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

The Baptist Church of Streetsboro, Ohio, Part 1

This is the Old Baptist Church at Streetsboro, Ohio, built about 1820.




Here are the HABS drawings.


 I wondered about its geometry. What framing traditions had the master builder brought with him to Ohio?
It looked linear, simple, obvious. Was it?



I explored the plan and elevation. While many forms of the Lines created by circles and squares worked pretty well, nothing quite fit.  
I went back to the basics, the construction: What did the carpenter do? In what order?


He was asked to build a church about 'so big'  - here about 36' x 50'. He laid out a rectangle using the 3/4/5 Triangle.  The HABS drawings are blurry and tiny. The dimensions appear to be 38'-4.5" wide by 51' long,  3 units wide by 4 units long. (The length is about an inch too short.)

The triangles are ABC and ADC. They could also be ABD and BCD. The 2 layouts cross in the center.
The carpenter could check his diagonals, just as workers do today. When the diagonals were the same length the floor frame was square.

 The bents for the frame were naturally the same width as the floor. It seemed possible that the framer used the floor of the church for his layout. I had seen this in an upstate NY barn. I wrote about it here: https://blog.greenmountaintimberframes.com/2014/12/04/geometry-in-historical-frames-a-guest-blog/

The elevation of the front of the church appears to be 2 squares wide. But the pediment did not come easily from that form - slightly too big.

However when I laid out the frame based on Lines laid on the inside edge of the sill and posts, everything fit and the peak of the bent, the location of the ridge of the church was the center of the rectangle. So simple, so easy!


How was it to the framer's advantage to lay out the frame from within the frame, not outside?  
He needed at least 3 bents, probably 5 or more. He needed consistent marks for lengths and widths of all members and for each mortise and tenon. The Lines laid inside the frame would not be disturbed while the frame was laid out and marked. The timbers could be moved off the floor to cut the joints; another bent could be laid out.  Or the bents could be stacked on each other.
Modern framers using timber and dimensional lumber stand within their work, measure, mark, and check from inside. Then they cut the lumber someplace else. Why not this earlier framer too?




After the bents and the roof trusses came the walls and the windows.
The spacing of the windows and their width comes from the rectangles that are within the original larger rectangle.
The green lines are 2 of those rectangles, the dashed lines with arrows on the left show the window frame locations.  The green dashed line with an arrow on the right ( top left) is the width. 



  



The geometry of the bents determined the shape of the facade, the height of the pediment. The front elements of the church - the  pilasters and a grand door -  were designed after the frame. The front windows were in place, therefore the pilasters needed to be equidistant on each side.
The door went in the middle, that's custom. Then there was the left over space in between. (See more about this below.)

The framers also had to provide support for the steeple. I have only photographs to show where the steeple sits. Was it directly over the front wall? a few feet back?  I would assume a bent supported the front and back walls of the steeple. The diagrams do show how the width of the tower and the size of the clipped corners were determined: the plan is a square with its corners cut off. 

Carpenter squares began to be manufactured in the States - not imported from Britain - around 1820. They had true 90* corners and consistent dimensions. 3/4/5 triangles and rectangles were easy to lay out accurately. An inexperienced carpenter could erect a  simple frame without much worry. A master carpenter working with church members as a volunteer crew could expect his crew to build a reasonably accurate frame.

Part 2,  the design of the exterior of the church is here: https://www.jgrarchitect.com/2018/04/the-baptist-church-of-steetsboro-ohio.html

 

7/27/22: I wrote this post in 2018. When I reviewed it recently, I saw how much needed to be revised, simplified; how much I'd learned about using geometry in construction during the last 4 years.  Understanding Practical Geometry (the name Asher Benjamin and Peter Nicholson used) is an on-going exploration.

 

The Baptist Church of Streetsboro, Ohio, Part 2

The Streetsboro Baptist Church, built c. 1820: the second phase of its construction - its decoration - the front facade and the steeple.

 

 

The first post* discussed how the framer used the geometry of the 3/4/5 Triangle to layout the floor, the bents, the walls and  windows, the roof and steeple. After the framers made the building 'tight to the weather',  joiners would often be responsible for the finish work: window sash, doors, molding.  Different trades had different skills and tools.

I think this division of labor happened here.

 

The church front on a cloudy day in October. It is a handsome building. It is also a box decorated with boards and moldings. That's what I am looking at in this post.

The HABS drawing is below.





 

 

The windows had been set by the framer when he laid out the floor plan, the walls, and the roof frame. The black lines show what the front wall would have looked like when the joiner began his work. Holes for windows, a space - perhaps a larger framed opening - for a door, a triangular gable. 

The congregation expected that this box with a roof would become a modern Greek Revival church. 

Of course the joiner was considering the pediment, the frieze, the architrave. the water table. He also needed to lay out a facade which has grace and rhythm as well as symmetry.

 

Here is the geometry of the facade as the framer knew it: 3 bays with their height from the floor to the roof trusses, their width between the corner posts, and a door, centered but of undetermined dimensions. The windows are centered within the  3/4/5 rectangles of  the frame's rhythm. Their shape is 2- 3/4/5 rectangles.
 

 

I think, the joiner chooses to balance the windows first, to set them as supporting wings to the central door. The corner boards grew to become paired columns balanced by 2 more columns on the other side of the windows. Note that the columns are not on the lines of the bays, therefore the center bay is slightly wider than the side bays. The window bays became back drop to the central bay with its  double door and paneled transom.The joiner 'adjusted' the geometry; but the window bays' symmetry is so strong it is hard to catch. The tall, broad main door, recessed  in the main bay, then surrounded by the columns and the frieze, becomes the focus. 

The joiner 'fooled the eye' and created a dynamic facade, much better than 3 equal rectangles would have been.

 

The framer built the base which supported the steeple. Its dimensions at the roof are based on the 3/4/5 Triangle.

 

The steeple uses neither the geometry of the frame nor that of the front facade. It is a series of blocks, decreasing in size, with their corners clipped. The design uses the square and the circles that fit within and without it. Was it the work of the same joiner? **



The HABS drawing shows the steeple sections.

Here I have added the circles  - In 'A' the red circle is outside, the green inside. In 'B' that green circle is now outside, a new smaller red circle inside. 'C' continues the progression with the red circle from 'B' now the outside. The green circle of  'C'  is the base of the spire.

 The steeple layout follows the  drawings of James Gibbs in his book "On Architecture", published in England in 1728. Copies were in the Colonies, available to builders.  I have written about Gibbs' steeples here: https://www.jgrarchitect.com/2022/02/james-gibbs-steeples.html  

 These  HABS measurements are too simple for an in depth study of the steeple geometry.

The shapes that make up the tower are a series of blocks with related faces all derived from the simple manipulation of the square: a complete square, 2 squares, one square, half a square (the base for the spire).
The spire's height uses the width of the steeple's base as its unit of measure: it is 1.5 times as tall as the base is wide.
 

The paneling, edge moldings,  and the series of roofs as the tower extends create the steeple.

The  door itself is approximately square, the transom: half a square. They are the same size as the section of the steeple which holds the bell.

The wall of that bay acts as a setting,  a frame for the door.  The columns and architrave are a second frame.

 
   

Look again at the photographs.

The church's grace and presence come from simple proportions in the design and the understanding of how light and shadow give life to the parts themselves and thus to the whole building. 

Here is what Asher Benjamin wanted the joiner - and by extension, we who see the church - to understand about moldings :  

"...the bending, or turning inward, of the upper edge of the Grecian, or quirk ovolo, when the sun shines on the surface [and] causes a beautiful variety of light and shade, which greatly relieves it from plane surfaces, and if it is entirely in shadow, but receives a reflected light, the bending or turning inward, at the top, will cause it to contain a greater quality of shade in that place, but softened downward around the moulding to the upper edge."   ***

 

* Part 1:  https://www.jgrarchitect.com/2018/04/the-baptist-church-of-streetsboro-ohio.html

 
** The Sandown, NH, Meeting House and Gunston Hall in Virginia are good examples of this separation of craft. At Sandown a skilled joiner built the main door and the pulpit, perhaps the wainscotting and box pews. George Mason of Gunston Hall brought William Buckland from England to create the porches and interiors for his new brick house.

**Asher Benjamin, The American Builder's Companion, 6th edition, 1827, R.P. & C. Williams, Dover Publications reprint, Plate IX, Names of Mouldings.

 

James Gibbs' Of Architecture, Draughts for a Menagery, Part 2 of 2

 

The  second menagery in James Gibbs' On Architecture* was never built.  In Part 1 of 2,  I wrote about the first one which was built at Hackwood Park, an estate near London. It is still standing.

Here is Gibbs' portrait with his compass, the mark of his profession. 

Gibbs' expected "any Workman who understands Lines" to be able to execute his designs. What would a workman have seen in Gibbs' drawings? What could I see? Did I understand Lines?

The first option was so simple. I was looking for an equally direct layout for this second design. I could find only complex solutions. They worked, but they were not direct. For 3 months the obvious design  was right there and I couldn't see it. I put the puzzle aside several times.  

This 'menagery' was to be a welcome destination for those strolling through the Hackwood Park estate grounds. Built of stone, the menagery would not have been as dark as this image.The 'draught' elevation accentuates the quoins and articulated arches, to give the workmen the necessary information. 

Like the design which was built, the pavilion required a gracious porch with a room on each side: one for serving and drinking tea, one for the quiet perusal of books about nature, especially birds. The living quarters for the staff who took care of the estate's pheasants were around the back. 

Today a plan usually lays out the exterior dimensions. Here the exterior is to be stone, perhaps ashlar or split, with rusticated, oversized  arches and  quoins (the corner blocks). Using inside dimensions to layout the plan allowed the exterior dimensions to vary.  

Note that Gibbs' drawing of the exterior stone facade is structural. The blocks interlock on the corners; the arches and key stones interlock with the wall. 

 

The floor plan begins with the central form: the porch and caretaker's quarters. Its size is determined by 2  3/4/5 rectangles overlapped at their mid-points. 

Gibbs assumes the workmen who might copy his 'draught' know how to build walls; he is not providing a construction document.

The wall between the porch and the living quarters is set beside the center of the rectangle found by drawing the diagonals of the main pavilion. This makes the porch large and gracious. It is also the way a mason sets lines today, building beside his lines. 

 

A workman could true his rectangle to center the doors on the walls.  Note the red line.

 

 

 

 The lines from the corners to the center locate the center lines for the arched columns. 

I have left out many lines here for clarity. They could be added to check the work.

 I call this 'The rule of Thirds' because artists who use these lines as a design tool call it by that name.  It's the 3x3 pattern that appears when we edit cellphone pictures. **

 

The wings of the menagery are set back 1/4 of the depth of the main block in the front and the back - note the red line with arrows As the geometry here is the 3/4/5 rcctangle it is fitting that the wings' length is proportional to the main pavilion's length: 6/8 or 3/4. 

 The wings are themselves both 3/4/5 rectangles.

 

The 3/4/5 rectangIe was a common way to add a wing to an existing building. If the mason set his length against the central form at 4 units and his width at 3 units, his wing would be square against the main block. His stone work would be true.  I have drawn the 3 x 4 units here. I have also left my pencil marks for further information (enlarge the drawing!) 

 While this menagery design is more complex to write about than Gibbs' other design*, it is quite easy to lay out with a compass and straight edge. A trained workman would have known the steps. The 3/4/5 rectangle and the Pythagoras Theorem are used today.

 The elevation? It's 4 squares and the same pediment layout that  Gibbs used on the menagery design which was built. The inside dimensions govern. The red arcs drawn show the floor width of the rooms are also the height of the elevations.    

A note: the windows are centered on the rooms' inside wall, but not on the exterior width of the wing.  The quoins are such a strong visual vertical that they appear as an anchor. The windows were centered on the rest of the wall.

The pediment is drawn  following the rules described by Serlio. For step by step instructions, refer to Part 1 of 2 of this post of the draughts for the Menagery.*

 * Gibbs' book On Architecture, published in 1728, includes 150 plates: plans, elevations, sections and perspectives of buildings Gibbs had designed and built. The quote is from his introduction, page i. A reprint is available from Dover Publications.

Part 1 of 2, the post for Gibb's design of the Menagery which was built is here: https://www.jgrarchitect.com/2021/12/james-gibbs-book-of-architecture.html

** I've posted about The Rule of Thirds in more detail here: 

https://www.jgrarchitect.com/2020/08/lesson-6-rule-of-thirds-part-1_21.html

https://www.jgrarchitect.com/2020/08/lesson-6-rule-of-thirds-part-2-serlio.html 

Rockingham Meeting House, Rockingham, VT

The Rockingham Meetinghouse  was begun in 1787, dedicated in 1798.

 After some preliminary analysis of the design and frame I realized in 2014 I needed to see it. I wondered if it would be as spare as the Rocky Hill and Sandown meetinghouses which preceded it.

It is.

 

 

 The site, on top of a hill with a view all around, emphasizes the simplicity of the structure. One can only come to it from below, and like many 18th c. New England buildings it sits upright and confident. It is very impressive.

I returned in 2018 and early 2020,  I updated the geometry as I learned more. I revised the drawings again when I gave a Zoom presentation in spring, 2021. What I first saw as a complicated geometry became simple and direct. 

This is not a dramatic design created by a London architect like Robert Adam to wow his rich patrons. It is a meeting house for a rural community. It is a straightforward layout planned by master builder, John Fuller, for a simple timber frame to be erected by a crew of local citizens.


The Town Fathers specified a building 44 ft. by 56 ft. The HABS drawings read 44'-4" x 56'-6". The difference could easily be the addition of the sheathing and siding to the frame. The porches (the end staircases) are square: 12'-2" x 12'-2". 
The difference could also be that the rule used then and the one we use today differ slightly. I am not sure they had a 'rule'. Poles of various lengths, 4ft., 5 ft., 10 ft., are in some illustrations.

 

'General' John Fuller, the master builder, was also the architect, engineer, framer. He knew the meeting house required an open space in the middle so everyone on the floor and in the balcony could see the preacher in the pulpit - and be seen by him. The pulpit was centered high on one wall, a window behind,the balcony on 3 sides.

 

He laid out a 3/4/5 rectangle. noted here in red. Then he laid out a square in the middle which defined the open space and divided that into thirds to set the columns for the balcony and the posts for the frame. See the black square and columns.


He extended the column spacing - the dashed black lines - to place the posts on the front and rear walls.This made the balcony the same depth all the around.
The porches are squares set in the middle of the west and east walls. The exterior posts were set at the porch corners, not at the 1/3 points of the wall. This also allowed for 2 windows on each side of the porches. 

 

 

 

Those 4 closely spaced posts in the center support the 4 attic trusses which are braced together to span the width of the church and allow the center of the meetinghouse to be an unobstructed space.

Walter Wallace, standing in the joined trusses under the ridge, gives a sense of how big the framing is.

 

 

The HABS prints of the End Elevation and the Interior Section for the Rockingham Meeting House are hard to read but their basic dimensions are clear.

The End Elevation is composed of 2 squares. The roof is framed using the 3/4/5 triangle. 

The notation to the right of the rafters says the pitch is a 9/12, modern language for the same thing.

 

 

The porches, the name for the stair towers, are set in the center of the end walls. The diagram shows that if the overall width of the wall is 8 modules (each square divided into 4 equal parts) the porch  is 2 modules wide.  

The proportions are 3-2-3, a graceful rhythm. If instead the massing had been 3-3-3 - all the widths equal - it would have felt dull. 

John Fuller, Master Builder, understood how to create with those simple shapes.  

 

 

The Interior Section shows the roof trusses, all using the 3/4/5 geometry. I've highlighted them in black for visibility

The meeting house height is divided in half horizontally. The columns which support the balcony divide the width of the meeting house in thirds.

If those columns has been the posts in the exterior walls only 1 window would have been possible on each side of the door. As I described above, the posts in the exterior walls were set differently (the black dot/dash line). 

 

 

 

 

With his frame laid out, John Fuller now needed to place the windows. The 6 posts on the front elevation were fixed. To allow any visual space* between the 2 windows on either side of the main door had to framed against the posts.  Here you can see how they were placed; there is no room for casings. 

* 'Visual space': the windows needed to be viewed as separate shapes, not as pairs.    

 

 

 

The red lines on the front elevation show the locations of the 6 bents for the meeting house.

 

One more window was needed on either side of the main entrance.

Where would they fit so that they were part of the whole, not call attention to themselves, and enhanced the main entrance ? 

Fuller used geometry to place the outer windows in relationship with the others. 

On the right side of the entrance is the front elevation as it was built.

On the left the outer windows are shown set in relationship not to the posts, but to their next closest windows and the left side of the elevation. The entrance is flanked, but not crowded by the windows.

 

 

The 'empty' wall to the left becomes part of the geometry. It shares the  proportions, being 1/4 of the wall. It is not 'left over'.

 

You can see the design succeeded. The uneven spacing between the windows is interesting and enlivens the facade, but it does not detract from the main door with its pediment. On either end the stretches of wall without a window anchor the meeting house to its site. 

 

 

Here is the main door. Its height is the determining dimension. Half the height is the radius for a circle and its square, drawn in red. The rotated square is drawn in black. The intersections determine the width of the architrave, the columns. The location of the plinth blocks and the depth of the moldings in the architrave, over the door are governed by the sides of the smaller square to the original circle.

The pediment follows Serlio's instructions:
 half of the width dropped below the base of the pediment - black lines - becomes the point for an arc whose radius -dashed red line - is the distance to the edge of the pediment. The dropped line is extended up to the arc; that marks the height of the pediment.

 

 

 

 

 

 

This way of laying out a pediment is shown in Asher Benjamin's 1797 pattern book:

 

 

The door itself came after the frame was in place. It was built to fit the opening. 

First the door's rails and stiles were laid out. 

Then the Rule of Thirds divided the remaining space in half and sized the panels and the stiles between the panels.

 

 

 

 

Last picture: The windows on the sides of the meeting house were framed against the posts as they also were on the front and rear elevations.

Just as at the Rocky Hill Meeting House in Amesbury, MA, the eaves on the porches bump into those windows. Neither master builder had solved that problem.

 

For excellent information about trusses in meeting houses and churches see Historic American Roof Trusses, Jan Lewandoski, et al., published by the Timber Framers Guild, 2006. www.tfg.org.

The Rockingham Meeting House is not included but the theory, practice, and evolution of the trusses used for similar meeting houses is laid out with clear photographs and Jack Sobon's drawings.

If you do not know how to use of the 'Rule of Thirds' square as a design tool, see: https://www.jgrarchitect.com/2020/08/lesson-6-rule-of-thirds-part-1_21.html