Monday, July 12, 2021

The Geometry of Gunston Hall's North Porch

This post is about exploring Practical Geometry, getting lost, and finding the simple answer. I have used my working  drawings to show the process. Faint lines show where I erased possibility that didn't work.


Gunston Hall was the home of George Mason, a Virginia planter, with a big family, lots of land, and many enslaved people. He was one of the delegates to the Constitutional Convention in Philadelphia is 1787.

He was also a mason: he advised George Washington about mortar mixes. 

When he had his house built, in 1754, he made the formal dining room and parlor (on the right side of the house in this picture) larger than the family parlor and chamber (on the left side of the picture).  


That meant that the house was not symmetrical around the door.  (To see this look at the spacing between the windows on the left and right sides.) The lack of balance might have distracted those arriving to either the north or the south entrance. However the small door and windows at the entrance were probably more jarring.

William Buckland, a young architect just come from England, solved the problem. The porches he designed are so inviting they make the asymmetry is almost invisible. More importantly they enveloped the existing entries.

In  2014, I wrote about the house here:

The geometry I suggested for the north porch never quite fit. I tried other solutions.  Nothing was much better. I could see the geometry of Buckland's design but I couldn't draw it. 

Here is the HABS drawing of the north porch.




 Here is my attempt to understand what geometry had guided the proportions of the design.

It's neat. It seems to work until you consider that the intersections of the lines do not tell the carpenters how to layout the porch, where the parts should go. The diagram is interesting, but gives no useful information.

Here is a later exploration when I still wanted the diagram to describe the porch design.

Perhaps my initial square was too large, not based on the right width or height?

Would a  circle within its square work?

How about a daisy wheel ?  or an octagon?

The pencil lines, the red and blue inked lines are finally just confusing.

6+ years later I changed the question. Not, "What geometry did Buckland use?" Rather, "What was he given?" 

I knew this. I'm an architect who works with existing houses. My first questions about a house are, "What's here? What are the existing conditions?" 

The HABS drawing of the Hall show the north wall, a door with a fanlight and small side windows, a baseboard, a chair rail and an expanse of upper wall with some crown molding.  This is what existed, and seen from the outside,  just too little.

I realized that the main problem was not the lack of symmetry, it was the dinky entrance. The door and lovely fanlight are dwarfed by the expanse of brick, the windows are minuscule. 

Refer to my first picture of the house (above) to see how little the windows are compared to the windows on either side.  

Buckland couldn't change this; he had to work with what was there. 


What were the existing dimensions?
One was the height from the sill of the door to the eaves of the roof. Buckland could add inches by adding a step down, but he couldn't easily go higher.

The second dimension was the width. His width couldn't be so large that it drew attention to the uneven spacing of the left and right windows. And: the porch needed to be centered on the door,to  enhance it and its fanlight. The side windows had to be integral to the design.

He used the given height for his width. He drew a square and added the diagonals and the mid-lines. horizontal and vertical.

He added his Lines - upper center of the square to left and right corners.

The square was divided into thirds where the Lines crossed the diagonals. Placing the columns within the 1/3 of the width gave more importance to the door while keeping the rhythm, as well as keeping the steps wide and gracious, the porch airy and open. If the columns had been set on - not beside -  the 1/3 lines the design would have been static, staid.

The points where the Lines crossed the diagonals also marked the edge of the frieze (also called the architrave).

The mid-lines divided the square into 4 smaller squares. When  diagonals were added to the upper squares, where the Lines crossed the diagonals located the height of the frieze, the beginning of the gable, the roof. In the lower squares the Line crossed the diagonals at the top of the hand rail.

I have drawn these  Lines only the right side. The layout is hard to read  when all the Lines are added. On a framing floor. all the Lines would be marked and used to layout the frame.

The arch draws our attention. The half-round shape makes the porch open and welcoming. It frames the main door and its fanlight - inviting you to the house.

Its diameter is 1/3 the width of the porch. It is part of the whole.

 I've drawn it with its cascading circles because it's fun. I also wanted to note how the circle can be a tool for laying out a plan.

Palladio used a circle as his unit of measure: he called it a 'diameter'. His diameter was usually the column in his drawing. Although it's possible that the circle here was also the unit of measure, I think it more likely that it was one element Buckland knew how to use, one that would enhance and unify the porch with the house, especially as its diameter came from the basic geometry of the porch.    

For an introduction to the Rule of Thirds see: 


An after thought: George Mason used the geometry of the 3/4/5 triangle to layout Gunston Hall, including the dimensions of  his windows. If William Buckland had used the geometry of the square and its circle, its proportions would not have complimented the Hall.


Saturday, May 1, 2021

Saltbox Geometry


I have been thinking about simple house forms and their straightforward geometry. 

I was asked about window placement for a modern saltbox. I had no simple answer.  A traditional saltbox has a door in the middle and a room on each side. The windows are evenly spaced because vernacular construction in the western world was evolving from medieval to renaissance design.  A modern saltbox?  Is it an oxymoron and if not, what would be right? That's the background. Below is the geometry for a vernacular saltbox.

Settlers in the Colonies included a good number of carpenters; men who had finished at least their 7 year apprenticeships. Still every family needed a house. A straightforward plan was required; one that was easily laid out with available tools, like twine and chalk or charcoal - a Line. A carpenter square was useful, but not always truly square; it needed to be checked by geometry.  

The form that developed was a simple vernacular American house - 2 rooms over 2 rooms with a center entrance. Common all over the Eastern Seaboard, it went west with the settlers. The lean-to - its sloping roof the signature of a saltbox - was regularly added to the back, first for storage, later to expand the house.  

 My diagrams here are for a simple generic New England saltbox.

The carpenter needed to decide how wide, how big the rooms would be. He choose the same length for the depth and width of the parlor and the hall. That first length governed all the choices, the placement, the patterns, the dimensions that followed.

He knew the fireplaces and chimney stack would be placed in the middle so he made space for them. Then he laid out a square on one side, and another on the opposite side. This became the guide for his timber frame.


 Here is the sequence that begins with a length and ends with a square. A carpenter knew practical geometry. He knew how to use a straightedge and a Line. He had no ruler or tape measure. He probably began with the 5th diagram. He didn't need a physical compass. His Line could be pinned at one end and then swung in an arc to mark the corners. The resulting square could be checked (trued) by matching its diagonals.


My mythic carpenter choose a room depth of 16 feet. 20 feet was a common length in later houses. The diagram is to scale; it is 16 ft. deep and 40ft. wide.  I have allowed 8 ft. for the fireplaces and chimney. 





The same square - 16 ft. x16 ft. - was used for the height. Here it is divided in half for the first and second floors.

The ridge of the roof frame is half the height of the box - 8 feet.

Sometimes the dimensions for the framing began at the foundation. Sometimes the dimensions began after the sill was laid, made level and true. 

Here are the posts, the beams (called girts), and the rafters laid out. Next will come the summer beam, and then the joists.

For reference I am using Abbott Lowell Cummings' framing from his booklet, Architecture in Early New England. His first diagram is entitled "Typical framing details.." . The second, "seventeenth century house plan"  shows the early fireplace and chimney configurations. (see below)





The windows were centered in the shape, right in the middle. Glass was a luxury in early houses; windows were small.

There might not have been one in the attic.


 Here is the diagram for easily finding the center line of the square. Swing the arc of the length as shown in the first square -dashed red lines, solid black lines. The crossing points are centered on the square as shown with the solid and dashed lines  in the second square,


The front elevation of the house was as simple as the floor plan and the side elevation: 2 squares with the chimney stack in the middle which also gave room for a stair and a entry door.

I've added shading to the roof.


The layout shows the post and beam frame, ready for the summer beam and the floor joists.  All of this could be laid out with Lines, made true and square by the diagonals, which are also Lines. A Line might be chalked to leave a mark ( a Line) on a framing floor, or it was a length of twine pinned in place by an awl, or tied to a stake.




 Here's the front elevation with  one window centered in each room.


 As glass became more available more windows were added.

There were 2 ways to place the windows using the geometry of the Rule of Thirds. Here on the left the square is divided into thirds, the windows centered on that Line. 

On the right the inside edges of the windows are on the Line. 


 Here is a diagram for the Rule of Thirds. The diagonals for the square are crossed by the center line. Then new diagonals - red - are added to the rectangles on either side of the center line. The diagonals of the square cross the diagonals of the rectangles at points which divide the square into thirds.

I have only shown the square's diagonals and the red diagonals in the drawing which shows window placement.  The square with all its Lines can be visually overwhelming.


This diagram shows the squares divided into thirds - the dashed lines. On the left the windows are placed on the center line - dot-dash lines. On the right the inside edge of the window is on the Line -  solid lines

In a new community settlers from different areas  brought different framing traditions. 2 houses side by side might use different patterns, reflecting the carpenter's background.

The lean-to was an obvious expansion: just an extended roof covering the new space. The space did not always include a fireplace. When a fireplace was added it was laid up against the existing masonry.

 Here is the diagram  showing the carpenter's Lines.

Note that while in a diagram the roof would meet an 8 ft deep and 8 ft high room at the upper far corner, in reality the width of the posts and beams and rafters often made for a lesser height.



The back wing was useful. It often was 10 feet deep.  The diagram shows how the roof pitch changed. If the lean-to were added after the main house was built, the rafters might join the frame under the roof. That would also lower the roof pitch. 


Here is the window placement;  all are centered on their interior spaces.

The photograph at the beginning of this post shows how owners adapted and updated the basic house. More windows were added on the sides. The front windows were sometimes enlarged. Columns and an architrave with molding were added to the front door. After 1780 front entries were added to many houses.


The rhythm. pattern, proportions - the geometry, including the window sizes, of a Georgian saltbox came from its construction, the available materials, and its function. It used timbers and hand tools to create shelter. It did not need to accommodate bathrooms or closets, nor provide much privacy. I think a modern saltbox, built with modern materials and tools for 21st Century life, will need its own rhythms, patterns, and proportions.




This is Abbott Lowell Cummings' first illustration in his pamphlet, showing the frame I have laid out  - a center chimney with a room on each side on each floor.


The main floor plan shows how the lean-to was added and used.  This plan, common on the New England seacoast, came with settlers to western Massachusetts,Vermont, and upstate New York. It is often inside what appear on the outside to be Federal and Greek Revival houses.  The chimneys move, the ceiling are taller, the stairs more gracious, but the floor plan remains.





Abbott Lowell Cummings' plan is not as 'square' as my diagrams. Here is the geometry. The Hall on the right is a square room. The dashed red arc is the width of the room transferred to the length.  

The width of the Parlor matches the Hall, but its length is shorter. It is determined by where the arcs of the width, the red dashed lines, begun at each corner of the fireplace, cross.   

The layout for the house, its rooms, begins at the chimney stack. It seems to have been placed first, the house framing against and around it.

The back wing is set true with the house using the 3/4/5 triangle which will always have a square corner,  red dotted lines. Here the wing is square, 'true', with the existing house.

* Abbott Lowell Cummings, Architecture in Early New England, Old Sturbridge Village Booklet Series, Sturbridge, MA, printed by the Meriden Gravure Company,  Meriden, CT. 1974.

 In this post I have capitalized Line because those who wrote pattern books capitalized it, and because the Line creates the design.
The name 'saltbox' was given to these houses in the late 1800's, by New Englanders who had salt boxes of  a similar shape in their kitchens. In the southern US, these roofs were/are referred to as 'cat slides'. The saltbox houses whose geometry I have studied have wonderful variations and quirks. Often these are due to the changes in  fireplace, bake oven, smoke chamber, flue, and chimney construction technology. 

The photograph of the Kimball House comes from the archives of the Andover Center for History and Culture. 


Monday, February 1, 2021

The 'Cube' in Albrecht Durer's Engraving, 'Melancholia', 1514

This is the kind of research that happens when one is self-isolating through a winter pandemic.   

Albrecht Durer's engraving 'Melancholia', was first printed in 1514. 

 I wrote about the carpentry tools scattered around the edges of the image in my previous post: "Albrecht Durer's 'Melancholia' and His Knowledge of Construction and Practical Geometry", posted on 1/17/2021.

On the left side in the middle is a polyhedron, the 'Cube'^, a 3-d shape.The angel is studying it intently. Why? And what is it?

And why the sphere? A pure geometric shape, white, abstract, perfect, set among real tools we can identify, and a skinny dog. It is the only other abstract thing besides the Cube in the engraving. 

I have some suggestions based on Durer's knowledge of and interest in geometry.


Durer wrote about geometry in his book, Underweysung der Mesang mit dem Zirckel und Richtscheyt,  published in 1525.*   The last 6 words of the title translate as Measurement with Compass and Straightedge


 One of his diagrams:


I copied the pattern of triangles, cut it out, and folded it into a  20 sided polyhedron, (an icosahedron,) fastened it with tape. 

Note: The 5 triangles on the right that would complete the shape are not taped together; it fits in my hand.

If the shape were made from wire it would look like the drawings beside the pattern - the view from the sides and then from the top or the bottom. The lines are the edges of the polygons. 



Another of Durer's diagrams is 12 joined pentagons which become a polyhedron. To the right are the side and top views. The views are not elevations; they are transparent.  


Again, I copied it and cut it out.




It folds up to be much like a soccer ball with edges.

Durer's book contains many of these polyhedrons. He mixes the kinds of polygons to make the 'sphere'. Here are 2 of his simple ones



The models led me to consider the Cube in Melancholia as a study about how to create a sphere from planes, from polygons. 


I extended the main sides of the polygon - red lines - so that the shapes became diamonds. Then I extended the edge of the almost invisible left side and found its tip met the main side at the top. 






Next I divided the length of 2 sides of the diamonds in half. When I joined the mid points with a red lines  I saw that they followed the edge of Durer's polyhedron.




Perhaps Durer was drawing a truncated cube in perspective.  I made a diagram: 6 squares which if folded would make a cube. I lopped off 6 corners  - red dashed lines




I added the triangles, printed it, and cut it out.





Cut and fastened with tape, it is flimsy, not  solid like Durer's block.





Tipped on its side, it has the planes of Durer's 'Cube' - his 8 sided shape.











Historians tend to title the book, Measurement with Compass and Ruler. In 1525, rods with 10 parts were common, so were boards with straight edges, used for drawing straight lines. Regular, agreed upon dimensions, such as would be on a ruler, did not exist. Note that there is no ruler among the tools Durer includes in his engraving, 'Melancholia'. 

^ Durer's polyhedron is not a cube. I use the name as an easy reference.

*Underweysung der Mesang mit dem Zirckel und Richtscheyt can be read online through the Warnock Library in Nuremberg, Germany.  A translation by Walter Strauss, published in 1977, is out of print, only available at museum and college libraries which currently are closed.

At any time, but especially in the year of Covid-19, to read a 500 year old book, housed 3800 miles away, while sitting in my office with my cat sprawled out beside me, is remarkable. That I can shift from one page to another and back again as I consider the images is an amazing gift.

Sunday, January 17, 2021

Albrecht Durer's 'Melancholia' and his knowledge of construction and practical geometry

Note: click on any image to enlarge it!

Albrecht Durer was a painter and print maker in Germany, 1471-1528. He also wrote books, and traveled widely in western Europe. He was a superb draftsman. I have enjoyed the compositions, the details, and the lines in his engravings and woodcuts for 50 years.

His woodcuts were for the people, most of whom were  illiterate. Their livelihoods did not require writing. They could read the images: here the rough stable, the well dressed men coming to see a baby, the angels and the star. 

The engravings were not so easy to make. They were for books, for people who could read.

These plates come from the Dover Publications reprints of Durer's wood cuts and engravings.* This is print #183.



He drew what he saw around him. His plates are full of the life he knew, including construction details. 

This detail from #183 shows a truss which includes a collar tie with angled tenon joints and pegs. The purlins and rafters are structurally correct.



The detail from Plate #190, shows the grain of the wood brace running in the right direction. The angled cuts for the joints could serve as templates for repair.


    Here in Plate #185, the brace is tied. a peg serves to tighten the tie as needed. The thatch for the roof is properly applied.

    When I began to write this post I was thinking about Durer's knowledge of construction and his use of geometry in his compositions. (More about that in another post.)
    I was side tracked as I began to read Durer biographies. The scholars who wrote them rarely knew about construction. To them the structures are allegories or useful for his compositions. 
    I saw practical and abstract geometry.
    This is my exploration, as a Geometer, of one of  Durer's most important engravings.

    'Melancholia', Durer's engraving about the temperament of  artists and  artisans, was made in 1514.*  

    Durer put many construction tools around his melancholy angel. She holds a compass.  Tucked beside her hand is a gauge. The putto sits on a mill wheel next to a ladder. Set beside the polyhedron is a pot for hot liquid metal, possibly lead, on a brazier. Under the angel's skirts is a pair of pliers.

    The tools are used in practical geometry. Above the dog is a hammer. Between his hind legs and the sphere: a Line and its plumb bob.

    In the lower left corner is a profile gauge.

    Then: a plane, a saw, and a straight edge that can be used to draw arcs. It may also be a level.

    Lastly: nails and a nail punch.

    The theory was that Melancholy, influenced by the planet Saturn, was part of the inherent character of artistic and philosophical people. There were 3 levels. The lowest was artists and artisans. The next was scholars, natural scientists, and statesmen. The highest level was theologians and those who studied the secrets of the divine.
    Here the putto is taking a nap after doing some numbers (perhaps) on a slate, the lowest level. The angel, on the other hand, is intently studying the polyhedron, an abstract shape. She is a scientist. The dog  - which I learned represents 'faithfulness' - is waiting. The sphere is an abstraction, perfect as the tools and mill wheel are not. They are all 'things', 


    The scales, the hour glass, the bell are said to refer to the knowledge of artisans: they understand weight, measure (of time), music (as an expression of geometry). 


    I have very little understanding of the Magic Square. I do know that all the lines add up to 34 and no number is used twice.

    I do wonder if in a largely illiterate society, the fairly recent acceptance of Hindu-Arabic numbers, as opposed to Roman Numerals, might be part of its purpose here: to illustrate the scholarship, the mathematics that numbers made possible.

    Try substituting Roman Numerals in the grid of the Square.
    Can you quickly add up the amounts? How would you do it as an arithmetic problem on paper? 

     16 +5+9+4 = XVI+V+IX+IV .


    I will write about the polyhedron in the next post. 


    *images from  The Complete Woodcuts of Albretch Durer, edited by Dr. Willi Kurth, 1963  and The Complete Engravings, Etchings & Drypoints of Albrecht Durer, edited by Walter L Strauss, 1972.  Both republished by Dover Publications: www.dovderpublications. com.

    Walter L. Strauss' analysis is the best I've found. I wish he had known more about geometry.