From a Circle to the Pythagorean Triangle via the Schifferstadt House.

The  geometry used to lay out the Schiefferstadt House, 1755, was the 3/4/5 rectangle. Probably.

'Probably' because Practical Geometry, the use of geometry in construction, was taught by doing, not by reading and writing. The drawings we have assume a knowledge of basic geometric patterns. Written records are rare and incomplete.

The stone walls for the House were laid one row after another, consecutively. Unlike wood frame structures which are form and infill, in masonry buildings the  form and the skin are one. 

This is the back of the house, showing not just the main stone house and the brick wing, but the extensive stone foundation.

Every wall of the House needed to be trued as it was built. Here is a wall in the cellar: laid up stone.  Consider how hard those slabs would have been to adjust later on. The walls were trued with a plumb line and the lines of 3/4/5 triangle as they rose.*  

 

The frame of a wood structures determines its size, its corners, its form. The parts for the frame, the studs and braces, are cut and assembled. The shape can be adjusted, changed, trued using lines, even after it is raised. This image of a barn frame is from Wm Pain's The  Carpenter's Pocket Dictionary, 1781, redrawn by Eric Sloane.**  

The stone and brick buildings I have studied use the 3/4/5 triangle. Chimney blocks are 3/4/5 rectangles. 

So, why didn't I immediately try the 3/4/5 triangle when I looked at the house geometry? Well, I wondered if the Schiefferstadts'  traditional building patterns, brought with them from Germany, would be different from those I'd studied before, the vernacular housing built by English, Dutch, and French immigrants. Those began with the circle and its square. I began there too, looking for differences. I missed the obvious: the stone. The 3/4/5 rectangle easily fits the plans, the simple solution. KISS***

 

Then, as I was playing with the circle and its square (left image), this happened.

I saw that when I begin with the square derived from the radius, its circle and lines (left image), I can easy to locate 6 other points around the circumference , making 12 equidistant points around the circumference, (center image). I saw that circle geometry 'finds' the 3/4/5 rectangle (right image); that the Pythagorean Theorem is a 'short cut' using the 3 and 4 units that are already there.

On the left: the 12 pointed daisy wheel.  On the right: the 3/4/5 rectangle with units, and the 3/4/5 triangle.

 

 

 

 

 

 

*The walls are 'kept in line'. I am often surprised to realize that a common phrase, such as '"staying in line", probably began as construction lingo.

** Wm Pain, The Carpenter's Pocket Directory, London, 1781.

     Eric Sloane, An Age of Barns, Voyageur Press, Minneapolis, MN, 2001, p.37. originally published by Funk&Wagnals, c. 1967.  

*** KISS: "keep it simple, stupid"

The earlier posts on the Schiefferstadt House:  

https://www.jgrarchitect.com/2024/08/a-closer-look-at-schiefferstadt-house.html

https://www.jgrarchitect.com/2024/07/the-geometry-of-schiefferstadt-house.html