The Geometry of Quatrefoils

 

 

Yes! What a quatrefoil! 

What amazing craftsmanship! What exuberant geometry! The interrelated series of curves are such fun to study. 

The fretwork is part of an Episcopal church in Cody, Wyoming,

The photograph, by Travis Wade, Cabinetmaker, is shared here with his  permission. His shop, the  Atlas Cabinet Shop. LLC , is repairing/reconstructing the fretwork. (It is not nearly as pink as this image.)   

I, of course, was especially interested in the  geometric shape -the quatrefoil - at the center of this fretwork, the basis of the design. 

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Here, because the layers of  'tendrils' are an important part of the design, the layout of the circles with space for 2 rows of tendrils had to be the first concern. (A). Would it fit? How?
Then the size and profiles of the tendrils. 

Lastly: the size and shape  of the quatrefoil: should the foils be circular, oval,or ellipse? (B)

 

 Here is a  simple quatrefoil:  a shape: 4 lobes on the ends of crossed lines. The word in French means '4 leaves'.  

 

 

 

 

This form is one variation, the lobes are 1/4 the width of the cross.  

 

 

 

 

Or: the lobe is a half circle which if completed would be a circle.The design is 2 circles in width and 2 in height. 

 

 

This 13th C. stained glass window in Chartres Cathedral is a quatrefoil with another inside it. The inner quatrefoil overlaps the outer lobes.    

There are about 40 quatrefoil windows in Chartres, some small spots of light and color, others part of the larger windows. These quatrefoils surround the story of Adam and Eve.*   

 

 

 

 

 

 

Here is another quatrefoil. Its layout is more complex.

 

 

 

 

It's used here in the fence at the Abbey of Saint-Denis, Paris, France.** 

  

 The Abbey has been there since the 5th C. CE,  but the structure we see today was mainly built in the 12th and 13th C., It's been rebuilt regularly since then.

I have no date for the fence. Maybe Second French Empire 1852-1870?  

 

I imagine the ironmonger measuring the length of the fence, dividing it into segments; then laying out his grill sections in each segment.  

He knows what he wants to create.

The square is the base for his geometry. The welding points on the fence  connections are at the centers of the sides of each square. All his pieces  - the leaves of his quatrefoil - will be the same.  He will attach each one at the center of the rails, and to each other at the arrows. 

 

 

 

 

He needs to lay out the shape of the arcs. Using the rule of thirds he divides his sections into 9 squares, 3 across and 3 tall.

The diagonal of the square is the diameter for the  circle he will use for his lobes. The lobes are bigger than the squares and would overlap. (See the next diagram.) So they are slid toward the center. 

 

 

Finally he  joins each square to its neighbor where they intersect and adds an arrow over the weld.

The design of this quatrefoil is much more complex than the first one. But visually it is much less intense. It is just a fence using 13C. visual language.  

 

The layouts comes with a caveat: The images from Chartres Cathedral, the Abbey of Saint-Denis and the church in Wyoming are all photographs.They were taken at an angle. I cannot check my geometry with  dimensions.  

Wikipedia has an excellent review of the shape's  religious connotations in many cultures.  But good images and  measured drawings of quatrefoils have been  difficult to find. 

*Chartre Cathedral, France, from the book Chartres Cathedral, text by Malcolm Miller, photographs by Sonia Halliday and Laura Lushington,  Riverside Book Co., NYC. 1996

**This image comes comes from the Buffalo, NY, Historical Museum website. The photographer is  Chuck La Chiusa.

 

The Practical Geometry of Boston's Old North Church

3/26/2026: Revised.  

The Old North Church, in Boston's North End, was built in 1723.*

The church fathers responsible for its construction had visited England where they had probably seen the churches in London built after the Great Fire in 1669. Those churches were built using the new Classic architectural style which included half circle windows - as are in the church - instead of Gothic pointed ones.**

I visit the church when I teach/share/explore Practical Geometry with the North Bennet Street School Preservation Carpentry students. We, along with the instructors, see its construction first hand, from the crypt with its tombs to the timber roof frame in the attic.    

 This year I explored the church's geometry. The Historic Structures Report for the church, written c. 1970, includes floor plans which trace the evolution of the church interior. These are not HABS drawings, not as detailed and possibly not as accurate. This is, therefore, a preliminary exploration.

The exterior is brick. The floors and and roof are timber framed. One of the church fathers seems to have overseen the master mason responsible for the foundation and walls. A master timber framer was responsible for the roof. All of these men would have known Practical Geometry.*** 

 

 

Drawing #1 shows the church interior layout in 1730. The entrance and apse are extensions of the main space. The pulpit with its curvy stair is drawn to the left of the apse. In later years the pew locations, the stairs and some exterior doors change. 

The geometry of the exterior walls is simple. It is the 3/4/5 rectangle. The 2 3/4/5 triangles and the rectangle are outlined in red.

 

 

 

 

 

In Drawing #2 the black diagonals of the Rule of Thirds cross the interior. Where they cross are the points**** which locate the red vertical lines which divide the space in thirds. These lines layout the inside width of  the apse and the entry.  

The apse plan is a half circle. The entry is square. Both are drawn in red. The  church wall is a little thicker at the entry to support the steeple.  

The distance of 2 of the columns from the back and/or the front walls is also set: see the dashed red line at the 2/3-1/3 division of the church interior.

Drawing 2A: 

Add the remaining lines for the Rule of Thirds, see the left side lines in red. The points 'C' - where the lines cross - are the columns' distance from the wall and from each other.  The lines are beside the columns so they can be reference points during construction.  

My line B, noted here and on Drawing #2 is nice but not necessary.

I drew only the left side, as too many lines at this scale are visually confusing. The right side matches the left. 

So, layouts #3 and #4 are unnecessary.  #5 is needed: it locates the windows. 

Drawing #3:  The square laid out in Drawing #2 by the Rule of Thirds can also be divided horizontally by the same Rule - see the black lines. The points allow 3 equal divisions across the church - the red lines, labeled A. 

 

 

 

 

 

Drawing #4: By adding  the Rule of Thirds' vertical lines  - again the black lines - the points are found on the red lines (A). These points locate the columns on both sides of the body of the church: the vertical red line which crosses the A lines at B to mark the distance between the 3 columns on each side along the length of the church. The fourth column, located in the space at the bottom of the drawing, uses the same spacing as the other columns

   

 

Drawing #5: The windows were centered between the columns. 

In a time where candles and  torches were the only artificial lighting available, placing the windows for maximum natural light was essential.   

The arcs in black  on the right side locate the center line (C) of the window. 

The center of radii for the arcs is at the columns. IE: place your compass point on the edge of one column; swing an arc. Swing a similar arc from the opposite column. The line drawn by the 2 points where the arcs cross will locate the center of a window. 

*The church is best known for its steeple where Paul Revere hug his lanterns on April 15, 1775, warning the citizens of Massachusetts Bay by which route the British soldiers would march to Lexington and Concord.

 **This drawing is by Suzanne Carlson, from  John Fitzhugh Millar's book, The Architects of the American Colonies, Barre Publishers, Barre, MA, 1968. Used with permission from John Fitzhugh Millar.  

*** In 1723 someone who designed buildings was not called an architect. James Gibbs who designed and supervised the construction of St. Martin's in the Fields in London, was chosen by Commissioners to be "the Surveyor of that work" . See Gibbs' Book of Architecture, London, 1728, p. iv. There seems to be no use of a similar word in the church record.

**** Remember that in geometry, Practical or otherwise,  a Line cannot be drawn without 2 points. To learn about the Rule of Thirds, see  https://www.jgrarchitect.com/2020/08/lesson-6-rule-of-thirds-part-1_21.html

 

 

 

 

  

The layout of an Italianate window pediment, Part 2

Last September I wrote about duplicating this ogee molding  - the continuous double curved casing above the triple windows in the center  of the photograph. 

 

Here's the link to the post: 

https://www.jgrarchitect.com/2025/09/the-layout-of-italiante-window-pediment.html 

  

 

Nathan Goodwin, the master carpenter for the reproduction, and I worked out the geometry. We didn't quite agree about the steps. That was interesting but not important. He laid out the curves and created the pediment.

 Later Nathan found an article, written c. 1850, by the Episcopal Bishop of Vermont for his clergy, recommended the use of this curved pediment for Vermont parish churches. The Bishop included a drawing of the geometry. It seems unlikely that a church man, even one very well educated, would have designed this. Where had he found it?  

Perhaps the Bishop had seen Peter Nicholson's  Carpenter's New Guide, 10th Edition, published in Philadelphia in 1830. 

Peter Nicholson was a British engineer and scientist. He wrote numerous popular books for master carpenters and builders.  To learn more see this post: 

https://www.jgrarchitect.com/2016/08/practical-geometry-as-described-by_16.html 

The frontispiece of the 10th edition says that the New Guide is "a complete book on lines, for carpentry and joinery, treating fully on Practical Geometry".  

Then after 6 lines describing what the book includes, it say,  "THE WHOLE FOUNDED ON TRUE GEOMETRIC PRINCIPLES"  

This page is fascinating in itself. It reads like an advertisement for the book. And there are 11 different fonts in several different sizes.   

 

 

 

Our pediment is on Plate 10, Page 25. The Plate is described as "The Sections of a Globe....also, the section of any figure standing on an irregular base". 

 

Our diagram is part of Figure 6, in the lower right corner, the geometry used 'To find the Ribs of a Gothic Niche, being the Plan, and No. 1, the Front Elevation" (p.27) The elevation (above) and the plan (below) are in the lower  right corner. The 4 diagrams to the left  describe how to layout each rib. 

 

 

  

 

 Here is a larger image of the niche elevation.

 

 

Nathan and I  were only laying out a molding to go over windows, not a niche. But the upper diagram is what we 'discovered'.   

 

 

 

 

 

 

Here's my rough diagram.  

I teach Practical Geometry but I am still an apprentice compared to Nicholson's mastery.

 

Update 2/26: Thanks to 'old pro millwork' on Instagram, we know this geometric layout goes back to before 1758.This diagram is in The Carpenters and Joiners Repository, by William Pain, published in London, in 1758. Pain prints it again in his later pattern books.  

I will add the diagram.

Asher Benjamin's pattern book in Bennington, Vermont

 

In 1838, Judge Luman Norton, his daughter Louise and her husband Christopher Fenton built this 2 family house in Bennington, Vermont. The side entrances are original; their porches, contemporary.* 

Hiram Waters, Master Carpenter in Bennington, probably designed and built the House.  The record shows that he bought a copy of Asher Benjamin's The American Builder's Companion, 6th edition, published in 1827.** 

Based on my research I think he also owned Benjamin's next book, The Architect, or Practical House Carpenter, 1830**.

Waters used Benjamin's molding profiles as well as his drawing for columns for his own house and carpentry shop. He also used them for this house.

 

The porch roofs and columns in front of the side doors  are 1950's +/-additions. Except for the storm doors, the entries are original. 

 

 

 

 

 

 

The pilasters on either sides of the doors match those in Benjamin's  in The American Builder's Companion.  All the parts are there.

 

 

 

 

This is Plate E,  Ionic Columns 

 

 Here is one of the main doors from the inside. 

  The Norton-Fenton House is currently closed to visitors. These interior photographs were taken in 2017 or earlier. 

 

 

 

This is Plate XLVII from Benjamin's The Architect, or Practical House Carpenter .   The lower casing profile matches that of the house.                                                           

 

 

 Here is the  base of the casings around the front door and its sidelights.

 

 

 

 

 

And Plate XLVIII from
The Architect, or Practical House Carpenter.

 

                       

 

 

 

 

 Finally,  one of the two curved staircases in the Norton Fenton House.

Benjamin uses 9 pages in his 1827 pattern book, 11 in his next book, to describe how to layout and finish these stairs.  

 

 

 Here are details on cutting the face moldings for the sides of the planks used to build the stairs. PLATE LVI, The American Builder's Companion. 

 

 

 

I am always impressed that a carpenter could 'read' these drawings and build from them. Hiram Waters was surely a superb Master Builder.   

 

 

 

 

 

*I've written about the design of this house in an earlier post:

https://passingbyjgr.blogspot.com/2017/05/the-4-front-doors-of-norton-fenton-house.html 

 *I've also written about Hiram Waters, 1797-1890:  

https://passingbyjgr.blogspot.com/2020/10/hiram-waters-workshop-monument-avenue.html 

 

** The Bennington Museum's Wallomsack Review has an excellent introduction to Hiram Waters. https://benningtonmuseum.org/wp-content/uploads/Walloomsack-Review-14-Autumn.pdf

 The unabridged 6th edition of The American Builder's Companion and The Architect, or Practical House Carpenter have been republished by Dover Press.

 

The layout of an Italiante window pediment

 

The curvy Italianate molding above that window in the center?  How did they do that? 

How did carpenters in south eastern Massachusetts in c.1850 lay out the picturesque pediment so it could be cut and assembled?  

 

 

This was the question posed to me by Nathan Goodwin of H.I.S. Construction. He was asked to lay out a copy of that molding for installation above two garage doors. 

Nathan and I posted ideas back and forth. Nathan needed to draw and define the curves, especially how the arc over the shuttered windows evolves into the arc which ends in a point above the main window.

While we focused on that curve I wondered how the master builder laid out the whole design. What geometry might he have used so that the pediment complimented/completed the windows? So it is integral, not just stuck on the top?  

 

 

 

 

 

 

The first task was to see the form that was there: 

A center window flanked by 2 skinny windows, now shuttered. Together with their casings they form a rectangle. The 3 windows' tops are half circles. Around and above them is the embracing molding which follows the semi-circles before it swoops up with a reverse curve to meet at the center in a point. The molding seems to be the same width throughout.  

We saw that the pediment was derived from the windows. The windows and their casings were a rectangle, 4 units wide. Note the red  rectangle and lines below the window sill.  

 

The curve for the section of the pediment over the side windows was easy to see and draw. It's the extension of the small windows' half circles. The red dashed line and the dotted lines drawn here radiate from the center of those shuttered windows. I shared this with Nathan. 

 

The pediment's height over the center window was also easy to find - see the dashed red arc on the right. The half circle's radius is half the width of the windows.  Note the black dashed line. It's a reverse curve. Where was its center? How was it generated?

The white-out on the image comes from us exploring and rejecting options.   

Nathan and I shared ideas back and forth. He posted this suggestion: a layout based on the width divided into 4 units.  

 He extended my 4 units into rectangles, divided the rectangles themselves into 4 parts and used 3 units as the radius for the molding hoods over the shuttered windows. 

The sides of the rectangles cross the semi-circle. He added diagonals from that point to the center of the top of the main window. The crossing point became the center of several exploratory circles for the reverse arc curving to the pediment's center.

 

 

Nathan ended up with this diagram. It worked for the width of the garage doors; he could lay out the curves and cut the parts. 

 
 

 

 

 

I wanted to know about the original design - the layout of windows, casings, pediment with moldings. How might the master builder/ joiner/carver (I don't know his title) have laid out the design? 

 Here's what I saw.  Nathan's division of parts works across the width of the window. The whole width is 16 parts/units.The center window is 6 parts wide; the casing on each side is 1 part. The side windows are 3 parts wide; with one part on each side for the casing.  

Nathan's geometry for the reverse arc over the main window also works. The radius of the center window's arched top is 3 parts, with the casing: 4 parts. This doubled is the diameter of the circle which draws the arc of the  pediment.

More or less! The craftsman left us no notes. And: I am analyzing from a photograph of the window, not the real thing.   

 

 

   

   

 

 

 

 

 

 

 

Asher Benjamin explains how to make stucco

 

First, a brief introduction to stucco. 

Are stucco and plaster the same thing? Not now.Today stucco is used outside. Its recipe will be different from plaster used inside. In 1814, the names referred to the same thing.  

  

These cherubs are plaster. They cavort in the dining room of the Canfield Casino of the History Museum in Saratoga Springs, NY.   

 

 

 

 They are forever in motion on the frieze of the entablature, 

 just below the deep cornice, 

a counterpoint to the circular window ringed with energetic vegetation centered under the coffered barrel vault above.  

Lots and lots of white plaster. Made from lots of molds. Mass production allowed this extravagance in 1902.

 

 

 

100 years earlier moldings were made from wood. Plaster sealed the wall. It was held in place by lath.   

Plaster was lime, sand, hair and water, mixed by hand. 

Its keys fit into the spaces between the lath, holding the plaster in place,  allowing the plaster to create a 'wall'. That plaster kept out drafts, reflected light and created a space, a place. It still does.  

 This plaster is applied to lath which was split by hand, thus the uneven shadows.   

 

This image of the back side of lath shows the plaster keys pushing though the spaces between the lath.  This lath is even because it was cut by a circular saw. This plaster wall would be much smoother than the image above. 

 

 

 

Asher Benjamin shared the latest uses of plaster with aspiring Master Builders in his book, The American Builder's Companion. 

It was first published in 1806, then updated in 1814 to include stucco ornaments (Plate 36,  prefaced by 3 pages (74-76) of instructions, titled Plate XXXVI.

 

 

 

 

Benjamin was a Master Builder with apprentices. He also ran a school for carpenters in Boston. He was skilled at describing how to execute the work. 

 

 

 

 

Here is his footnote describing how to make the stucco. He says to  just mix it up, beat it well every day, and let it cure for four or five days on a brick wall.

Not too much a stretch from plastering a wall, but seemingly far removed from the Canfield's yards of dentil and corbel moldings and the carefree cherubs.

 

He describes the best way to make an ornamental stucco ceiling, by hand. 

Then he explains how to do it more quickly, cheaper. What he describes is the beginning of mass production, ie: the Industrial Revolution, starting us on the path to those cherubs.

For the rest of Benjamin's instructions, please read his notes. He wishes "to give the student every information in my power... and be of no injury to those who are well acquainted with the art of stucco working."*  I think he would be pleased that we are reading his instructions 200 years later.

 

In 2009  I wrote 2 posts about  how Asher Benjamin wanted the builders who would read his pattern books to see and enjoy what moldings could create. He clearly loved the "beautiful variety of light and shade". 

I included the portrait of Benjamin in the first post. Historic Deerfield recently cleaned it. To see it please google 'Asher Benjamin'.


http://www.jgrarchitect.com/2009/01/beautiful-variety-of-light-and-shade.html

http://www.jgrarchitect.com/2009/12/strong-mouldings-and-falling-water.html