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Symmetry of Assemblies

The outside of the 6-piece burr has considerable symmetry. There are 23 different ways to rotate and/or reflect the shape which end up looking the same. To these 23 symmetries add the 'identity' symmetry, which does not move the object at all, and the result is a mathematical group of order 24.

When a 6-piece burr assembly is analyzed, one can ask whether the internal cube arrangements have any symmetry. There are 11 possible answers to this question, which are listed below. The syntax of the symmetry designations should become clear.

Possible symmetries:

1 no symmetry of assembly
2MA reflective symmetry through some plane
2MB reflective symmetry through the center of the burr
2R 180-degree rotational symmetry about some axis
3R 3-way rotational symmetry which interchanges the axes
4MA 4-fold symmetry including 2 reflective symmetries through different planes and 1 180-degree rotational symmetry.
4MB 4-fold symmetry including 1 reflective symmetry through a plane, 1 180-degree rotational symmetry, and reflective symmetry through the center.
4R 180-degree rotational symmetry about all 3 axes
6M 3-way rotational symmetry and reflective symmetry through the center, giving 6 symmetries.
8M reflective symmetry through each orthogonal plane, resulting in 8 symmetries total, 4 of which are reflective.
24M all symmetries. This is possessed by only one assembly, which is constructed from 6 piece #2's (JRM numbers). This assembly has 8 internal cube voids in the center and cannot be taken apart.

For those interested in the mathematics, these symmetry types represent all subgroups of the order-24 group of external symmetries except the subgroup which would be designated as 12R. Using the standard cubic-cut pieces, the only assembly with these symmetries is the assembly which has symmetry type 24M.

Examples of assemblies with each of these symmetry types are in the Appendix in Table 6.

Choice of Construction Method

There are two options for constructing all the 6-piece burr assemblies with a computer program:

  1. Have the program determine all different sets of 6 physical pieces (including duplicates), and for each of these determine all possible assemblies that can be constructed from them. Then analyze each of these assemblies to see if it is a solution, and how interesting it is.
  2. Have the program construct the assemblies individually, analyzing each one after it has been constructed.

The advantage to 1) is that the uniqueness of solutions will be immediately known, as all other assemblies using the same pieces will be done at the same time.

The advantage to 2) is that the program logic for constructing assemblies will be executed essentially once, rather than being re-executed for each new set of pieces.

Approach 2) was chosen because of the huge savings in computer time for constructing the assemblies. This then required that all "interesting" assemblies be reexamined to check for uniqueness.

The Basic Logic

The goal of the program logic is to do the following:

  1. Determine all possible ways in which the 32 cubes in the interior of a 6-piece burr can be assigned to one of the 6 pieces, or to empty space, so that all 6 pieces are connected pieces formed by removing cubes from Figure 1.
  2. Rotations and/or reflections of the internal cube construction should not be duplicated - that is, of the 24 possible rotations and reflections, one should be recognized as the "first" one, which is analyzed; and all the others should be recognized as not being the "first", and so should be discarded.
  3. If some of the 24 rotations/reflections are the same as the original, record this information so that assemblies can be counted by symmetry type.

This section describes how the above requirements were carried out. The logic was used for the NOTC and HB6 analysis programs. The early general, holey programs, called GB6, which were used to analyze 1, 2, 3, 4 and 5-hole assemblies used a more complicated logic which will not be described here.

The first task is to order the 2225 piece orientations. The following operations can be applied to each piece orientation:

These operations correspond to cube switchings as follows (numbers as in Figure 1):

When rotating or reflecting an assembly, each piece orientation may change. The 3 permutations above, plus the original orientation, are the only ways a piece orientation may be changed when an assembly is rotated and/or reflected. Rotations about the long axis for ambiguous pieces cannot occur during R&R of an assembly.

Each of the 2225 piece orientations is grouped with the 3 (or fewer) piece orientations resulting from applying SIDM, FLIP, and EFEM. The result is as follows:

Each group is represented by the piece orientation of the group which has the smallest MATR. MATR is computed with the following formula:

     MATR = 4096 - X(1) - 2*X(2) - 4*X(3) ... - 2048*X(12)

     where X(i) = 1 if cube #i (Figure 1) is present, and
                  0 if cube #i is missing.

     For example, the solid piece has piece orientation number:

     MATR = 4096 - 1 - 2 - 4 - ... - 2048 = 1

and the piece which has 10 cubes cut out (number 3 in the JRM notchable list) has two piece orientations with MATR values 1024 and 3328.

The sorting of the 2225 piece orientations is done as follows:

  1. The solid piece is placed first
  2. The groups of 4 are placed next, so that the group representative has an order number of the form 2 + 4*N , and the remaining members of the group have the next 3 positions in the order SIDM, FLIP and EFEM. Those groups whose pieces have the fewest removed cubes come first, and within groups where this number is the same, the group containing the lowest MATR number comes first. The sort numbers for these groups are from 2-2137.
  3. Next are placed the groups of 2 piece orientations each, with the same rules as 2) for sorting these groups with each other. These sort numbers are from 2138-2221.
  4. Finally, the remaining 4 piece orientations which are in groups by themselves are placed in positions 2222-2225.

A listing of the 2225 piece orientations in this order is in Table 3.

The 534 Cases

The vast majority of the assemblies can now be constructed in 534 different groups as follows:

  1. Let N be a number from 1 to 534.
  2. In piece position 1 from Figure 2 place the representative of the Nth group of 4 piece orientations. The sort order for this piece orientation will be 2 + 4*(N-1).
  3. Construct all assemblies in which the other 5 pieces are occupied by piece orientations with numbers from 2 + 4*N to 2225. This is easily done with a computer program with 5 nested loops, with the loop indices taking all values from 2+4*N to 2225. Each time a loop index is changed, the piece orientation specified by the loop value is checked to see if it can fit with the pieces already placed. If there is interference with another piece, the next value is tried without executing the inner loops. The program places pieces in Figure 2 in the order 1-3-5-2-4-6. This is so that as much interference as possible occurs early on to save loop overhead.

It is easy to see that all assemblies constructed by this procedure have no internal symmetry, and that no 2 such assemblies are equivalent to each other by rotation and/or reflection. In addition, the only assemblies which are not constructed by this method are those in which either:

  1. One piece is solid
  2. None of the 6 piece orientations belongs to one of the 534 groups of 4, or
  3. Two or more of the 6 piece orientations belong to the same group of 4, and the others belong to groups higher in the sort order.

Table 9 contains information on each of these cases and who ran the program for this case.

The Remaining Cases

Eight more specialized programs were written and run to handle the small number of assemblies not handled by the 534 cases. These results are summarized as cases 535-542. The eight cases are listed below:

  1. One solid piece. The solid piece (MATR = 1) is assigned to piece #1 in Figure 2. The other pieces may take on any of the other piece orientations, but comparisons of pieces 3 & 4 and pieces 5 & 6 are made so two assemblies which are rotations or reflections of each other are not both analyzed. Some assemblies have symmetry of order 2 or 4, and these are counted separately. No disassembly analysis is run as all these assemblies have a partial solution of level-1. The results are recorded as case #535.
  2. Two pieces are from the same group-of-4 and the other four pieces have higher sort numbers. In addition, these two pieces are not parallel in the assembly. This is handled as follows: For each N from 1 to 534, piece #1 in Figure 2 is assigned number 2+4*(N-1) and piece #3 is allowed to have all values from the group, that is 2+4*(N-1) through 5+4*(N-1). The other four pieces may take on any values greater than 5+4*(N-1). The program is then also run with piece #1 given value 4+4*(N-1). All the resulting assemblies are different and have no symmetry. The results are stored as case #536.
  3. Two pieces are from the same group-of-4 and the other four pieces have higher sort numbers. In addition, these two pieces are parallel and have the same piece orientation number. The program handles this by assigning 2+4*(N-1) to both piece #1 and piece #2. About half of the assemblies are skipped, and a small number have rotational symmetry about the long axis of pieces 1 and 2. The results are stored as case #537.
  4. Same as 3), except that piece #2 is assigned number 3+4*(N-1), or SIDM of piece #1. Some of the assemblies have reflective symmetry through the plane between pieces 1 & 2. The results are stored as case #538.
  5. Same as 3), except that piece #2 is assigned number 4+4*(N-1), or FLIP of piece #1. Some of the assemblies have rotational symmetry about a third axis. The results are stored as case #539.
  6. Same as 3), except that piece #2 is assigned number 5+4*(N-1), or EFEM of piece #1. Some of the assemblies have reflective symmetry about the center. The results are stored as case #540.
  7. Three or more pieces are from the same group-of-4 and the other pieces (if any) have higher sort numbers. The program has elaborate symmetry-checking code to eliminate assemblies which are R&R equivalent to previously analyzed assemblies. The assemblies can have a variety of symmetries which are recorded separately by the program. The results are stored as case #541.
  8. No piece is from a group-of-4, and no piece is solid. All pieces have sort orders 2138 or higher. This program also includes elaborate symmetry checking and recording. The results are stored as case #542.

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