Polyhexes
Contents of this Page
What are Polyhexes?
Making of Tetrahexes
Playing with Tetrahexes
Tetrahexes, expanded
Pentahexes
Polyhexes on the Internet
References
..
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What are Polyhexes?
Polyhexes are figures which you can form with at least two hexagons. 

You order and call them by the number of the hexagons. 
 
You can build one figure by two hexagons and three trihexes by three hexagons. 
......


There are seven tetrahexes. ...

Furthermore there are 22 pentahexes. Computers found 82 hexahexes, 333 heptahexes, 1448 octahexes or 8-hexes, 6572 9-hexes, 30490 10-hexes, 143552 11-hexes, 683101 12-hexes,  ... (Book 2, page 152 and 160).

Martin Gardner described polyhexes in the magazine Scientific American and made them popular. He repeated that in book 2 (Page 149ff.) and added readers' results.

Making of Tetrahexes  top
If you want to play with polyhexes, you must build them yourself.
There are several methods.
1st method: 
Print a picture with equilateral triangles (6 triangles=1 hexagon), mark the seven pieces, glue them on cardboard and cut them out. 
Here is a drawing of equilateral triangles  for downloading. 
2nd method: 
The densest sphere packing in the plane corresponds with a hexagon pattern. Glue balls to each other so that you get tetrahexes.
3rd method: 
It is said that you can also glue nuts (which belong to screws) together.


You could buy a set of tetrahexes.
...... The firm Kipfer CH-3303 Jegensdorf  offered the set on the left with the name HEXAGON. 
It is described in the packing as follows:

"Puzzle containing 7 parts: You can create more than 900 symmetrical shapes! 92 examples and solutions on coloured cards are enclosed." 



You also could find a set of tetrahexes in the catalogue of the Swiss firm Naef. It is shown on the Japanese web site below. 

Playing with Tetrahexes top
Here are the seven tetrahexes once again. The names under the pictures are from book 2. 

........
bar, worm, wave, arch, propeller, bee, pistol 
I suggest to replace the name pistol by locomotive. 

There are several questions according to the pentominos, the figures with five squares.
Parallelogram
Seven tetrahexes have 28 hexagons altogether. The question is whether there is a parallelogram 4x7.
There is one....and more:
Source: book 3, page 125

Triangle
...... 28 is a triangular number. That is 28=1+2+3+4+5+6+7. 
Thus you can form a triangle of 28 hexagons (on the left). You can't solve the figures with tetrahexes (Torsten Sillke, URL below). 

You must be content with an "alternate triangle" (on the right). 

......

Figures of Tetrahexes 
It is not easy to lay figures. Best is to keep it up and wait and see. You must not use all pieces. Symmetric figures are always nice. 
You better recognize the shapes if you only use one colour. 

Figures of all Tetrahexes

My bird, my boat, my...

You can solve the following figures.
Source: Book 2, page 151

Ring
There is the problem of forming a ring including as many hexagons as possible. 

34 hexagons are included. 

35 hexagons are possible. (Source: Livio Zucca, URL below).
This is a nice aspect of mathematics: Better solutions lead to more handsome figures. 
Are there more than 35 hexagons? 

Figures with as many holes as possible
Can you break the number 6?

Tiling the Plane
...... You can fill the plane with only one tetrahex. The propeller is one example. You form a figure with four propellers, which repeat in the plane. 
(Steven Dutch, URL below). 
Tiling a Triangle
...... As mentioned above you can't lay the triangle on the left with all tetrahexes. It is possible to choose one, the locomotive. This is the only piece which, itself, can fill the triangle. 
(Book 3, page 124)

Tetrahexes, expanded  top
It is useful not to adhere at seven tetrahexes, but to widen the set. The ways of playing are larger and the puzzles simpler. 

You can reflect the asymmetrical pieces, so that you get 10 pieces with 40 hexagons. They are called "one-sided polyhexes". You find rectangles 5x8 and 4x10 on Andrew Clarke's web site  (URL below).


You get 12 pieces and 40 hexagons if you add pieces with one, two and three hexagons. 

You find a ring formed by all pieces on Kate Jones's web site (URL below).


Pentahexes    top
22 Pentahexes:

The number is too large to give an interesting puzzle. Pentahexes is a topic more for computers. Many people have worked on these pieces, as you can see in book 2 and several web sites. I restrict myself on a parallelogram 10x11 on this page. The puzzle pieces of the series "Beat the Computer" from the 1970s are packed in this shape.

Polyhexes on the Internet top

German

Andrew Clarke (The Poly Pages)
Polyhexen

Peter F.Esser
Puzzle aus Muttern

Torsten Sillke
Triangles and Pyramids with equal Polyspheres



English

Andrew Clarke (The Poly Pages)
Polyhexes

Eric W. Weisstein (World of Mathematics)
Polyhex

Erich Friedman (Math Magic)
Triangle tilings

Joseph Myers 
Polyomino tiling

Kate Jones (Kadon Gamepuzzles)
A New Puzzle Genre: Polyforms

Miroslav Vicher
Polyhexes

Peter F. Esser
Front and Back Colored Tetrahexes

Steven Dutch (Professor Dutch's home page)
Polypolygon Tilings

Torsten Sillke
The Impossible Tetrahex Triangle

Wikipedia
Polyhex (mathematics)


Dutch

NN
Polyhexes


References    top
(1) Karl-Heinz Koch: ...lege Spiele, Köln 1987 (dumont taschenbuch1480)  [ISBN 3-7701-2097-3] 
(2) Martin Gardner: Mathematischer Zirkus, Berlin 1988 (ISBN 3550076924) 
(3) Solomon W.Golomb: Polyominoes, Princeton, New Jersey 1994 (ISBN0-691-08573-0)


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URL of my Homepage:
http://www.mathematische-basteleien.de/

©  2003 Jürgen Köller

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