A Cabinetmaker’s Sketchbook using
Freehand mechanical drawing.
Part 2
During the Renaissance, the great architects set out lists of their favored room proportions. They would start at the Unity Circle, proceed to the Unity Square, and end at the double square. For example, Vitruvius recommends the Unity Circle, the Unity Square at 1x1, 5x6, 4x6, 3x4, the square root of 2, 2x3, 3x5, ending in the double square. Palladio’s recommendations are the Unity Circle, 1x1, 3x4, root 2, 2x3, 3x5, ending in the double square. By the way, Palladio began his career as a stone mason and carver. Serlio gives 1x1, 4x5, 3x4, root 2, 2x3, 3x5, again ending in the double square. One of my teachers said that Serlio also seems to have come up through the trades; he said he had the feel of the workbench in his writings. Alberti suggests the Unity Circle, 1x1, 3x4, 2x3, and the double square. So let us do cuts and expansions of the Unity Square leading to the double square.
The first cuts are the simplest, as it is the division by two, with the continuing regression of the diagonals starting with lines EZG and FZH.
The second set of cuts uses the diagonals EZG and FZH, as well as the half diagonals AH, AG, BE, and BF. Where these lines intersect a set of squares can be drawn that divides the Unity Square into nine equal squares. The cuts and extensions will proceed as before. I personally find this division the most useful, for example the 3x5 makes a great start for a free standing bookcase, either vertical or horizontal.
The third construction is based on the Root Rectangles. For example, the Unity Square has a side of One; therefore the diagonal is the Square Root of Two by the Pythagorean rule of Right Triangles. If the diagonal of the Root Two rectangle is extended we get a rectangle One by the Square Root of Two with a diagonal of the Square Root of Three, and so on. You will notice that when we get to the Double Square our diagonal is the Square Root of Five, this becomes very important in the final constructions, as it was in the last, for example lines BE, BF, AH, and AG are all diagonals of the internal double squares of the Unity Square.
The forth construction is based on the octagon. The octagon is drawn by setting the compass at the radius of the Unity Circle, and drawing the arcs centered on the corners of the Unity Square through point Z, then connecting the points where the arcs touch the sides of the square so a regular octagon can be generated.
The power of the Octagon is that it can generate divisions by Ten. If you look at the right side of the drawing you can see that we have two squares on either side of a rectangle. If the squares are divided by three, and the rectangle is divided by four we get 3+4+3=10, the divisions are not perfectly matched but are close enough for our purposes. For those of you familiar with the drawer divisions of the base of an eighteenth century highboy the use of this proportion is obvious, it leaped out to me the first time I saw it demonstrated. Another use for the octagon is the sizing of room moldings. Since many of these architects call for the room to be divided vertically by five the octagon can give us the heights of moldings for any room based on the Classical orders, very useful in designing libraries. I’ll go into that method in a future post.
The final construction gives us the Golden Section, or Golden Mean, one of the most contentious areas in this field. You can explore the Internet and find many sites set up to deny that the Golden Section was ever used for design in antiquity, yet it still turns up. As one of my teachers, Steve Bass would say, “If your within 5% you’ve got a definite, if you’re within 10% your closing in.” Anyone who has spent time building knows how true this is. Now let’s move into the arena of the Great Pyramid of Cheops, where the side of the base square is One and the height is close to Phi, this sort of “close enough for government work“ makes sense doesn’t it? Especially if you’re working stone with stone 5000 years ago, got to hand it to them. If that doesn’t affect you maybe take a look at the plan of The Forbidden City in Beijing, or the internal geometric relationships in the Tage Mahal. This is as close to Plato and the Socratic dialog known as The Timaeus as I will get here, I’m no philosopher or mathematician; so let’s get back to the sketchbook and the workbench.
In this set of divisions and extensions we find that the division of Unity into the segments Phi and Phi squared, Plato’s uneven break, (this Phi squared is sometimes known as Phee) produces a ratio between Phi and Phi squared that is the same ratio between Unity and Phi. By moving these geometric cuts up and down we can find a large amount of dimensions all with a Golden Section relation to each other. We will look into that in the next drawing.
So how can we use Phi to geometrically generate a series of related dimensions? We can through the use of the Phi Scale.
For example, in my old note book I have a design for a small side board, based on a mahogany plank that I had at the time. The plank was 16” x 55” x 5/4; this was to be the top. Starting at a Unity Square with a side of 52” and letting Phi be the height of the project I was able to carry through the design using the Phi scale to generate the thickness of the turned legs, the details of the carvings on the legs, and any other detail that was necessary, including the depth of the unit. The only problem is I never got to build it, the usual excuse of woodworkers and cobblers. By the way, the Phi scale joined with some simple divisions allows for the graduated proportioning of drawers in a four drawer chest. I’ll get to that again in a future post.
I think this is as far as I want to go today. In the next post in this series I’ll give the design for a set of proportional dividers to be used in photo analysis. By analyzing photos of preindustrial era furniture in magazines dedicated to the sale of antiques maybe we can tease out the uses of these manipulations of the Unity Square.
Further reading and drawing
I know my hyperlinking needs work.
I’m working on it.
After all I’m just a Boomer.
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