Sunday, February 28, 2021

Sidney Colvin's barn, now part of the Southshire Community School



Sidney Colvin's  c. 1850 farm in North Bennington, VT, included this barn. When the  railroad was routed through his original farm, he moved about a mile away and built his farm anew south of the rail line. 

His new barns were specialized as recommended by the agricultural journals of the day. This one, close to the house, was designed for horses and wagons. 



Colvin's new home, next door to this barn, now serves as Southshire Community School's main building.  

The barn yard is the playground and the site for annual school theater performances in which every child participates .



 Originally, the barn had stalls for driving and draft horses and perhaps oxen. One bay was for wagons; above was the hay loft.  The frame was - and is - the local hybrid: Anglo-Dutch with a 5 sided ridge beam.


This orthographic construction of the barn and the other as-built line drawings are by Goldstone Architecture, Bennington, VT, used with Jeffrey Goldstone's permission.

Sydney Colvin and his framer would have discussed the barn's site: near the house; its use and size: about 26' deep and 46 ' wide: deep enough for the wagons, wide enough for the animals and their needs. The long side with the main doors would face south to open to the sun. 

The first task was to clear the site for the foundation. Then the framer used the daisy wheel to set his dimensions.

Here is a double daisy wheel. One set on the horizontal axis, the other  on the vertical. It has 12 petals, 12 points.


For more information about daisy wheels:



Using the 26' length as his radius,  the framer drew a circle with a Chalk Line. He staked the double daisy wheel: 6 points beginning at the southern or northern edge of the circle, the second 6 beginning at the east or west edge. He probably didn't need to draw all the petals.


 This was all he needed to layout the foundation.

The radius of the circle, 26 ft., is also the length of the distance between the 2 points as seen on the west side (left). A circle's radius always divides its circumference into 6 equal parts. 

 Straight lines can be drawn using the corresponding points on the  east side (right).  These would be the south and north walls of the barn.





Lines (dashed red lines) extended from the 2 points on either side of those walls  cross the line of the wall. They position the east wall.





The foundation is staked.







And trued by checking the diagonals.No measurements were necessary. The layout could be done with a Chalk Line and stakes.


When the foundation was set, the interior foundation walls were placed. The barn had 2 main spaces: on the west side were stalls which would be a step up to protect the horses' feet. On the right, on grade, would be the stall for the wagon.  

Using the radius, the 26 ft. width of the barn, (dashed lines) the framer swung 2 arcs, laying out a square, and the wall between the 2 spaces.  

 The wall between the stalls for the draft and driving horses was easy to layout. It  lines up with 2 of the points (marked in red) from the daisy wheel begun from  the east-west axis.

The framer did not need to measure any length. All his dimensions came the geometry.



The measured drawing shows the floor plan as of 2010.

The 2 spaces on the left had floors; the floor joists and their pockets are still there. 

Thus the left-hand space could be used as a framing floor.




This is a cross section of the interior. The lower level is for the horses and wagons. The upper level is a hayloft which is, except for the posts which support the ridge beam, open east to west. 

The floor joists are shown running east to west, which is correct. The orthographic construction shows them running north to south.



The same geometry that was used for the plan was also used to layout the 4 bents of the barn.

The framing floor was a 26 ft. x 26 ft. square. The framer swung the arcs of the square's sides; the arcs crossed each other. Those points mark the mid-lines of the square. Joined they divide the square into 4 smaller squares.


I rotated the plan 90*, to show the layout of the walls. East is to the top.




The lower half of the square, laid out by the intersection of the arcs, is the wall frame for the barn's 4 bents: 2 for the ends, 2 for the partitions. 

The vertical line divides the lower rectangle in half and positions the interior posts.






Each square of the end wall is divided into 3 rectangles by The Rule of Thirds. The lower two rectangles are the wall on the main floor, the upper one is part of the end wall in the hay loft. 

For simplicity I drew the Rule of Thirds only on the left side of the end wall. The black arrow on the right points to the beam that carries the floor joists for the hayloft. 


For more about the Rule of Thirds see:


Here are the rectangles making up the wall frame  - and the roof system.

The red outlined posts, beams and rafters are the frame for the east and west ends of the barn. The hayloft (which is above the beam at the arrow) is open from east to west. The collar ties are only in the end walls.

The frame is called Anglo-Dutch because it combines elements of both framing traditions. Here following Dutch framing,  the bents'  posts 'stick up' above the north-south beams so that the beams from both directions don't join into the posts at the corners at the same level. The beams are staggered, one set above the other. The east-west beams are the plates which carry the roof.  

The roof frame  was laid out using the upper half of the framing floor square. Dividing it into 2 squares, the framer swung the arcs  of each square - dashed curved red lines. (I have only shown the arcs in one square. The framer needed both sets of arcs to draw his Lines )  

The point where the arcs crossed at the bottom of the square set the height for the bents' posts, and the plates which carry the roof - solid red line.

The point where the arcs crossed at the top of the square - straight dashed line - gave the framer the position of the ridge beam.Using those points I have drawn the gable outline in a solid red line.

The collar tie for the gable is located where the arcs cross on the sides - the mid-point of the square. (I did not mark these points.)

The elevation above is the barn's section through the center. It also is the barn's 2 bents for the end walls.  The framer now has the information he needs to cut his timbers.  

This barn, framed with practical geometry, was built up the road from 5 factories manufacturing carpenter squares. However, the 3/4/5 triangle, which comes naturally from a carpenter square, was not used to layout this frame. The lumber was also cut by a sash saw, not the modern, in 1850, circular saw. 

Sidney Colvin followed the advances in agriculture. His framer did not use new technology, at least not when laying out this barn. Dimensions were beginning to be standardized as machine parts needed to be exact so they would be interchangeable, easy to replace. However. the barn was a building, not a moving machine. Its repetitive parts were the braces, joists, rafters, made of wood. Replacements could be made by hand, on site. Geometry was a more practical tool than dimensions.

*                *              *              *              *              *              *              *              *             *              *        

     The Southshire Community School Barn Repair                  


 The Southshire Community School used the barn for seasonal and long term storage for 30 years.

 In 2019 the barn was repaired from foundation to ridge, funded in part by a grant for barn repair from the State of Vermont.

 I wrote the part of the grant request which focused on the architectural and construction history.


Some of the sills had rotted. The dry laid stone foundation had come apart. 

Both were repaired.


The frame was rebuilt as necessary.

The siding was reused and replaced in kind.

The cornice and frieze were replaced. This existing rotting section became a template.



The barn was painted and put back to use.

In the fall of 2020 it became new classroom space for the school because of the Covid-19 pandemic.

I will add a current photo of the barn with windows, doors and a ramp.

My thanks to Jeffery Goldstone, Architect.  He was part of the team that wrote the barn grant. His as-built drawings made this exploration of the barn's geometry much easier to explain.

Monday, February 1, 2021

The 'Cube' in Albrecht Durer's Engraving, 'Melancholia', 1514

This is the kind of research that happens when one is self-isolating through a winter pandemic.   

Albrecht Durer's engraving 'Melancholia', was first printed in 1514. 

 I wrote about the carpentry tools scattered around the edges of the image in my previous post: "Albrecht Durer's 'Melancholia' and His Knowledge of Construction and Practical Geometry", posted on 1/17/2021.

On the left side in the middle is a polyhedron, the 'Cube'^, a 3-d shape.The angel is studying it intently. Why? And what is it?

And why the sphere? A pure geometric shape, white, abstract, perfect, set among real tools we can identify, and a skinny dog. It is the only other abstract thing besides the Cube in the engraving. 

I have some suggestions based on Durer's knowledge of and interest in geometry.


Durer wrote about geometry in his book, Underweysung der Mesang mit dem Zirckel und Richtscheyt,  published in 1525.*   The last 6 words of the title translate as Measurement with Compass and Straightedge


 One of his diagrams:


I copied the pattern of triangles, cut it out, and folded it into a  20 sided polyhedron, (an icosahedron,) fastened it with tape. 

Note: The 5 triangles on the right that would complete the shape are not taped together; it fits in my hand.

If the shape were made from wire it would look like the drawings beside the pattern - the view from the sides and then from the top or the bottom. The lines are the edges of the polygons. 



Another of Durer's diagrams is 12 joined pentagons which become a polyhedron. To the right are the side and top views. The views are not elevations; they are transparent.  


Again, I copied it and cut it out.




It folds up to be much like a soccer ball with edges.

Durer's book contains many of these polyhedrons. He mixes the kinds of polygons to make the 'sphere'. Here are 2 of his simple ones



The models led me to consider the Cube in Melancholia as a study about how to create a sphere from planes, from polygons. 


I extended the main sides of the polygon - red lines - so that the shapes became diamonds. Then I extended the edge of the almost invisible left side and found its tip met the main side at the top. 






Next I divided the length of 2 sides of the diamonds in half. When I joined the mid points with a red lines  I saw that they followed the edge of Durer's polyhedron.




Perhaps Durer was drawing a truncated cube in perspective.  I made a diagram: 6 squares which if folded would make a cube. I lopped off 6 corners  - red dashed lines




I added the triangles, printed it, and cut it out.





Cut and fastened with tape, it is flimsy, not  solid like Durer's block.





Tipped on its side, it has the planes of Durer's 'Cube' - his 8 sided shape.











Historians tend to title the book, Measurement with Compass and Ruler. In 1525, rods with 10 parts were common, so were boards with straight edges, used for drawing straight lines. Regular, agreed upon dimensions, such as would be on a ruler, did not exist. Note that there is no ruler among the tools Durer includes in his engraving, 'Melancholia'. 

^ Durer's polyhedron is not a cube. I use the name as an easy reference.

*Underweysung der Mesang mit dem Zirckel und Richtscheyt can be read online through the Warnock Library in Nuremberg, Germany.  A translation by Walter Strauss, published in 1977, is out of print, only available at museum and college libraries which currently are closed.

At any time, but especially in the year of Covid-19, to read a 500 year old book, housed 3800 miles away, while sitting in my office with my cat sprawled out beside me, is remarkable. That I can shift from one page to another and back again as I consider the images is an amazing gift.

Sunday, January 17, 2021

Albrecht Durer's 'Melancholia' and his knowledge of construction and practical geometry

Note: click on any image to enlarge it!

Albrecht Durer was a painter and print maker in Germany, 1471-1528. He also wrote books, and traveled widely in western Europe. He was a superb draftsman. I have enjoyed the compositions, the details, and the lines in his engravings and woodcuts for 50 years.

His woodcuts were for the people, most of whom were  illiterate. Their livelihoods did not require writing. They could read the images: here the rough stable, the well dressed men coming to see a baby, the angels and the star. 

The engravings were not so easy to make. They were for books, for people who could read.

These plates come from the Dover Publications reprints of Durer's wood cuts and engravings.* This is print #183.



He drew what he saw around him. His plates are full of the life he knew, including construction details. 

This detail from #183 shows a truss which includes a collar tie with angled tenon joints and pegs. The purlins and rafters are structurally correct.



The detail from Plate #190, shows the grain of the wood brace running in the right direction. The angled cuts for the joints could serve as templates for repair.


    Here in Plate #185, the brace is tied. a peg serves to tighten the tie as needed. The thatch for the roof is properly applied.

    When I began to write this post I was thinking about Durer's knowledge of construction and his use of geometry in his compositions. (More about that in another post.)
    I was side tracked as I began to read Durer biographies. The scholars who wrote them rarely knew about construction. To them the structures are allegories or useful for his compositions. 
    I saw practical and abstract geometry.
    This is my exploration, as a Geometer, of one of  Durer's most important engravings.

    'Melancholia', Durer's engraving about the temperament of  artists and  artisans, was made in 1514.*  

    Durer put many construction tools around his melancholy angel. She holds a compass.  Tucked beside her hand is a gauge. The putto sits on a mill wheel next to a ladder. Set beside the polyhedron is a pot for hot liquid metal, possibly lead, on a brazier. Under the angel's skirts is a pair of pliers.

    The tools are used in practical geometry. Above the dog is a hammer. Between his hind legs and the sphere: a Line and its plumb bob.

    In the lower left corner is a profile gauge.

    Then: a plane, a saw, and a straight edge that can be used to draw arcs. It may also be a level.

    Lastly: nails and a nail punch.

    The theory was that Melancholy, influenced by the planet Saturn, was part of the inherent character of artistic and philosophical people. There were 3 levels. The lowest was artists and artisans. The next was scholars, natural scientists, and statesmen. The highest level was theologians and those who studied the secrets of the divine.
    Here the putto is taking a nap after doing some numbers (perhaps) on a slate, the lowest level. The angel, on the other hand, is intently studying the polyhedron, an abstract shape. She is a scientist. The dog  - which I learned represents 'faithfulness' - is waiting. The sphere is an abstraction, perfect as the tools and mill wheel are not. They are all 'things', 


    The scales, the hour glass, the bell are said to refer to the knowledge of artisans: they understand weight, measure (of time), music (as an expression of geometry). 


    I have very little understanding of the Magic Square. I do know that all the lines add up to 34 and no number is used twice.

    I do wonder if in a largely illiterate society, the fairly recent acceptance of Hindu-Arabic numbers, as opposed to Roman Numerals, might be part of its purpose here: to illustrate the scholarship, the mathematics that numbers made possible.

    Try substituting Roman Numerals in the grid of the Square.
    Can you quickly add up the amounts? How would you do it as an arithmetic problem on paper? 

     16 +5+9+4 = XVI+V+IX+IV .


    I will write about the polyhedron in the next post. 


    *images from  The Complete Woodcuts of Albretch Durer, edited by Dr. Willi Kurth, 1963  and The Complete Engravings, Etchings & Drypoints of Albrecht Durer, edited by Walter L Strauss, 1972.  Both republished by Dover Publications: www.dovderpublications. com.

    Walter L. Strauss' analysis is the best I've found. I wish he had known more about geometry.