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.

Practical Geometry - Drawing a Square with a Compass, Part 2

 4 more ways to draw a square with a compass.

For Part 1 see: https://www.jgrarchitect.com/2019/12/practical-geometry-drawing-square-with.html


How to draw a square with a compass  #3
Peter Nicholson wrote about Practical Geometry in 1793.  His first plates are introductions to the first rules of geometry: using a compass to bisect a line,

My blog post about him is : https://www.jgrarchitect.com/2016/08/practical-geometry-as-described-by_16.html
It includes images of Plate 2 and Plate 3.

Here I have copied just the image of a square. Nicholson includes instructions for finding the square 'abcd' by dividing the arc a-e (the black spot) in half then adding that half to a-e and b-e find d and c.

Asher Benjamin and Owen Biddle in their pattern books copy Nicholson.
They do change the order of the letters which makes the steps easier to follow: a and b are 2 corners of the square. The arcs of a and b create c. Half of arc a-c is d. Add the length b-c to the arcs of a-c and b-c to find e and f: the square has its 4 corners.




How to draw a square with a compass, #4


A 3/4/5 triangle always has a right angle (90*) where the lengths 3 and 4 meet.
2 3/4/5 triangles are a rectangle which is 3/4 of a square.
I have drawn this on graph paper for clarity.

When carpenter squares became widely available and accurate, the square corners were easy to establish. The compass was only needed to lay out the length.

Before that - before about 1830 - the carpenter could have laid out his square like this:

His length is laid out in 4 units.
He knows approximately where the 2 sides will be. He does not know if his angle is 90*.

Here I have drawn the arc of the length of 4 units - on the right side. Then the arc of 5 units with its center at 3 units  on the left side. where they meet will be the 3/4/5 triangle.











The carpenter did not need to layout the full arcs as I have drawn them.
If he held his Line at the right lengths he could have marked a bit of both arcs where he thought they cross, and then placed a peg where they did cross. He would have checked his square by matching diagonals.

The relationship between the 3/4/5 triangle and the square is good to recognize. However, the 3/4/5 triangle is usually the only geometry. Layout by a carpenter square, widely available in the 1840's, was simpler and took less training than using a compass.





This small, simple house, built c. 1840 for a cobbler, was probably laid out using a carpenter square. I've tried other geometries which almost fit. The 3/4/5 triangle does.


I wrote the original post in 2014. It's time to revisit and review.
Here's the link to the post:
  https://www.jgrarchitect.com/2014/10/the-cobblers-house-c-1840.html



How to draw a square with a compass, #5

 Lay out a perpendicular through a line. Draw a circle with its center where the lines cross.
Draw lines - here dash/dot lines - between the points where the circle crosses the lines.

This square, as a diamond, was often used by finish carpenters because it easily evolves into more complex layouts. 

Below is the entrance porch for Gunston Hall, designed by William Buckland, c. 1761. The rotated squares determine the size of the porch. They also locate the floor, the pediment, the roof pitch, the size of the arch, the height of the rail.
 My post on Gunston Hall is: https://www.jgrarchitect.com/2014/05/gunston-hall-ason-neck-virginia.html

Here the glass facade of
 the Mass. Ave. entrance to MIT. For more, see:











https://www.jgrarchitect.com/2018/04/a-little-bit-of-geometry-of-mit.html



 How to draw a square with a compass, #6

 On a line select a length - see the dots .
Using the length as the radius draw a circle using one dot as the center.
Now there are 3 dots. Draw 3 circles using all 3 dots as centers.
Drop a perpendicular line at the first circle's center.
Now there are 2 new dots for centers of more circles.
Connect the petals where the 4 circles cross.
A square.




This modest farm house, c. 1840, used the square crossed as the squares above are for the Gunston Hall porch.

https://www.jgrarchitect.com/2014/09/how-to-construct-square.html

One last note: the circle to square diagram #6 can also become the diagram for #5. 

Each master builder probably had his preferred way of using his compass, even when he practiced within a tradition.
Still, just as a 3/4/5 triangle is part of a square, these diagrams are also simply different choices, different perceptions of the same geometry.

  

Practical Geometry at MIT

 This is the curtain wall of the Mass Ave entrance at MIT,  the Massachusetts Institute of Technology, Cambridge, Mass.
The picture arrived in my mail box last week.



Immediately I saw the geometry. I knew what geometry the designer used and how it was manipulated.





I was writing posts for this blog and my local blog. None quite came together. Each had parts which require more drawing, thinking, and better choice of words.
Frustrating.
Then the latest MIT mailing to alumni/ae showed up with this picture.
I laughed. I walked beneath that wall of glass and columns for 4 years. In that time I paid a lot of attention to how we used the space it sheltered, how the shape and size of that 'entrance' directed what we did. The curtain wall was not part of my thinking, although the light it allowed into the rotunda was.
The wall's pattern, rhythm, proportions - or even the idea that it was geometry - was not part of my analysis, nor was it ever alluded to by others. 


Here is the pattern.
Upper left : A square and its diagonals.
Upper middle: The circle that comes from using the diagonal of the square as the circle's diameter.
Upper right: The square that fits around the circle.

How the pattern grows:
Top row: Overlap a circle of the same size, so that the perimeter of each circle touches the square inside the other.
Second row: This pattern can grow sideways as well as up and down.
4 squares on the right: Once established leave out the circle, continue the squares, add the diagonal, horizontal, and vertical lines.


The pattern could have started with the circle and the 2 squares fitted around it.
Using the circle as the unit  - the 'module' or 'diameter' in classical terms - is the traditional way to begin a design. (See Palladio through Asher Benjamin.)

From a photograph I cannot judge the diameter of the column.  Does it taper? have entasis? The pilaster in the corner on the right appears to be the same width as the unit I chose: the original square.
If instead the column is the unit, the module,  the circles of the curtain wall might be 3/4 or 2/3 of the module.

I have been told that the main buildings at MIT - dedicated in 1916, designed by William Wells Bosworth -  were designed using geometry. The drawings of those buildings would be well worth studying.

Practical Geometry has become an integral part of how I see buildings.  I was surprised to find that it has become a design tool for me as it was for those who used it for construction.

Meanwhile, this little bit of geometry was just plain fun.