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. 

Tuesday, November 20, 2018

Lines, in historic and modern construction

 How we build today that can be traced back to the ancient world, 
We may have forgotten how to use geometry, but we still use the concepts and tools.

Now called Chalk Lines

Every carpenter knows :1) how to use a Line,  2) why to lay it out,  3) how to pluck the twine at the right angle.

 We learned to snap the Line from someone else, not a book.  This is hands-on teaching - master to apprentice.

We've been educating the next generation this way long before it was written down.
Theo Audel tried in 1923.
Audel's Carpenters and Builders Guide has 2 illustrations showing how to 'pluck' the Line. Other illustrations show how to set the Line with an awl.

Here is his illustration of the Line with its reel and awl.

 He explains that "The line consists of a light string or cord"  made of cotton or linen; that it can come in 20 ft., 50 ft., and 84 ft. hanks, on up to 450 - 600 ft. long lengths. That's a awfully long cord!

In 1923, tape measures were still about 5 years in the future.
We did have 6 ft long folding rules.

This one is mine: it helps me catch nuances in an existing building because I am close to what I need to document.
The tape measure is better for overall dimensions.

That word, 'Line' - the L capitalized - is often used in Practical Geometry. Serlio"s diagram and explanation is the earliest example I have found. The word was understood, not requiring any special explanation. James Gibbs says about his drawings  that they are  "Draughts of useful and convenient Buildings ...which may be executed by any Workman who knows Lines,..."

Today we use Lines for setting a wall in a straight line. The Line is along, or off-set, from where the wall needs to go. We build to it.
Rectangular foundations are 'trued' by checking with a Line that the diagonals match.

How would builder without 20th Century tools use a a Line to lay out a building?

The slave cabin of Tuckahoe is an example.

The carpenter knows how big he will make the cabin and what it will look like: 2 square boxes with roof, lofts, chimney, 2 doors and 2 windows. 

He starts with a Line (C-D) and its Perpendicular (A-B) - basic geometry that he can easily lay out with his cord.  The Lines do not yet have length, just direction.
 He chooses his length  for the width of the cabin: A-B. Probably he has a rod marked off with 5 or 10 units. See the illustration at the end of the post.
He  swings his cord  in an arc using A-B as his radius from C through B to D. Now he has the width and the length of the cabin, and 2 corners.
To find the other 2 corners, he moves to B and swings his arc from E through A to F.
He doesn't yet know where E or F are...
For that  he will stand  at D and swing the cord from A to E. Where the cord intersects with the earlier arc will be the right rear corner, E. F will be determined the same way.
Then he will check that his rectangle is true by running his diagonals F-D and C-E  If they match, he is set. If not he will adjust his Lines.
At no point does he need to use numbers. 
The walls of the cabin would be laid out on a framing floor with cord set from point to point, just as timber framers who work by hand do today.

Here's a 14th c. wood cut showing a rod marked in 5 units. It may be longer; it may extend behind his body to end at the rectangle with the triangle at its end - which might be for plumbing a surface.
 C. 1800 pattern books refer to 10' rods. 16.5 ' rods were used to lay out acres.

See my Bibliography for the books of Serlio and Gibbs referred to here.
See also my posts on both men, and my posts on Tuckahoe Plantation.

Audels Carpentry and Builders Guide, a practical illustrated trade assistant, Theo. Audel & Co., NYC, 1923

The subject of the next post:

 How a daisy wheel fits into the use of Lines

This daisy wheel, drawn by a compass, was on the wall of a barn in Vermont. Close examination shows that the center and the tips of the daisy's petals were regularly pricked. The radius and the diameter were regularly used as  dimensions.