Friday, August 23, 2013

Cobbler's house north of Boston, c. 1840 - Part 3 of 4

I have just finished writing about the Arnold House: how it uses the 3-4-5 triangle as a regulating principal and how it is almost but is not the same as the Golden Section.

If you don't know what that is, please read the posts on the Arnold House. Thank you.

I said the same thing about this house: it 'almost used Circle Geometry', but didn't. I realized I needed to look again.

I do love the grace of the Golden Section and Circle Geometry.
But look at how the 3-4-5 triangle fits.
(A)  marks the diagonal of the rectangle created by the 3-4-5 triangle. It's half the house. The other side matches it.
(B) marks the corner of the smaller room on the right, its size determined by the triangle. The larger rectangle forms half of the house footprint. The room's proportions are determined by the proportions of the house itself.
The back wing, a rectangle that matches the 2 front sections, is a little longer. (C) shows how the extra space is determined by half a 3-4-5 rectangle.

I rather hoped the end elevation wouldn't fit the pattern!
It does.
(A) shows the rectangle created by 2  3-4-5  triangles. It fits the size of the house, easily giving the master carpenter the dimensions he needed for post and beams. It even determines the roof pitch! Not a 9/12 or a 10/12 pitch, ratios commonly used today, but the 3-4-5 triangle.

However, the floor heights, the window placement, the elevations still seem determined by both circle geometry and the Golden Section.

The house's outline - its footprint, height and roof pitch - do seem governed by the triangle.

How nice it would be if I could teleport back to 1840, watch the house-wright at work, and ask questions!

My great thanks to Jay Cougar White Cloud for his good questions that pushed me to consider the simplicity of  framing with the 3-4-5 triangle.

For the next post, and a more thorough analysis of the house click here:

Arnold House, Westfield, Massachusetts - Part 2

Here is the Arnold House around 1933.

And below is the measured drawing of the front done at the same time for the Historic American Building survey, a project of the WPA. The drawings and picture are available at the HABS website.

In the upper left corner of the drawing I have added the 3-4-5 triangle with its 90*, right angle. To see why I think this house was designed using that ratio see the previous post, Arnold House, Part 1. Click on the illustrations to enlarge.

On the left side I have drawn in red dashed lines the 3-4-5 triangle using the first floor as the base and the ceiling of the second floor as the top. A house-wright would have used that line to determine his top plate. See (A).

The center of the  resultant rectangle is marked by the red 'x'. The 'x' is the location of the second floor beam.
The right hand edge of the rectangle is the edge for the door, telling the framer where to locate the rough opening. The windows and doors do not appear to be determined by the elevation, but by the plan. See my earlier post on the Arnold House - Part 1.

As drawn on the right side - see (B) a red dash and dot line - the space on either side of the door is a square. But the windows are not symmetrical within the box. I tried using the Golden Section. It doesn't fit. Circle geometry - here a circle based on the square - almost fits: several inches too large.

Here I used the 'face' of the house clapboard from foundation to eaves, as I think the concern of the master carpenter would have shifted from framing to presentation - how the house appears to the community.

Two squares with the arcs from their circles would have fit across the front of the house, meeting at the middle of the front door IF the house had been 9" longer - 40 ft, not 39'-3", a true 3-4-5 rectangle. Was the foundation not true? The house-wright inexperienced or lax? If the wood was green would it have shrunk 9"? Or?

For reference I have added a diagram of a square with the circle whose radius is half the square's diagonal.

This house is no longer there. I would very much like to be able to see it,walk through the rooms, sense the proportions. I want to know if it feels differently because the ratio is mathematic, not geometric.

The 3-4-5 triangle is not the Root-Two Rectangle. The diagram here shows the difference geometrically : the triangle is marked off in red, the ratio (a square and the length of it's diagonal) in black.

 In algebraic language : (a x a) + (b x b) = (c x c)
                            (3 x 3) + (4 x 4) = (5 x 5)  or   9 + 16 = 25

For the Root-Two Rectangle to equal the triangle, if the side were 3 then  the diagonal of the square would have to equal 4. It doesn't.
                    (3 x 3) + (3 x 3) = 18   the square root of 18 is about 4.25

I think the evidence is too strong that the house-wright was using the  3-4-5 triangle to put together a traditional looking house. The fact that circle geometry almost works is just that - it is almost the same. But it isn't.

updated 12/31/15

Tuesday, August 20, 2013

Arnold House, Westfield, Massachusetts - Part 1

The Arnold House,Westford, Mass. c. 1800, was measured by Edward E. Jordan for the Historic American Building Survey.
On the first HABS sheet is written: "Note: house partly destroyed by fire during winter of 1933-34." I know very little about the house or the family. Probably the builder was Joseph Arnold who died in 1823.

While I was thinking about how this house was laid out, a friend asked if I had ever seen the 3-4-5 triangle used for design. Like the circle and the square it is independent of fixed dimensions. It is simply a ratio that always produces a right (90*) angle.

I had not seen it, but I began to pay attention.

So - look at this house. Click on illustrations to enlarge.
Note the outside dimensions -
 30'-6"  by  39'-3".
For a wood frame, post and beam, hand hewn house that's close enough to 30ft. and 40 ft. So for the 3-4-5 triangle the diagonal across the house, (A), should be about 50 feet long. It is.

Then look at (B), the diagonals of the front rooms. The same angle gives the dimension of the parlor. The diagonal (B) in the Living Room is not as clean, but still defines the size of the fire places and chimneys and the post location at the rear of the house. (C) determines the post location in the Dining Room.

The 3-4-5 ratio also gives the locations of the windows, at least one door and the size of the Room behind the Parlor.
Here I have added what I think was the original back wall of the Parlor for clarity (X). I have also used a red line with a dot for the rectangles created and the center lines of the doors and windows
(D) shows the Parlor window placement on the side directly on the center of the room, the front windows equidistant from the center.
(E) shows the window spacing in the Living Room determined by the rectangle turned on its side
(F) places the center line for the Dining Room window and door at the edge of the rectangle.
(G) determines the size of the Room behind the Parlor and the location of the stairs.

Next I will post the front elevation and add some ideas I have about the house design.