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. "

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