Archive - 2023
Thursday, February 2, 2023
Practical Geometry at Mud University, Cambridge, NY, March 3-4
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
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.
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.
I look forward to seeing you there. Jane
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.
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!
** 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.
Tuesday, January 10, 2023
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.