The Perfect Cube and its Sphere

 

 

This perfect cube and its square was drawn by Sebastiano Serlio, c. 1540.

It is an Euclidean solid: 6 square faces.  It is 'perfect': each side exactly like the others. A  perfect sphere would fit within it. A perfect circle fits its perfect square face. Another square is within that circle, and a smaller square within that. 

The shapes are bound by the diagonal Lines which create 2 points at the intersections for drawing the next square or circle.
That cube and sphere were not only theoretical ideals, they were practical, a layout tool, the pattern governing a design. The pattern book writers called this 'Practical Geometry'.

 

 

Serlio drew his tools in the lower right corner on the frontispiece to his book, 'On Architecture'. **

The perfect cube is in the lower right corner. His compass is in the middle surrounded by his straight edge, carpenter square, stylus. his line*. 

I've kept track of that pattern: the cube, its circle, the next smaller square and its circle, the diagonals. I want to understand how it is layout tool.

Some of what I've learned is posted here.

Hagia Sofia was built in the early 6th C. by Justinian I, the Byzantine Emperor in then Constantinople, now Istanbul. Earlier churches on the site had burned and the first dome of Hagia Sofia fell in.  The second is still standing 1500 yrs later.

 

 

 

 

 Here is what it looks like from the inside.

This is Bannister Fletcher's diagram** of the dome  formation for Hagia Sophia: the square with its circles. One is around it, the other within it. Their sizes are governed by the square.

This design can be seen in the mosques and churches built after Hagia Sophia in eastern Europe and around the Mediterranean.  The examples in western Europe which I have found are in Italy. 

The shaded areas are called 'pendentives'. There are several ways to build these, not to be discussed here.

 

A dome needs to be held up, of course. When it is the top of a silo, there is no structural problem - the dome is supported all the way around. 

 

 

 

 

But in a church or mosque - where people congregate - the dome needs to be on supports so it is visible. We need to be under it and in its space. 

The weight of the dome must be supported, and its thrust as well.

This diagram from Mosque,**by David Macaulay, explains the problem and show.s the solution in the Byzantium empire: columns (blue) support arches (green) with  cylinders (brown) adding weight behind each arch. At the bottom of the drawing is the floor plan, a square which fits within the circle of the dome.

Other domes had been built. The Pantheon dome with its oculus, c. 120CE, is the best known example. However, the Pantheon's geometry is circular. Hagia Sophia adds the circle's square. Or perhaps the square's circle. 

When Hagia Sofia was being built the Roman Empire was collapsing. Western Europe built little except in  those ports where there was political power, trading, and influence from the cultures around the Mediterranean.

Venice, with its location and port, did flourish. It began to build  St. Marks Cathedral*** in 1000 CE.

 

These drawing of the plan and the interior are from Bannister Fletcher. **

The design shows many circles within their squares.

The large circles are domes, seen here from above. The square bases are visible too.

 

The small circles are the arched stone work of the columns which are shown as 4 black cubes around each circle within its square.

 

The ponderous columns are divided into 4 piers which makes them appear less massive and intimidating. They join at the springing points to support the arches.

This photograph is from Laurie Smith's' book, The Geometrical Design of Saint David's Cathedral Nave Ceiling.**

 

It's the ceiling under the new (c. 1535) roof for the cathedral.

The pattern is squares and the circles set side by side but not in a simple repetition. Laurie's compasses show the layout.
This ceiling pattern is obviously geometric but it is not in the lineage of the designs of Hajia Sofia or Serlio. It is, to quote Laurie Smith, " ...an exceptional carpentry idea and one that was unique to Wales.

These are the geometries used  for the pendants in St. David's Cathedral, as documented by Laurie Smith. The first 4 are based on the use of a compass, the next 2 on a diamond and a square. The last is related to, and perhaps growing out of, the circles and squares in ceiling pattern. 

 

These dome elevations and plans are part of William Ware's book, American Vignola, published in the States in 1903.**

 He describes the dome on the left (C) as "being generally a full hemisphere, constructed with a radius less than that of the sphere of which the pendentives form a part."  

If the same dome is erected upon  a vertical cylinder, visually a band below the dome, it is a 'drum' dome. Here: the dome on the right (D).

I have wondered why he did not recognize the lineage of the perfect square and its circle. He knew geometry.

 

The drum dome is the plan and elevation of the main dome at Massachusetts Institute of Technology, built in 1916.

 The glass blocks which fill the oculus of the MIT dome are set in Serlio's  pattern: the circle is the outer shape with its square and its circle set within it.

 To see the glass of the oculus, please follow the link, as the photographs are under copyright. https://capitalprojects.mit.edu/projects/great-dome-skylight-building-10 

 

A similar dome was placed over the Massachusetts Avenue entrance at MIT, built the the 1930's.

The glass curtain wall that faces Mass Ave is naturally based on the square and its circle. However the overlap of the square within the circle is not a simple repetition of square set next to square. The band between the squares  is a simplification (no curved lines) of the complex pattern seen in Saint David's Cathedral.

 

 

 

 I will update this post as I learn more.

 

*The tangled Line with its plumb bob is in the lower left corner. It can be tied to something and held taut with a plumb bob on the other end. It is not perfect. It is how we attempt to build perfectly, with no mistakes.

**  frontispiece, On Architecture, Sebastiano Serlio

**page 281, 288, A History of Architecture on the Comparative Method, Bannister Fletcher 

** page 11, Mosque, David Macaulay. 

** pages 7,15 and 31, The Geometrical Design of Saint David's Cathedral Nave Ceiling, Laurie Smith. Laurie's book can be purchased through me, as well as through the Carpenters Fellowship in the UK. 

** page 88, The American Vignola, William R. Ware

For more information please see my Bibliography: https://www.jgrarchitect.com/2022/03/a-bibliography-for-my-traditional.html

***I have lost the name of the engraver for the image of St.Marks. I don't know where I found it. The pictures of Hagia Sophia also cannot be credited. If you recognize what publication they appeared in, or who made the images, please let me know.

How to Layout a Pediment: 350 years of instructions

A short history of classical pediments in the Western world,  c. 1540 to c.1903. 

Vignola's Rule was first laid out by Giacomo (Jacopo) Barozzi da Vignola in his Cannon, published in 1562.

This image was  published in 1903 in William Ware's The American Vignola.* 

 

Was it Vignola's  rule or did he just record it? 

It's possible that the Rule itself was already widely known.  

 

In 1540 Sebastiano Serlio drew this sketch in Book IV of his On Architecture.* 

He wrote, "...
having drawn the cornice, divide the upper line from one side to the other in the middle, between A and B; drop half of this plumb from the middle to make C; then placing one compass point on C and the other on the side of the cornice A, arc to the side B; the highest point of the curved line will mark the required height for the pediment. A curved pediment can also be made with such a rule."*

The diagram shows the actual twine held tight at Points A and B. It is a Line with its ends dangling and curling.

 

Palladio doesn't describe this, but the roofs in his The Four Books of Architecture (1570) use the same pitch. 

 

Vignola's book on architecture was translated, of course.

This image, c. 1600, attributed to Vignola, comes from a book published in Spain.

 

 

 

 

 

 

His 5 Orders of Architecture* were translated into English and published by John Leeke in 1669. Vignola's portrait (Plate I)  is surrounded by a decorative frame topped by an extravagant  pediment.  It does not quite follow Vignola's Rule. Follow the red lines.

 

 

 

 

 

 

John Leeke' book  included 3 pediments attributed to 'Michel Angelo'. This one, the simplest, does seem to use Vignola's formula. The angle is the proper 22.5*. Did Michael Angelo know of Vignola's work?

 

 

 

A complete English translation of Serlio's 5 Books on Architecture, including the sketch of the pediment's layout, was available in the UK about 1720. 

James Gibbs probably had read both Vignola and Serlio.  He uses the geometry in the pediment of a Menagery, in his book, On Architecture*. Perhaps Gibbs includes the knowledge of this diagram when he writes that his 'draughts ... may be executed by  any Workman who understands  Lines'. 

William Salmon's book, Palladio Londinensis*, published in the 1750's, was intended for the London builder. Pediments are included; their proportions are measured in parts. Perhaps Salmon considered the understanding of geometry by London's builders to be scant, or that it was not applicable to London's tightly set row houses.

 

 

The rule for laying out a pediment came to the States with craftsmen as well with their pattern books. The Rockingham Meetinghouse, finished by 1797, is a classic New England meeting house:  plain and unadorned...

 

 

until you look at its doors. The pediments of the classical frontispieces follow Serlio's layout. 

 

I began my diagram here on the lower edge of the pediment's frame. If the layout is moved to the upper edge of that plate, the arc marks the top of the ridge of the pediment roof, rather than at the underside.

 

 

At about the same time William Pain includes the same arcs (lightly dashed here) and describes how to draw them in The Practical House Carpenter,* a pattern book widely available in the States.

Here is Asher Benjamin's simplification of Pain's engraving in The Country Builder's Assistant*, published in 1797.

 

 

 

 

 

 

This door front in Springfield, VT, c. 1800, was probably inspired by Benjamin's illustration.

 

 

 

 

 

The Industrial Revolution brought new tools and materials. Galvanized metal allowed shallow roof pitches which didn't leak.

 

 

 

Here's an example of a shallow roof pitch from Samuel Sloan's pattern book, The Modern Architect*, published in 1852 .

 

 

 

 

Sloan's Plate XXXV shows the steeple structure. The frame spanning the building uses the traditional roof pitch of Serlio's pediment (22*). The pitch of the roof is much shallower (15*)

For the next 50 years architectural style was giddy with the designs made possible by the Industrial Revolution.

 

 By 1900, the design possibilities made possible by 60 years of industrialization were taken for granted. Architects looked to Europe, especially the Ecole des Beaux-Arts in Paris, for inspiration, and perhaps a re-grounding in tradition.

 

William Ware wrote The American Vignola in 1903 as a guide for his students at the Architectural School of Columbia University. 

Here's his Plate XVII with many pediments. The dotted lines compare measured/built pediments in Greece and Rome to Vignola's standard.

 

 

 

 

 

 

 

 The 2 small diagrams on the right side Plate XVII are his codification to Vignola's Rule. 

 

 

 As I compiled these illustrations I found 2 caveats:

Owen Biddle* published his pattern book in 1804. Instead of a general rule for 'a pitch pediment frontispieces', Biddle wrote that the roof angle was 2/9 of the span. He also wrote that for 'the townhouse with a narrow front... the true proportions of the Orders may be dispensed with..." p. 34

William Ware noted that "...if a building is high and narrow, the slope needs to be steeper, and if it is low and wide, flatter." p. 45

And finally, this advice from William Salmon: "...when you begin to draw the Lines,... omit drawing them in Ink, and only draw them with the Point of the Compasses, or Pencil, that they may not be discovered when your Draught is finished..." p. 103.

 

*The books from which I have copied illustrations and quotes for this blog are listed in alphabetical order by author.

Asher Benjamin, The Country Builder's Assistant, Greenfield, MA., 1797, Plate 10. 

Owen Biddle, The Young Carpenter's Assistant, Philadelphia, 1805, Plate 15. 

James Gibbs, On Architecture, London, 1728, introduction and Plate 84.

William Pain,The Practical House Carpenter, fifth edition, London, 1794,  Plate 38.

Sebastiano Serlio, On Architecture, , c. 1545, Book IV, page xxviii.

Samuel Sloan, The Modern Architect, 1852,  Plates XXVII and XXXV.

WilliamWare, The American Vignola, W.W. Norton & Co., NYC, 1903, Plate XVII.

 

 

Teaching Practical Geometry

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

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

Here is what we did:

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

 

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

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

 

 

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

 

 

 

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

 

 

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

 

 

 

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

 

 

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

 

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

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

 

 

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

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

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

 

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

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

 

 

 

 

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

 

This was more than enough for one 75 minute session. 

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

 

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

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

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

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

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

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

 

Serlio Studies a Roman Temple

 

 

 

Sebastiano Serlio, 1475-1554, wrote 7 books 'On Architecture and Perspective' .

A contemporary of Palladio and Vignola, he spent much of his career in France working for King Francois I. The first part of his treatise was published in 1537.

Here he is with his compass.

 

 

 

The cover of Book I includes this drawing of  builders' tools across the bottom, including a tetrahedron and a cube with diagonals, squares and circles on its face, in the right corner.*

What's the cube about? I didn't know, but I am beginning to find out.

 

 

Book III, On Antiquities includes the illustration and measured plan of this temple outside of Rome, now thought to be the Sepulchre of the Cercenni. 

He writes that it was "built partly of brick, partly of marble and to a large extent ruinous."

To read the geometry look at 

1) the square, its diagonals which mark the outside of the temple including the bays;

2) the circle which fits within the square which mark the corners of the temple itself;

3)  the square which fits within the circle locating the outside of the walls.

Rotate the first square 90* to make an 8 pointed star. The intersections of the stars points mark the outside corners of the bays.

The inner square (barely visible in red) was not used.

 

Here is the how the  master mason could have used geometry to lay out the plan on site.

The red square was probably the beginning. It is the foot print for the walls.Then the diagonals, the circle around it, and the next square were added. These set the depth of the bays.

Next the outer square was rotated. It crossed the large square at 8 points. Those points when joined laid out the width of the bays.

The lines set the perimeter of the plan. The mason had his foundation plan and could set his lines.

 I usually find that the interior geometry of a masonry building is laid out from the inner side of the walls. This is practical: reaching over the walls with lines would not have been easy nor accurate. 

The square and its circle neatly locate the columns which support the vaulting.

* Serlio is my favorite writer/architect from the Italian  Renaissance. I have posted about him in this blog several times.  The cover of his Book I included the tools he and his contemporaries used: www.jgrarchitect.com/2020/08/lesson-6-rule-of-thirds-part-2-serlio.html

Try this one too: https://www.jgrarchitect.com/2022/10/serlios-lines.html

 His books are listed in my bibliography : https://www.jgrarchitect.com/2019/06/bibliography-includiung-websites.html