Building symmetric figures from Polycubes: Torsten, July 1996 -update- 23. Nov. 1996 -update- 09. Dec. 1996 -update- 15. Nov. 2000 I have found (24.04.96) a figure with central-symmetry (Punkt-Symmetrie), which contains only 17 F-pentacubes. Is this the smallest odd number of F-pentacubes? General Problem: Given a polycube without a certain symmetry. Find a figure which can be tiled with the smallest odd number of this polycube having this symmetry. For polyominoes one can analyze the 2-dim case too. If odd boxes are known for a polycube one has upper bounds for several symmetries. How can one get lower bounds? If you consider other polyforms, the condition gcd( #pieces, order(symmetry group) ) = 1 should hold. For polyominoes and symmetry group not trivial this gives: #pieces odd. But for polycubes or polyspheres we have symmetry group which have odd factors. Therefore . 1 3 1 3 3 1 1 2 3 . 2 2 2 . is uninteresting. Refs: - Daniel A. Rawsthorne, Tiling complexity of small n-ominoes (n<10) Disc. Math. 79 (1988) 71-75 Tiling of the plane by congruent polyominoes under (1) translation and rotation (2) translation, rotation, and reflection The tiling complexity is the smallest number of tiles building a superblock which tiles the plane by translation. - Karl Scherer, A Puzzling Journey to the Reptiles and Related Animals, Self-published and distributed (1987) 2.5 Odd Reptiles: Let us call a reptile an odd reptile, if there is an odd n such that the reptie is rep-n. In a similar way we may define even reptiles. [...] The point is that it is mostly much harder to show that a reptile is odd than to show that it is even. a) There are many reptiles which are not odd (easy) b) There are reptiles which are not even (hard problem) ------------------------------------------------------------------ o o o o : (Dec. 1996) ----- sym point: order=2 (3 pieces) 1 1 1 2 1 2 3 3 3 1 2 2 1 2 2 2 3 2 2 3 3 3 3 2 1 1 2 3 3 1 2 3 3 1 1 1 sym |: order=2 (3 pieces) 1 2 1 1 1 1 1 1 1 1 2 2 1 2 1 1 1 1 2 1 1 2 2 1 1 3 2 3 2 2 1 2 3 3 3 2 2 1 3 2 3 3 3 3 2 3 2 2 3 2 3 3 3 2 3 3 3 3 3 2 sym -: order=2 (3 pieces) 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 1 2 2 2 3 3 3 2 . 3 3 3 2 3 3 3 2 3 2 2 3 2 2 3 2 2 3 3 3 3 2 2 2 sym /: order=2 (3 pieces) 1 1 1 2 . 1 1 1 <- two layer figure 2 2 1 1 1 sym +: order=4 (5 pieces) 1 1 1 1 1 1 1 1 2 1 2 1 1 2 2 2 2 2 2 2 2 1 3 2 3 3 3 4 3 3 3 4 3 3 4 2 3 4 4 3 4 4 3 4 4 5 4 5 4 5 5 5 4 5 5 5 5 5 5 5 1 1 1 1 2 2 2 1 3 3 1 <-- 3*4 roof 2 2 2 1 1 1 1 sym C3: order=3 (4+3 pieces) . x x . x x x . . . x . . . x x . . . x x . . x x x x x . . x x x . x x x x x x x x x x x . x x . . . x . . . . x x x y y x . . x <- 2 layer figure with 3 holes y y y x . x y y x x sym C6: order=6 (impossible) the number of pieces must be a multiple of 3. figures with this symmetry have planar components. sym D3: order=6 (7 pieces) x x x x x x x x x x x x x x x x x x x x x x x x x x x x sym D6: order=12 (impossible) the number of pieces must be a multiple of 3. figures with this symmetry have planar components. x x x x x x . x <- 3 pieces x x x x x sym S4: order=24 (5 pieces) tetrahedron of order 4. sym S3 w S2 (Octahedralgroup): order=48 (11 pieces) octahedron of order 4. ------------------------------------------------------------------ small pattern with symmetry Torsten Sillke, 10.10.96 This are examples only. No minimality have been shown. odd number of I1. (Nov. 1996) ----------------- sym point: (3 pieces) x . . x . . x sym |: (3 pieces) x x . x sym /: (3 pieces) x x x sym +: (3 pieces) x x x sym X: (3 pieces) x x x sym C4: (5 pieces) . x . . . . . . . x . . x . . x . . . . . . . x . sym D4: (1 piece) x odd number of I2. (Nov. 1996) ----------------- sym point: (3 pieces) x x x x x x x x x x x x x x x x x x sym |: (3 pieces) x x x x x x x x x x x x x x x x x x sym /: (impossible) Assume having a figure with symmetry /. Color the board alternating (chess board coloring). Let the diagonal be white. The number of black squares on both sides of the diagonal is equal. The total number of black squares is even. The number of dominoes is equal the number of black squares and therefore even. sym +: (1 piece) x x sym X: (impossible) As sym / is impossible. sym C4: (impossible) Number of squares is a multiple of 4 (for even area). sym D4: (impossible) As sym C4 is impossible. odd number of I3. (Nov. 1996) ----------------- sym |: (3 pieces) x x x x x x x x x sym point: (3 pieces) x x x x x x x x x sym /: (5 pieces) x x x . x x x x x x x x x x x x sym +: (1 piece) x x x sym X: (7 pieces) x x x x x x x x x x x x x x x x x x x x x sym C4: (7 pieces) x x x . x . x x x x x x x x x x x x x . x . x x x sym D4: (3 pieces) x x x x x x x x x odd number of L3. (Nov. 1996) ----------------- sym /: (1 piece) x . x x sym |: (5 pieces) . 1 . 2 . 3 1 1 2 2 3 3 4 5 5 . 4 4 5 . sym point: (5 pieces) . 1 1 . . . 1 2 2 . 4 4 2 3 3 . 4 5 3 . . . 5 5 . sym X: (5 pieces) . . 2 2 . . . 2 3 3 4 4 5 5 3 4 6 5 . . . 6 6 . . sym C4: (7+4 pieces) (A) + 4 sym +: (7+4 pieces) (A) + 4 sym D4: (7 pieces) . 1 2 2 . 1 1 2 3 3 4 4 5 5 3 = (A) 4 6 5 7 7 . 6 6 7 . odd number of I4. (Nov. 1996) ----------------- sym point: (3 pieces) x x x x x x x x x x x x sym |: (3 pieces) x x x x x x x x x x x x sym /: (impossible) Proof for I(2k): Assume having a figure with symmetry /. Color the diagonals parallel to / with distance p*k alternating. The other diagonals are ignored. Example coloring k=3. / w . . s . . w . . s each I6 fits exactly on s. . . s . . w . . s . . s . . w . . s . . s . . w . . s . . w / Let the diagonal be white. The number of black squares on both sides of the diagonal is equal. The total number of black squares is even. The number of I(2k) is equal the number of black squares and therefore even. sym +: (1 piece) x x x x sym C4: (impossible) Assume having a figure with point-symmetry. Tile the board alternating with 2*2 square (black-white). Let the point of rotation be the middle of one 2*2 square. The orbits of symmetry are of order 4 and monochromatic. Every I4 meets exactly 2 squares of one color. Therefore the number of I4 is even. sym X: (impossible) sym D4: (impossible) odd number of L4. (Nov. 1996) ----------------- sym /: (3 pieces, Mike Reid) a a a a b c . b b b c c c (7 pieces) y y y y y y y y x x x x 3 3 3 x x x x 2 2 3 1 2 1 2 1 1 sym |: (impossible) Assume having a figure with symmetry |. Color the board with alternating strips which are parallel to |. There are two cases now: Case1: The symmetry axis is in the middle of a white strip. The number of black squares on one side is equal the number of black squares on the other. The total number of black squares is even. Case2: The symmetry axis is between two strips. The number of black squares is equal the number of white squares, as the each white is paired with a black by symmetry. The total number of squares is a multiple of 4 (L4 is a tetromino). The total number of black squares is even. Each L4 touches an odd number of black squares. The number of L4 pieces is even. sym point: (impossible) Assume having a figure with point-symmetry. Color the board with alternating strips. There are two cases now: Case1: The symmetry point is in the middle of a white strip. The total number of black squares is even, as they come in pairs by the symmetry. Case2: The symmetry axis is between two strips. The number of black squares is equal the number of white squares, as the each white is paired with a black by symmetry. The total number of squares is a multiple of 4 (L4 is a tetromino). The total number of black squares is even. Each L4 touches an odd number of black squares. The number of L4 pieces is even. sym +: (impossible) sym X: (impossible) sym C4: (impossible) sym D4: (impossible) odd number of T4. (Nov. 1996) ----------------- sym |: (1 piece) x x x x sym \: (impossible) Assume having a figure with symmetry \. Color the board alternating (chess board coloring). Let the diagonal be white. The number of black squares on both sides of the diagonal is equal. The total number of black squares is even. Every T4 fits an odd number of black squares and therefore the number of T4 is even. sym point: (impossible) Assume having a figure with point-symmetry. Color the board alternating (chess board coloring). There are two cases (for even area). Case1: The symmetry point is in the middle of an edge. Then the symmetry maps every square to a square of opposite color. So the number of black and white squares is equal. But then the number of T4 of (3,1) and (1,3) type must be equal. So the number of T4 is even. Case2: The symmetry point is at a corner. The symmetry gives pairs of black and pairs of white squares. So the number of black squares is even. Every T4 fits an odd number of black squares and therefore the number of T4 is even. sym +: (impossible) sym X: (impossible) sym C4: (impossible) sym D4: (impossible) odd number of N4. (23. Nov. 1996) ----------------- sym point: (1 piece) x x x x sym \: (impossible) Assume having a figure with symmetry \. Tile the plane with 2*2 squares and color it alternating. Let the diagonal of symmetry be white. Then the number of black cells on each side of the diagonal is equal. Therefore the number of black squares is even. Every N4 fits an odd number of black squares and therefore the number of N4 is even. sym |: (impossible) Assume having a figure with symmetry |. There are two cases (for even area). Case1: The symmetry axis is in the middle of a strip. Then color the plane as follows: | . 1 . 1 . 1 . 1 . . 2 . 2 . 2 . 2 . . 1 . 1 . 1 . 1 . . 2 . 2 . 2 . 2 . . 1 . 1 . 1 . 1 . . 2 . 2 . 2 . 2 . | The number of cells labled 1 on both sides of the symmetry axis is equal. Therefore the number of cells labled 1 is even. Every N4 fits an odd number of 1 labled cells (exactly one). and therefore the number of N4 is even. Case2: The symmetry axis is between two strip. Tile the plane with 2*2 squares and color it alternating. Let the symmetry axis halve the 2*2 squares. | b b w w b b w w b b b b w w b b w w b b w w b b w w b b w w w w b b w w b b w w b b w w b b w w b b b b w w b b w w b b | The number of black squares on both sides of the symmetry axis is equal. Therefore the number of black squares is even. Every N4 fits an odd number of black squares and therefore the number of N4 is even. sym C4: (impossible) Assume having a figure with symmetry C4. As the area of the figure is even the center of symmetry is at a corner. Tile the plane with 2*2 squares and color it alternating. Let the center of symmetry be in the middle of a white 2*2 square. The orbits of symmetry are of order 4 and monochromatic. Therefore the number of black squares is even. Every N4 fits an odd number of black squares and therefore the number of N4 is even. sym +: (impossible) sym X: (impossible) sym D4: (impossible) odd number of P5. (Oct. 1996) ----------------- sym \: (3 pieces) . x x x x x x x x x x x x x x x . . x x . . x x x x x x x x x x x x x sym point: (5 pieces) . x x . . . . x x x x x x x x x x x x x x x x x x x x x . . . . x x . sym |: (7 pieces) . 29 26 26 29 29 26 26 26 29 29 27 27 24 24 30 30 27 27 24 24 24 30 30 27 25 25 25 30 28 28 25 25 . 28 28 28 . . 20 16 16 . . . 20 20 16 16 16 . . 20 20 17 17 17 . . 21 18 18 17 17 . 21 21 18 18 15 15 15 21 21 18 19 19 15 15 . . 19 19 19 . . sym C4: (9 pieces) 28 28 . 25 23 23 23 28 28 28 25 25 23 23 29 29 29 25 25 22 . 29 29 26 26 26 22 22 . 30 27 26 26 22 22 30 30 27 27 24 24 24 30 30 27 27 . 24 24 figure with unique solution sym +: (11 pieces) . . . . 20 16 16 . . . . . 22 22 20 20 16 16 16 14 14 . 22 22 22 20 20 17 17 17 14 14 14 23 23 . 21 18 18 17 17 . 13 13 23 23 21 21 18 18 15 15 15 13 13 . 23 21 21 18 19 19 15 15 13 . . . . . 19 19 19 . . . . sym X: (7 pieces) . . X X X . . . . X X X X . 15 15 15 X X X 10 15 15 13 13 13 10 10 16 16 16 13 13 10 10 . 16 16 14 14 . . . . 14 14 14 . . sym D4: (17 pieces) . . . 24 24 24 19 19 . . . . . . 24 24 19 19 19 . . . . . . 26 26 20 20 16 . . . 29 29 26 26 26 20 20 16 16 14 14 29 29 27 27 27 21 20 16 16 14 14 30 29 28 27 27 21 21 17 17 15 14 30 30 28 28 23 21 21 17 17 15 15 30 30 28 28 23 23 18 18 17 15 15 . . . 25 23 23 18 18 . . . . . . 25 25 22 22 18 . . . . . . 25 25 22 22 22 . . . odd number of L5. (Nov. 1996) ----------------- sym point: (9 pieces) 37 37 37 37 40 34 34 37 32 40 38 34 33 32 40 38 34 33 32 40 40 38 34 33 32 32 38 38 33 33 35 39 35 35 35 35 39 39 39 39 36 36 36 36 36 sym /: (11 pieces) . . . 42 42 42 42 35 35 . . 45 42 38 38 38 38 35 . . 45 43 43 43 43 38 35 . . 45 43 39 39 39 39 35 . . 45 45 39 36 36 36 36 48 48 48 48 40 40 40 40 36 48 . . 44 44 44 44 40 . 49 . . 44 . . . . . 49 49 49 49 . . . . . sym |: (11 pieces) 57 x1 x1 x1 x1 x1 x2 x2 57 x1 x1 x1 x1 x1 x2 x2 57 55 55 55 55 51 x2 x2 57 57 . 58 55 51 x2 x2 58 58 58 58 56 51 x2 x2 59 56 56 56 56 51 51 52 59 59 59 59 52 52 52 52 figure with unique solution sym X: (21 pieces) . . . . 46 46 46 46 40 40 . . 57 54 50 50 50 50 46 42 40 39 . 57 54 54 54 54 50 47 42 40 39 . 57 55 55 47 47 47 47 42 40 39 57 57 55 51 51 51 51 42 42 39 39 58 58 55 52 52 44 51 43 43 43 43 59 58 55 52 48 44 44 44 44 41 43 59 58 56 52 48 48 48 48 45 41 . 59 58 56 52 49 45 45 45 45 41 . 59 59 56 53 49 49 49 49 41 41 . . 56 56 53 53 53 53 . . . . sym +: (17 pieces) 23.11.96 TS A A . . . . . . . 49 46 A 59 55 55 53 53 49 49 49 49 46 A 59 57 55 54 53 50 50 48 48 46 A 59 57 55 54 53 50 51 48 46 46 59 59 57 55 54 53 50 51 48 47 47 A 57 57 54 54 56 50 51 48 47 A A 58 56 56 56 56 51 51 52 47 A A 58 58 58 58 52 52 52 52 47 A A A . . . . . . . A A sym C4: (17 pieces) 22.11.96 TS . . . . . . A . . . . . A 54 54 49 49 A A A A . . A 54 52 49 45 43 43 43 43 . . A 54 52 49 45 45 45 45 43 . A A 54 52 49 46 44 44 44 44 . . 55 55 52 52 46 46 46 46 44 . . 56 55 53 53 47 47 47 47 A A . 56 55 53 50 50 50 50 47 A . . 56 55 53 50 48 48 48 48 A . . 56 56 53 51 51 51 51 48 A . . . . . 51 . . . . . . sym D4: (25 pieces) 22.11.96 TS B A A B B B B A A A A A A A 54 54 49 49 A A A A A A A 54 52 49 45 43 43 43 43 B A A 54 52 49 45 45 45 45 43 B A A 54 52 49 46 44 44 44 44 B B B 55 55 52 52 46 46 46 46 44 B B B 56 55 53 53 47 47 47 47 A A B 56 55 53 50 50 50 50 47 A A B 56 55 53 50 48 48 48 48 A A A 56 56 53 51 51 51 51 48 A A A A A A 51 B B B B A A B odd number of N5. (Dec. 1996) ----------------- sym X: (25 piece) . . . . . . . X X X . . . . . . . . 97 97 97 X X 82 82 82 . . . . . . 97 97 94 94 94 84 84 84 82 82 . . . . . 98 94 94 89 89 89 85 84 84 80 78 . . . . 98 98 89 89 90 90 85 83 80 80 78 X . . . X 98 90 90 90 85 85 83 80 78 78 X . . X X 98 91 91 91 85 83 83 80 78 X X . . X 99 95 95 95 91 91 83 86 86 79 X . . . X 99 96 96 95 95 86 86 86 81 79 79 . . . . 99 99 96 96 96 87 87 87 81 81 79 . . . . . 99 92 92 92 88 88 87 87 81 79 . . . . . . 93 93 92 92 88 88 88 81 . . . . . . . . 93 93 93 . . . . . . . sym C4: (25 piece) . . . . . 94 94 . . . . . . . . . . . 94 94 94 90 90 83 83 83 78 . . . . . 96 96 90 90 90 83 83 81 78 78 75 . . . . 97 96 96 96 89 89 89 81 78 76 75 . . . . 97 97 91 89 89 85 81 81 78 76 75 75 . . . 99 97 91 91 85 85 81 84 84 76 76 75 . . . 99 97 92 91 85 84 84 84 79 77 76 . . . 99 99 95 92 91 85 86 86 86 79 77 77 . . . 99 98 95 92 92 86 86 87 87 79 79 77 . . . . 98 95 95 92 87 87 87 80 80 79 77 . . . . 98 98 95 93 93 93 Z Z 80 80 80 . . . . . 98 93 93 Z Z Z Z Z Z . . . . . . . . . . . Z Z . . . . . sym +: (31 piece) . . . . . 85 85 78 78 78 . . . . . . . . 90 88 88 85 85 85 78 78 74 . . . . 98 94 90 90 88 88 88 79 79 79 74 72 69 . . 98 94 94 90 86 86 79 79 76 74 74 72 69 . 98 98 95 94 90 89 86 86 86 76 74 72 72 69 69 98 96 95 94 89 89 87 87 87 76 76 72 73 71 69 99 96 95 95 89 87 87 80 80 80 76 73 73 71 70 99 96 96 95 89 91 91 81 81 80 80 73 71 71 70 99 99 96 91 91 91 82 82 81 81 81 73 71 70 70 . 99 97 97 97 92 92 82 82 82 77 77 77 70 . . 97 97 92 92 92 83 83 83 77 77 75 75 75 . . . . 93 93 93 84 84 83 83 75 75 . . . . . . . . 93 93 84 84 84 . . . . . sym D4: (37 piece) . . . . . Y Y Y Y Y . . . . . . . X X X 94 94 Y Y Y Y Y X . . . X X 94 94 94 90 90 83 83 83 78 X X . . Y 96 96 90 90 90 83 83 81 78 78 75 X . . Y 97 96 96 96 89 89 89 81 78 76 75 X . Y Y 97 97 91 89 89 85 81 81 78 76 75 75 Y Y Y 99 97 91 91 85 85 81 84 84 76 76 75 Y Y Y 99 97 92 91 85 84 84 84 79 77 76 Y Y Y 99 99 95 92 91 85 86 86 86 79 77 77 Y Y Y 99 98 95 92 92 86 86 87 87 79 79 77 Y Y . X 98 95 95 92 87 87 87 80 80 79 77 Y . . X 98 98 95 93 93 93 88 88 80 80 80 Y . . X X 98 93 93 88 88 88 82 82 82 X X . . . X Y Y Y Y Y 82 82 X X X . . . . . . . Y Y Y Y Y . . . . . odd number of (61) (Aug. 1998) ------------------ sym point: (7 pieces) a a a a a d d e e d d e e b b c b f e d f f b b c c g f c c g f g g g odd number of (U5) (Nov. 2000) ------------------ sym C2xC2xC2: (7 pieces) a a b a . b a a b c c b . . . c c b c d d . d . c d d c c e . . . c c e f f e f . e f f e -- mailto:Torsten.Sillke@uni-bielefeld.de http://www.mathematik.uni-bielefeld.de/~sillke/