Practical Geometry at Mud University, Cambridge, NY, March 3-4

 

ANNOUNCEMENT

 

 I'm giving a class on Practical Geometry at Mud University, in Cambridge, NY, on March 3-4, 2023. Their website is at the end of this post.*

Just in time for mud season!  Come learn about practical geometry at Mud University.  

FREE and FUN! With a fabulous instructor: me. 

Anyone who's curious is welcome, no math or drawing skill needed. 

 

March 3, First meeting: I will introduce geometry as practical knowledge well understood until about 1950. We will use compasses to layout daisy wheels.

March 4, Second meeting: we will draw the patterns, hands-on, with compasses.

 

 

 

 

 

Here's a diagram - the square and its circle.

 

 

 

 It is the language for the pattern of a quilt (dated 1847) 

 

 

 

 

 

and the roof structure for St. David's Cathedral in Wales (c. 1550).  

 

You will learn what Practical Geometry was/is, and see many examples of the use of geometry in construction and design.

 You will see how our ancestors, weavers, quilters, cabinet makers, and builders used geometry for design and construction. I will mention drawings, paintings, and illustrations, including how our cell phones superimpose the 'rule of thirds' over our snap shots.

Ask me if you have questions. Or just sign up.                       *  https://www.muducambridge.org/

I look forward to seeing you there.  Jane 

 

St. David's Cathedral and geometric pattern :

Smith, Laurie, The Geometrical Design of St. David’s Cathedral Nave Ceiling, A Geometer’  Perspective,  

The Geometrical Design Works, 2017, printed Exeter, UK. and others.

 

 

 

 

 

 

 

 

 

 

A Lancaster Clock Case: its geometric design

 The Dietrick American Foundation published an article about a Lancaster clock case, researched and written by Christopher Storb, in July 2022.* It was forwarded to me by Craig Farrow, Cabinet Maker.**  He knew I would be interested.

The Foundation wrote that it "intended [such articles] as a type of crowd sourcing, where responses and information shared by readers can inform research."  I am happy  to respond, to try this way of sharing information, to see if it can be successful.

As my research on the use of geometry in construction - Practical Geometry - is not well known or understood, I have written this post as an introduction.

I will write to Christopher Storb when I publish this analysis. I look forward to his reply and the information from others who have responded.

The article is fascinating with clear images and explanations.

I especially liked the the medallion at the base of the clock case and was delighted with the cabinet maker's tilt of the knot. I appreciated Christopher Storb's clear analysis of the geometry of the knot as intertwined hourglasses rotated to "create the illusion that the design is in motion, mirroring the actual rotation of the hands of the clock dial above."

 

Here is the geometry as drawn by Christopher Storb: the daisy wheel and its 6 outer circles, the lines of the parts of the geometry used on the clock outlined for clarity and then the pattern rotated to fit the diagonal, upper left to lower right.

 

 I saw that the geometry governed the design of the whole lower panel, not just the knot. 

I decided to map it.

 

 

 

The photograph in the article is not quite square. The image of the knot and its panel is slightly skewed; the diagrams drawn over the image are not quite true. Therefore I have drawn the geometry separately. 

See the lower edge: there is a space below image on the lower left corner, but almost none on the lower right. That's enough to skew the geometry.

 

I began with the panel which is the front of the base. It is a square. 

I added the diagonals.

 

 

 

Then I divided the sides in half, vertically and horizontally. ***

Here is the geometry as laid out by compass and straight edge. 

The cabinet maker did not need to use numbers. Each line came from the basic shape, that first square.

.

The horizontal and vertical lines bisect the scalloped edge.

Every line crosses the others in the center.

 

From those lines several others are easy to add. The sides of the smaller rotated square run from center point to center point.

A small circle - with a diameter the distance from the center of the design to the inner rotated square - can be added.

 

That small circle is the first circle of the knot, located by the geometry of the face  - its diameter is determined by the squares.

The rest of the knot can be laid out with a compass as shown in Christopher Storb's diagrams.

 

 

 

The cabinet maker did not need to rotate his diagram. The knot began on the slope of the diagonal.

 

Here is the vertical hour glass image, in the center, as it is in the upper right of Christopher Storb's diagrams.  6 more circles can be added around the outer ring of the hour glass layout. I have only drawn 2.  Their arcs are the inner curves of the hour glass shape.

The square shield surrounding the knot comes from the circles that radiate out from the knot on the diagonals. Their centers are the corners of the square. The cabinet maker did not need to draw those circles. His compass was already open to the circle's radius and could mark the corners of the square.

I have added the upper left circle in red, then the 3 other centers of the circles  - the corners of the square - as red points on the diagonals.

 

Here is the layout of all the squares of the design. 

They are governed by the lines which begin with the exterior square of the clock case's base. 

 

The square (as drawn here) of the shield confines the knot. It is static but the knot is fluid. 

 

 

 

 

 The shield's circle which surrounds the square, is also a perfect, stable shape. The knot implies movement and change while the circle is constant, never changing. The circle constrains the knot's curves just as the square does.

The diagrams are the forms for the design. They were the beginning. Then the cabinet maker played with the shapes.

His solution was to compliment the knot by curving the corners of the square with the circumference of the circle.  He loosened that circumference by adding the scallops and fins.

The  flourishes - the scallops and fins - are laid out by the arc of the radius of the inner circle of the knot, a smaller circle which comes from the width of the wood and pewter bands.

The shield  becomes not a constraint but a backdrop, a commentary. Both the square with its softened corners and the circle behind it with its horns and scallops present to the viewer that remarkable knot with its pewter ribbon. 

What knowledge and skill this cabinet maker had!

 

* Dietrick America Foundation, An Extraordinary Lancaster Clock Case, by Christopher Storb, July 22, 2022.  https://www.incollect.com/articles/dietrich-american-foundation-an-extraordinary-lancaster-tall-case-clock

** Craig Farrow, Cabinet Maker, https://www.craigfarrowmasterfurnituremaker.com/

 

*** dividing a square in quarters using a compass: 

 This diagram was published in pattern books written to instruct apprentices. The square with its arcs has 2 points both horizontally and vertically. The lines of these points divide the square into quarters.

 

 

 

 

Geometry in Construction = Practical Geometry

Geometry in construction = practical geometry.

Does that seem strange, a philosophical stretch?  As recently as the 1930's it was widely understood, commonplace.  Since the 1950's, geometry has been taught as precise, logical, beautiful, magical, amazing.  But practical? Barely. Today the idea is usually met with skepticism.

However, you who read my blog know this is what I study: what those builders know about geometry and how did they use it? 

 

Euclid's geometry starts with a Point which has no dimensions.  Two points make a Line - 1 dimension.  3 make a Plane - 2 dimensions.

 

4 points make an object  - 3 dimensions.  

 

How can this geometry be practical? 

A Line laid out between 2 points will always be straight. 

A Line drawn by hand might curve; a Line marked by snapping a length of twine cannot curve. This is the beginning: it will be true.  If the geometry is not accurate it will not be practical.

The Line A-B can become a radius. The radius can draw a circle. 

Whether the circle is drawn with a compass set to the length of the radius. or by hand with a length of twine, it will close if the the work is accurate. If the circle does not close upon itself it is not true.        At every step of the layout if the geometry doesn't hold, the designer will know to stop and correct the drawing.

 

The radius of the circle always divides the circumference of the circle into 6 parts. If the points on the circle, marked by swinging the arc of the radius, are not spaced accurately they will not end exactly where they began. They will not be true. The work cannot proceed. These 6 points on this daisy wheel are not quite accurate.  Note that the daisy petals' shapes are not identical; the points are not equidistant. If I measured the diameters, petal to petal, they would not match. I was not careful enough.        

 

 

 

 The 6 points, joined with lines, can be used in construction.

 

The rectangles that come from the 6 points can be proved by their diagonals. If they match, the rectangle will have 90* corners and be true. If the diagonals do not match the shape is not a  rectangle. 

A building needs to be stable, whatever materials it is made from, whatever form it takes. For simple vernacular housing the circle was the practical geometry needed to erect a stable, sturdy dwelling.  

The layout tools available to the builder of the Lesser Dabney House* in rural Virginia, c. 1740, were twine, some pegs, a straight edge, some chalk or soot so the twine could mark a line, perhaps a scribe, a compass.

He could have laid out this house with the first 4. A peg could have served as a scribe to mark a point. Twine with a loose knot around a peg turns as a compass does.

 

 

 

Here is the floor plan as it was recorded by Henry Glassie, c. 1973: 3 rooms with 2 chimneys and a stair to the attic.  3 windows, 4 doors. The door to the left may have gone into another shed.

 

 

The builder stood where he wanted the main wall of the house to be. He pegged the width he chose with twine A-B. That length became his radius. He drew his arcs to find the center of his circle C. Then he drew his circle.  And found it true. The circle's radius steps off 6 times around its circumference. The arc create the 'daisy wheel'.

 

A-B in the diagram above became 1-2,  the width of the house. The arcs 1-3 and 2-6 of that width crossed at the center of the circle with its 6 points: 1,2,3,4,5,6. 

The Lines 1-5 and 2-4 laid out the side walls; 6-3 locate the back wall. Diagonals across the rectangular floor plan proved the layout to be true.

The main block is about 20'x17'. The 2 doors  welcomed cooling through breezes in the summer. The wall room on the right may been a later addition to create a parlor, more private and warmer in winter.

Later the builder added the shed. He made his twine the length of the house, folds it in half and then in half again. He then knew what was 1/4 the length of the house (x). He laid out that length (x) 3 times to get the depth of his shed. He stretched his twine diagonally from one corner to the other. If the twine measured5 (x) his shed walls were a 3/4/5 rectangle; the corners 90*, and  true to the main house. The shed roof framed cleanly against the house and was weather tight.

The circle and the 3/4/5 triangle - Practical Geometry -  were the only measuring systems necessary to construct this house.

 

*The Lesser Dabney House, Fig. 45, Type 3, p, 105; the photograph: p.104. 

Henry Glassie, Folk Housing in Middle Virginia, U Tennessee Press, 1975; plans, drawings and photographs by Henry Glassie.

Henry Glassie recorded floor plans and what history he could find, He photographed. He did not make measured drawings like those in HABS at the Library of Congress. website

 

 

Virginia Folk Housing, Part 2 of an update

 

The Moore House* photographed by Henry Glassie, built before 1750.

 

This house has 2 rooms up and down, 2 fire places, 2 chimneys, and a shed on each end. The main block  is double the size of the house I wrote about in Part1: https://www.jgrarchitect.com/2022/11/virginia-folk-housing-update.html

The geometry begins as it did in Part 1, using the width as the circle's  radius.   

 

Here is the floor plan: 2 rooms with fireplaces, and sheds on both ends.

The daisy wheel progression begins with a length A-B which becomes the radius of a circle here lettered C-A.

The daisy wheel for this house begins with the left wall of the main house.

That wall's width  is the radius, 1-6. A is the center of the circle. The daisy wheel lays out the other 4 points, 2, 3, 4, 5.  

 Lines 1-3 and  6-4 are the sides of the house. 2-5, the diameter of the circle, lays out the interior wall.

 

Lines 1-5 and 2-4 can extend forever. Where is the right end wall of the house located? Where is C? 

It's at the end of the circle, but that's only a point, not a line. 2 points are necessary to draw a line to mark the right end of his foundation and the floor of the house.

If the carpenter extends his arcs he can quickly find the missing points. 

Extend the arc centered at 3 (2-A-4) to B.  The arc centered at 4 (5-A-3) crosses the earlier arc at B.   He has 2 points: A and B, And can draw line A-B. 

Now C is fixed at the intersection of A-B. C is the center of a new arc, (7-A-8). The extended arc from 5 (6-A) crosses at 7. The arc 2 (1-A) crosses at 8. 7-C-8 locates the right wall.

C also locates the center of the fireplace and the chimney. 

The daisy wheel is often dismissed as a design tool. It is flexible, quickly drawn, and accurate. 

The geometry comes from the first length - the width chosen by the owner and builder for this house. That width, and the house, could be bigger or smaller to suit the owner's needs and budget, as well as to the lumber available for joists and rafters. 

Once the carpenter decides on a width he uses one compass setting, one radius, for the whole layout. Every point is checked. As the lines are marked, the diagonals can prove the layout to be  true.

If he drew a layout at a smaller scale, he could easily step off to full-sized construction dimensions with his compass. He could also draw the layout on the ground, stake the points and mark the wall locations with twine just as framers and masons do today. 

Consider how the plan would be laid out if the circle is not used. Use a 10' pole - a common tool of the time.  Each corner would need to be figured independently;  every dimension stepped off separately, and with what accuracy? 

The daisy wheel locates all angles and lengths quickly. It has built-in checks from the beginning and as the layout progresses: if the circle doesn't close, the 6 points will be uneven, the arcs won't cross, the diagonals will not match. The layout will not be accurate.

 Both wings are 3/4/5 rectangles. See the left shed. The floor plans of wings were usually 3/4/5 rectangles so that they would sit square to the existing house. All the joists would then be the same length; as would be the rafters.  

 

 

My earlier complex geometry 'works'; the lines are there. But they don't give the basic information the builder needs: the dimensions of the foundation, the floor plan, the size of the house.

 

*The Moore House, Fig. 31, Type 5, p, 77; the photograph: p.76. 

Henry Glassie, Folk Housing in Middle Virginia, U Tennessee Press, 1975; plans, drawings and photographs by Henry Glassie.

Virginia Folk Housing, Part 1, an update

The house recorded by Henry Glassie in Folk Housing in Middle Virginia * were basic shelter for people with few resources. They may have been the first house for someone homesteading, built by a sharecropper or by someone enslaved.   

This is Fig. 35, The Parrish House, a "small mid-eighteenth-century house of sawed logs", p. 84 in Glassie's book.*

 

The geometric diagrams I drew in May 2014,** were accurate but much too complex for these houses. More importantly they didn't begin as a carpenter would: with the size of the foundation and the floor plan.

 

 A carpenter's first question is, " Why?" Then he asks, "How big? How long? How wide?" 

The red line across the bottom of the floor plan is 'how long', about 21 ft. That distance can be the beginning of the layout, the first Line that determines all the others.

 

That Line can be the radius for a circle:

The arcs of the Line A-B cross at C. That's the center of the circle for the layout of this house.

In the diagrams below: 1) B-C is the radius of the circle. 2) Beginning with B on the circumference  the arcs of the daisy wheel are added. The 6 even spaced points around the circle A, B, D, E, F, G  are located.

 

 

 

 

Connect the Lines. A-F and B-E are perpendicular to A-B. G-D is the diameter. They mark the width and length of the rectangle for the house plan.  If there is a question about accuracy, diagonals can be used to true the shape.

 

 

Here is the plan within its circle, the circle that begins with the carpenter's choice of width, his 'module'.

 

 

The masonry block for the 2 chimneys is square, centered, and 1/3 of the width the house. Glassie's photograph shows a shed sheltering that fireplace.

 

*Henry Glassie, Folk Housing in Middle Virginia, U Tennessee Press, 1975. The book includes more information, drawings, and a photograph of the house. It no longer exists.

** The original post is here:  https://www.jgrarchitect.com/2014/04/18th-c-virginian-folk-houses.html. Its companion, here: https://www.jgrarchitect.com/2014/05/18th-c-virginian-folk-houses-part-2.html

I considered deleting the 2 posts, but their existence brought a comment and question which prompted this update.

Also:

As I read them I realize how much I have learned about geometry since 2014. I saw it and tried to explain it, just as Henry Glassie did in his Rules, Chapter IV, The Architectural Competence.

When I began to study Practical Geometry there were no books, no one for discussions or critiques. I was teaching myself, reading early pattern books line by line. Laurie Smith was the only person I knew who saw geometry as I did, and he was in the UK. Later that year he came to the States; I took a workshop with him. I was able to work with him until his death last year.  

I don't want this information to be lost again. I want others to find it, question it, reject and/or improve upon my analysis, their own analysis, expand our understanding.

  

 

Serlio's Lines

 Lines

That's the word they used.   Lines.  An important word, often capitalized.         

Sebastiano Serlio writes, ".... if the architect wants to build a temple door which is proportional to the place, he should take the width of the central body of the temple, that is the floor space - or between the walls if it is small, and between the pillars if it has transepts. From this width he should draw the same height which will make a perfect square. 

... He should draw two diagonal lines and then the two other lines from the bottom corners to the top [center.] The "lines will form the opening of the door, and they will also enable the ornaments to be carved, as is shown... If 3 doors... were to be built in the face of a temple, the same proportions could be used in the smaller sides." *

We use this word: "line".  Usually we add helpful adjectives. 

 Metaphoric lines: "toe the line", "step over the line", life line; or bus and subway lines. 

Demarcation lines: fence line, property line, finish line, white line, sight line.

Rope that becomes a line: tow line, clothes line, fishing line, electric line.   

And in construction: chalk line, plumb line.  

The line shown here can be either. a chalk line that can be rewound into the case, or a plumb line by hanging the line on a peg and using the case as the plumb bob.

 We check that a foundation, a frame is true with matching diagonal lines.

There is also 'straight line', an oxymoron in geometry. 

 

Serlio's definition is geometric; a line is "a straight and continuous representation from one point to another, having length without width."  Here is his diagram shown above, rotated and then all 4 diagrams overlaid to make one 'star'.

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." 

 Do we, in 2022, know what these words, the various intersections of lines mean? What results from them?

The easiest answer is the lines can divide a rectangle or trapezoid in half, vertically or horizontally, or in 3, 4, 5, 6 (etc.) equal parts.

Any 2 points can establish a line, so lines can create simple or complex patterns.

Here is one of Serlio's designs that begins with a square and its diagonals. Every dimension on the plan comes from that initial diagram.

This villa comes from Serlio's Book  VI: On Geometry, titled: Treatise: On Domestic Architecture, written c. 1545-9, Plate XXXVIII, Project 28, of 73 plates.

Serlio drew in the lines for his readers.They were not laid out first. The lines come from the geometry. The placement of columns, walls, openings come from the "various intersections of lines".

He also gives dimensions: those little hatch marks in the center. 

 

Here's the geometry:

The plan is a square. It is divided into 4 parts horizontally and vertically - 16 equal squares. the top row is: 1 square, 2 squares, 1 square. The  bottom row matches it.  The vertical rows also repeat the pattern: square, double square, square. The center space is 'left over' - 2 squares x 2 squares.  That space is divided into 9 equal squares. The columns mark the intersections.

 

All of this can be easily laid out with Lines. The diagonals neatly cross the corners of the structure and the 4 columns in the central room.
The Lines Serlio used (shown above) to locate the door can also be laid out here. They cross at the center of the walls.

Serlio drew the lines that show the widths of walls, openings, and columns. How did he knows where those lines should be?  I followed his lead.

His design is a square: I drew squares. I located the center of each side of the plan and added the Lines from center point to center point. This divided the plan into 4 smaller squares. Now there were 4 points of intersection, plus the one in the center.

 

 

These points allowed more Lines to be added.

The Lines laid out the wall locations. They are at the back of the niches and the fireplaces. They told the mason where to begin his work. He could add the decorative niches, pilasters and mantles in front of the structure. The fireplace flues would line up.   

 

Don't miss the wonderful details: the circular stair - lower left - is at an intersection. The main stair fits neatly into the lower left square, the octagon room in the lower right.
 

So, the columns?

 The Lines  - the diagonals that Serlio used in his drawing for locating the door  - locate the columns. They are on the 'third points'  in the space: dividing the center hall into 9 squares.

This is  a simplified version of the star I drew above. It is the 'Rule of Thirds' that we use when we compose images on our cell phones, that artists consider when they compose a painting.

 

 

 Here is a detail showing how the Lines of the column locations are extended into the loggia 'G'. The Lines determine the placement of the outer side of the square columns. The center space - where G is on the center of 5 Lines - is divided into 4 spaces, easily done using the star. 2 of the spaces equal the opening between the columns, 1 is the width of the columns.  Note that the width of the columns is also the width of the walls.  

Each dimension comes directly from the first geometry, the square and its diagonals.

 

The Front Elevation! 

 

That simple layout creates the structure of the villa. Visually the walls became a backdrop for columns, arches, niches, friezes, lintels, dormers, balconies. However it is the geometry which holds all those pieces together.

Note that those square columns, here on the front of the house, have round pilasters added to their front sides, with doubled pilasters on the corners. And don't miss the chimneys!

* Sebastiano Serlio, Book I: On Geometry, See my bibliography: https://www.jgrarchitect.com/2019/06/bibliography-includiung-websites.html

                                Book  VI: On Geometry, On Domestic Architecture,  A Dover Publication edition, 1996, of work originally published by The Architectural History Foundation, NY, and the MIT Press, Cambridge, MA, 1978.