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.

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.    

http://www.jgrarchitect.com/2015/06/geometry-for-cobb-hepburn-house-part-1.html
http://www.jgrarchitect.com/2015/07/geometry-of-cobb-hepburn-house-part-2.html

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.