Making a Stomachion

The Stomachion, or Ostomachion as pointed out by the Ancient Roman poet Ausonius, was a puzzle game from the Ancient Greek, similar to the Chinese Tangram.

The original game was made of 14 pieces made of bone. The Ancient Greek Archimedes studied the game and wrote a mathematical treatise about its geometric and combinatorial properties, known as the Archimedes Palimpsest.

Little is known about the rules of the game. It seems that Archimedes used it as a combinatorial test asking how many different arrangements of the puzzle are possible. This question was solved in October of 2003 by the mathematician Bill Cutler, showing that there are a total of 17.152 solutions, but only 536 of them are truly unique.

However we can arrange the pieces to get some other shapes.

You can use the attached template (PDF and SVG) to transfer it to your preferred material.

In this instructable I'm going to guide you on how to make the Stomachion template using only a pencil, a ruler and a compass, so you can make your own game set on any material you cannot easily transfer the template.

Materials

• A piece of plastic. I have used part of a damaged DVD case.

Tools

• A pencil.
• A ruler or a set square.
• A compass
• A utility knife or utility scissors.

Some steps are easier using a set square. In any case I'm going to guide you how to do all the steps just with one ruler and a compass.

First draw a line, and mark the length of the desired square with the compass. Let's call this two points A and B.

Draw a perpendicular line from B. You can use the compass to mark two equidistant points to it. From this two points, keeping the same width in the compass, you can draw two arcs. The line from B to the point where the two arc cross is perpendicular to the segment AB.

Then using the width AB in the compass draw an arc from B that cut the new line at a point that we are going to call C.

With the same width draw an arc from A, and another arc from C that crosses the previous one. This point is going to be D.

Finish the square joining C to D and D to A.

If you get lost you can follow the steps given at the Math Open Reference.

Draw a diagonal line like in the picture, that is, from D to B.

Draw a line from the middle of AB to the middle of CD as in the picture.

This can be done by obtaining the bisection of AB (or CD, the result is the same). So with the compass draw an arc beyond the middle of AB from A, and another arc from B. These arcs will cross in two points. Draw a line between those two points and you have the desired line.

Call the middle points E and F.

Draw the diagonals as in the picture, that is, from A to F and from F to B.

Repeating the procedure of step 4, draw the line from the middle of AE to the segment AF. The point at the middle of AE will be called G.

This step is a bit tricky.

First determine the middle point of AD. As you already know E, just draw an arc from A that crosses AD, and with a width equal to AE. From this point draw a draft line to E, and mark where it cross AF. The final line is from this point (G in the picture) to E.

Determine the middle point of the segment DH (see the picture), and draw a line from that point to A.

Draw a line from the middle of BC (I) to BF. Call this point J.

Draw a line from J to E.

Math Open ReferenceThe final line and the most tricky one using just a compass and a ruler. You have to draw a line from J to 2/3 from C to I.

Draw an arbitrary line from C. With an arbitrary compass width, draw an arc from C that cuts this line. From this cut draw an arc, an repeat it one last time. From this last point draw a line to I.

Now with the compass witdh about half this last point and the previous mark, draw an arc from the point that crosses both lines.

Move the compass with the same witdh and repeat from the last mark.

Take the distance od the first arc with the compass, and translate it to the point where the previous second arc crosses the line, drawing a new arc that crosses the previous one. Join this point to the last mark and continue until crossing BC. This point is at 2/3 from C to I.

Draw a line from this point to J and you have finished your Stomachion template.

PS: If you got lost with this last instruction, you can see hoy to draw a parallel line to another passing through a given point at Math Open Reference.

Now cut the pieces with no particular order.

Once you have them cut, you can start playing. You can rearrange the pieces to shape again a square...

... you can rearrange them to shape a right triangle...

... and other shapes like a rombhus, an hexagon, a trapezoiz (trapezium), a diamond, and a bird.

I challenge you, fellow maker, to find another shapes that can be arranged using the Stomachion.

Some extra shapes will be added to this instructable after the end of the Puzzle Contest 2016 (the crown and the elephant among others).

And don't forget to vote if you liked it!

As promised you have here some other possible shapes you can arrange with a Stomachion set.

Hope you enjoyed this game!