Wednesday, September 19, 2018

Corn Crib as Geometry

A corn crib.
Little. 17 ft by 21 ft , about 18 ft to the peak.
New roof, south facade, and paint: c. 2015. Painting on going.
New concrete blocks for piers; old  concrete piers formed in wicker baskets.
Moved, maybe by oxen.
Door relocated for easier access on foot. The main door was on this elevation, about 3 ft above grade, ideal for access from a wagon.

This south wall was rotting away.
The corn crib deserved better.
During the repairs we found that it is timber framed, scribed, the main framing beams hewn, others cut with a sash saw.
It has log rafters. and may predate the c. 1810 the house.

The scribe marks are similar to others found in the neighborhood - a common framer? Or a common teacher of corn crib builders?

Click on the photographs to enlarge them and read 'II', 'IIII', and 'III' on the posts and the beams. 'III' shows how water in that joint wore it away. 

I measured it - the width, length and height; the size of the posts and beams, and their locations; the roof pitch, the slant of the walls. I drew it up and checked my notes for clarity. 

The plan and elevation are below. The sides were for corn, the center aisle for work, the back (lower) section for grain, tools, equipment.

The walls slope out to shed the rain and snow, and to keep the corn from locking in place. The beam running below the beam at the eaves supports the floor of the small loft above the back storage section. There is a ladder built into the wall for access.

This is a normal corn crib for this section of the Hudson River water shed where New York, Massachusetts, and Vermont meet.
It does not seem to be the usual way corn cribs are built in other parts of the country.  I plan to look more carefully.

How I 'found' the geometry:
Note: the geometry is right there. It does not need to be found. I need to recognize it!
Looking at the  floor plan I could see the posts in the upper section forming a square. I added the diagonal and then using the diagonal as a radius I swung an arc. It landed on the lower left corner of the crib.
This is a simple easy plan. The elevation should be as simple.
I tried a square based of the width of the floor (about 17 ft. wide). It didn't quite fit.
I tried a square based on the width of the walls as they meet the roof  (about 18 ft. wide). Again, not good enough.

The utilitarian corn crib's frame needed to be as straightforward as its plan. The geometry tell the carpenter where to put his frame - exactly. Not within  4-5 inches.  He will have timbers laid out on a framing floor. He will want lines (chalk lines or taunt twine) that tell him position, lengths, mortises.  He needs no frills here.

This is when I have a cup of coffee, clean up the office, walk to the mailbox. And come back to try something else, experiment, play. "What if I drew the 'square' using the angled walls? Why not?"

It works. The Lines cross above and beside the framing members. The trapezoid ends below the ridge telling the framer where to cut his joints on the rafters so they will lap each other. The side braces (dashed line) are easily located.
The Lines on the framing floor will be beside the posts and beams, just as they are outside the edge of the plan.
The Lines needed to erect the crib are as minimal as those needed to layout the floor.

Tuesday, September 18, 2018

Archimedes' Stomachion

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 least as far back as the 6th century BCE when it is mentioned in the Bible.

Archimedes in Syracuse, a geometer and engineer, would have known Practical Geometry as it was applied to the construction around him in the 2nd and 3rd Centuries BCE.

Here is a basic Practical Geometric  diagram, the Rule of Thirds,  drawn in 6 steps.

The points where two lines crossed were used to determine both design and structure: the size of a building and its frame, its ornamentation.

This drawing is messy... I will replace it.

I have drawn the Stomachion as a square. It could also be a rectangle or a trapezoid - but for those shapes there would be many less solutions.

In the second square I have extended the Lines in red. The dashed lines show how the small lines on the lower left corner and the middle right side were laid out.
The Stomachion uses both the division of the square into thirds and the division of the square into quarters.
The shapes are clearly based on the Rule of Thirds.

This drawing by Sebastiano Serlio, is from his book Architectura, published in France in 1537. He is discussing how to place a door in an existing facade. These Lines match the ones in the Stomachion.

Perhaps someone else has seen how the Stomachion relates to Lines, the Rule of Thirds, and Practical 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. 

Monday, September 17, 2018

The Vail House, c. 1805, Bennington, Vermont


The Vail House was deconstructed this past summer for repair and reconstruction in another town.


It was once one of the most stylish houses in Bennington, its architraves and columns more complex than most local houses, its fanlight and surround unique to this part of Vermont. 
Similar details exist on a few houses across the border in New York.

The Victorian updating can be seen here - the double windows on the first floor, right, and the porch with curly brackets   Well executed at the time and then let go.
  I measured and photographed it about 4 years ago. I wish I had documented it more carefully. I have no image of the front of the house!

On September 16, I will include its  geometry as part of my presentation  'Practical Geometry' for the Bennington Historical Society lecture series at the Bennington Museum.

The family wanted a broad front hall with space for a sweeping staircase. This was the new style. The framer's answer was to  add 1/3 of the width to each side. The red square in the center shows how this worked. It was divided into 3 equal parts using the Rule of Thirds.
The house was to be 3 parts  deep and 5 parts wide. 

As you can see the division into 3 is not quite where the posts and beams are.
While the size was set by an addition of proportional lengths, the rooms were set by a different application of the Rule of Thirds . I call it 'Crosses Squares' .

 Each side is a square, the Rule of Thirds applied to each side makes the front rooms square, the back rooms long and skinny, The posts and beams are set where the walls will be. 
Usually the front hall will be the width of the extra third. Here you can see that it is wider.  Or perhaps the house is wider... slide those 
squares on each side towards each other about a 12" and the  crosses squares would mesh.

The floor plan is traditional for this part of Vermont: 2 square front rooms, a long skinny space in the back divided into smaller rooms, the plan of a salt box. I wrote about this in an earlier post:

 This is the west elevation. The shutters are a later addition.

Here is the Practical Geometry: a square in the middle, with the left and right sides 1/4 of the whole. The Lines locate the windows' size and placement. The sash themselves are squares, which is in keeping with the layout. The  decorative architrave's height is determined by the half of the square.

As I did not measure the exterior extensively I have not tried to layout the geometry of the corner boards or the frieze.
The photographs show that I have not accurately located the quarter circle vents in the eaves.  They are farther apart than I drew them, The proper location is probably on the 1/4 line of the square.
I think the roof pitch matches the Lines which divide the square into quarters - or the dash dot line I use to call out the left quarter of the house. This would be a logical choice:  a natural choice, using proportions the framer already is working with and also complementing the design of the house.  

Sunday, September 2, 2018

Practical Geometry - what our ancestors called this geometry

Practical Geometry.
It's what our ancestors called these diagrams I draw.

 Here is Peter Nicholson who wrote about Practical Geometry. His writings make clear that geometry was once an expected and necessary part of construction, used both by the designer and the artisan.

His first book, The Carpenter's New Guide, published in London in 1792.

He begins with a Preface, some of which I quoted in an earlier post:

Page 2 is copied here.

The use of geometry in construction was so accepted that Peter Nicholson waits until his third paragraph before he shares that geometry is useful in mathematics and science too.

By the time of his death in 1844, Nicholson had published 27 books in London, New York City, and Philadelphia.  More than 10 years later his books were still in print.

This portrait is in his updated book The New and Improved Practical Builder, published in 1837.

This time he writes a whole paragraph explaining Practical Geometry. Has he been asked to be more thorough? Have the new uses of geometry in science changed the perception of what geometry is? Have men become carpenters by necessity - especially in the New World - rather than by apprenticeship, and thus desire to educate themselves?

Here is the Introductory Chapter

The second paragraph describes the 2 branches of Geometry: Theoretical and Practical.
Now  the Theory of  Geometry is carefully described, including a reference to Euclid, but it is still one of the 2 branches of Geometry.
The other branch, Practical Geometry,
 allows "the architect to regulate his designs and the artisan to construct his lines". 

Later, on page vii, he writes, "There is no mechanical profession that does not derive considerable advantage from it."

first portrait: by James Green, 1816, now in the National Portrait Gallery, London
second portrait: the frontispiece of The New and Improved Practical Builder. Don't miss his compass.

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.