Sunday, March 31, 2019

A Cabin on Magnolia Plantation, Louisiana

I will be in Natchitoches, Louisiana, April 23- 25, teaching Practical Geometry, in a full day workshop on Thursday, and 2 hr. workshops on Friday and Saturday. 

This image - the cross section of a cabin, built c, 1845, on the Magnolia Plantation, its geometry and a compass - is my on-line poster for the Thursday workshop. It and a description of the workshop is now posted on the Preservation Trades Network website: 

I have the HABS drawings for this cabin, but no picture, as I have not seen yet it.
I hope to visit when I am at the conference.
It's too easy to miss something essential when I work on the geometry of a building I have not be in and around. On the other hand, including local buildings in my workshops and power point talks is important, and researching their geometry will help me see them more clearly when I am there. 

Corn cribs and cabins were utilitarian, built using ordinary construction. Studying these simple structures helps me understand what geometries the local carpenters used.

This cabin has brick walls. It has 2 rooms with back to back fireplaces in the center. The floor plan begins with a square space and its diagonal used as a radius to draw an arc. Then another square space was added on the other side. The space created by the arc is for these fireplaces - possibly including their foundations.  
Follow the  black  line and arc with arrows and the red line and arc with arrows.


The section of the cabin shows a floor set above the ground.  The height of the cabin is half the width, here shown by 2 squares, beginning at  grade, not at the floor.  
The diagonals of the squares becomes the radius for arcs; the point where they meet is the cabin's ridge.  Follow the  black  line and arc with arrows and the red line and arc with arrows.

The squares could be moved up; base on floor, top on the rafters. But then the arc would be above the roof line. 

The end elevation matches the section and locates  the window - see arrows.

The side elevation continues the pattern: 2 squares on either side of the center partition, the windows centered on the double square. The red square shows both the partition location, left, and the window location, right.
Note that the end walls are in addition to the squares, just as the center partition is.

There are 2 issues I can't resolve without more knowledge of the building and its construction.

1) The window locations on the floor plan do not match those on the end elevation. Which should I use for my analysis? 
Most of the brick houses I have studied use geometry to mark where an opening begins rather than a center line. Why the difference here?

2) I do not know enough about brick construction in Louisiana to make educated assumptions about framing.
How do the floor and the walls join? Is there a sill?  There seems to be a plate. How is the roof framed? What is the reason the end walls are higher than the sides?
The site appears level in the drawing,  with brick footings maybe 18" deep, slightly wider than the walls. Is that the whole foundation?  Is there a foundation for the chimney?

All the buildings whose geometry I've analyzed have been no farther south than Virginia. Their geometry begins with a flat base, a foundation, which can be set true and level. Where is that here? Has this to do with climate or topography?

I look forward to my visit.

The website for Magnolia Plantation is: 


The National Park SErvice has several articles about slavery in Louisiana including this:

The HABS drawings call this a "slave cabin".  I refer to it as a cabin.
The first people who lived here probably were enslaved. They were people first, with skills and families. Maybe their names have been recorded, possibly connected to this cabin. Later free blacks and Creoles lived in these houses at the plantation.
Maybe a mason or a brick maker lived here and I am studying his skill and knowledge.
I have learned that on some plantations tutors and overseers who were not enslaved often  had similar housing. 
This is a "cabin".

Saturday, March 9, 2019

The Geometry of Fences, c. 1830

Asher Benjamin's The Architect or Practical House Carpenter, published in 1830, includes a plate with 3 designs for fashionable fencing,  2 for gates. The lower drawing also includes a post.

While Benjamin includes a scale between the middle and the lower illustrations, he gives no other dimensions or information. He assumes the reader will know how to lay out the design.

The 2 right hand designs are repeated diagonals. quite simple to draw: cross your rectangle, lay in the horizontal and vertical center lines, embellish as you wish.
The bottom drawing is the fence for the gate above and shows the post and its ball.

What about the fence with curved balusters and the gate below it with rectangles and crosses?

First: the fence with curved balusters: 
The center of the rail is the center of the arc. The extended arc becomes a semi-circle  whose radius appears to be the distance to the edge of the bottom rail from the center and the height to the post below the ball and its base.

The arc determines the curve of the baluster. The circle using the same radius, centered on the baluster, follows the reverse curve. Both balusters are shown in red. 
Using Benjamin's scale the balusters could be cut from boards about 3 feet long and 8 inches wide. They are all the same shape.

The circles intersect  - see the vertical dashed lines in black. That intersection gives the spacing for the balusters.
Drawing the next circle using the point where the first circle's circumference crosses the center line, adding the vertical at the intersection, the placement for the balusters, noted in red. continues.

The gate seems at first glance to be more complex.

In my exploration I decided the size and the structure of the gate was known - its width, the framing members, the depth of the bottom rail. The black rectangle outline is what's left - the space for the fret work.
The diagonals are easy to draw - corner to corner. The space can be easily divided in half horizontally and vertically. I have not noted those lines.
But what about the inner rectangle?  The little corner squares?

Two radii are drawn; both A and B are arcs the height of the space, their centers are in the bottom corners, they arc from the top to the bottom - follow A and B's arrows.  They cross at C which is both the vertical center of the rectangle and the depth of the cross brace.

Reverse the arcs.  Now A and B land on the bottom rail  and mark the placement of the vertical braces. C marks the horizontal brace .

The horizontal and vertical braces are noted in red.

The rectangle of the space could be larger or smaller without changing the way the design developed or the general appearance. The proportions would still relate to each other.

The reader in 1830 - probably a house carpenter - would not have required my explanation to copy or adapt the  designs for his own use. He would have learned Practical Geometry as an apprentice. He would have read the design development automatically; he needed no words of explanation.
He might have drawn  his own small diagram on a board. Then he would have drawn the arcs full size on his framing floor - or table, as this is not very big - and found his lengths. The diagram would have remained until all the parts were made.

Plate XXXIII, Asher Benjamin's The Architect or Practical House Carpenter, 1830, L. Coffin, Boston. From the Dover Reprint first published in 1988.

This Asher Benjamin pattern book especially interests me because a diary was written by a local farmer in this period. Its author notes that his friend, a carpenter, traveled twice to Albany to buy his copy of the pattern book when it was first published. The details in several local houses seem to indicate that the joiner worked directly from Benjamin's plates.

Tuesday, January 22, 2019

the Geometry of the Kirkland Temple

.A reader of this blog asked me to look at the geometry of the Kirkland Mormon Temple in Kirkland, Ohio.  He saw similarities between the geometry of the Cabin at Tuckahoe Plantation and the Temple.

I have not seen the Temple, but HABS drawings are available on the Library of Congress website; and the Kirkland Temple has good exterior and interior pictures on their website.

The Kirkland Temple, built 1833-6, has a design specific to its use, not a traditional church form adapted to a new way of worship. I am not referring to belief, but about how the religious group planned to meet together.The Temple has 3 floors, each for a specific use: the Church floor, the Apostolic floor, the School and Quorum floor,  and the accompanying a Vestibule and Stair Well. This is different from the churches the people who built this would have known.
However, the red organizing diagram for the frame was not new; it was the ancient pattern that the craftsmen had learned as apprentices. They used the Square and  its Lines to plan the facade. The diagonals mark the placement of the Palladian window, the Lines encompass the ellipse in the pediment.

The builders did not use the division of  Lines into thirds. They seem to have preferred dividing in half and then in half again.

I have marked the 'third points' with red circles; the design does not depend on them. I could have left them out of the diagram.
Half the square also determines very little - maybe the sash location of the Gothic windows.  However, the body of the facade is 3/4 of the square, the pediment 1/4. The pitch of the roof is the diagonal from the center to the upper corner.

The  body divided in half - or the square divided into 3/8 and 5/8 - determines the 2nd floor location - the horizontal red dashed line.

For clarity I  have only laid out one quarter of the possible facade Lines - the red square on the lower left.  Half the quarter seems to set the height for the Gothic windows on the first floor. 3/4 of the small square seems to locate the door with its fanlight. The location of the Gothic windows does not quite work - the vertical red dashed lines.
The west elevation is  much the same, not quite symmetrical. The plans show that the windows were set to accommodate the stairs in the Vestibule and the seating the Church and Apostolic spaces.   


The plan is a square, solid red lines, and an overlapping 2nd rectangle whose length is determined by the overlapping arcs of its width, dashed red lines.

Again I would like to compliment my analysis by the experience of being there, walking through it as well as around the outside. For example here the 2 overlapping squares seem to include the platforms and stairs in front of the Temple. If I were there I might understand if the stairs had been part of the intention of the original design.

The division of the square into quarters locates the  columns and the beams along the length of  the whole structure. The spacing of the columns across the width probably is 1/4, 2/4, 1/4. The columns in the  interior elevations look wider than they are drawn in plan here.

The 5 columns at the east end (bottom of the  drawing)  support the tower.

The Gothic windows and the Federal doors also use  squares, and their division into halves as the initial layout.  The interior dimensions  - the panes, the panels -  do not seem to follows the same pattern.


The Church sanctuary and the Apostolic floor both have a central square flanked with smaller ones on each side. The regular spacing of the columns, the square side aisle bays between them, and the central naves with arched ceilings facing Palladian windows create 2 dramatic spaces. 
I do wish there was more information about the framing. Look at that blank space above the side aisles!

I was curious about the Temple partly because Joseph Smith, Jr. was born in Vermont where I live. I wondered if the framing traditions I see here were used in Kirkland, Ohio. I was curious about what forms the early Mormons used.  I wanted to compare it to the Streetsboro Baptist Church - built about 15 years earlier - near by.
I found a use of Practical Geometry that was very basic. Perhaps it allowed untrained members of the community to help with the construction.
From the photographs on the Kirkland Temple website the community seems to have created a striking building with effective spaces. 

The structure was measured in the 1930's for the US Dept of the Interior; the drawings are now part of the HABS collection in the Library of Congress. 

Sunday, January 6, 2019

Finding a Simpler Way to Layout a Building

The builders knew how they were using Practical Geometry - where they would begin, how they would use the diagrams. 
I don't.  I look at their frames, their finished buildings. I see a rhythm, a pattern. I try to discover the steps. When I can record a plausible geometry I draw it and write about it.

This year, fine tuning the power point presentations I would give 4 times, I thought that my diagrams were too complicated. Maybe not for a joiner building the main door of a church; but for the timber framers in a framing yard, I thought, "Too complex, too many lines!"

 Asher Benjamin's design for a "House Intended for the Country" belonged in my presentation. My audience usually knew about Benjamin. If they didn't, I needed to introduce them to him and his designs.
As I added it, I saw how the center square was the main idea - everything else came from it. The width of the wings comes from the Arc of the Diagonal of the Central Square.  So simple!

Then I drew the Central Square on the plan  - inside the walls as this was to give information to the men framing the interior.  The square, its Diagonal, and its Lines locate the entry hall width, the back edge of the curving stair and the wall for the upper left room, as well as the fireplace locations. The left and right sides of the house beyond the square are 1/3 the width of the square.

These are simple, small illustrations in a pattern book,  about 3" square, necessarily generalized. Even so, they convey the information needed by a builder to layout a similar house.

I then revisited the framing layout for the Rockingham Meeting House in Rockingham, Vermont, begun in 1780, finished by 1800.

Here is the geometry for the  plan. A center Red Square with its Black Posts  on the half and  thirds. The width is determined by extending each side 1/2 the square. The length is also extended 1/2 the square.  The Dashed  Red Lines show how the line extended located the posts on the exterior. 
Note on the left how the stair wing is laid out on the opposite side of the Line.

This is a much simpler layout than what I drew in  2014. 

The geometry for the front elevation of the Rockingham Meeting House seems to be most easily laid out as I drew it originally. 

That post is at:   
Details about the stair wings are included. 

The  size of the meeting house could be seen as a 3/4/5 rectangle: 3 deep, 4 long. The post placement does not follow that pattern. Perhaps that is why the window placement on the front and rear elevations is not regular.

The post and beam placement required that the 2 windows on either side of the door be framed against the posts - a little tight.  

The photograph shows hows the windows come against the posts. The exposed beams here holds up the gallery. That curved ceiling results from the slope of the floor above.

The Vail House geometry follows this pattern. I wrote about it here:

I have no name for this pattern. The previous name I used to describe this way of designing, "crossed squares",  is retired as interesting but too complex: not Practical.

Asher Benjamin, The American Builder's Companion, 1804,  Plate 55, 'Designs for a House intended for the Country'.


Saturday, December 29, 2018

Framing a Barn with Practical Geometry in 1791

 I wrote this post for Green Mountain Timber Framers' website blog in Dec. 2014.
As I often refer to it and use the diagrams when I lecture and teach, I copied it here for easy accessibility - with a few edits for clarity.

I invited myself to a Green Mountain Timber Frames barn dismantling in the fall of 2014. I wanted to watch it come down. I also wanted to investigate its geometry.

Here's what I saw.

The three barns sat beside the road  on the uphill slope of a valley, connected in an L shape.
None of them faced the road on their west and windy side. Instead they faced south and east, creating a protected barnyard, a sun pocket.

In the middle, protected from storms and wind, was the corn crib. Other farm buildings repeated the pattern, facing south, no doors on the west.

The main barn also had a door to the north, directly across from the one facing south. It fronted the farm road and looked at the house across the way. Two doors across from each other allowed for easy moving of machinery, ventilation and threshing. A north facing door was for bringing in hay and grain on the shady side of the barn in the summer.

After we had climbed up to and down from the rafters, Dan McKeen (who then owned Green Mountain Timber Frames) handed me prints o f the measured frame.
To have a sense of the building I checked some of the dimensions. The framers really did make his barn 30'-1" wide. He also made it 42'-6" long.

Why those dimensions? Laurie Smith, the English Geometer, suggested that a layout using the diagonal of the square was the reason.
The diagram shows how a framer would have used that set of proportions (which is the square root of 2) to layout the floor. This is easily drawn.
The rest of the barn frame comes directly from this diagram .

Both the extra inch and the square root are indications that the master carpenter for this barn used geometry to determine its size and framing. The ruler the carpenter used was not accurate by today's standards. Because he used Practical Geometry for his layout -  proportions and relationships between parts, not fixed dimensions - it didn't matter.

The second diagram shows the floor plan of the barn.

The height of the  new rectangle on the end of the square was a good height for the barn wall. So the framer drew a square in each corner. Using the diagonals for those squares he swung an arc on both sides. Where they met marked the ridge for the roof.

I have drawn the diagram as if the framer used the barn floor for his layout. Carpenters today use the floor of a house to layout the walls  and the rafters for the roof above, so this is a reasonable assumption.

The measured drawings of the barn show how the diagrams were applied to frame the west end wall.
The red X on the right is the diagonals of the original square. The DASHED LINE is the arc of the diagonal locating the ridge.

The green DIAGONAL of the SQUARE on the left is cut by the green ARC of the length of the square. That intersection is the location of the left interior post.

The east end uses the same geometry as the west end.

Here is the diagram for he diagonal cut by the arc. It is easy to draw and based on dimensions already being used by the framer. Locating the posts is straight forward and simple, easy to do with a straight edge, some twine and a way to hold the twine taut.
The north and south walls also use matching diagrams.

 Shown here is the north wall. The RIGHT SIDE matches the layout of left end of the west wall shown above: the divided by its DIAGONAL AND ARC INTERSECTION,.
Then comes the SQUARE door opening; its lintel determined by the INTERSECTION OF THE TWO ARCS of the square.

The left side was divided in half as shown by the DIAGONALS


Note that the braces and the poles are also located using the same geometry: just turn the diagram above upside down.  The diagram here shows all 4 arcs within the square. 
Green Mountain Timber Frames website is

The  measured drawings used here were produced by James Plastteter in May 2014. Platteter is a master furniture maker. His website is down, but his fine work can be seen by searching by his name. 

Sunday, December 2, 2018

The Daisy Wheel - a Module, a Diameter, a Part

This post follows my post on Lines: how we designed and framed using compasses and twine.

I used the Tuckahoe Plantation cabin as a my example.
Its floor plan is composed of 2 squares. Its elevations come from the division of the square into thirds, easily done on a framing floor with cords anchored in place on each end.

As long as the original length, here A-B, was on site to use as a reference, the cabin frame would fit neatly together.

The windows however might be made by a joiner, off site. He could take a length of twine with him that matched A-B. But his windows would be smaller. How does he figure out the needed window size?

He would refer to a daisy wheel drawn by the master carpenter.

The diameter of the circle is a fixed length; the daisy wheel shows the craftsman where that diameter is. It is the Module, the Diameter, the Part described by Vitruvius and Palladio, referenced by the pattern book writers.
'P' on a drawing can also refer to the Latin 'pes' or the Italian 'piede', meaning 'foot'. 

Vitruvius, (Book I, Chapter II, Symmetry,) says. "Symmetry is a proper agreement between the members of the work itself, and relation between the different parts and the whole general scheme, in accordance to a certain part selected as a standard. ... In the case of temples symmetry may be calculated from the thickness of a column, from a triglyph or even from a module."

Asher Benjamin divides the lower chord in his truss diagrams into 4, 7, and 9 Parts. 
The Country Builder's Assistant, Greenfield, MA, 1797, half of Plate 29.

Owen Biddle adds a line below his fireplace mantle which divides the width into 5 Parts; and one of those  parts into 4 smaller parts.
Owen Biddle, Biddle's Young Carpenter's Assistant, Philadelphia, 1805, half of Plate 21

The master carpenter chose his circle diameter - often a hand's breath, about 8", or  from thumb to first finger, about 6".
He drew his circle on a board and stepped the radius around the circle 6 times, swinging an arc each time. The pattern is a daisy wheel.
Always, in every circle, the tips of the petals mark the diameter of that circle.  The other carpenters could measure the diameter with a compass whenever they needed.

The cabin width might be the daisy wheel stepped off 3 times, then that length stepped off 8 times. If the daisy wheel was about 8", the width would have been about 16 ft, a common size for small houses before 1850.
The windows might be 3 daisy wheels wide. The joiner fashioning windows needed only take the daisy wheel's diameter with his compass and transfer that length to his work to make the window fit the cabin.

*   *   *   *           *           *            *           *           *            *           *           *           *           *
Here I have counted off 3 units, then used that dimension to count off 10 lengths.

Other daisy wheels have been found on roof and wall sheathing boards. After a building was framed the daisy wheel was no longer needed but the board still was.

Daisy wheels drawn for practice or perhaps to alleviate boredom also exist.

This pattern is on a bedroom wall where it is known that someone was confined due to illness for a long time. It shows no signs of being used as a reference.

The daisy wheel at the beginning of this post is on this board leaning against my breezeway wall. The 9 ft. tall sheathing board was part of an 1780 Vermont barn. The wheel was about 4 ft off the floor - easily accessible. It was drawn by compass; the center and the tips of the daisy's petals were regularly pricked. The radius and the diameter were used as dimensions. As it was in a protected and easily seen location it was probably also used for other buildings nearby.
Its owner gave it to me.

Vitruvius', Asher Benjamin's, and Owen Biddle's books are listed in my post of my bibliography.

If you do not know how to draw a daisy wheel, the steps are shown here. 

#1 Draw a circle
#2  using the same radius, place your compass on the circumference - the line of the circle - and draw an arc.
  #3 Where your arc crosses the circle's circumference, place your compass and draw an arc. Do this a total of 6 times.

 #4 When you get all the way around the circumference you have made a daisy wheel.

Monday, November 26, 2018

Archimedes' Stomachion - Dissecting its Geometry

Updated and reconsidered

A copy of The Archimedes Codex was recently loaned to me by a friend who found it interesting.
I agreed. I enjoyed the discovery, the history, the math and the science.

I especially appreciated the chapter on the Stomachion, a puzzle I had not seen before. My grandson and I had fun with all the solutions.

I already knew the square and the Lines of the Stomachion.  It is a geometric diagram used for layout and framing, part of Practical Geometry which was commonly used for construction at least as far back as the 6th century BCE when it is mentioned in the Bible. Practical and Theoretical Geometry were co-equal branches of the same mathematics, Vitruvius writes of how one informed the other.

I cannot comment about Archimedes' understanding of the multitude of  combinations possible for constructing the square. I can, however, easily see how to transpose triangles from one place to another in the diagram.

He wrote about shape:
"So then, there is not a small number  made of them, because of it being possible to rotate them into another place of an equal and equiangular figure, transposed to hold another position; and again also with 2 figures, taken together, being equal and similar to two figures taken together --- then out of the transposition, many figures are put together." *

Archimedes was a geometer and engineer as well as a mathematician. He would have known and used Practical Geometry as it was applied to the construction around him in the 2nd and 3rd Centuries BCE.  His understanding of geometry, theoretical and practical, should be part of the discussion.  Perhaps he was thinking about the lines as well as the shapes. A person of his ability could have considered both effortlessly.

The Stomachion reminded me of this drawing by Sebastiano Serlio, from his book Architectura, published in France in 1537. He is discussing how to add a door to an existing facade.
The diagonals and the lines from center top to the lower corners determine the  size and placement of the door and its 'ornaments' - Serlio's word.

In 1821, the same lines were used to layout the Weathersfield, VT, church.
Here is its Palladian window. I have added the Stomachion lines which apply to its proportions in red.

Practical Geometry used lines to determine both design and structure: the size of a building and its framing, its ornamentation.

These drawings show how the lines of the Stomachion were determined. They are not random.

The first 3 squares focus on the  right side which is half of the square.  The red lines are the diagrams extended. First, the Stomachion.
Second, the square divided in half using its center line.  Third, the location and layout of the small triangle. Note that all the lines depend upon 2 points.
I really enjoy the 3 similar triangles flipping back and forth along the line.

In the second  3 squares focus on the left side.
First, the Stomachion.
Second, the diagonals of left hand half, and then that half divided in half again. 
Third, part of the original from the upper left corner which determines where the left-hand angled line (also a diagonal) stops.

Archimedes knew the shapes were proportional to each other. He must have know the lines. Were they so commonplace that he doesn't mention them? And we do because we've forgotten them?

Perhaps someone else has seen how the Stomachion relates to Lines, how these lines come from dividing a rectangle into parts, how this is Practical Geometry - perhaps even Theoretical Geometry.  I would like to meet that person.

                            *                               *                          *                            *                           *

The Archimedes Codex, How a Medieval Prayer Book is Revealing the True Genius of Antiquity's Greatest Scientist, Reviel Netz & William Noel, De Capo Press, Great Britian, 2007.

* quote from page 255 of The Archimedes Codex.