Ostomachion or bone-fight (9th of March 2010)
Ostomachion, or translated literally from greek, "bone-fight" is a puzzle that is tightly tied with Archimedes, although it may be older than Archimedes (the puzzle is also known as loculus Archimedius or Archimedes' box). In any case, ostomachion is apparently most known from the mathematical study by Archimedes. It is a puzzle made of 14 polygons, 13 of which are different and two are the same. Those 14 polygons (11 triangles, 1 pentagon and 2 quadrangles, originally made of bone) can be assembled in a square, as is shown in the image above. The image was obtained from the ostomachion scheme taken from Wikipedia, printed on a piece of paper that was then glued to a cardboard from which the pieces of the puzzle were cut out. Very nice for kids I think, I also made it for my nephew. But, the problem of assembling the elements in a square is fairly tough and perhaps not best suited to pre-school children. I'll check it. Once one alows the "flipping" of the polygons and assembling those so that their back side comes to the front (see the image below), many additional possibilities for assembly arise and the problem becomes simpler.
Of course, for the mathematically inclined mind, a question regarding the number of ways in which one can assemble the square from the given elements immediately arises. The answer is "536 without rotations and reflections". It is obvious that any assembly of the square can be rotated three times by 90 degrees, so that one gets the same square each time, formally differently assembled. Therefore, the rotational multiplicity of the assembly is 4. Ostomachion is, besides for interesting mathematics it represents, perhaps better known (and can be understood in such a way) as a set of elements that one can use to form interesting shapes. Such interpretation of ostomachion is probably best suited to children. Anyhow, here is my contribution to the shapes that I am sure are inummerable (in the mathematical sense also). I begin with some kind of bird-dinosaur, ancestor of all the birds (below).
I continue with the sail (below).
Some kind of a crab in defensive position. Perhaps it is in retreat and threatens with its claws (below).
And I finish the sequence with T-Rex. To the contemporaries of Archimedes the image would mean nothing I suppose.
Web page bibliotheca Augustana offers some more interesting shapes/assemblies that are shown below (sword, dog and elephant). So try it if you like it...
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Last updated on 9th of March 2010.