Drawing the Square and its Circle - Ad Infinitum

 

Note: updated for clarity 9/26/2026 

Serlio's cube:  It's on the frontispiece of Sebastiano Serlio's On Architecture *, published c.1550, translated into English in 1611. The geometry shows the structure for adding a circle within a square - for example, a dome above a square space. 

 

The lower section of the frontispiece shows the tools a master builder would have used for design c. 1550.  This cube is a tool.

 Serlio's geometric diagram: a base of a Tuscan column. **

 

And his diagram for a Corinthian column. **

 

 

How can these layouts be drawn with a compass and a straight edge?

 

The first step is the geometry of the square.

The diagram is in Audels' Carpenters and Builders Guide.  published in 1928***. 

Next: add the circles that are larger and/or smaller than the square. Below is a sequence.

1) the square with its diagonals

2) Half the length of the square's diagonal is the radius for a circle around the square.  

3) The mid-point of a side of the square gives the length of the  radius for the circle within the square. 

4) The points where the inner circle crosses the square's diagonals determine a square within the first square.                               

 Reader beware: This could go on ad-infinitum. 

 

 

What if the beginning dimension is the diameter of a column?  How can a square with the right proportions - the pedestal, for example - be drawn around that circle?  

The diagram is from Edward Shaw's Civil Architecture* Plate 2. 

 

The length of the sides of the square is the diameter of the circle. But where does that length 'sit' on the circle? 

The radius of the circle (C-A) is half its diameter (D-E). By definition it makes a point at the circumference of the circle (A).     

1) C is the center of the circle. C-A is its radius.

2)  C'-A is also the length of the radius. C'-A-C' is the length of the diameter.  It can be 'trued' using step 3. 

3) The diameter has a length, therefore a point at each end   Both C's can be one end of a line which extends to the point C, (the center of the circle). If they are equal, the line C'-A-C' is true, sitting symmetrically on the circle's circumference.  

4) Those lines, extended, create 2 more C'-A-C's. The length of those 2 lines is the diameter of the circle: 2 more points. The points can be joined to become the square around the circle.    

5)  C- C' becomes the radius for a circle around the square, around the circle.

Too many words! It's much easier just to draw it together.

I've written other blog posts about the use of this geometry in real construction.  

Practical Geometry at MIT, April 9, 2018.

Serlio studies a Roman Temple, April 1, 2023

The Perfect Cube and its Sphere, July 16, 2024 

James Gibbs' Steeples, February 25, 2022    This is squares, not circles.

* Audels Carpenters and Builders Guide, # 2, NYC, 1923, p.494

** Sebastiano Serlio, On Architecture, c. 1550; translated from Italian into English by Hart and HIcks,1996.  illustrations on pages 125, 257, 341 

*** Edward Shaw, Civil Architecture, 6th edition, published in 1852

 

The Geometry of Quatrefoils

 

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

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 was interested in the  geometric shape - the quatrefoil  - at the center of this layout. The word in French means '4 leaves'. It's the basis of the design.

 

The size of the circle was probably the first decision. The circles  needed to fit neatly across the space and be big enough for 2 rows of tendrils inside the lobes. 

 

 

 

Then, how could the tendrils to 'grow' within the geometry? The circles nestled above the row create 'triangles' and paths between the circles which are organic to the design. 

 

This quatrefoil pattern could have been used:  a circle with 4 inner circles, each of which is half the circle's diameter, set on the vertical and horizontal axes of the circle . 

The  half-circle lobes (A and B) themselves are 1/4 the width of the cross.  

 

 

 

 

 

Each lobe completed would be a circle.The design is 2 circles in width and 2 in height. 

 

The size and profiles of the tendrils are fitted into the design. They 'grow'  within the lobes and then up and down along the quatrefoil itself.   

 

And another look at the Cody, Wyoming design : 

 

 

 

 

 

 

 

 

 

 

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. Here the quatrefoils surround the story of Adam and Eve. The center quatrefoil is often left empty, signifying a path to the unknown, existence before birth and after life.  Here it shows God creating Adam. 

 

 

The fence at the Abbey of Saint-Denis, Paris, France.** uses quatrefoils.  

  

 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 welds on the fence  connections are at the center 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 center square ( see left diagram) is the diameter for the  circle he will use for his lobes. The lobes are bigger than the squares and would overlap. (See the right diagram.) So they are slid toward the center. 

 

 

Then  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.

 Because the quatrefoils seen on the internet are mostly diagrams,  I will add the real ones which I find here . 

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. 

  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 side lights.

 

 

 

 

 

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.