Saturday, October 29, 2016

Practical Geometry, Drawing the Diagrams #2, the 3/4/5 Triangle

Here's the second diagram I taught at the 2016 PTN Workshops.

I did not lay it out as I have done here. Today I think this diagram would have been a good handout.I could have drawn it; the participants could have followed along and had a cheat sheet to take home.

4/18/2017: this diagram is awkward. I will redraw and simplify.

Using the  3/4/5 triangle for construction

 3/4 5 triangles always have a 90* angle where the side with 3 units meets the side with 4 units.

Draw a line and mark off your unit.

Lay out lines of 3 units, 4 units and 5 units.
On my diagram:  A-B = 3 units
                            A-C = 4 units
                            B-D = 5 units

Swing an arc from either end of A-B; one arc with a 4 unit radius, one arc with a 5 unit radius,
Where the arcs cross is E.

Draw lines from A to B  to E to A.
This is a 3/4/5 triangle. The corner at A is 90*

For fun I have laid out another triangle beginning with 5 units, use 3 and 4 units for the radii of the arcs. It is another 3/4/5 triangle with a 90* corner.

We used Gunston Hall, built of brick by George Mason from 1755 to 1759, as an example. Mason  was a real mason; he gave George Washington advice about mortar recipes. He would have used the 3/4/5 triangle when he built walls or, as a Master Mason, instructed others. The triangle was/is one way to keep brick square and true.
It would have been ordinary for him to use 3/4/5 geometry to design his house.

The base of the brick work at Gunston Hall is 4 units. The height of the brick work of the end wall at Gunston Hall measures 3 units. The diagonal is 5 units.

The floor plan is also laid put using the 3/4/5 triangle. See my post for more information and drawings:

I asked the participants at the PTN session to divide the width of the Gunston Hall side elevation into 4 equal parts. I wanted them to draw the geometry for themselves, to see it come to life.

Again a handout with step by step instructions would have been helpful.   
Not everyone knew how to divide a line into parts; but those who did showed those who didn't. It was a excellent group.

One of the first figures in the pattern books on Practical Geometry is the division of a line by a perpendicular. Here is  Figure 5, Plate II, of  Asher Benjamin's The American Builder's Companion, first edition published 1806.

 Asher Benjamin's Figure 3, Plate II,  shows a
simply drawn 3/4/5 triangle expressed with units 6/8/10
with short arc lines at c, the top, to show the use of a compass to make a circle with the radius determined.

His description assumes a familiarity with the language of geometry and compasses.

"To make a perpendicular with a 10 foot rod. Let b a be 6 feet; take eight feet in your compasses; from b make the arch c, with the distance ten feet from a; make the intersection at c, and draw the perpendicular, c b. "

Thursday, October 6, 2016

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  -



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.



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.

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