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What is a Polyiamond?
Polyiamonds develop while putting equilateral triangles together in
such way they must have at least one side in common.
The Scottish mathematician T.H. O'Beirne suggested the name polyiamonds
in "New Scientist" 1961 [(1), page 164]. He called the figures after the
diamond.
Simple Polyiamonds top
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You can only form one figure with two or three triangles.
There are three figures of four triangles, the tetriamonds. |
Pentiamonds top
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There are four figures of five triangles. |
... ... |
If you want to play with them, you should build a set.
Therefore you print a pattern of equilateral triangles, mark the four
figures in a size you like, glue them on cardboard and cut them out.
Here is a triangle pattern for downloading. |
Though the number of pieces is small, you can design figures:
You can recognize:
1 intercity, 2 sphinx, 3 crooked tower, 4 trapezium without a corner,
5 terraced houses, 6 motorboat, 7 motorboat with a peephole.
Even symmetric figures are possible:
Hexiamonds top
It is worth working on the pieces with six triangles, the hexiamonds.
You have more pieces than with pentiamonds and therefore more ways of playing.
You can build them in the same way as described above.
There are 12 hexiamonds.
The names of the figures go back to O'Beirne mentioned above.
Playings on Hexiamonds top
Pentominos are decribed on another place
of my homepage. Pentominos are pieces of five squares. You can solve different
problems like forming rectangles, new figures, figures with holes, enlarged
pentominos or rings. You can transfer these problems to hexiamonds.
1st Problem: Parallelograms
... ...
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The Parallelograms 6*12 und 9*8 are possible and solved on the left. |
2nd Problem: New
figures
If you design your own figures, you first must
find out, whether you can solve them. You use the chessboard method:
You alternately colour all pieces and count the triangles of each colour.
If you colour all 12 pieces, then 10 pieces have 3 white and 3 black
(grey) fields, 2 pieces have 4 or 2 black fields. There is a statement
for all pieces: They have the distribution 38+34 or 36+36. The second sum
comes, if you exchange the colours of the two pieces on the right.
When you have designed a figure and have coloured it like a chessboard,
the distribution of the single pieces must be transfered to the whole figure.
The following figure has the distribution 38+34. You can solve it.
It isn't sure that a figure is always possible, if the distribution
is 38+34 or 36+36.
You only can say, maybe it is soluble.
More designs:
Is there a solution? What is it?
3rd Problem: Rings
... ... |
You build a ring of all hexiamonds.
Then you should surround as many connected (white) triangles as possible.
Is the number 91 the maximum? |
4th Problem: Enclosing single
triangles
... ...
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You must use all pieces and surround as many single triangles as possible.
I found eight triangles in a first attempt. |
5th Problem: Small figures
You needn't use all 12 pieces for building new figures.
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You form a star of eight hexiamonds or 48 triangles. |
6th Problem: Duplicating
... ...
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You build a hexiamond with double magnification using four pieces.
Eight pieces are left.
Question: Can you build all hexiamonds like this? |
7th Problem: Tripling
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You build a hexiamond with triple magnification using nine pieces.
Three pieces are left.
You can only do it with nine hexiamonds (Origin: Instructions "Verhext") |
8th Problem: Figures of the
same hexiamonds
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You can build a larger hexiamond ("Sphinx") with four equal pieces. |
... ...
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The small and large hexiamonds needn't be the same.
Example: Four pieces ("yacht") form a larger hexiamond ("rhomboid"). |
The piece "yacht" can cover the whole plane.
Question: Which pieces also tile the plane?
Verhext top
... ... |
There was a famous puzzle with the name "Verhext" (Bewitched) in Germany
in the 1960s. It used all the 12 hexiamonds.
Professor Heinz Haber developped it and presented it in detail on television
and in his magazine "Bild der Wissenschaft".
The pieces had the names Kamm, Kirche, Pfeil, Feile, Revolver, Haken,
Hexagon, Segelboot, Schlange, Tanker, Pfeffermühle, Dach.
Manufacturer: Herbert Zimpfer, Metallwarenfabrik, 7586 Altschweier /
Baden |
Heptiamonds top
There are 24 heptiamonds.
Furthermore there are 66 octiamonds, 160 figures of 9 triangles, 448 figures
of 10 triangles, and 1186 figures of 11 triangles.
Polyiamonds on the Internet
top
German
Steffen Mühlhäuser
Rhomba
Andrew Clarke (Die Poly-Seiten)
Polyiamonds
Wikipedia
Heinz Haber
English
Andrew Clarke (Die Poly Pages)
Polyiamonds
Col. George Sicherman (Polyform Curiosities)
Mixed
Polyiamond Compatibility
Ed Pegg Jr. (mathpuzzle.com)
iamonds, octiamonds
and beyond
Eric W. Weisstein (MathWorld)
Polyiamond
N. J. A. Sloane (The On-Line Encyclopedia of Integer
Sequences!)
Number
of triangular polyominoes, Number
of one-sided triangular polyominoes
Steven Dutch
Polypolygon
Tilings
Stewart R. Hinsley
Polyiamonds
Wikipedia
Polyiamond
References top
(1) Martin Gardner: Mathematisches Labyrinth, Braunschweig 1971 (ISBN
3-528-08402-2)
(2) Karl-Heinz Koch: ...lege Spiele, Köln 1987 (ISBN 3-7701-2097-3)
(3) M.Odier, Y.Roussel: Trioker mathematisch gespielt, Braunschweig,
Wiesbaden 1979 (ISBN 3-5 28-08394-8)
Feedback: Email address on my main page
This
page is also available in German.
URL of
my Homepage:
http://www.mathematische-basteleien.de/
©
2003 Jürgen Köller
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