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.

Geometry of the Old First Church Fanlight

 

 

This is the fanlight over the main door to the Old First Church, built in 1803-5 in Bennington, Vermont. Lavius Fillmore was the Master Builder; Oliver Abel, his Master Carpenter, and Asa Hyde, the Joiner and carver.  

 The fanlight design consists of 2 parts: the 'scallops' around the curve and the 'leaves' coming up from the base. It is simple, graceful.

How was it laid out? In 2012 - when I first wrote about this fanlight - I knew the geometry for the scallops around the curve - expanded daisy wheels on the horizontal and the vertical axis. 

The 3 leaves below the scallops?  I was lost.

Laurie Smith - English timber framer, historian, geometer, the most knowledgeable person I know about the use of circle geometry in medieval design and construction - provided an answer.
Here was our geometry for the fan light in 2012.  Now, in 2021, I think it is probably not how Lavius Fillmore laid out the pattern. 

I post it because we are teaching ourselves. We are constantly learning more about how to use Practical Geometry (or 'Architectural Geometry'). One solution is not the only one. 

A square can be  derived from a circle in many ways - as can a 3/4/5 rectangle - both done using only a compass, straightedge and a marking tool. To see how the geometry can be followed and bring different designers seeing different paths, arriving at the same solution, is valuable.

 

 

 The  circle, its 6 points around the circumference laid out by the radius of the circle, is set on a line which defines the shape of the fan light.

 

 

The circle is surrounded by 6 circles which have their centers on the 6  points. The center pattern is a daisy wheel with 'petals'.

 The circles expanded.

 

 

This set of circles around the original circle adds petals 
to the exterior of the first circle. Add the fanlight shape and the petals  become scallops around the arc of the fanlight.

 

 

 

 

 Rotate the circles 15*  - or 1/2 a petal - and the fanlight's scallops' locations change.

Overlapped, the daisy petals create the double scallops around the Old First Church fanlight. See the photograph above.

 

 

The overlapped petals are also the pattern of the 'leaves' in the  fanlight: too big, not in the right location, but crossed as are the leaves. 

This pattern is much easier to layout than the geometry in the next 5 drawings. I include those drawings to show another way which is equally valid but more complex. 

However, as I have explored the Old First Church geometry over the past 10 years,  I have gained a sense of how Lavius Fillmore designed. I do not think he used the sequences laid out here. He understood what the simple act of turning an expanded circle on its axis could create. 

The simple solution is here :

https://www.jgrarchitect.com/2021/10/geometry-of-old-first-church-fanlight.html 

 

 

Here is how the leaves could have been added. These steps are our 2012 solution.

Add regulating lines from the center of the circles to the second ring of circles and center lines in the petals.

 

Connect the center points of the scallops to each other. Where they cross the petals is the center of the small circles which form the leaves. 

 

 

 

 

 

 

The radius of the circles is the distance from the center of the petal to the scallop.

 

 

 

 

 

 

 

 

I've drawn it in red to make it more visible. It is a complex layout for a seemly quiet, unassuming design.

This pattern was drawn at about 3/8" = 1'-0".  A scale of 1"=1'0" might have been easier. However it would still be tiny here on the page. For clarity I left out the overlapping scallops.

 

 

my drawing in 2012:

The real fanlight was laid out full scale - 5+ feet across -  on a framing table or floor. The proposed design sketch would have been studied, the arcs drawn with a compass using chalk or charcoal, the lines checked, redrawn, the points pinned.  Finally, when the regulating lines were erased, the simple, clean design was visible.

I would like to have been there, listening, watching. as the men drew this.  I think they were pleased as they derived the pattern and settled on a design.  It's not structural at all. It's one of the first things you see, an introduction to the church. It's also their signature.

Then -  I realized that this derivation is not simple enough. 

My Addendum, my next post, is my current solution, my current understanding of how Lavius Fillmore, Oliver Abel, and Asa Hyde designed the fanlight. 

https://www.jgrarchitect.com/2021/10/geometry-of-old-first-church-fanlight.html

ARCHITECTURAL GEOMETRY, A Rare Geometrical Record from Rural Devon, by Laurie Smith

 

 

Laurie Smith has written a new book: ARCHITECTURAL  GEOMETRY  A Rare Geometrical Record from Rural Devon.  

Here's the cover.

 

 

 

 

 

 

 

 

   

 

The book is about the many daisy wheels and other geometry found on the walls of a Devon threshing barn.
The barn, shown here, is owned by Richard Westcott, editor of The Three Hares, a Curiosity Worth Regarding. 
The image is #3, page 4.  

Richard Westcott, Laurie Smith, and their friend, the photographer and film maker Chris Chapman, examined and recorded the geometry on the barn’s walls -  over 169 separate geometric shapes.

This photograph is of a "divider scribed daisy wheel from the wall's inner surface." 

The image is #4; the quote is from page 4.

They researched the barn’s history, took measured drawings and photographs, and explored the geometry.

Then Laurie wrote this book. 

 
Like all of Laurie’s books it has beautiful diagrams. Clear descriptions accompanying the diagrams explain how the daisy wheels still visible on its walls governed the siting, layout, and frame  of the barn.

Image 39, page 26

This is 1of 5 of Laurie's illustrations showing the development of the barn section.     

He includes examples of similar geometries give context and nuance. 

Here is one of 4 daisy wheel drawings for the geometry of the Barley Barn, Cressing Temple, Essex, UK

 Image 27, page 19.

 

 

 

 

Along the way Laurie explains terms and forms which we rarely use today, including the use of a perch, pole and rod as measuring devices.  

He introduces the reader to the Trivium, the Quadrivium, and Whirling Squares. 

Part of Image 63, page 50 

 

He writes thoughtfully - and with humor -  about apotropaic symbols.  

 

 

At the end of the book Laurie considers how all of that geometry - 169 separate images - came to be scribed on the interior walls of a rural threshing barn. He suggests a 'geometry school'.  I agree with his theory: I have also found incomplete geometries drawn on plaster walls here in the States.

His descriptions encourage the reader to examine the image, think about what he's written, look again, and understand the geometry.

Here is what he says about this double daisy wheel:

"The image shows the geometrical precision of the divider-scribing, the scars of the divider pin at the twelve points around the primary circle and the compound damage caused at the symbol's axis by the passing of twenty four arcs."

Image and quote, page 59 

 

The book's copyright page includes this introduction: 

"Laurie Smith is an independent early-building design researcher, specializing in geometric design systems. Because the medieval educational curriculum included geometry he uses geometric analysis to excavate and recover the design systems of the past, a process he thinks of as design archaeology. He lectures, writes, runs practical workshops  and publishes educational articles on geometrical design that are available from his website."

www.historicbuildinggeometry.uk  

e laurie@historicbuildinggeometry.uk  

 I highly recommend this book to all who are interested in historic construction and geometry. Copies can be purchased from Laurie Smith in the UK or from me ($20.00 postage paid) in the USA. 

 

*All the photographs: the barn, the hand and the daisy wheel, and the 12 pointed daisy wheel in the barn shown here are by Chris Chapman, copied by me from the book for the purpose of this review. 

The geometry is by Laurie Smith, also copied with his permission.