To cover a solid

In an old problem the pentominoes are folded to cover a cube. Below there are two my solutions. Wen-Shan Kao has many others. The question now is: Are there other regular polyforms sets that cover a regular polyhedron?


The 12 pentominoes cover cubes



In the pictures and in the table below you could read how many squares a square (or a cube area) contains exactly, and how many triangles are contained in a triangle or on an area of a tetrahedron, an octahedron or an icosahedron.

Squares cover a square, triangles cover a triangle


Nr. L H Square Cube Triangle Tetrahedron Octahedron Icosahedron
1 0 1 2 12 3 12 24 60
2 0 2 8 48 12 48 96 240
3 0 3 18 108 27 108 216 540
4 0 4 32 192 48 192 384 960
5 0 5 50 300 75 300 600 1500
6 0 6 72 432 108 432 864 2160
7 0 7 98 588 147 588 1176 2940
8 0 8 128 768 192 768 1536 3840
9 0 9 162 972 243 972 1944 4860
10 1 0 1 6 1 4 8 20
11 1 1 5 30 7 28 56 140
12 1 2 13 78 19 76 152 380
13 1 3 25 150 37 148 296 740
14 1 4 41 246 61 244 488 1220
15 1 5 61 366 91 364 728 1820
16 1 6 85 510 127 508 1016 2540
17 1 7 113 678 169 676 1352 3380
18 1 8 145 870 217 868 1736 4340
19 1 9 181 1086 271 1084 2168 5420
20 2 0 4 24 4 16 32 80
21 2 1 10 60 13 52 104 260
22 2 2 20 120 28 112 224 560
23 2 3 34 204 49 196 392 980
24 2 4 52 312 76 304 608 1520
25 2 5 74 444 109 436 872 2180
26 2 6 100 600 148 592 1184 2960
27 2 7 130 780 193 772 1544 3860
28 2 8 164 984 244 976 1952 4880
29 2 9 202 1212 301 1204 2408 6020
30 3 0 9 54 9 36 72 180
31 3 1 17 102 21 84 168 420
32 3 2 29 174 39 156 312 780
33 3 3 45 270 63 252 504 1260
34 3 4 65 390 93 372 744 1860
35 3 5 89 534 129 516 1032 2580
36 3 6 117 702 171 684 1368 3420
37 3 7 149 894 219 876 1752 4380
38 3 8 185 1110 273 1092 2184 5460
39 3 9 225 1350 333 1332 2664 6660
40 4 0 16 96 16 64 128 320
41 4 1 26 156 31 124 248 620
42 4 2 40 240 52 208 416 1040
43 4 3 58 348 79 316 632 1580
44 4 4 80 480 112 448 896 2240
45 4 5 106 636 151 604 1208 3020
46 4 6 136 816 196 784 1568 3920
47 4 7 170 1020 247 988 1976 4940
48 4 8 208 1248 304 1216 2432 6080
49 4 9 250 1500 367 1468 2936 7340
50 5 0 25 150 25 100 200 500
51 5 1 37 222 43 172 344 860
52 5 2 53 318 67 268 536 1340
53 5 3 73 438 97 388 776 1940
54 5 4 97 582 133 532 1064 2660
55 5 5 125 750 175 700 1400 3500
56 5 6 157 942 223 892 1784 4460
57 5 7 193 1158 277 1108 2216 5540
58 5 8 233 1398 337 1348 2696 6740
59 5 9 277 1662 403 1612 3224 8060
60 6 0 36 216 36 144 288 720
61 6 1 50 300 57 228 456 1140
62 6 2 68 408 84 336 672 1680
63 6 3 90 540 117 468 936 2340
64 6 4 116 696 156 624 1248 3120
65 6 5 146 876 201 804 1608 4020
66 6 6 180 1080 252 1008 2016 5040
67 6 7 218 1308 309 1236 2472 6180
68 6 8 260 1560 372 1488 2976 7440
69 6 9 306 1836 441 1764 3528 8820
70 7 0 49 294 49 196 392 980
71 7 1 65 390 73 292 584 1460
72 7 2 85 510 103 412 824 2060
73 7 3 109 654 139 556 1112 2780
74 7 4 137 822 181 724 1448 3620
75 7 5 169 1014 229 916 1832 4580
76 7 6 205 1230 283 1132 2264 5660
77 7 7 245 1470 343 1372 2744 6860
78 7 8 289 1734 409 1636 3272 8180
79 7 9 337 2022 481 1924 3848 9620
80 8 0 64 384 64 256 512 1280
81 8 1 82 492 91 364 728 1820
82 8 2 104 624 124 496 992 2480
83 8 3 130 780 163 652 1304 3260
84 8 4 160 960 208 832 1664 4160
85 8 5 194 1164 259 1036 2072 5180
86 8 6 232 1392 316 1264 2528 6320
87 8 7 274 1644 379 1516 3032 7580
88 8 8 320 1920 448 1792 3584 8960
89 8 9 370 2220 523 2092 4184 10460
90 9 0 81 486 81 324 648 1620
91 9 1 101 606 111 444 888 2220
92 9 2 125 750 147 588 1176 2940
93 9 3 153 918 189 756 1512 3780
94 9 4 185 1110 237 948 1896 4740
95 9 5 221 1326 291 1164 2328 5820
96 9 6 261 1566 351 1404 2808 7020
97 9 7 305 1830 417 1668 3336 8340
98 9 8 353 2118 489 1956 3912 9780
99 9 9 405 2430 567 2268 4536 11340

Now we could look for the number that coincides exactly with the number of squares or triangles in a polyominoes or polyiamonds set. I suggest the shapes below, without to guarantee the existence of a solution. However I'm optimist because no parity problems are present.


Hexominoes
Cube of line 23 (if you don't use the cross)

Hexominoes
My solution



Hexominoes + Pentominoes
Cube of line 33 (35x6+12x5=270 exactly)

Hexominoes + Pentominoes
Sept 2001 - Solution by Alessandro Fogliati



Fractal
Cube covered by 6 fractals on a grid of infinite number of squares.

Fractacube



Sexominoes
The 6 full-sexed Sexominoes cover the cube nr. 10 exactly (solution)

The 6 full-sexed sexominoes cover a cube



The 24 one-side Sexominoes cover the cube of line 20 exactly (solution)

The sexominoes cover a cube



Sexominoes Number Five

Sometime my problem is to name the new polyforms: in this case I inspired by "Chanel nr.5" and "Mambo nr.5".

They are the 120 Order 5 squared flippable Polysexes. They cover the cube nr.22 exactly. (We must work with this interesting tile, also on the plane! I had supposed his existence some months ago, but it passed unnoticed. They are 120, that is a wonderful number for a geometric puzzle! Click here if you want to see all the Sexominoes Number Five.)

Polysexes cover a cube
Sept 2001 - Solution by Alessandro Fogliati




Tubominoes

The 30 two Flanges for edge Tubominoes cover exactly the cube nr.11. Do you want to minimize or to maximize the number of tubes? (I think it's a hard problem)

The tubominoes cover a cube




Polyarc

Here there are the 5 Tetrominoes with the 7 Biarc and the 22 Triarc by Henri Picciotto for a total of 60 squares! Do you want to cover the cube nr.21? I think it's possible.

Tetrominoes and Polyarc
Sept 2001 - Solution by Alessandro Fogliati


.When I have seen the Picciotto's Page, I drew immediately the 24 Tri-Arc-iamonds.. I think they are nice but I don't know if they could cover the octahedron nr.30 (24x3=72 exactly).



Tiling

Sometime it isn't easy to tile a cube by a polyomino.

Polyominoes tile a cube area

Many Polyominoes can to tile a cube area. The Domino and each Triomino tile the elementary cube. The cube nr.1 (12 squares) is tiled by a Triomino, a Tetromino and a Hexomino. Many Pentominoes (not all!) tile the cube nr.11 (30 squares) that is tiled also by a 10-omino and a 15-omino. The cube nr.2 (48 squares) is tiled by 2-3-4-6-8-12-16-24-ominoes! Perhaps it's not easy to tile the cube nr.21 (60 squares) with a Hexomino. The Sexomino "FMFM" tiles all the cubes.




Hexiamonds
Tetrahedron of line 12 and Octahedron of line 30

Hexiamonds
At left a solution by Alessandro Fogliati



Order 8 Isoperimetric Polyiamonds
Tetrahedrons of line 8 (exactly) and line 44 (with 4 holes)
You could find the Isoperimetric Polyforms on The Poly Pages by Andrew Clarke.

Isoiamonds8
Sept 2001 - Solution by Alessandro Fogliati



Eptiamonds
Octahedron of line 31 exactly

Eptiamonds
Sept 2001 - Solution by Alessandro Fogliati



Eptiamonds + Hexiamonds
Icosahedron of line 2 (12x6+24x7=240 exactly)

Eptiamonds + Hexiamnds
Sept 2001 - Solution by Alessandro Fogliati



Octiamonds
Tetrahedron of line 54 (66x8+4=532)
You could find many information about octiamonds on Mathpuzzle.com by Ed Pegg Jr.

Octiamonds
Sept 2001 - Solution by Alessandro Fogliati




Sexiamonds
The 4 Order 2 Triangular Polysexes cover the tetrahedron nr.10

Sexiamonds cover a tetrahedron
Solution




Apartheid

These are the Order 14 two-side Triangular Polysexes. They are obtained coloring with 7 colors the 4 forms below. Each edge could be colored with a different color. The complete set has 560 pieces [see Chris Hartman's email]. The rules of the apartheid (terrible!) impose that only the edges with the same color can to join. The set could cover exactly the Icosahedron nr.22.

Apartheid puzzle
I'll die before see the solution of this puzzle.


Any racial tolerance (red with yellow, etc.) facilitates the solution. (The tolerance does easy the things!). If you use only 3 colors, you obtain a set of 56 pieces that cover exactly the Octahedron nr. 11. The Order 7 two-side Triangular Polysexes could cover exactly the Tetrahedron nr.31.




Polyhexes & SexeHexeS

This could be an other chapter. The study below shows that regular hexagons can to cover a regular octahedron, but not a tetrahedron or an icosahedron. There are many combinations.

Hexagons cover an octahedron



Nr.LHHex on Octahedron
1 1 0 4
2 1 1 12
3 2 0 16
4 2 1 28
5 2 2 48
6 3 0 36
7 3 1 52
8 3 2 76
9 3 3 108
10 4 0 64
11 4 1 84
12 4 2 112
13 4 3 148
14 4 4 192
15 5 0 100
16 5 1 124
17 5 2 156
18 5 3 196
19 5 4 244
20 5 5 300
21 6 0 144
22 6 1 172
23 6 2 208
24 6 3 252
25 6 4 304
26 6 5 364
27 6 6 432
28 7 0 196
29 7 1 228
30 7 2 268
31 7 3 316
32 7 4 372
33 7 5 436
34 7 6 508
35 7 7 588
36 8 0 256
37 8 1 292
38 8 2 336
39 8 3 388
40 8 4 448
41 8 5 516
42 8 6 592
43 8 7 676
44 8 8 768
45 9 0 324
46 9 1 364
47 9 2 412
48 9 3 468
49 9 4 532
50 9 5 604
51 9 6 684
52 9 7 772
53 9 8 868
54 9 9 972


Tetrahexes
Octahedron of line 4 (my solution)

The tetrahexes cover an octahedron



Order 6 Hexastrips
Octahedron of line 21 (24x6=144 exactly)
The Polystrips are an invention of Miroslav Vicher.

The Hexastrips cover an octahedron



SexeHexeS

Do you want to solve the octahedron nr.2 with 12 of 13 full-sexed sexehexes? Or the octahedron nr.11 with 84/92 sexehexes?

The full-sexed sexehexes cover an octahedron




_________________

First edition: Aug 15th, 2000 - Last revision: Sept 20th, 2001

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