digplanet beta 1: Athena
Share digplanet:

Agriculture

Applied sciences

Arts

Belief

Business

Chronology

Culture

Education

Environment

Geography

Health

History

Humanities

Language

Law

Life

Mathematics

Nature

People

Politics

Science

Society

Technology

The 108 free heptominoes

A heptomino (or 7-omino) is a polyomino of order 7, that is, a polygon in the plane made of 7 equal-sized squares connected edge-to-edge.[1] The name of this type of figure is formed with the prefix hept(a)-. When rotations and reflections are not considered to be distinct shapes, there are 108 different free heptominoes. When reflections are considered distinct, there are 196 one-sided heptominoes. When rotations are also considered distinct, there are 760 fixed heptominoes.[2][3]

Symmetry[edit]

The figure shows all possible free heptominoes, coloured according to their symmetry groups:

  • 9 heptominoes (coloured red) have an axis of reflection symmetry aligned with the gridlines. Their symmetry group has two elements, the identity and the reflection in a line parallel to the sides of the squares.
Reflection Symmetrical Heptominoes-90-deg.svg
  • 7 heptominoes (coloured green) have an axis of reflection symmetry at 45° to the gridlines. Their symmetry group has two elements, the identity and a diagonal reflection.
Reflection Symmetrical Heptominoes-45-deg.svg
  • 4 heptominoes (coloured blue) have point symmetry, also known as rotational symmetry of order 2. Their symmetry group has two elements, the identity and the 180° rotation.
Rotation Symmetrical Heptominoes.svg
  • 3 heptominoes (coloured purple) have two axes of reflection symmetry, both aligned with the gridlines. Their symmetry group has four elements, the identity, two reflections and the 180° rotation. It is the dihedral group of order 2, also known as the Klein four-group.
  • 1 heptomino (coloured orange) has two axes of reflection symmetry, both aligned with the diagonals. Its symmetry group also has four elements. Its symmetry group is also the dihedral group of order 2 with four elements.
Rotation and Reflection Symmetrical Heptominoes.svg

If reflections of a heptomino are considered distinct, as they are with one-sided heptominoes, then the first and fourth categories above would each double in size, resulting in an extra 88 heptominoes for a total of 196. If rotations are also considered distinct, then the heptominoes from the first category count eightfold, the ones from the next three categories count fourfold, and the ones from the last two categories count twice. This results in 84 × 8 + (9+7+4) × 4 + (3+1) × 2 = 760 fixed heptominoes.

Packing and tiling[edit]

Although a complete set of 108 heptominoes has a total of 756 squares, it is not possible to tile a rectangle with them. The proof of this is trivial, since there is one heptomino which has a hole.[4] It is also impossible to pack them into a 757-square rectangle with a one-square hole because 757 is a prime number.

Tiling the plane with copies of a single polyomino[edit]

All but four heptominoes are capable of tiling the plane.[5]

Heptominoes incapable of tiling a plane, including the one heptomino with a hole.

References[edit]

  1. ^ Golomb, Solomon W. (1994). Polyominoes (2nd ed.). Princeton, New Jersey: Princeton University Press. ISBN 0-691-02444-8. 
  2. ^ Weisstein, Eric W. "Heptomino". From MathWorld – A Wolfram Web Resource. Retrieved 2008-07-22. 
  3. ^ Redelmeier, D. Hugh (1981). "Counting polyominoes: yet another attack". Discrete Mathematics 36 (2): 191–203. doi:10.1016/0012-365X(81)90237-5. 
  4. ^ Grünbaum, Branko; Shephard, G. C. (1987). Tilings and Patterns. New York: W. H. Freeman and Company. ISBN 0-7167-1193-1. 
  5. ^ Gardner, Martin (August 1965). "Thoughts on the task of communication with intelligent organisms on other worlds". Scientific American 213 (2): 96–100. 

Original courtesy of Wikipedia: http://en.wikipedia.org/wiki/Heptomino — Please support Wikipedia.
This page uses Creative Commons Licensed content from Wikipedia. A portion of the proceeds from advertising on Digplanet goes to supporting Wikipedia.
119 videos foundNext > 

Daniel Corral: Heptomino @ ASTO Museum 04-17-2010 (4)

Written for, and performed by Erin Barnes (perc.) and Tara Boyle (fl.) This performance is from a concert of chamber music by Daniel Corral at the ASTO Museu...

Conway's game of life files pattern 42: B-heptomino

The b-heptomino is a common seven cell methuselah that stablizes in three blocks, two gliders and a ship after 148 generations. It is very similar to the her...

Conway's game of life files pattern 9 Pi Heptomino And A Hexomino Pi Parent

The pi heptomino is a fairly commoly occuring pattern in conway's game of life, either as a stand-alone pattern or as debris created by part of a larger patt...

Conway's game of life files pattern 31: Herschel

The herschel (or rarely called the d-heptomino or the j-heptomino), is a VERY important pattern in conway's game of life, perhaps second to the glider. Its n...

Paralotnia 001

Pierwszy lot.

Joshua mini concert2

Joshua mini concert3

Block Puzzle For Iphone

http://itunes.apple.com/us/app/block-puzzles/id511736378 Block puzzle is a addictive polyomino puzzle game. There are domino ,tromino, tetromino, pentomino, ...

Conway's Game Of Life Files pattern 106: Gourmet

The following is quoted from the link below: "Gourmet is a period 32 oscillator that was discovered by David Buckingham on March 4, 1978.[1] It consists of four cis-boat with tails and four...

Conway's game of life files pattern 35: Bi-gun

To quote the lifewiki article I got the information from "The bi-gun is a double-barreled glider gun that was found by Bill Gosper in the early 1970's. Much ...

119 videos foundNext > 

We're sorry, but there's no news about "Heptomino" right now.

Loading

Oops, we seem to be having trouble contacting Twitter

Talk About Heptomino

You can talk about Heptomino with people all over the world in our discussions.

Support Wikipedia

A portion of the proceeds from advertising on Digplanet goes to supporting Wikipedia. Please add your support for Wikipedia!