Geometry in Construction = Practical Geometry

Geometry in construction = practical geometry.

Does that seem strange, a philosophical stretch?  As recently as the 1930's it was widely understood, commonplace.  Since the 1950's, geometry has been taught as precise, logical, beautiful, magical, amazing.  But practical? Barely. Today the idea is usually met with skepticism.

However, you who read my blog know this is what I study: what those builders know about geometry and how did they use it? 

 

Euclid's geometry starts with a Point which has no dimensions.  Two points make a Line - 1 dimension.  3 make a Plane - 2 dimensions.

 

4 points make an object  - 3 dimensions.  

 

How can this geometry be practical? 

A Line laid out between 2 points will always be straight. 

A Line drawn by hand might curve; a Line marked by snapping a length of twine cannot curve. This is the beginning: it will be true.  If the geometry is not accurate it will not be practical.

The Line A-B can become a radius. The radius can draw a circle. 

Whether the circle is drawn with a compass set to the length of the radius. or by hand with a length of twine, it will close if the the work is accurate. If the circle does not close upon itself it is not true.        At every step of the layout if the geometry doesn't hold, the designer will know to stop and correct the drawing.

 

The radius of the circle always divides the circumference of the circle into 6 parts. If the points on the circle, marked by swinging the arc of the radius, are not spaced accurately they will not end exactly where they began. They will not be true. The work cannot proceed. These 6 points on this daisy wheel are not quite accurate.  Note that the daisy petals' shapes are not identical; the points are not equidistant. If I measured the diameters, petal to petal, they would not match. I was not careful enough.        

 

 

 

 The 6 points, joined with lines, can be used in construction.

 

The rectangles that come from the 6 points can be proved by their diagonals. If they match, the rectangle will have 90* corners and be true. If the diagonals do not match the shape is not a  rectangle. 

A building needs to be stable, whatever materials it is made from, whatever form it takes. For simple vernacular housing the circle was the practical geometry needed to erect a stable, sturdy dwelling.  

The layout tools available to the builder of the Lesser Dabney House* in rural Virginia, c. 1740, were twine, some pegs, a straight edge, some chalk or soot so the twine could mark a line, perhaps a scribe, a compass.

The builder could have laid out this house with the first 4. A peg could have served as a scribe to mark a point. Twine with a loose knot around a peg turns as a compass does.

 

 

 

Here is the floor plan as it was recorded by Henry Glassie, c. 1973: 3 rooms with 2 chimneys and a stair to the attic.  3 windows, 4 doors. The door to the left may have gone into another shed.

 

 

The builder stood where he wanted the main wall of the house to be. He pegged the width he chose with twine A-B. That length became his radius. He drew his arcs to find the center of his circle C. Then he drew his circle.  And found it true. The circle's radius steps off 6 times around its circumference. The arc create the 'daisy wheel'.

 

A-B in the diagram above became 1-2,  the width of the house. The arcs 1-3 and 2-6 of that width crossed at the center of the circle with its 6 points: 1,2,3,4,5,6. 

The Lines 1-5 and 2-4 laid out the side walls; 6-3 locate the back wall. Diagonals across the rectangular floor plan proved the layout to be true.

The main block is about 20'x17'. The 2 doors  welcomed cooling through breezes in the summer. The wall room on the right may been a later addition to create a parlor, more private and warmer in winter.

Then the builder added the shed. He made his twine the length of the house, folds it in half and then in half again. He then knew what was 1/4 the length of the house (x). He laid out that length (x) 3 times to get the depth of his shed. He stretched his twine diagonally from one corner to the other. If the twine measured 5(x) his shed walls were a 3/4/5 rectangle; the corners 90*, and  true to the main house. The shed roof framed cleanly against the house and was weather tight.

The circle and the 3/4/5 triangle - Practical Geometry -  were the only measuring systems necessary to construct this house.

 

*The Lesser Dabney House, Fig. 45, Type 3, p, 105; the photograph: p.104. 

Henry Glassie, Folk Housing in Middle Virginia, U Tennessee Press, 1975; plans, drawings and photographs by Henry Glassie.

Henry Glassie recorded floor plans and what history he could find, He photographed. He did not make measured drawings like those in HABS  now in the Library of Congress and available on its website.

 

 

Virginia Folk Housing, Part 2 of an update

 

The Moore House* photographed by Henry Glassie, built before 1750.

 

This house has 2 rooms up and down, 2 fire places, 2 chimneys, and a shed on each end. The main block  is double the size of the house I wrote about in Part1: https://www.jgrarchitect.com/2022/11/virginia-folk-housing-update.html

The geometry begins as it did in Part 1, using the width as the circle's  radius.   

 

Here is the floor plan: 2 rooms with fireplaces, and sheds on both ends.

The daisy wheel progression begins with a length A-B which becomes the radius of a circle here lettered C-A.

The daisy wheel for this house begins with the left wall of the main house.

That wall's width  is the radius, 1-6. A is the center of the circle. The daisy wheel lays out the other 4 points, 2, 3, 4, 5.  

 Lines 1-3 and  6-4 are the sides of the house. 2-5, the diameter of the circle, lays out the interior wall.

 

Lines 1-5 and 2-4 can extend forever. Where is the right end wall of the house located? Where is C? 

It's at the end of the circle, but that's only a point, not a line. 2 points are necessary to draw a line to mark the right end of his foundation and the floor of the house.

If the carpenter extends his arcs he can quickly find the missing points. 

Extend the arc centered at 3 (2-A-4) to B.  The arc centered at 4 (5-A-3) crosses the earlier arc at B.   He has 2 points: A and B, And can draw line A-B. 

Now C is fixed at the intersection of A-B. C is the center of a new arc, (7-A-8). The extended arc from 5 (6-A) crosses at 7. The arc 2 (1-A) crosses at 8. 7-C-8 locates the right wall.

C also locates the center of the fireplace and the chimney. 

The daisy wheel is often dismissed as a design tool. It is flexible, quickly drawn, and accurate. 

The geometry comes from the first length - the width chosen by the owner and builder for this house. That width, and the house, could be bigger or smaller to suit the owner's needs and budget, as well as to the lumber available for joists and rafters. 

Once the carpenter decides on a width he uses one compass setting, one radius, for the whole layout. Every point is checked. As the lines are marked, the diagonals can prove the layout to be  true.

If he drew a layout at a smaller scale, he could easily step off to full-sized construction dimensions with his compass. He could also draw the layout on the ground, stake the points and mark the wall locations with twine just as framers and masons do today. 

Consider how the plan would be laid out if the circle is not used. Use a 10' pole - a common tool of the time.  Each corner would need to be figured independently;  every dimension stepped off separately, and with what accuracy? 

The daisy wheel locates all angles and lengths quickly. It has built-in checks from the beginning and as the layout progresses: if the circle doesn't close, the 6 points will be uneven, the arcs won't cross, the diagonals will not match. The layout will not be accurate.

 Both wings are 3/4/5 rectangles. See the left shed. The floor plans of wings were usually 3/4/5 rectangles so that they would sit square to the existing house. All the joists would then be the same length; as would be the rafters.  

 

 

My earlier complex geometry 'works'; the lines are there. But they don't give the basic information the builder needs: the dimensions of the foundation, the floor plan, the size of the house.

 

*The Moore House, Fig. 31, Type 5, p, 77; the photograph: p.76. 

Henry Glassie, Folk Housing in Middle Virginia, U Tennessee Press, 1975; plans, drawings and photographs by Henry Glassie.

Virginia Folk Housing, Part 1, an update

The house recorded by Henry Glassie in Folk Housing in Middle Virginia * were basic shelter for people with few resources. They may have been the first house for someone homesteading, built by a sharecropper or by someone enslaved.   

This is Fig. 35, The Parrish House, a "small mid-eighteenth-century house of sawed logs", p. 84 in Glassie's book.*

 

The geometric diagrams I drew in May 2014,** were accurate but much too complex for these houses. More importantly they didn't begin as a carpenter would: with the size of the foundation and the floor plan.

 

 A carpenter's first question is, " Why?" Then he asks, "How big? How long? How wide?" 

The red line across the bottom of the floor plan is 'how long', about 21 ft. That distance can be the beginning of the layout, the first Line that determines all the others.

 

That Line can be the radius for a circle:

The arcs of the Line A-B cross at C. That's the center of the circle for the layout of this house.

In the diagrams below: 1) B-C is the radius of the circle. 2) Beginning with B on the circumference  the arcs of the daisy wheel are added. The 6 even spaced points around the circle A, B, D, E, F, G  are located.

 

 

 

 

Connect the Lines. A-F and B-E are perpendicular to A-B. G-D is the diameter. They mark the width and length of the rectangle for the house plan.  If there is a question about accuracy, diagonals can be used to true the shape.

 

 

Here is the plan within its circle, the circle that begins with the carpenter's choice of width, his 'module'.

 

 

The masonry block for the 2 chimneys is square, centered, and 1/3 of the width the house. Glassie's photograph shows a shed sheltering that fireplace.

 

Part 2: https://www.jgrarchitect.com/2022/11/virginia-folk-housing-update-part-2.html

Another introduction to the geometry: https://www.jgrarchitect.com/2023/01/geometry-in-construction-practical.html

 

*Henry Glassie, Folk Housing in Middle Virginia, U Tennessee Press, 1975. The book includes more information, drawings, and a photograph of the house. It no longer exists.

** The original post is here:  https://www.jgrarchitect.com/2014/04/18th-c-virginian-folk-houses.html. Its companion, here: https://www.jgrarchitect.com/2014/05/18th-c-virginian-folk-houses-part-2.html

I considered deleting the 2 posts, but their existence brought a comment and question which prompted this update.

Also:

As I read them I realize how much I have learned about geometry since 2014. I saw it and tried to explain it, just as Henry Glassie did in his Rules, Chapter IV, The Architectural Competence.

When I began to study Practical Geometry there were no books, no one for discussions or critiques. I was teaching myself, reading early pattern books line by line. Laurie Smith was the only person I knew who saw geometry as I did, and he was in the UK. Later that year he came to the States; I took a workshop with him. I was able to work with him until his death last year.  

I don't want this information to be lost again. I want others to find it, question it, reject and/or improve upon my analysis, their own analysis, expand our understanding.