Polyiamonds
 What is a Polyiamond? Simple Polyiamonds  Pentiamonds Hexiamonds Playings on Hexiamonds Verhext Heptiamonds An Email Polyiamonds on the Internet  References.
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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
 You can only form one figure with two or three triangles.  There are three figures of four triangles, the tetriamonds.

Pentiamonds   top
 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.

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
 ...... 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
 ...... 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.
 You form a star of eight hexiamonds or 48 triangles.

6th Problem: Duplicating
 ...... 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
 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
 You can build a larger hexiamond ("Sphinx") with four equal pieces.

 ...... 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

Puzzle from different polyiamonds
 ... Name and Manufacturer unknown Sent by Bodo Schnell

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.

An Email    top
Craig Knecht sent me the following interesting email.
Hi Jurgen Koller,
I have ventured into the world of polyiamonds.  I have made some cartoons and tried to humanize the subject ...
In the cartoon below -
The fat bird cannot be tiled with the sphinx tile.
The hexagon with and without the fat bird can be tiled with the sphinx tile.
I just find that to be interesting ...

I found a proof that all hexagons can be tiled with the theoretical maximum number of heptiamonds....
https://oeis.org/A298267
https://oeis.org/A291582
Cheers !
Craig
PS https://en.wikipedia.org/wiki/User:Knecht03/sandbox

Polyiamonds on the Internet   top

German

Andrew Clarke (Die Poly-Seiten)
Polyiamonds

Gerd Müller
Hexiamonds interaktiv

Steffen Mühlhäuser
Rhomba

Thimo Rosenkranz
Hexiamond-Figuren

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

Johannes Hindriks
Heptiamonds

N. J. A. Sloane (The On-Line Encyclopedia of Integer Sequences)
Number of triangular polyominoesNumber of one-sided triangular polyominoes

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)
(4) Zusammenlegspiele mit Quadraten und Dreiecken, Bild der Wissenschaft 11/1965, Seite 946ff., Fortsetzung in Heft 12/1965
(5) Noch einmal: "Verhext", Bild der Wissenschaft 3/1967, Seite 238ff.

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