Archive - 2023

This is my daisy wheel and its 9 ft. long board. It was part of an outbuilding of a barn in Vermont.

The center of the wheel is 46" from the floor, a good height for a workman setting his basic dimension for his compass width (the diameter of the daisy wheel) as he began work and checking it as he needed throughout the day.

It was also useful information for those who came later, expanding or repairing the farm complex.

My daisy wheel was easy to find. However, many are discovered in obscure places, on beams and roof sheathing.

This daisy wheel was on roof sheathing of a barn in upstate New York, built c. 1795.

Unlike the daisy wheel above which has deep holes at its center and at the outer points of the petals as they meet the circumference of the circle, this wheel is cleaner, probably used only for this barn, not the larger barn complex around it.

Here is a tracing of it.

Both of these daisy wheels measure 8" + a smidgen. The depth and age of the scribe's grove makes precision difficult. However: 8.25" x 2 = 16.5". 16.5" x12 is a rod, 16.5 feet.

The rods laid out by stepping off these 2 daisy wheels' diameters might differ from each other by several inches. But each frame would be consistent within itself.

A rod was a common dimension in England. It was also called a 'pole' and a 'perch'. Land surveyors still measure land in rods today.

This daisy wheel is on the interior side of the sheathing on the second floor of the Gedney House in Salem, Massachusetts, built in 1665, expanded in 1712, and 1800, The house is now owned by Historic New England.

The daisy wheel is quite small.

Its diameter is 5.5". 5.5" x 3 = 16.5". 16.5" x 12 = a rod.

This one comes from the Beatty-Cramer House in Maryland.The wheel diameter is 3". It and the 'eye' below it might be Masonic symbols of God and Truth, as geometry was known to be true.

The understanding that geometry is 'true' was not part of my high school geometry class. I learned that from carpenters.

Palladio wrote c.1570 that he would use the diameter of a column as his 'module', his measure for his work. Here's his drawing:

The column is a circle. It is laid out with a compass, Its diameter, stepped off using a compass, is the module: it measures the distance between the columns. The scribed daisy wheels that we find today are also modules.

Here he is, holding his compass.

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.

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.

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). 

I'll talk about what Practical Geometry was/is using a Power Point presentation to show the history of the use of geometry in construction and design.

Ask me if you have questions. Or just sign up.

I look forward to seeing you there. Jane 

https://www.muducambridge.org/

 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!

*** 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.

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.