Practical Geometry - drawing the diagrams, #1

The participants at the hands-on sessions I taught on Practical Geometry  at the 2016 PTN Workshops, asked me to post the diagrams for the basic geometries they worked with.
Here is the first.

How to divide a square into thirds:

We used graph paper for the first geometry so everyone could see the lines develope into a pattern. Everyone could count the squares to be sure they were following directions.

1.  Draw a square 12 units wide and 12 units long. label the corners A, B, C, D.

Add the diagonals - the lines from one corner cross the center to the far corner. A to C; B to D.
The lines will cross in the center of the square. Count the units to prove this to yourself.
Label the center of the square E.

Divide the square in half vertically, F to H - follow the line the graph paper.
Divide the square in half horizontally, G to I - follow the line on the graph paper.

This is the basic pattern. The square can now be divided into 3, 4, or 5 (or more)  equal rectangles as needed.


2. To divide the square into thirds:
Add a line from each corner to the middle of the opposite side. A to G and B to I.
These lines cross the original pattern.

K and L, if extended parallel to A-B, would define a rectangle that is 1/3 of the whole square.




A rule in geometry is that there must be 2 points to establish a line.
 Below is a diagram of  how the diagonals from the corner of a square to the middle of the opposite side give 2 points for the lines which divide a square into three rectangles of equal size.



This division of the square into thirds is often found in pre-Industrial Revolution design.
I do not think framers drew out the whole diagram on a sheathing board or a framing floor. Rather because the diagram was common knowledge they just drew the parts they needed.


An example:

At the workshop I taught the application of this geometric pattern using the plans and elevations from a cabin at Tuckahoe  -   http://www.jgrarchitect.com/2014/06/cabin-tuckahoe-plantation-goochland.html.

 

 

The end wall of the cabin is  2/3 of a square. The roof  begins on the 2/3 line. Its pitch follows the diagonals of the upper square. the windows, doors and fireplace are centered on the square. That's all.

I then showed the group how Owen Biddle used the same geometry to tell a mason where windows and doors were to be placed.
The  elevation and floor plan are both composed of 2 squares. On both the window placement is one side of the center line. The  diagonals from corner to center call out the window width ( on the elevation and the interior partition on the floor plan.
In the floor plan I have used a dashed line to note the lines dictating the window width.  
http://www.jgrarchitect.com/2015/11/owen-biddles-plan-and-elevation-for.html
 

 

 

I did not include diagrams showing how master joiners laid out squares bounded by circles, bounded by squares to set out the dimensions and relationships  between parts of doors and architraves for Georgian meeting houses.
Shown here is how the surround of the main door for Rockingham Meeting House, Rockingham, VT, may have been laid out.

https://www.jgrarchitect.com/2014/04/rockingham-meetiinghouse-rockingham-vt.html




The carpenters and masons called these geometries 'lines' They would have have been explained verbally as a master taught an apprentice.  Sebastiano Serlio and James Gibbs both refer to 'lines' - see my post  http://www.jgrarchitect.com/2017/04/serlio-writes-about-practical-geometry.html

A barn built in the 1830's

Green Mountain Timber Frames http://www.greenmountaintimberframes.com/ measured this barn before they dismantled it to use its frame anew.

Due to the wood used - poplar, beech, hemlock - the layout and the construction we think this barn was built by a farmer without an extensive background in framing. We think it dates to the  1830's.

 The floor was dirt, the head room under the hay loft not quite 6 ft.
What was it used for? Sheep perhaps? Sheds, windows and a silo were added over the last 180 years, making the original purpose hard to read.


I start with the farmer.
He had some wood of a certain size and length he could use for posts and beams for a barn. He knew how the barn would be used and where it would go.

Probably he had a carpenter square - they were readily available. But maybe not, as his dimensions don't quite fit. And he was much more comfortable with the old-fashioned geometry of the 'whirling square'.

He started with the width - 18 feet. He made a square: 18 ft wide, 18 ft. high - first diagram.\
Or so it seems. Today that height is 17'-10", 2". I think originally the width was also 17'-10". His inch seems to have been just a bit smaller than today's inch

He could have started with a string about 18 ft. long. He could have used a compass with a 27" radius, stepped it out twice for 4'-6", twice more for 9' and doubled that for 18'; or a pole 4'-6" long.

In his square he laid out his center lines and then the star that joins the points - the second and third diagrams. This is a medieval framing system which came to New England with the English colonists.

I have added circles to mark how the lines of the star cross at the locations of the girts, I've added a green dashed line to show how the height of the wall is 2/3 the height of the end wall. Almost. It's off by 2"

Using a carpenter square to layout a 3/4/5 triangle does not work as well. The wall height isn't high enough. The lower girt can be determined - see the green circles - but not the upper one.

While the frame appears governed by the traditional English framing geometry, the frame itself has dropped girts - a Dutch traditional way of framing. The girts are mortised into the posts below the upper beam. This combination of framing methods is sometimes referred to as 'American'.


The floor plan is simple: three 3/4/5 triangles. If the width is 18'  the length should be 40'-6" . It was 40'-2" measured on site. The men repairing the frame tell me it is 40'-1"; that the 2 interior bents are at 13'-4 1/2" from each end.

If one arm of the 3/4/5 triangle is 17'-10", the other is 13'-4 1/2".
3/4 of one side of a 17'-10" square = 13'-4 1/2". So either framing system fits the floor plan.

Dan McKeen, GM Timber Frames, also tell me 3 girts are beech, one poplar.The top plates are poplar and in good shape. The posts are sawn hemlock and hewn beech. The ties are sawn hemlock.

I looked at how did this farmer/framer laid out his girts in the side walls.
Here I tried - on the right in red - 3/4/5 triangles. The intersections - red circles - using a triangle that includes the rafter tails, are close, but convoluted. Not simple.
However, a square laid out inside the frame - on the left in green - neatly divides the space in thirds - green circles.

The star of the square used for the gable end laid out along the side also notes the placement of all the  girts.








The numbers on the early carpenter squares were engraved by hand. It is possible that this farmer/framer owned a square that was very slightly off. Or he used his own measure.

The man who built this frame comes alive as I study it; I've met him. Now I want to ask how he learned to frame - who taught him? what tools did he like? where did he start? were we right about his choice of materials?

Timber frame from a carriage house

Green Mountain Timber Frames of Middletown Springs, Vermont, http://www.greenmountaintimberframes.com/
erected this 17 ft. by 26 ft. frame in their workshop.
Originally a carriage house, it had last been used as a garage and was for sale.

I asked Dan McKeen if I could visit and watch his crew work. I learn a lot from watching.
Dan welcome me, give me a tour, let me watch, and was happy to have me  measure this frame.
I measured both sides but only made notes on the rest.The ridge is 5 sided, the rafters are 36"o.c. The pitch? I don't know.The front and rear sides were not assembled. There was no sill.


I noted that the length and height of each side was almost equal, about 2" off; that the braces were set at 45* and that the cross beams though different sizes were set at the same height. I wondered if it had been laid out from the foundation or from the sill, as I have seen both.

The date of construction was c.1840. Based on the date I tried a 3/4/5 triangle layout. Even when I allowed for the sill it was not successful. Nothing lined up nor gave the framer any information.

The frame itself said, 'Square'. It also said 'Simple', not a tour d'force. I tried squares.



The square which begins on the outside of the bent and ends on the center post works. It and its star divide the square into thirds - red dashed lines -, locate the braces  and the top of the cross beam - red circles.

After the square was laid out snapped lines on a framing floor would have laid out the star.  6 of the lines of the star - noted here in solid red lines - mark the important intersections. They divide the upper half of the square into thirds and cross at the top of the cross beam.

The frame sold before I could return to check the pitch, the front and the back.  I had measured the mortises in the end posts which were for the front and rear walls. They are drawn on the right post.



The same square determines their locations. Here I have extended a green dashed line from the intersections of the star (green circles) to the placement of the mortises.

The 5 sided ridge and the use of the square lead me to think the carriage shed was built before 1820 or that the framer was using old-fashioned methods and did not own a 'new fangled' carpenter square.


 My earlier posts show how to layout a square - http://www.jgrarchitect.com/2014/09/how-to-construct-square.html -
and how to divide the square into thirds and the ensuing star -  http://www.jgrarchitect.com/2014/04/sandown-meeting-house-revisited.html.