Monday, November 26, 2018

Archimedes' Stomachion - Dissecting its Geometry

Updated and reconsidered

A copy of The Archimedes Codex was recently loaned to me by a friend who found it interesting.
I agreed. I enjoyed the discovery, the history, the math and the science.

I especially appreciated the chapter on the Stomachion, a puzzle I had not seen before. My grandson and I had fun with all the solutions.

I already knew the square and the Lines of the Stomachion.  It is a geometric diagram used for layout and framing, part of Practical Geometry which was commonly used for construction at least as far back as the 6th century BCE when it is mentioned in the Bible. Practical and Theoretical Geometry were co-equal branches of the same mathematics, Vitruvius writes of how one informed the other.

I cannot comment about Archimedes' understanding of the multitude of  combinations possible for constructing the square. I can, however, easily see how to transpose triangles from one place to another in the diagram.

He wrote about shape:
"So then, there is not a small number  made of them, because of it being possible to rotate them into another place of an equal and equiangular figure, transposed to hold another position; and again also with 2 figures, taken together, being equal and similar to two figures taken together --- then out of the transposition, many figures are put together." *

Archimedes was a geometer and engineer as well as a mathematician. He would have known and used Practical Geometry as it was applied to the construction around him in the 2nd and 3rd Centuries BCE.  His understanding of geometry, theoretical and practical, should be part of the discussion.  Perhaps he was thinking about the lines as well as the shapes. A person of his ability could have considered both effortlessly.

The Stomachion reminded me of this drawing by Sebastiano Serlio, from his book Architectura, published in France in 1537. He is discussing how to add a door to an existing facade.
The diagonals and the lines from center top to the lower corners determine the  size and placement of the door and its 'ornaments' - Serlio's word.

In 1821, the same lines were used to layout the Weathersfield, VT, church.
Here is its Palladian window. I have added the Stomachion lines which apply to its proportions in red.

Practical Geometry used lines to determine both design and structure: the size of a building and its framing, its ornamentation.

These drawings show how the lines of the Stomachion were determined. They are not random.

The first 3 squares focus on the  right side which is half of the square.  The red lines are the diagrams extended. First, the Stomachion.
Second, the square divided in half using its center line.  Third, the location and layout of the small triangle. Note that all the lines depend upon 2 points.
I really enjoy the 3 similar triangles flipping back and forth along the line.

In the second  3 squares focus on the left side.
First, the Stomachion.
Second, the diagonals of left hand half, and then that half divided in half again. 
Third, part of the original from the upper left corner which determines where the left-hand angled line (also a diagonal) stops.

Archimedes knew the shapes were proportional to each other. He must have know the lines. Were they so commonplace that he doesn't mention them? And we do because we've forgotten them?

Perhaps someone else has seen how the Stomachion relates to Lines, how these lines come from dividing a rectangle into parts, how this is Practical Geometry - perhaps even Theoretical Geometry.  I would like to meet that person.

                            *                               *                          *                            *                           *

The Archimedes Codex, How a Medieval Prayer Book is Revealing the True Genius of Antiquity's Greatest Scientist, Reviel Netz & William Noel, De Capo Press, Great Britian, 2007.

* quote from page 255 of The Archimedes Codex. 

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