Tiling and Packing results of Torsten Sillke

Polycubes Problems


Pentacubes with Künzell's numbers

Ekkehard Künzell wrote a book (in German) on games played with pentacubes. There he explains his pentacube numbering system. The book is sold by Ingo Uhl GmbH in Germany. Look after the reservat book. There is another human-friendly pentacube naming system from Kate Jones of Kadon Enterprises, Inc.. She made also a hexomino naming system.

Here is my new systematic numbering scheme for polycubes. It is the lexicographic numbering of the coordinates. Mirror images a noted by negative numbers.

Packing Boxes with like Pentacubes

Box packings of Pseudo-Pentacubes. These need only be edge-conected. Further one can consider not connected polycubes. Packings of handed Pentominoes. These problems have only been solved recently.

Packing Cubes

Cubes packed with identical pieces. Cubes packed with mainly different pieces.

Packing Boxes with like Tetracubes

Packing Boxes with like Hexacubes

Packing Boxes with like Heptacubes

The following P-Heptacubes a simple box packers. But the rectangles are hard. See Mike Reid.

Other Polycubes Problems

Packing Polyspheres

Tilings with Matching Conditions

The Who is Who in Tiling and Packing

Tiling Programs

Tiling and Packing Links

Playing with Tetracubes

Playing with Pentacubes

Proofs and Techniques

You can find impossebility proofs in most of the box lists of like polycubes. Most of these proofs are by bounds or modular (e.g. parity) restrictions. Typical proofs are REHM's cubes and the SOMA cube secrets. Sometimes you analyse possible local configurations. The best example of this type is Yuri Aksyonov's proof Handed Y5 rectangles. These type of proof are considered simple proofs. I don't say they are simple to find. I am especially proud of finding the 3-Bone Impossibles. Sometimes you can show that counting is not enough to show impossibility. For 2-dimensional problems you can than try to use the word-group method. It had been used the first time by John H. Conway. You need some group theory to see that such a proof does indeed proofs anything. And you must be familiar with it (of a program like GAP) to find such proofs of yourself. Mike Reid used this method since april 1999 with good results. I hope he will publish these results. I rank this method as advanced.
Torsten Sillke,
Home | FSP Mathematisierung | Fakultät für Mathematik | Universität Bielefeld

last changed 2009-01-04