Wednesday, June 6, 2018

Presentation and Workshop at Hale Village and Farm, 2 PM, June 22.

Here's the link:

Here's the announcement:

Wondering what these diagrams have to do with historic structures? They are 'Practical Geometry'.
Consider that those buildings we love were often built before we had tape measures. This is what we used. Curious? Find your compass and show up at Hale Village and Farm outside Cleveland on June 22.

I'm giving a lecture with lots of illustrations and a hands-on workshop. No math ability required. I will have compasses to share

I will use the Streetsboro Baptist Church as an example. We can look at it inside and out as it is now in the Hale Village. I'd like to see the framing, of course. Maybe there's a hatch into the  attic.

I will also show the geometry of the brick Jonathan Hale House, begun in 1810.
The house plan and elevations are squares with a straight forward application of the Rule of Thirds determining window and door placement.

The window geometry is a little more subtle; it uses the intersection of the arcs of the sides of 2 squares for the sash size.
A framer - or in this case a mason - would build the house and then turn the finish work over to a joiner whose knowledge of practical geometry would be more sophisticated.
The pane sizes were determined by dividing the width into quarter and the height into thirds,

I will take my daisy wheel. It is incised into a 9' length of sheathing from a 1780 barn built for/by a Quaker farmer in Vermont.
It will help explain how geometry was a practical way to layout and measure parts.  

                If you come to the talk, please introduce yourself.  

Sunday, April 29, 2018

The Baptist Church of Streetsboro, Ohio, Part 2

This post about the Streetsboro Baptist Church, built c. 1820, is about 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.

Often in pre-Civil War construction, after the framers made the building 'tight to the weather', the joiners did the finish work: window sash, doors, molding.  Different trades had different skills and tools.

I think this division of labor happened here. The geometry for the front of the church is not the 3/4/5 Triangle but the square and its division by the Rule of Thirds and the Rolling Star

The diagram on the right describes how a square divided in half with its diagonals (noted in red) can be divided into 3 equal rectangles by adding lines from the center of one side to the corners on the opposite side (noted in green). The layered lines make a star that is continuous (follow one line and see!) This pattern is sometimes called 'the rolling star'. We do not know what name the carpenters gave it.

Here is a close-up of the church front on a cloudy day in October.

The HABS drawing is below.


The joiner saw his facade as 2 squares, or as one square in the middle and a half on either side - see the next diagram.

The windows had been placed by the framer. The pilasters on each side needed to be equidistant (see the red arrows). He had to include that in his layout. He needed to decide:

How deep would the frieze be? The architrave? the capitals?
What is the size of the door?
How does he integrate the pilasters' placement with the door?

He used the control points (where 2 lines in the Rolling Star intersect).
(A) shows the double pilaster width as part of the star.
(B) locates the upper corners of the door, its height and width.
The door is set back just enough to be framed by the shadow of the wall.
(C) locates the division between the architrave and the frieze. That band of molding circles the church. It and the molding which outlines the pediment make the building whole. Without them the front would seem just plastered on the front of a box, the roof unimportant. 
(D) gives the height of the pilasters' shafts and the beginning of the capitals.

The bases of the pilasters extend as far down as necessary to cover the sill and sit proud of the foundation. They are actually the water table working its way around the ends of the pilaster shafts.

Providing a base to support the steeple was the job of the framers. 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 above 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 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 moldings on those edges and the series of roofs as the tower extends create patterns of shadow and light.

The  height and width of the door comes from the geometry of the Rule of Thirds. But a 3/4/5 triangle is part of a 4/4 square; so the door - 3units by 4 units rectangle below and a half above - is also part of the dimensions of the church itself. See the green diagonals at the door.
 The section of the steeple which holds the bell is also a 3 x 4 rectangle. 
The windows, not marked here, are double 3x4 rectangles.

Look again at the first picture. The church's grace and presence come from a simple manipulation of proportion in the design and the use of light and shadow to emphasize its character.

* 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.

Tuesday, April 17, 2018

The Baptist Church of Streetsboro, Ohio

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 looks linear, simple, obvious. Is 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 they match the floor is square.

 The bents for the frame were naturally the same width as the floor. It seems 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:

The elevation of the front of the church was 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 . 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. 
 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 outside dimensions of the church plan seem to have been used to place the windows. This makes sense: the wall needed to be flush with the outside of the posts. Pockets for the studs would be cut in the sill and plate in relationship to the outside surface.


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 window 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. As I have no drawings of the side elevations, nor do I know the location of the bents. I do not know quite where the steeple sat: directly on 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: it was a square with its corners cut off.

The steeple grew from the proportions already in use for the church itself. The 2 rectangles made up of the 3/4/5 triangle were laid out.

Look at the left side rectangle first.
The lines of the Rule of Thirds*  - the diagonal and the line from the upper corner to the middle of far side cross under the steeple outer edge.
The lines cross just above a pilaster; the dashed red line with arrow labeled A locates the inside of the tower wall.

Now to the rectangle on the right side: On the right if the rectangle is divided into its 4 internal rectangles, the center of the small upper left rectangle determines the outer edge of the tower front corner. 
The cross sections of the tower were drawn on the original HABS drawings. They show the clipped corners. 

This change from delineating the inside of the frame to the outside clapboard and molding of the church is probably due to who was in charge: the framer for the construction, the joiner for the visual effect, the finish carpentry.  The joiner was responsible for making that central left over space fit!

The door and the steeple use the square for layout and design, not the 3/4/5 triangle. This diagram is to remind us that the triangle is just one way of dividing the square, that it is part of the same vocabulary.

The next post will show the front door and the steeple.   

Saturday, April 14, 2018

Of Course Geometry is Magic!

I am often told that daisy wheels used in Practical Geometry are magic.  Here is my response

Yes, geometry is magic.

The technical word is 'apotropaic': these shapes are protective symbols.

The basic shapes of geometry are perfect. They never change.
So is it any wonder that we think these perfect shapes that we humans can not just imagine but also draw, are paths to the supernatural? Of course we see them as holy, sacred, mystical.

A circle of any size always comes back upon itself. Its radius, diameter, and circumference are always in the same ratio to each other. If they aren't  - it's something else, NOT a circle.
 Pi (the ratio between the circle and its diameter) is real and easy to see. Its arithmetic equivalent is infinite; it has been computed to more than 1 million digits with no end yet. (Google the number, just for fun!)

A triangle with sides 3, 4 and 5 units in length always has a 90*, square corner - seen here on the lower right side.

The square is always made up of 4 sides of equal length with 90' right angles. If any of those definitions is not present, it's not a square.

These shapes are part of each other: Here I've drawn a circle, some of its triangles, the squares that come from it.  These are the simplest forms, combined they can become endlessly varied and complex.

Geometry is science. On the grand scale geometry is the double helix of our DNA, the rotation of the planets.
It is a basic in our natural world, the small scale: the bee's honey comb, the crystals in a geode, the reflection in a mirror, the ripples of a pebble in a pond.
The Golden Section seen in the sun flower and conch shell is the expansion of these basic forms.

Too many of us found only relentless logic in our high school geometry class. We didn't twirl compasses, make daisy wheels, stars, hexagons, pentagons, octagons... using just arcs, points, and lines.
We rarely delighted in learning the magic of this world of patterns, proportions, rhythms that we are part of, that does not need words or numbers. Ionic volutes, daVinci's man, the tile in the Alhambra didn't grace our walls.

Practical Geometry is the use of geometry for construction: the arch of a Roman aqueduct or the cantilever of a suspension bridge, the vaulted ceiling and the rose window of a cathedral, the timber frame of a barn, the placement and size of architectural elements. It was used to build the pyramids, noted in the Bible. It is ancient, now mostly forgotten due to the Industrial Revolution.

I study and blog about this practical geometry, part of our heritage which we no longer perceive, to help us recover it.
I hope we will learn again to see it and use it.

For more on this c. 1830 house see:

For the use of geometry in the past several thousand years see:

Monday, April 9, 2018

A Little Bit of the Geometry of MIT

 This is the curtain wall of the Mass Ave entrance at MIT,  the Massachusetts Institute of Technology, Cambridge, Mass.
The picture arrived in my mail box last week.

Immediately I saw the geometry. I knew what geometry the designer used and how it was manipulated.

I am currently writing three separate posts for this blog, one for my local blog. None has quite come together. Each has parts which require more drawing, thinking, and better choice of words.
Then the latest MIT mailing to alumni/ae showed up with this picture.
I laughed. I walked beneath that wall of glass and columns for 4 years. In that time I paid a lot of attention to how we used the space it sheltered, how the shape and size of that 'entrance' directed what we did. The curtain wall was not part of my thinking, although the light it allowed into the rotunda was.
The wall's pattern, rhythm, proportions - or even just the idea that it was geometry - was not part of my analysis, nor was it ever alluded to by others. 

Here is the pattern.
Upper left : A square and its diagonals.
Upper middle: The circle that comes from using the diagonal of the square as the circle's diameter.
Upper right: The square that fits around the circle.

How the pattern grows:
Top row: Overlap a circle of the same size, so that the perimeter of each circle touches the square inside the other.
Second row: This pattern can grow sideways as well as up and down.
4 squares on the right: Once established leave out the circle, continue the squares, add the diagonal, horizontal, and vertical lines.

The pattern could have started with the circle and the 2 squares fitted around it.
Using the circle as the unit  - the 'module' or 'diameter' in classical terms - is the traditional way to begin a design. (See Palladio through Asher Benjamin.)

From a photograph I cannot judge the diameter of the column.  Does it taper? have entasis? The pilaster in the corner on the right appears to be the same width as the unit I chose: the original square.
If instead the column is the unit, the module,  the circles of the curtain wall might be 3/4 or 2/3 of the module.

I have been told that the main buildings at MIT - dedicated in 1916, designed by William Wells Bosworth -  were designed using geometry. The drawings of those buildings would be well worth studying.

Practical Geometry has become an integral part of how I see buildings.  I was surprised to find that it has become a practical design tool for me as it was for those who used it for construction.

Meanwhile, this little bit of geometry was just plain fun.

Thursday, February 15, 2018

The geometry of a 1870's barn

 This Vermont barn was built in the 1870's . It has been used for storage for the last 20 years.

I prepared a report on its history and structure for its owners so they could consider repair and reconstruction with some real knowledge - not just good memories and/or worry about costs.

The barn was well built by a farmer who knew his land and a framer skilled at his trade.
The frame is regular, much of it still sturdy. Its mortises,  tenons, and pegs are still secure.

Its bents use dropped girts and posts to purlins which support  common rafters, a framing system regularly used in the Hudson Valley watershed, not often seen in this area of Vermont.

While I was not asked about the barn's geometry, as I laid out the plan and the frame I could see the geometry clearly - not complex, quite simple, repetitive, and straightforward.

Here is the 3rd bent and the lower level floor plan.
The bent is one of the 4 timber frames across the barn that are then fastened together with plates and girts. Walls and flooring have been left out.
The plan is mainly the post locations. I have not included the exterior wall girts.  The braces which are visible in the photograph to the right are barely noted.

The floor plan could easily have been laid out using circle geometry.

I have added Laurie Smith's diagram for drawing a square beginning with a circle. It is a very clear description.

For his websites see:


Here is my drawing of the floor plan with its posts laid out using circles. The first  (top) 2 bays are of equal depth and width.  The dashed green line shows the layout determined by the circles.

The lower bay (between bent 3 and 4) is not as deep. Perhaps the land dropped off too steeply, or the lumber available was not as long. The dotted red line in the lower right rectangle shows how the crossing of the arcs of the square determined the depth of the bay. 

The base of bent 3 is vague on purpose. I don't really know the depth of many of the lower level posts. The land slopes west to east. The floor on the east end has been built up over the years with layers of discarded boards.  The right end has been reconfigured for cows; the left end has a false ceiling.
The main  barn level of the bents is divided into thirds. The  posts are the height of a third of the bay's width - the space they outline is a square. I've drawn it in red. The dropped girts are set at the point where the arcs of the square cross. Also drawn in red.
This is similar to how the lower level east bay's depth was determined.
The posts that support the purlins ( the roof beams ) are centered on the squares below. The height of the ridge is also determined by where the arcs of the loft square cross.

Lastly the location of the lower girt which becomes the plate for the wing is determined by the Rule of Thirds.

Such basic practical geometry tools! They are  those described by Serlio, Palladio, and Asher Benjamin - circles, arcs, lines - applied in very simple ways with impressive results.

Well thought out, straightforward without fancy flourish, the space and the frame speak to me. But I am simply the one who documented this, sharing the power, the grace, that I found.

The barn, after 150 years, is no longer essential. It is very possible that it may not survive until a new purpose discovers it.


Wednesday, February 7, 2018

How Practical Geometry is practical

This is a sequel to my previous post:

Do I think the carpenter who laid out the small simple cabin at Tuckahoe actually drew the arcs on the  ground? or on the floor  - once he had squared the foundation and set the sills?

No, I think he knew the geometry. Someone had already taught him what I drew.
I think he swung the arcs but marked only the foot or so where he  knew the crossings would be. He knew that he wanted to locate the center of each wall, and  - by basic geometric rules - he needed 2 points to draw a line perpendicular or parallel to the wall in question.

Here is a lithograph of Pere Soubise, patron saint of the Campagnons, French carpenters who have finished their apprenticeships and begin traveling from town to town, from job to job to learn new skills. (In English an Apprentice becomes a Journeyman at this stage of his training because he 'journeys'. When he has gained enough experience he is then eligible to become a Master.)

Pere Soubin is probably mythical. But the date of his portrait is known: 1863. Click on the print to read the attribution. 
In 1863 a portrait of an important man included the tool of his trade: Pere Soubise holds a compass.

I have enlarged that part of the image. He holds his hand in  a way that he would if he were using the compass to measure a distance based on the drawing held in his other arm. Or as he would to  mark joist pocket locations on a beam, stepping off from one to the next.

Today a carpenter marks stud spacing with a tape measure that has multiples of 16" highlighted in red. The carpenter doesn't count 16" each time, he uses the tape's marking as a shorthand.
Similarly the framer in 1860 did not need to swing the arc from one point to the next, he used the compass to keep his spacing consistent.

As I was writing this a timber framer who did a lot of repair of old barns mentioned that he often found common rafters laid out at 39.5". 
I laughed and told him he had given me a challenge: Why 39.5"?

Here's the arithmetic: Many of the barns were about 40 ft long. 40 ft = 480" . 12 x 39.5 " =  474", 6" shorter than the barn's length. 3" each end for the end rafters.
However, that begins with the solution. It doesn't address how the framer found his answer.

Here's the  geometry.
The framer knows he will use  3" wide rafters on each end of his 40 ft long  barn, so he will have 474" in between for his rafters.
He wants to figure out what distance will work so he can tell the men working with him where to set the rafters and cut the pockets in the plate. The plate is sitting right there in his framing yard -  which might be the floor of the barn he is building.

He could make a scale drawing on a board and scale up to the plate using his compass, like this:

Or he could stretch his line the length of the plate between his end rafters. Then fold the line in half and and then half again. Now he had the length of the plate divided into 4 equal Parts. ( #1 , # 2)
 He thinks 12 rafters should do it. That means 3 rafters for each Part. But what's the spacing? On the framing floor he draws out a square using the Part as the side.  The handy Rule of Thirds quickly divides the square into 3 equal rectangles and the Part into 3 equal lengths. (#3)
4 parts x 3 divisions = 12 rafters. Good to go.
He doesn't care that the length of each is 39.5". He cares that he has divided his plate evenly. (#4)

Note that the framer does not add, subtract, multiply or divide. He could show this system to someone who spoke a different language. Neither would need know how to read words or compute. They would need to be able to think logically and reason visually.  Geometry is a language in itself.

By the 1860's  - the time of the Pere Soubise portrait - both France and England had standardized dimensions (meters in France, feet and yards in England). Tape measures existed  but were not widely used. Wooden folding rules were popular after the Civil War,  but carpenters still understood and used compasses for layout and design. 

I have met young timber framers who journey as Compagnons.
 For more information about the French Compagonnage historically and today:
And note the compass leaning against a beam in the first engraving.