Monday, November 30, 2015

Owen Biddle's 'Plan and Elevation for a Small House'

Owen Biddle published his Young Carpenter's Assistant in 1805 in Philadelphia, PA. He included 2 house designs as teaching tools. Biddle wrote that his drawings were not meant to be "eligible for the builder". Instead they were "aiming at instruction for the student".

In 1797, Asher Benjamin published The Country Builder's Assistant in Boston, MA. It also included house designs. Benjamin used crossed squares and the rule of threes in his designs,*

What geometry did Biddle use?
Here is his "small house" - Plate 36.

The layout is 2 squares side by side for the plan and the elevation. The elevation measures the square from the  first floor to the top of the brick coursing. He writes that the student "suppose the building to be raised just above the principle floor, and the wall made level all around."

I enjoyed this because a contractor today still wants his foundation and first floor level before he starts his partitions, of course!

Note on the plan  that the depth of the front porch is determined by extending the diagonals.

Usually I draw both diagonals to denote a square. Here since I will be adding other lines I decided to simplify for legibility.

Biddle designs using the square divided into quarters. On the elevation the top of the first floor windows is determined by the horizontal center line. The placement of the windows is determined by the vertical center line.

The floor plan follows the same geometry. The edge of the window frame is determined by the vertical center line.

Still using the squares, Biddle divides one half in half again, vertically.  This locates the columns - and the size of the front porch - at the 3/4 mark of the squares on each side. Note that the horizontal center line dictates not the top of the fan light casing but the height of the brickwork. These are structural dimensions, information for the brick layer, the sash maker.
The curve of the entrance stair follows the diagonal.
The width of the window is determined by the intersection of the diagonals.

On the floor plan the right hand  side of the square is divided in half; it determines the location of the wall between the rooms and the hall.

The porch is 1/4 of the square.

Here I have drawn the lines that layout the portico in red and given the pattern of the division across the bottom: 2-1-1-1-1-2.
The architrave and the roof are 1/3 of the square of the porch.

When I first saw this I thought the portico and the doorway used the rule of thirds. I was wrong. The door itself may be 1/3 of the porch, but the casing around the door is placed at  1/4 the width of the porch.  The height of the fan light is 1/4 of the square; the door below 3/4.

Note the shadow  to the right of the porch. Biddle wanted his book to teach drawing and presentation. He tells the student (reader) to "enliven the drawing by giving the appearance of shadow". (Plate 36).

This house was an exercise, just as Asher Benjamin's were. The geometry he uses comes naturally and easily, Although he knows the rule of thirds he barely uses it. He does not use crossed squares. Benjamin uses both.

This is not the geometry seen in New England nor in the folk houses documented by Glassie. It might be the geometry of Philadelphia.

 *To compare see my post: 

Owen Biddle, Biddle's Young Carpenter's Assistant; or A System of Architecture Adapted to the Style of Buildings in the United States, Benjamin Johnson, Philadelphia, and Ronalds & London, New York; 1805
Dover Publications, Inc. edition, 2006, unabridged republication

Friday, November 13, 2015

Tessellations and Geometry, teaching kids

Perhaps I should let the photographs speak for themselves.

The picture is from our first class.

The teacher, Jude, drew the circle and swung the arcs you see on the blackboard when  he realized we needed to watch the circle and its daisy wheel come into being. He knew how to help each student as they learned to twirl a compass - not easy for some children. Jude was essential to our success. .

When we looked for the triangles in the circle they were with me, as you can see.

I introduced the wall tiles and carvings from the Alhambra at the end of class. Everyone knew immediately why I had brought the pictures; they was fascinated.
I showed them how carefully the craftsmen had made every joint as perfect as possible. They understood.
Jude taped them to the blackboard. He invited me to come to school earlier for the second day so that we could do more

The next day we expanded the circle . That was easy. Extending the segments of the hexagon, making a new bigger star was an obvious step.

 So we tackled the 'rolling circles' of the church windows.   Much harder... not making the circles but finding the muntin pattern within the jumble.

The older children were the most successful.

Some children explored on their own and showed me their work.

I brought in 2 star dodecahedrons* for them to hold -  a great hit.

I had fun, learned a lot. The kids practiced and understood.
And they were patient: The compasses I brought were accurate but delicate and hard to adjust. The students shared and adapted. I will find them better tools.  

 * That may not be the proper name  for a dodecahedron studded with pyramids - I haven't found a good reference. Please advise if you know.

Saturday, November 7, 2015

Geometry of the Cobb-Hepburn House, Part 3, front elevation

Here's the  Cobb-Hepburn c. 1780, in Tinmouth, Vermont, as it was  being dismantled last winter.

This front elevation feels bare and stark, less  sophisticated than similar houses built at the same time  in New England.
The 4 closely paired windows on either side of the facade, and the wide expanse of wall between the windows and the door seems 'not quite right'. As I travel, though, I see the same spacing on other houses near by. Am I seeing a local variation? the same framer working on many houses?

The frame is well built even though it was completed in stages.

The geometry, however, is rudimentary.  The plan for the posts and beams begins with squares, crossed to create a rectangle. The distance they are crossed is based on the arcs used to lay out the square - one of the first manipulations of practical geometry that an apprentice would have mastered.

The first plan shows the posts and beams with the crossed squares in red. The second plan shows how the width the squares are crossed was determined by the crossed arcs - dashed red lines - of the squares. 

(The center beam is off set to allow the chimney to pass and exit the roof at its peak.)
Was the framer never taught the geometry? He was capable of quality timber framing; he must have served a apprenticeship. Was his training interrupted by the American Revolution?
What he uses here are only the very elementary forms of practical geometry.

Here is the first page of  Biddle's Young Carpenter's Assistant. published in 1804. Biddle wrote his book for carpenters like the man who framed this house.
To see the bibliographic information about Owen Biddle's book please see the links at the end of this post.

After explaining how to make a drafting board, fix paper upon it and make a T square  - A,B, and C,  - Owen Biddle lays out solutions "to some of the most useful geometrical problems, which every Carpenter ought to be acquainted with."   
E: how to raise a perpendicular,  F: how to let fall a perpendicular, G:how to add a perpendicular at the end of a line.
And then H:  which I have marked with a red square 
how to layout out a square.
 I shows how to draw a 3/4/5 triangle which will always have a right angle. J  shows how to divide a circle into 12 equal parts.

Very simple work with a compass - and the geometry used in the design  of the Cobb- Hepburn House.

I have labeled the floors, the rooms, and the windows and door on the frame for easier understanding.  

The floor plan used the intersection of the arcs  of the square based on the width of the house for the placement of the interior beams ( BII and BIII).
To read the 2 previous posts which discuss this please see the links at the bottom of this post.

The front elevation uses the same geometry - the intersection of the arcs derived from the height of the house is both the edge of the posts for BII and BIII and the top of the 2nd floor plate. See the black dots where the arcs cross.

The framer next needed to place the windows and the front door. He 'crossed' the rectangles (BI to BII on the right, BIII to BIV on the left) on either end of the front wall. They cross in the center of the shape, which is also the 2nd floor plate. Upstairs and downstairs windows are symmetrical to that  crossing.

 I have outlined the right side with a red dashed line and added the diagonals.

Then it was easy for the framer to 'cross' the lower half of the rectangle. I drew it with black dashed lines. Where the red diagonals and the black diagonals cross is the center of the window frames.

The location of the door is similarly found by dividing the left over center space in half.    

Friday, November 6, 2015

Tessellations and Geometry

How to teach geometry.

NOT the high school class of logic and proofs that comes after algebra.
Instead the real understanding and use of form, proportion, rhythm. Or: how a circle, a square, tells itself how to divide or grow.
Sounds mythical, magical, doesn't it? No wonder people call it 'sacred geometry'.

But it's just points, lines, circles, rectangles, triangles; shapes that are part of nature, part of the earth, something our ancestors understood and used for thousands of years,

Where do I begin?

Maybe with an elementary school math class.

My grandchildren (7 and 9) are working with tiles and making patterns - tessellations - at their ungraded school.

 Here is yesterday's tessellation pattern on their dishwasher. Another pattern, mostly in green, is on the refrigerator. I've just ordered 200 more tiles because sharing is difficult when you're working out a pattern!

I asked their teacher if I could show the class how to use a compass and how to divide a circle into its 6 parts, how the pattern expands.
I sketched some diagrams on a paper scrap using a plastic cup as my compass. He liked the circles. He liked introducing the students to the words 'radius' and 'circumference', to Islamic art through the tiles of the Alhambra.
I am invited to share circle geometry in math class next week, two 30 minute sessions.

When I told my grandchildren that I would be coming, the younger asked me what that word 'geometry' meant. The older told me that shape I called a diamond was really a rhomboid.

Clearly this will be interesting.
 Whatever lesson plan I take into class may have to be discarded. So my goals are simple: practice drawing a circle with a compass; see what happens.

I'll start with: why a point and a lead, how to set a radius, how to turn the compass by its knob.

When we master that we can make some circles and then divide the circumference in parts. I want them to see that no matter where they start dividing the circle its outside edge will be 6 equal pieces.

Then we can do daisy wheels - that will please everyone.

I hope we are making patterns in less than 30  minutes.
To help them see how a pattern can grow I will have this pattern  along with others and a coloring book based on the Alhambra designs.

This is the painted and carved wood ceiling of the Hall of the Blessing in the Alhambra. It uses the circle divided into 8 parts, not 6 as I will be teaching. I am prepared for a simple discussion and illustration about dividing a circle into 4 parts instead of 6!

If every student has enough manual dexterity (!) in the second sessions we may be able to see how the curved muntins for the arched window at the Old First Church in Bennington come from  a circle.

I think I will have on hand my copy of Norton Jestor's The Dot and the Line, a romance in lower mathematics.