Poly cube

A poly cube (or poly cube ) is a body that consists of connected cubes . For small the names are single cube ( ), Biwürfel ( ) Triwürfel ( ) Tetra Cube ( ), Penta cube ( ), Hexawürfel ( ) Heptawürfel ( ) Oktawürfel ( ) common. ${\ displaystyle n}$ ${\ displaystyle n}$ ${\ displaystyle n = 1}$ ${\ displaystyle n = 2}$ ${\ displaystyle n = 3}$ ${\ displaystyle n = 4}$ ${\ displaystyle n = 5}$ ${\ displaystyle n = 6}$ ${\ displaystyle n = 7}$ ${\ displaystyle n = 8}$

The number of different poly cubes grows very quickly with an increasing number of cubes : 1, 1, 2, 8, 29, 166, 1023, 6922, 48311, 346543, ... ( OEIS , A000162). They are divided into the sequence ${\ displaystyle n \ geq 1}$

of the flat (planar) poly-cubes, which correspond to the polyominos : 1, 1, 2, 5, 12, 35, 108, 369, 1285, 4655, ... (OEIS, A000105 ) and
of the spatial (stereometric) poly-cubes: 0, 0, 0, 3, 17, 131, 915, 6553, 47026, 341888, ... (OEIS, A006759 ).

Applications

The poly cubes are used on the one hand in mathematics lessons at primary and secondary level, where they are mainly used to train spatial imagination and to investigate simple properties, and on the other hand in geometric games such as the Herzberger cuboid , where free and creative design when developing and inventing There are practically no limits to shapes and structures.

Tri-cube

There are two different tri-cubes, namely the I- and L-shape corresponding to the triominos.

Tetra cube

The 8 tetra cubes: I, L, T, S, square, tripod, left-hand screw, right-hand screw

There are eight different tetra cubes, namely 5 flat (tetrominos) and 3 spatial.

Tetra cube	volume	surface	Edge sum	# Corners	# Surfaces	# Edge
I.	4th	18th	24	8th	6th	12
L.	4th	18th	26th	12	8th	18th
L1	4th	18th	28	15th	9	21st
L2	4th	18th	30th	17th	12	24
L3	4th	18th	28	15th	9	21st
N	4th	18th	28	16	10	24
O	4th	16	20th	8th	6th	12
T	4th	18th	28	16	10	24

Euler's polyhedron theorem applies to the flat tetra-cubes : # corners + # surfaces = # edges + 2.

The Soma Cube - a (3 × 3 × 3) cube - is from the seven irregular tri- and tetra cubes, ie those with re-entrant edge composed.

Penta cube

The 29 penta cubes

A total of 29 different penta-cubes can be formed from five unit cubes, namely the 12 flat (planar) penta-cubes, which represent the spatial counterpart to the 12 pentominos , as well as the 17 spatial (stereometric) penta-cubes, 5 of which are symmetrical and 6 with one corresponding each Mirror image.

The mathematician David A. Klarner found that 25 Y-penta-cubes can be combined to form a (5 × 5 × 5) -cube. There are 1264 different solutions.

If you leave out the four of the 29 penta cubes that have 4 or 5 unit cubes in one direction (pentomino form I, L, N and Y), you can use the remaining 25 parts to create the so-called Dorian cube - one named after its inventor Joseph Dorrie (5 × 5 × 5) -cube - join.

Twelve penta cubes and one tetra cube that can be put together to form a cube

The Bedlam cube - a (4 × 4 × 4) cube - invented by the British puzzle inventor Bruce Bedlam can be built from 12 penta cubes and 1 tetra cube. There are 19,186 different solutions.

Another (4 × 4 × 4) cube can be composed of ten mirror-inverted (L2, L4, S1, S2, V1) and two (L3, T1) penta-cubes as well as the L-tetra-cube.

The computer game BlockOut is based on poly cubes from mono cubes to penta cubes.

Hepta cube

One disassembled and several assembled Diabolic Cube.

A (3 × 3 × 3) cube, known as the “diabolic cube”, can be assembled from one di-, tri, tetra, penta, hexa and hepta cube. It is one of the oldest cube decomposition puzzles and was first mentioned in 1893 by the lawyer Angelo John Lewis (1839-1919) - under the pseudonym Professor Louis Hoffmann - in Puzzles Old and New . There are 13 different solutions.

Octa cube

Geometrically speaking, a subgroup of 261 of the 6553 spatial octa-cubes represent the three-dimensional network of a tesseract , i.e. a four-dimensional hyper - cube , as it is delimited by 8 cube-shaped cells. One of these possibilities was used artistically by the Spanish painter Salvador Dalí in his painting Crucifixion (Corpus Hypercubus) from 1954 .

literature

CJ Bouwkamp: David Klarner's Pentacube Towers. In: David Wolfe; Tom Rodgers (ed.): Puzzlers' Tribute. A feast for the mind. Natick (MA): AK Peters, 2002, pp. 15-18.
Solomon W. Golomb: Polyominoes. Puzzles, Patterns, Problems, and Packings. With more than 190 diagrams. Princeton (NJ): University Press, 1994. ISBN 0-691-08573-0 .

Web links

Commons : Polycubes - collection of images, videos and audio files

Ronald M. Aarts: Pentacube. (MathWorld - A Wolfram Web Resource.)
Andrew L. Clarke: The Poly Pages.
Stewart T. Coffin: The Puzzling World of Polyhedral Dissections. Cape. 3: Cubic Block Puzzles.
Jürgen Köller: Tetra cubes. (Math tinkering.)
Torsten Sillke: Tiling and packing results. (University of Bielefeld, Faculty of Mathematics.)
Eric W. Weisstein: Polycube. (MathWorld - A Wolfram Web Resource.)
Dice games.
Demo software for the Bedlam Cube.

Individual evidence

↑ A000162 Number of 3-dimensional polyominoes (or polycubes) with n cells. ( English ) The OEIS Foundation. Retrieved September 21, 2019.

↑ Chris J. Bouwkamp; David A. Klarner: Packing a Box with Y-Pentacubes. In: Journal of Recreational Mathematics 3 (1970), No. 1, pp. 10-26.

↑ Chris Bouwkamp: The Cube-Y Problem. In: Cubism For Fun ⟨Nederlandse Kubus Club⟩ 25 (December 1990), Part 3 (Arresting Arrangements), pp. 30–43. (contains the list of all 1264 solutions)

↑ See Scott Kurowski: Bedlam / Crazee Cube Solved. ALL 19,186 Solutions. ( Memento of the original from January 9, 2009 in the Internet Archive ) Info: The archive link was automatically inserted and not yet checked. Please check the original and archive link according to the instructions and then remove this notice. @1@ 2
↑ See Stewart T. Coffin: Geometric Puzzle Design . Wellesley (MA): AK Peters Ltd., 2016, ISBN 978-1-56881-499-5 , p. 45 (The 3 × 3 × 3 Cube). Online at: The Puzzling World of Polyhedral Dissections. , Chap. 3: Cubic Block Puzzles. Oxford University Press 1991.
↑ See PD Turney: Unfolding the tesseract. In: Journal of Recreational Mathematics 17 (1984/85), No. 1, pp. 1-16.

[1] A000162 Number of 3-dimensional polyominoes (or polycubes) with n cells. ( English ) The OEIS Foundation. Retrieved September 21, 2019.

[2] Chris J. Bouwkamp; David A. Klarner: Packing a Box with Y-Pentacubes. In: Journal of Recreational Mathematics 3 (1970), No. 1, pp. 10-26.

[3] Chris Bouwkamp: The Cube-Y Problem. In: Cubism For Fun ⟨Nederlandse Kubus Club⟩ 25 (December 1990), Part 3 (Arresting Arrangements), pp. 30–43. (contains the list of all 1264 solutions)