From a Circle to the Pythagorean Triangle via the Schifferstadt House.

The  geometry used to lay out the Schiefferstadt House, 1755, was the 3/4/5 rectangle. Probably.

'Probably' because Practical Geometry, the use of geometry in construction, was taught by doing, not by reading and writing. The drawings we have assume a knowledge of basic geometric patterns. Written records are rare and incomplete.

The stone walls for the House were laid one row after another, consecutively. Unlike wood frame structures which are form and infill, in masonry buildings the  form and the skin are one. 

This is the back of the house, showing not just the main stone house and the brick wing, but the extensive stone foundation.

Every wall of the House needed to be trued as it was built. Here is a wall in the cellar: laid up stone.  Consider how hard those slabs would have been to adjust later on. The walls were trued with a plumb line and the lines of 3/4/5 triangle as they rose.*  

 

The frame of a wood structures determines its size, its corners, its form. The parts for the frame, the studs and braces, are cut and assembled. The shape can be adjusted, changed, trued using lines, even after it is raised. This image of a barn frame is from Wm Pain's The  Carpenter's Pocket Dictionary, 1781, redrawn by Eric Sloane.**  

The stone and brick buildings I have studied use the 3/4/5 triangle. Chimney blocks are 3/4/5 rectangles. 

So, why didn't I immediately try the 3/4/5 triangle when I looked at the house geometry? Well, I wondered if the Schiefferstadts'  traditional building patterns, brought with them from Germany, would be different from those I'd studied before, the vernacular housing built by English, Dutch, and French immigrants. Those began with the circle and its square. I began there too, looking for differences. I missed the obvious: the stone. The 3/4/5 rectangle easily fits the plans, the simple solution. KISS***

 

Then, as I was playing with the circle and its square (left image), this happened.

I saw that when I begin with the square derived from the radius, its circle and lines (left image), I can easy to locate 6 other points around the circumference , making 12 equidistant points around the circumference, (center image). I saw that circle geometry 'finds' the 3/4/5 rectangle (right image); that the Pythagorean Theorem is a 'short cut' using the 3 and 4 units that are already there.

On the left: the 12 pointed daisy wheel.  On the right: the 3/4/5 rectangle with units, and the 3/4/5 triangle.

 

 

 

 

 

 

*The walls are 'kept in line'. I am often surprised to realize that a common phrase, such as '"staying in line", probably began as construction lingo.

** Wm Pain, The Carpenter's Pocket Directory, London, 1781.

     Eric Sloane, An Age of Barns, Voyageur Press, Minneapolis, MN, 2001, p.37. originally published by Funk&Wagnals, c. 1967.  

*** KISS: "keep it simple, stupid"

The earlier posts on the Schiefferstadt House:  

https://www.jgrarchitect.com/2024/08/a-closer-look-at-schiefferstadt-house.html

https://www.jgrarchitect.com/2024/07/the-geometry-of-schiefferstadt-house.html

Teaching Practical Geometry

 Several educators, curious about Practical Geometry, have asked me how I would share this geometry in the classroom. This post is an introduction to how I would begin.

In September, 2023, I presented 3  workshops at IPTW, the International Preservation Trades Workshops.* The last day was open to the public. About 10 people, aged 10-70+, came to learn about Practical Geometry. Some had never held a compass.  

Here is what we did:

We drew circles with compasses. Then we divided the circumferences into 6 equal parts and connected the points to make rectangles and squares. We used no numbers. 

 

We  explored the  design and layout tools a carpenter would have had before the Industrial Revolution: the compass, a line and a scribe. We talked about how those tools were used and are still used. We compared cubits (the length from your elbow to your longest finger).  We set carpenter's dividers for a day's work by the radius or the diameter of a daisy wheel. One of the participants taught the others how to snap a chalk line.

I brought my daisy wheel with me. It was scribed into a 9 ft tall board which was once sheathing on Vermont barn, c.1780. The barn was deconstructed about 10 years ago. The deconstruction contractor gave me the board.

 

 

I showed them the floor plan of one of the early Virginia folk houses recorded by Henry Glassie,** which used the geometry we had drawn. 

 

 

 

I shared a few pictures including this house whose plan we had just laid out.  

 

 

That image introduced the class to the chimney wing. Its plan would have used the 3/4/5 rectangle to make sure the wing was parallel to the house so that all the roof framing could be cut the same length.  

 

 

 

I showed the group a Menagery, a retreat intended for an English gentleman's estate, designed by James Gibb's ***, c. 1720.  

 

 

 The wings are laid out in the same way, using the 3/4/5 rectangle. Here it is because the rough laid stone on the exterior would have made an accurate layout and construction difficult.

 

Then the class learned about the 'star', the Lines, in the center of the Menagery. Those are also the lines on our cellphones which help us edit images, known by artists as the Rule of Thirds.  

Here is the geometry: the diagonal of the square and the Lines from the ends of one side (the corners) to the middle of the opposite side. The  pattern is turned 4 times.

 

 

Where the lines cross are points. 2 points connected are a line. That line is always straight.

Here, the points divide the large square into 9 small squares - the diagram used on cellphones - or 3 equal rectangles.

There are also 4 squares within the large square. If their diagonals are drawn, the large square can be divided into 16 small squares or 4 equal rectangles.

 

The Lines on the elevation for this brick house tell the mason where the sides of the door and window openings are. On the plan the Lines show the fireplace edges and the placement of the interior walls. 

The drawing is Plate 56 in Owen Biddle's pattern book, The  Young Carpenter’s Assistant,  published in 1805, by Benjamin Johnson, Philadelphia.

 

 

 

 

 I ended with these Lines in Sebastiano Serlio's Book I, c. 1540. It explains where to place a door in a castle wall. He ends Book I: On Geometry, " However, honest reader, although the things resulting from the various intersections of lines is infinite, to avoid being long-winded I shall come to an end."

 

This was more than enough for one 75 minute session. 

Several shorter lessons would have been easier for everyone. There was very little time for questions, more examples, or in-depth understanding.  

 

For more information: In 2020,  I wrote 7 posts entitled 'Lessons' for students of all ages. https://www.jgrarchitect.com/2020/04/lessons.html .

*The 25th International Preservation Workshops were held this year in Frederick, MD, at the Hargett Farm which will become the Historic Preservation Trade Center for the National Park Service.          See the Preservation Trades Network website, ptn.org, for more information.

** Henry Glassie,. Folk Housing in Middle Virginia, U of Tennessee Press: Knoxville, 1979

*** James Gibbs,  Book on Architecture, London, 1728, Dover reprint

**** Sebastiano Serlio . On Architecture, Lyon, France 1530, translated in1611,  on-line and translated by Vaughan Hart and Peter Hicks, 1996, Yale University Press, New Haven

To read more about this diagram see https://www.jgrarchitect.com/2022/10/serlios-lines.html 

 

Practical Geometry at Mud University, Cambridge, NY, March 3-4

 

ANNOUNCEMENT

 

 I'm giving a class on Practical Geometry at Mud University, in Cambridge, NY, on March 3-4, 2023. Their website is at the end of this post.*

Just in time for mud season!  Come learn about practical geometry at Mud University.  

FREE and FUN! With a fabulous instructor: me. 

Anyone who's curious is welcome, no math or drawing skill needed. 

 

March 3, First meeting: I will introduce geometry as practical knowledge well understood until about 1950. We will use compasses to layout daisy wheels.

March 4, Second meeting: we will draw the patterns, hands-on, with compasses.

 

 

 

 

 

Here's a diagram - the square and its circle.

 

 

 

 It is the language for the pattern of a quilt (dated 1847) 

 

 

 

 

 

and the roof structure for St. David's Cathedral in Wales (c. 1550).  

 

You will learn what Practical Geometry was/is, and see many examples of the use of geometry in construction and design.

 You will see how our ancestors, weavers, quilters, cabinet makers, and builders used geometry for design and construction. I will mention drawings, paintings, and illustrations, including how our cell phones superimpose the 'rule of thirds' over our snap shots.

Ask me if you have questions. Or just sign up.                       *  https://www.muducambridge.org/

I look forward to seeing you there.  Jane 

 

St. David's Cathedral and geometric pattern :

Smith, Laurie, The Geometrical Design of St. David’s Cathedral Nave Ceiling, A Geometer’  Perspective,  

The Geometrical Design Works, 2017, printed Exeter, UK. and others.

 

 

 

 

 

 

 

 

 

 

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.

  

 

ARCHITECTURAL GEOMETRY, A Rare Geometrical Record from Rural Devon, by Laurie Smith

 

 

Laurie Smith has written a new book: ARCHITECTURAL  GEOMETRY  A Rare Geometrical Record from Rural Devon.  

Here's the cover.

 

 

 

 

 

 

 

 

   

 

The book is about the many daisy wheels and other geometry found on the walls of a Devon threshing barn.
The barn, shown here, is owned by Richard Westcott, editor of The Three Hares, a Curiosity Worth Regarding. 
The image is #3, page 4.  

Richard Westcott, Laurie Smith, and their friend, the photographer and film maker Chris Chapman, examined and recorded the geometry on the barn’s walls -  over 169 separate geometric shapes.

This photograph is of a "divider scribed daisy wheel from the wall's inner surface." 

The image is #4; the quote is from page 4.

They researched the barn’s history, took measured drawings and photographs, and explored the geometry.

Then Laurie wrote this book. 

 
Like all of Laurie’s books it has beautiful diagrams. Clear descriptions accompanying the diagrams explain how the daisy wheels still visible on its walls governed the siting, layout, and frame  of the barn.

Image 39, page 26

This is 1of 5 of Laurie's illustrations showing the development of the barn section.     

He includes examples of similar geometries give context and nuance. 

Here is one of 4 daisy wheel drawings for the geometry of the Barley Barn, Cressing Temple, Essex, UK

 Image 27, page 19.

 

 

 

 

Along the way Laurie explains terms and forms which we rarely use today, including the use of a perch, pole and rod as measuring devices.  

He introduces the reader to the Trivium, the Quadrivium, and Whirling Squares. 

Part of Image 63, page 50 

 

He writes thoughtfully - and with humor -  about apotropaic symbols.  

 

 

At the end of the book Laurie considers how all of that geometry - 169 separate images - came to be scribed on the interior walls of a rural threshing barn. He suggests a 'geometry school'.  I agree with his theory: I have also found incomplete geometries drawn on plaster walls here in the States.

His descriptions encourage the reader to examine the image, think about what he's written, look again, and understand the geometry.

Here is what he says about this double daisy wheel:

"The image shows the geometrical precision of the divider-scribing, the scars of the divider pin at the twelve points around the primary circle and the compound damage caused at the symbol's axis by the passing of twenty four arcs."

Image and quote, page 59 

 

The book's copyright page includes this introduction: 

"Laurie Smith is an independent early-building design researcher, specializing in geometric design systems. Because the medieval educational curriculum included geometry he uses geometric analysis to excavate and recover the design systems of the past, a process he thinks of as design archaeology. He lectures, writes, runs practical workshops  and publishes educational articles on geometrical design that are available from his website."

www.historicbuildinggeometry.uk  

e laurie@historicbuildinggeometry.uk  

 I highly recommend this book to all who are interested in historic construction and geometry. Copies can be purchased from Laurie Smith in the UK or from me ($20.00 postage paid) in the USA. 

 

*All the photographs: the barn, the hand and the daisy wheel, and the 12 pointed daisy wheel in the barn shown here are by Chris Chapman, copied by me from the book for the purpose of this review. 

The geometry is by Laurie Smith, also copied with his permission.