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

A closer look at the Schiefferstadt House practical geometry

Please see my update:  https://www.jgrarchitect.com/2024/09/from-circle-to-pythagorean-triangle-via.html
I am not deleting this post because of my last paragraphs: I find the ways the Lines and layouts in Practical Geometry overlap need more consideration.* 

The lay out of the Schiefferstadt House,* uses a geometric pattern that was well known at the 1750's: the rule for drawing a square starting with a radius and a circle.

The diagram begins with the daisy wheel, scribed by a compass or a divider.  The 'petals' created by the 6 arcs of the radius around the circle make 6 points on the circumference.

The length of the radius for the circle is the width of the house.

 

 

 

When those points of the daisy wheel are joined they create Lines - dashed lines in the diagram. (Basic Euclidean geometry : 2 points are required to create a Line.)  The arcs of the radii cross those Lines to lay out a square. **

When one point of the divider, still opened to the width of the radius of the circle, is set on each of the 2 upper corners of the square, and the arcs swung, the arcs cross the circumference at the top of the circle.  Stepping off the radius around the circumference, will locate 6 more points.  All 12 points are equidistant from each other; all can be used for layout and design.

There is also a short cut to those upper 2 points; the place where the arcs cross the daisy wheel petals are points. 2 points = a Line. That Line extended is the same Line shown in my next diagram.  

 

The carpenter of the Schiefferstadt House could have used this geometry to step off  a rectangle about 18 units wide x 26 units long.  If his compass was open to a 2 ft span, the floor plan would have been 36 ft.wide  x 52 ft long. He would have trued his rectangle by checking that his diagonals matched, just as builders do today.

*

However the carpenter could also have used the square and its diagonal to lay out the plan. Those arcs would cross the circumference at the same place (dashed line), but they would cross the vertical lines of the rectangle about one unit higher than if the 12 points had been used (see the points where the dashed and red lines cross the circumference).  

This would make the floor plan 36 ft wide x 54 ft long. That's not much longer,  probably of little consequence to the design. However if the mason and the the framer were not using the same geometric progression (both using the first diagram or both using the second) the stone foundation and the interior wood frame would not have fit together. 

 

The drawings made c. 1978 for the restoration of the Schiefferstadt House may give me more information. The Frederick County Landmarks Foundation is sending prints. 

I will be looking for the simplest and quickest layout. I find that a builder tends to use the same same geometric progression for his plans and elevations. The geometry is one of his tools. The repetition of one pattern and one unit of measurement would be efficient and leave fewer chances for mistakes.  

If another layout is introduced it is usually the work of a craftsman whose work comes later - the finish carpenter adding a mantle, or the mason building a firebox and flu. Each might prefer a different system.

* The Schiefferstadt House, Frederick, Maryland, built in 1755, owned by The Frederick County Landmarks Foundation.  See my previous post for the geometry of the floor plan: https://www.jgrarchitect.com/2024/07/the-geometry-of-schiefferstadt-house.html

 

**2  basic practical geometry diagrams:

The diagram laying out how the radius of a circle can become the side of a square.

 Audel's Carpenters and Builders Guides , published in 1923, shows this diagram.

100 years ago, this geometry was common and practical knowledge.

The geometry of the Schiefferstadt House, Frederick, MD, 1758

The Schiefferstadt House 

 

 

This stone house, in Frederick, Maryland, was built c. 1755 for Elias Bruner, the son of German immigrants.

In the spring of 2023 the board of the Preservation Trades Network toured the house when we were in Frederick to plan the 2023 International Preservation Trades Workshops. The guides of the Schiefferstadt House showed us the house from cellar to attic, sharing both the original construction and the on-going work of maintenance and restoration. 

Our visit was well worth our time; many thanks to the Schiefferstadt staff and volunteers.

We saw many practical built-in systems for cooling and heating. 
The black blob in the lower left corner of this picture is one of several vents for the very effective basement cold storage vault. 

Under one kitchen window is a very useful drainage sill. 

 

 

 

 

 A close up of the trough with its lip and spout.  

The kitchen help need not carry the dirty dish water over to the door; instead it can be efficiently discarded out the window. 

 

I have seen a trough like this only once before, in medieval military barracks
in Switzerland.

The HABS drawing of the first floor plan shows the original stone house at the top of the drawing. The brick wing was built later. 

That first, 1637, house is outlined in red.

Stone walls are usually built between 2 lines, set to keep the walls straight. Here either the exterior line or the interior line could have set the governing dimensions.   

In both cases the practical geometry of the Line and its arc created the plan. First: The exterior geometry:

Using the width of the main house (red arrows) as a radius for a circle, I drew the daisy wheel with its 6 'petals',  noting the points around the circle where the arcs cross the circumference. The points connect the lines which lay out the long walls - see the red dots and vertical dashed lines at the top of the drawing.

The arcs of this daisy wheel create 6 points  If I rotated the wheel to begin the arcs not on the corners of the house, but at the center of the lower wall, the petals will be perpendicular to the side walls, parallel to the front and back walls. This adds 6 more points to the circumference - noted here as black dots, 12 points in all equally spaced*.

 2 of the new points lay out the location of the 4th wall, here at the top of the drawing, noted with red dots and a horizontal dashed line.

 

However, the plan for the house could also have been laid out from the inside. The exterior sides of the stone foundation walls could have been irregular, sloping away from the house below grade, only becoming straight  once they were above grade. I have seen this often in houses built before 1900. As in the first layout the width is the governing dimension. The geometry, a square and its diagonal, lays out the interior walls.

The builders dug into the side of the hill to set the foundation. When they set their governing lines which layout did they use? The exterior or the interior plan? Perhaps the exterior plan was concept, drawn during a consultation with the owner, then staked on site, perhaps with offset lines like those we use today. Then the interior dimensions could have been used when the workmen were on site, in the future cellar.

On the drawing are faint pencil lines of diagonals and an arc, using the exterior width as the side of the square. They don't quite fit. They are an exploration,  an essential part of discovering what geometry the builder used.

 

 

The house has a center entrance, a room on each side. Within both walls on either side of the center hall are fireplaces and flues. Here is the kitchen fireplace.

 

 

 

 

 Fireplaces on both sides of the center hall itself send heat to iron boxes on the first and second floor. This picture is a 2nd floor heating box, set into the wall between 2 rooms so it will radiate heat into both rooms. It's an early version of the radiator.

 

The flues of those first floor fireplaces join to become one chimney as they exit the roof.

Here 2 members of the Preservation Trades Network board and a staffer for the House stand in the first floor hall under the arch created by those flues as they come together overhead. One fire box is visible on the left. The one on the right has been closed.

This 3D drawing of the House is helpful even if not quite accurate. The flues come together over the first floor, not the second  as shown here.

However it explains how 'wishbone' masonry chimney blocks were located and the flues joined to become one chimney at the ridge of the roof.  

Each wing of the chimney required its own foundation which would have been built as the stone walls of the house were laid up.

I found as I studied the layout that while I could easily layout the exterior dimensions, I did not have enough experience with stone construction to understand how the masons would have worked once they set up the lines for the foundation walls. How did the masons measure where to set the chimney foundations?

A timber framer can begin the house frame with a sill set on the foundation after it is complete. I know those frames and foundations well. The joist pockets for the wood interior frame of the Schiefferstadt House had to be set in the stone as the wall was built. How did they know where to place them?

I asked Joe Lubozynski for help. He lives in the area, is an excellent architectural historian as well as craftsman. He knows historic stone construction and this house in particular. And he was willing to advise me. 

He and I reviewed how the house would have been built, beginning with digging into a slope (see the Architectural Cross Section above),  placing the footings, laying the stones for the walls and chimney foundations.  Then constructing the basement cold storage space with its vaulted cellar ceiling and placing the first floor frame.  

We considered whether the builders would have measured from the outside or the inside of the stone walls to place the joists. Our conclusion was obvious: it's much easier to set Lines from the inside of the walls and check them. The work would be done more accurately as well as more quickly.

At this point we drew the Rule of Thirds* diagram using the inside of the stone foundation as the rectangle.

 

  
The diagram easily located the floor joists. Note  how the inner sides of the floor joists are on the intersections of the diagonals. the diagonals cross, making 2 points. The points layout out the location of the joists and the joist pockets. See the 2 black lines with arrows on each end.The geometry is as simple as the layout of the foundation.

The chimney bases are white in this drawing in order to make the lines of the Rule of Thirds clear. 

 

 

The Rule of Thirds also locates the 2 foundations which support the fireplaces and chimneys. See the red arrows.

The measured drawings were done before the restoration of the house began. The fireplaces had been altered. The location of the masonry for the fireboxes and chimneys is an educated estimate. I have not measured the rebuilt and restored masonry. 

 

My conclusion? The technology of the Schiefferstat House is elegant and sophisticated for that period in the Colonies. The grand houses in New England and Virginia did not have the conveniences of this little house.  However, the practical geometry of the house is simple, traditional: very similar to what I've found from the same era in New England, New York, Virginia, and Louisiana.

* For information about Daisy Wheels,  drawings squares and rectangles with compasses, the Rule of Thirds, try:

https://www.jgrarchitect.com/2020/04/practical-geometry-lesson-4.html

https://www.jgrarchitect.com/2022/11/virginia-folk-housing-update.html

 https://www.jgrarchitect.com/2022/10/serlios-lines.html

 

 

The Perfect Cube and its Sphere

 

 

This perfect cube and its square was drawn by Sebastiano Serlio, c. 1540.

It is an Euclidean solid: 6 square faces.  It is 'perfect': each side exactly like the others. A  perfect sphere would fit within it. A perfect circle fits its perfect square face. Another square is within that circle, and a smaller square within that. 

The shapes are bound by the diagonal Lines which create 2 points at the intersections for drawing the next square or circle.
That cube and sphere were not only theoretical ideals, they were practical, a layout tool, the pattern governing a design. The pattern book writers called this 'Practical Geometry'.

 

 

Serlio drew his tools in the lower right corner on the frontispiece to his book, 'On Architecture'. **

The perfect cube is in the lower right corner. His compass is in the middle surrounded by his straight edge, carpenter square, stylus. his line*. 

I've kept track of that pattern: the cube, its circle, the next smaller square and its circle, the diagonals. I want to understand how it is layout tool.

Some of what I've learned is posted here.

Hagia Sofia was built in the early 6th C. by Justinian I, the Byzantine Emperor in then Constantinople, now Istanbul. Earlier churches on the site had burned and the first dome of Hagia Sofia fell in.  The second is still standing 1500 yrs later.

 

 

 

 

 Here is what it looks like from the inside.

This is Bannister Fletcher's diagram** of the dome  formation for Hagia Sophia: the square with its circles. One is around it, the other within it. Their sizes are governed by the square.

This design can be seen in the mosques and churches built after Hagia Sophia in eastern Europe and around the Mediterranean.  The examples in western Europe which I have found are in Italy. 

The shaded areas are called 'pendentives'. There are several ways to build these, not to be discussed here.

 

A dome needs to be held up, of course. When it is the top of a silo, there is no structural problem - the dome is supported all the way around. 

 

 

 

 

But in a church or mosque - where people congregate - the dome needs to be on supports so it is visible. We need to be under it and in its space. 

The weight of the dome must be supported, and its thrust as well.

This diagram from Mosque,**by David Macaulay, explains the problem and show.s the solution in the Byzantium empire: columns (blue) support arches (green) with  cylinders (brown) adding weight behind each arch. At the bottom of the drawing is the floor plan, a square which fits within the circle of the dome.

Other domes had been built. The Pantheon dome with its oculus, c. 120CE, is the best known example. However, the Pantheon's geometry is circular. Hagia Sophia adds the circle's square. Or perhaps the square's circle. 

When Hagia Sofia was being built the Roman Empire was collapsing. Western Europe built little except in  those ports where there was political power, trading, and influence from the cultures around the Mediterranean.

Venice, with its location and port, did flourish. It began to build  St. Marks Cathedral*** in 1000 CE.

 

These drawing of the plan and the interior are from Bannister Fletcher. **

The design shows many circles within their squares.

The large circles are domes, seen here from above. The square bases are visible too.

 

The small circles are the arched stone work of the columns which are shown as 4 black cubes around each circle within its square.

 

The ponderous columns are divided into 4 piers which makes them appear less massive and intimidating. They join at the springing points to support the arches.

This photograph is from Laurie Smith's' book, The Geometrical Design of Saint David's Cathedral Nave Ceiling.**

 

It's the ceiling under the new (c. 1535) roof for the cathedral.

The pattern is squares and the circles set side by side but not in a simple repetition. Laurie's compasses show the layout.
This ceiling pattern is obviously geometric but it is not in the lineage of the designs of Hajia Sofia or Serlio. It is, to quote Laurie Smith, " ...an exceptional carpentry idea and one that was unique to Wales.

These are the geometries used  for the pendants in St. David's Cathedral, as documented by Laurie Smith. The first 4 are based on the use of a compass, the next 2 on a diamond and a square. The last is related to, and perhaps growing out of, the circles and squares in ceiling pattern. 

 

These dome elevations and plans are part of William Ware's book, American Vignola, published in the States in 1903.**

 He describes the dome on the left (C) as "being generally a full hemisphere, constructed with a radius less than that of the sphere of which the pendentives form a part."  

If the same dome is erected upon  a vertical cylinder, visually a band below the dome, it is a 'drum' dome. Here: the dome on the right (D).

I have wondered why he did not recognize the lineage of the perfect square and its circle. He knew geometry.

 

The drum dome is the plan and elevation of the main dome at Massachusetts Institute of Technology, built in 1916.

 The glass blocks which fill the oculus of the MIT dome are set in Serlio's  pattern: the circle is the outer shape with its square and its circle set within it.

 To see the glass of the oculus, please follow the link, as the photographs are under copyright. https://capitalprojects.mit.edu/projects/great-dome-skylight-building-10 

 

A similar dome was placed over the Massachusetts Avenue entrance at MIT, built the the 1930's.

The glass curtain wall that faces Mass Ave is naturally based on the square and its circle. However the overlap of the square within the circle is not a simple repetition of square set next to square. The band between the squares  is a simplification (no curved lines) of the complex pattern seen in Saint David's Cathedral.

 

 

 

 I will update this post as I learn more.

 

*The tangled Line with its plumb bob is in the lower left corner. It can be tied to something and held taut with a plumb bob on the other end. It is not perfect. It is how we attempt to build perfectly, with no mistakes.

**  frontispiece, On Architecture, Sebastiano Serlio

**page 281, 288, A History of Architecture on the Comparative Method, Bannister Fletcher 

** page 11, Mosque, David Macaulay. 

** pages 7,15 and 31, The Geometrical Design of Saint David's Cathedral Nave Ceiling, Laurie Smith. Laurie's book can be purchased through me, as well as through the Carpenters Fellowship in the UK. 

** page 88, The American Vignola, William R. Ware

For more information please see my Bibliography: https://www.jgrarchitect.com/2022/03/a-bibliography-for-my-traditional.html

***I have lost the name of the engraver for the image of St.Marks. I don't know where I found it. The pictures of Hagia Sophia also cannot be credited. If you recognize what publication they appeared in, or who made the images, please let me know.

A Daisy wheel is a Module

Andrea Palladio wrote that he chose to use a Module to lay out the columns he drew.*
 

All the dimensions which he noted were derived from the diameter
 of the column measured at the bottom,  ie: the height of the column, the capital, architrave, frieze and cornice.

He wrote, " ..in the dividing and measuring the said orders, I would not make use of any certain and determinate measure peculiar to any city, as a cubit, foot, or palm, knowing that these several measures differ as much as the cities and countries; but imitating Vitruvius, who divides the Doric order with a measure taken from the thickness or diameter of the columns, common to all, and by him called a module, I shall make use of the same measure in all the orders." 

 

 

 

The cross section, the diameter, of a column is a circle.

 

 

 

 

A daisy wheel is a circle.

It's easy to draw with a compass or dividers.

 

 

 

 

 

 

 

 

It is a module.

 

 

 

This sheathing board leans against my corn crib. It was siding for a timber framed shed, part of a farm complex in Danby, Vermont. The board is 10' tall, angled at the top to fit under the roof eaves. 

 

 

Today it folds in 2 places. It fits in my car; it stands on its own at a conference, ready to be seen and examined.

 

 

The daisy wheel was cut into the board at a height of 42"  above the floor, a good height for the builder and his crew who would have set their dividers to its width. All of them needed to be using the exact same dimension (their module) as they laid out their work.  

 

This daisy wheel's diameter is slightly more than 8 inches.

The wheel is just above the center of the photo. In the images below the board has been laid down. 

 

 

Dividers set from one petal to the other across from it, the diameter of the daisy wheel.

Note the holes on the circumference and the center left by previous users.

 

 

 

Here the dividers, set open at the same angle, are rotated 60* farther around the circumference. They are at the points of another pair of daisy wheel petals, the circle's diameter. The distance is the same as it was before. 

Note the holes are not quite on the petals tips, rather they are on the circumference of the circle. Many daisy wheels were not precise.  

 

 

 

Here the dividers points have been slipped into the holes drilled by all the previous carpenters' dividers.

 

My dividers slid right in place, so secure they stood by themselves.

I had never tried this before - I was surprised and awed: my placement was one that many had done before me. 

 

The black marks on the board above the daisy wheel are the holes left by rusty nails.

 

* Andrea Palladio, Four Books on Architecture, 1570, Isaac Ware English Translation, 1738, Dover reprint, 1965. Palladio's statement about modules is on page 13, First Book. The image of the Doric Order is Plate XII, First Book.