5-D magic hypercube.