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. 

Ephrata Cloister 1733 cabin

I find - in upstate New York and Vermont - different geometries. The steps that led from a line and a length to a post and beam frame seem to depend upon nationality. In general builders with  Dutch and German heritage began their layouts with circles. Carpenters whose ancestors came from English usually worked with rectangles. However, my data base is not large.  Therefore:

When Voices of the Turtledoves, The Sacred World of Ephrata, by Jeff Bach, was recommended to me at the Preservation Trades Network Workshops this summer, I bought it. And I read a lot of it, especially Chapter 5: "God's Holy Point of Rest": Ephrata's Mystical Language in Space and Time.

Bach writes that the Cloister's adaptation of "familiar patterns" of construction to sacred, mystical proportions "contributed to Ephrata's anticipation and awareness of God's presence." p. 115.
He continues that in Ephrata's beginning period, in 1733, a cabin was built for 2 women solitaries, Anna and Maria Eicher, which was "probably typical". Page 116.
The footnote says a record of the time notes, "The house was 25' long, 20' wide and  8' 6" in height under the joists." #3, page 251.

Here is my sketch of the house based on the dimensions and the houses illustrated in Fig 7, Jacob Hibshman's survey of Ephrata. page 122. It is not particularly handsome, just serviceable.

I was, of course, interested to see what geometry, what 'familiar patterns', had been used in the construction of a cabin, patterns which might been applied 50+ years later in New York and Vermont.

Secondly I was, and am, curious to see how the vernacular proportions might have become  mystical symbols and forms. That question is not answered here.



The mystical geometry which interested the Cloister's founder, George Conrad Beissel, is illustrated in Figures 9 - page 125 - :  a box composed of pyramids which meet in its center and

a circle bisected in both directions and its hexagrams -as copied here.

My question was:
How would a carpenter with a pair of dividers lay out a cabin?








I began with the diagram in Figure 9 because it is a very simple geometry, quite easy to lay out: a circle divided into 4 equal parts and the star that is determined by its radius. A carpenter with only a little training could have drawn this diagram to use as a reference for the cabin's dimensions.


Assume that the circle's diameter - drawn at small scale - could be the cabin's length, 25 feet  - shown in red.

Move the 25 ft. length to the right side. This way of manipulating the circle is implicit in the crossing of the diameters - the division of the circle into halves and quarters.

The width of the cabin, 20 feet from the outside of the circle - the red line on the right -  is determined by the intersection of the triangles on the left.
 The floor plan of the cabin is outlined in red.







The description of the cabin says the height to the joists was 8'-6".
However, the joists were most likely let into the plate as seen here in another house frame.
A standard depth would have been 6". The height of the cabin from floor to plate probably was 9 feet.

The dotted red line on the right side is 9 ft from the perimeter of the circle.

The cabin's dimensions were determined by 3 lines laid over 2 simple easily drawn forms organically related to each other.








This analysis is unusual for me. I usually write about buildings I know personally. I think, however the diagrams are worth sharing - so direct!  simple!

I should also note that the diagram would have been 'drawn' on a board with the dividers, then used as reference; the lengths scaled up from the drawing. Carpenters do this today when working out a framing problem, using a pencil  - or sometimes a nail. Boards with such markings are regularly found during repair and restoration in the sheathing of houses and barns. We often miss the marks because they can best be seen in a raking light - and carpenters tend not to look for them on old roof sheathing.



Jeff Bach, Voices of the Turtledoves, The Sacred World of  Ephrata,   The Pennsylvania State University Press, University Park, PA, 2003.


Patrick W. O'Bannon st al., Ephrata Cloister: A Historic Structures Report, vol.1, The History and Archaeology of Ephrata Cloister, submitted to the Pennsylvania Historical and Museum Commission.

10/21/2015:
There are 85 entries in HABS/HAER  for Ephrata. I have looked at some. The geometric relationship between the hexagram, cube and mystical numbers and the Saal, for example, is not obvious. The plans and elevations, the 3 dimensional shapes of the rooms do not easily correspond with the diagram, nor with circles and triangles which are not overlaid.

Learning from a Workshop

The IPN Workshops at the Shelburne Farms Coach Barn were superb.
The barn is magnificent. To be able to be in and around it for 4 days was true luxury.

Above left is the main entrance from the court yard. Right is the dormer for the hay loft door above the stable. Below is half of one barn door showing its hinges and brick work.



The food was plentiful and excellent - local and fresh.
The company and the workshops couldn't be beat.
Of course I plan to go to the 20th annual IPT Workshop to be held in Virginia, autumn of 2016 . www.ptn.org


My presentation was almost derailed by the cheap school compasses I brought. The compasses did not hold their angle, so the diagrams we drew weren't true. I had not anticipated that the participants might not know how to draft: they needed basic instructions and better tools.

Luckily people bore with me and I presented twice. Many people talked with me about geometry between sessions.




Here is what worked best:

The daisy wheel: As people found the rectangle created by 4 points  they easily understood the geometry of the Old First Church in Bennington.








Making a square:  beginning with a line and a circle.

The hardest part for people to figure out was how to draw the arcs for the vertical line. I felt very successful when I heard one person explaining how to do it to another.










The 1830's farmer's cottage pleased everyone. They could see how to use what they had drawn.


A few people were able to rotate the square 45* to complete the diagram as shown

I brought the pictures from my post on  Asher Benjamin, Owen Biddle and Peter Nicholson. http://www.jgrarchitect.com/2015/07/geometry-of-cobb-hepburn-house-part-2.html

 We were to draw squares based on their diagrams  - as shown here:
This is where, especially, the compasses were not up to the task. The squares were not true;they were cock-eyed. I loaned my good compass out - so much easier to draw with good tools!
So, I explained and demonstrated. People practiced.
The pattern books' first pages of geometry turned out to be an adventure.


I thought to show 4 different ways to grow a layout from one dimension.
Instead I used the different buildings and diagrams as illustrations as people asked questions,
Good, thoughtful questions.

What a good time we had!

ME: giving a IPTN Workshop, July 22 - 24!

July 22-24, 2015 in the Shelburne Farms Coach Barn, Burlington, Vermont

                                                                             http://www.iptw.org/iptw_2015_home.htm


My workshop is called

"Line, Point, String: Scribe"

I want everyone to draw. So there will be

24 school compasses
2 packs of unlined paper
1 pack of grid paper
a pencil sharpener
some straight edges - not the "thin ivory scale or box rule" recommended by Owen Biddle
erasers - although I want people to explore, not correct mistakes

And of course, photographs and drawings, some posters.

I hope to help people be comfortable with geometry, to be able manipulate the forms,  design their own frames -
and thus create buildings whose parts are proportional to each other. Or maybe just understand how people did once upon a time.

Running  a power point presentation on geometry and construction may be possible.

I hope to be able to schedule a working session for all of us who are exploring geometry.
We have met at other conferences by chance. Maybe this time we can share together what we know.

If you are there too please come find me and introduce yourself.

Geometry of the Cobb-Hepburn House, an aside for skeptics

For previous posts on this house please read
http://www.jgrarchitect.com/2015/02/baring-bones-of-house.html
http://www.jgrarchitect.com/2015/06/the-cobb-hepburn-house-frame-tinmouth-vt.html
http://www.jgrarchitect.com/2015/06/geometry-for-cobb-hepburn-house-part-1.html


Here is the basic geometric shape used for the Cobb-Hepburn House.






When the house was built in 1780, the town of Tinmouth was less than 10 years old. It was the frontier. Paper would have been precious, not generally available for drawing house plans.


The master framer probably used dividers to layout the frame. We can see that he used them to draw the 2'  off set marks on the posts. Look to the bottom right of the post - 2 half circles above a line.

Sheathing was commonly used for diagrams.
I describe one such board found in a barn here:  http://www.jgrarchitect.com/2015/01/a-barn-and-its-daisy-wheel.html

Click the pictures to enlarge them.




25 years later when paper mills had become common, pattern books were popular teaching tools - beginning with basic geometry.


Here is Owen Biddle's Plate I in  Biddle's Young Carpenter's Assistant, 1804:


A and B are illustrations of how to attach paper to a board. C is the T Square.
(E,F,G are diagrams for perpendicular lines and right angles.  J is a 3/4/5 right triangle.K is the circle defined by 3 points not on a straight line.)


Just under the T Square is
H -  the layout of a square using the length of one side.


Biddle describes these engravings as " some of the most useful geometric problems which every carpenter ought to be acquainted with."
He explains that a student should have "a bow-pen or compass". 






 Asher Benjamin's  The American Builder's Companion, 1806, Plate II

has similar diagrams on basic geometry for carpenters.

All figures are explained on the accompanying page.
Fig.  12  is the same diagram as Owen Biddle's  H.

Benjamin writes in his Preface to the Third Edition:
"I have first laid down and explained such problems in Geometry, as are absolutely necessary to the well understanding of the subject."
He begins with

                           Plate I.
                  Practical Geometry.
                       Definitions. 

GEOMETRY, is that Science which treats the descriptions and proportions of magnitudes in general. 











Peter Nicholson's Guide, first published in 1792, in England, begins with geometry. It was updated and reprinted many times in London, New York and Philadelphia.
In his Preface  Asher Benjamin writes that he is "indebted to P. Nicholson's excellent books".

Figure 2 matches Benjamin's Fig.12 and Biddle's H.

This a a print of the actual page, Plate 3 - wear, age spots, and water stains included - in the 10th Edition, 1830.

I have the book in my library - on a long term loan.

.








I  have written this post because of the skepticism I encounter from academics as well as craftsmen.
The use of geometry in construction is often viewed as somehow made up. I suggest doubters read what the master carpenters themselves wrote.


Owen Biddle, Biddle's Young Carpenter's Assistant, originally published 1805, by Benjamin Johnson, Philadephia. Dover (2006) unabridged republication, Dover Publicatons, Inc., Mineola, NY

Asher Benjamin, The American Builder's Companion, first edition published 1806, This print taken from the 6th Edition, 1827; unabridged republication by Dover Publications, Inc., 1969.

Peter Nicholson, The Carpenter's New Guide: Being a Complete Book of Lines for Carpentry and Joinery, Treating Fully on Practical Geometry... 10th edition, John Griggs, Philadelphia, 1830.

Geometry for the Cobb-Hepburn House, Part 1

When does geometry enter into the design and construction of a building?
Not at first.
Only when the basics are answered can layout and design begin, can geometry be considered.

The design of any building begins with need, ‘What?’ and ‘Why?’.

Next comes, ’How big?’, ‘Where?’

Then, ‘What material?’, ‘What will it look like?’  

Of course people often start with the vision, what they want the space to look like, materials they hope to feature. They may focus on a specific use for the space.

Sometimes they just begin with ‘Bigger than this!’ The planning then must cycle around to answer the other questions.

 

The Cobb-Hepburn House began with most of those questions already answered.

About 1780 the Charles Miles family needed a house on their land in Tinmouth, Vermont.

The house would look like houses they knew: a 2 story box with a gable roof and a center chimney, with a layout that was also familiar: a hall, a parlor on the front and a kitchen with service spaces behind. 

Here is their house after the modern siding was removed. The original house was probably not painted. It would have had small paned windows and a larger chimney.

The house would be framed of wood of which they had plenty. The foundation and chimney of stone – which also ‘grew’ right there. The frame would be 4 bents long and 2 bays wide.  

14 feet square +/- seemed good sizes for the Parlor and Hall. The space for the chimneys needed not be as large.

Here began the geometry.

Please read left to right. I have shown in the first 5 diagrams 2 points for each straight line. After that I have assumed the geometry is reasonably clear. I have also added a description even though I often find it easier just to read the drawings. 

1 -- The width of the house was laid out: 28’-6”: the length of the first bent.
 Just a line:  A -B

2 and 3 -- A square was drawn, using the line as an arc:  A -B - C - D
All the sides are equal.

E marks the intersection of the arcs.

4 -- Diagonals were added: F is the center.

5 -- The square divided in half both ways: the second bent. 

6 -- At E a vertical line was drawn: the third bent.

7 -- E is the center of the second square. The length of the sides of the square could have been stepped off to match the first square or laid out with geometry.

The fourth bent is the right side of the 2nd square.   

8 -- is the plan for the house, showing the locations of the posts - almost.

The as-built plan for the Cobb-Hepburn House, shown here, is not quite symmetrical. 

Here is the plan showing the posts and  beams for the 2nd floor with the 2 crossed squares in red.

I have seen the use of 'Crossed Squares' in many New England floor plans.Before I explored the geometry of this house I had not seen layouts use the intersections of the arcs as determining dimensions.

The framer used his intersections as the outside of the post location for the center bents. He seemed to set the center posts and beams back about 8" so that the chimney could more easily exit at the ridge of the roof. 

Here is an elevation of the bents from front to back for the house.
It fits within a square.

If the square is divided into its 4 smaller squares and the arcs of the length of the smaller square are drawn the intersection is the location of the top of the plate - see the green square on the upper left side and intersection.

Once the height of the house frame was set, the rafters could be laid out.
The second floor location was determined very simply: the top of the beam was located at mid point of the distance from the first floor to the plate  -  shown with a dotted green square on the lower right side.  

In both cases the framer used his intersections as the outside, not center line, of his beams.

The elevations of the house used the same geometry. That's next.

  

The Cobb-Hepburn House frame , Tinmouth, VT

Here is the Cobb-Hepburn House coming down.

Glenn Tarbell recorded its dimensions as he and his crew dismantled the frame this past winter. I drew the framing diagrams. My measured drawings of the house before de-construction served as a reference.

The drawings tell a lot about how the house evolved.

Local records show that Charles Miles came to Tinmouth from western Massachusetts. He built this house about 1780. When he moved to Ohio about 1810, he sold the farm to Amos Brown. In 1821 Brown sold it to his son-in-law, Edward Cole. His daughter, Jane Cobb, inherited the farm and house when Cole died in 1852.
Hod Hepburn was the last owner who lived in the house. After he died the new owner asked Green Mountain Timber Frames to take down the house.
.

The house is 28'-6" wide by 39'-0" long, 2 bays (3 bents) wide and 3 bays (4 bents) long.
Its layout is derived from some of the earliest houses built in colonial New England

Center chimney floor plans were rarely built in seacoast New Hampshire and Massachusetts after 1760.  However, the first floor plan often appears in 2 story houses in New Hampshire and western Massachusetts up to 1770 and is a common plan for 2 story houses in Vermont through the 1840's.

The drawing is by William Lawrence Bottomley from his introduction  The Architectural Heritage of the Piscataqua, John Mead Howells, Architectural Book Publishing Company, Inc. 1937. His 10 page essay is one of the best introductions I know on early construction on the New England Seacoast from Salem, MA, to Portland, ME, .

Here is the probable floor plan for the Tinmouth House. I have labeled the rooms to match Bottomley's drawing.

The kitchen and the  pantry/dairy had been divided into smaller spaces by the time I measured the house in 2015. The other rooms still existed.
The first floor joists and sills were too rotted to be saved. The actual location of the original fireplaces is educated conjecture. A bake oven may have been beside the kitchen fireplace - we uncovered a mantle and cabinet door set in that wall.



The frame is massive, 10"x 10" posts rising to 10"x 14" and 10"x 16" gun stocks. The beams are 6"x 9", the plates 12" x 9". This is the 2nd floor SE corner seen from the 1st floor.
The roof rafters are of similar heft. The basic house frame was erected all at one time. Completing the interior frame took about 50 years.

.

The way the framer set back the longitudinal center bent -  about 12" from the center - allows the chimney (drawn in red) to rise through the roof at the ridge.



.





The frame for the north wall has missing and added studs as well as blocked windows, showing  how the house was changed through the years. The original 1st floor windows were directly below those on the second floor.








The west wall framing also shows window openings where we found no sash.
A door was added at some point and then closed off. A stud pocket remains in the beam above.









Empty joist pockets at the stair opening in the front hall indicate that the frame was reconfigured to allow space for the narrow, steep stair to the 2nd floor.




The second floor joists were made at the same time as the original frame.

The  Parlor, Hall, Bedroom and Kitchen joists are regularly spaced. The pantry/dairy joists aren't.
Quite a few of the joists have bark and wane; those above the parlor and the pantry/dairy are more logs than hewn.
The larger space between joists beside BII may indicate a stair.



Here is a possible explanation:


Charles Miles framed the house. He finished the Hall, Parlor and Kitchen; the Bedroom (which was usually reserved for the infirm or new mothers). Then he ran out of money, time, or energy. He finished the last joists with rougher wood. His family lived on the first floor.
Living in a partially finished house was not uncommon. Sections of a house were often used for storage and then added into the living space, just as we today add dormers to attics and insulation to  porches.



The attic framing was a different pattern: joists 2'-0" oc with intermediate 12" x 9" plates.
The joists all match: 3"x 6", cut by a sash saw at a mill.
The stair was relocated to where we found it.

As the frame was exposed we saw that bedrooms #1 and #2 were once one room, that bedroom #4 had neither lath nor plaster while the other rooms were finished.



Which owner installed the attic floor? The answer might depend on when sawn joists were readily available from a local mill.


The photograph shows the attic floor joist pattern. The window to the right had been blocked, but its outline was visible in the plaster wall. The window to the left was not original:the cut stud above and the lack of a stud on the left side were the signs of later construction.




Probably the Coles eliminated the fireplaces. Cast iron stoves were being manufactured in the 1820's.They were widely used by 1840.

 After stoves came central heat.We dismantled a modern cement block chimney serving a furnace and a modern wood stove.
Soot and char on beams implied that the framing around the chimney coincided with the installation of wood stoves.


For views of the house as dismantling began please see the previous post: http://www.jgrarchitect.com/2015/02/baring-bones-of-house.html