Proportioning systems

Simple Diagonals divide this space into halves, thirds, and fourths

Ever since the first caveman or cavewoman drew the first saber tooth tiger on the wall, we have been exploring, discovering, and rediscovering proportions. I watched one of those “how to paint” artists on PBS recently. The instructor was working on a painting that featured a sand hill crane. He started off by saying  “To pull this off you must get the proportions right.” I perked up and listened, then but he didn’t elaborate. Evidently everyone but me must know how to proportion a sand hill crane.

When it comes to woodworking, understanding proportions are one of the foundation blocks of design. I’ll go so far as to say that anyone who produces a large body of work which is recognizable has worked out an approach to proportions even if it’s something completely freehand in their mind. I suspect that’s that case with many accomplished designers. I’ve questioned many builders who produce outstanding work and often they cannot explain how they proportion their work. Obviously they have mastered it. I put them in a category of someone able to play music by ear but unable to read notes. My hats off to anyone who has achieved this whether through keen observation and study, talent, or most likely a combination of both. Down through history there have been numerous ways to use proportions to arrange a composition and link the parts together into a unity. I primarily focus on the use of simple whole number ratio proportions but I thought I’d share briefly another approach that was once used. Some refer to it as a simple geometric approach and in the fine arts it’s referred to as the armature of rectangle. Take any square or rectangle and with simple diagonals you can divide up a space proportionally both vertically and horizontally. Connecting the opposite corners divides the space in half (Point A). Connecting the midpoint down to a lower corner divides the space into thirds (Point B). Finally, connect two midpoints and it divides the space into fourths (point C). You can continue adding diagonals and dividing the space into smaller and smaller units. I have no idea if furniture builders used this method, but I do know they frequently used and studied simple geometry. It’s also a handy thing to know regardless.

Using simple geometry to design a doorway by Serlio

This approach was used by architects as this drawing from Serlio illustrates the layout of a door opening on a wall. Renaissance painters are most famous for using this approach to organize a composition on a canvas. If you would like to read more about that you might check out a copy of “Classical Painting Atelier” by Juliette Aristedes at your library.

George R. Walker

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19 Responses to Proportioning systems

  1. Jeff Burks says:

    Proportional diagrams like this always remind me of Archimedes stomachion. I just finished reading a book on the Archimedes Palimsest that goes into some detail about the puzzles that evolved into the mathematics of combinatorics, and this makes me wonder if the same game of subdividing squares into their internal triangles also evolved into this method of visualizing proportion in design. Great post.
    http://en.wikipedia.org/wiki/Stomachion

  2. joe sullivan says:

    George:

    I’ve decided to panel my entryway in white oak cut from my parent’s (and grandparent’s) place. While the sentiment will carry me a long way, I want to get the proportions right. If they are wrong it will bother me every time I see it. My eye for this stuff is sharp and unforgiving, but I don’t know how to make it happen.

    There are two facing walls to be paneled. The ling one is probably 9ft and the short one 4 ft because of a wide door into another room. Because of that door there is a short stub on the same side as well.

    Any guidance as to where to start in laying this out?

    • walkerg says:

      Joe,

      I’d be glad to give you some input but may need a better idea of what you are trying to accomplish. You can e-mail me direct at gwdesignmatters@gmail.com
      with a little more detail or perhaps a rough sketch of what you have in mind. Also, my first DVD touches on wall treatments and how the classic orders were used to layout the elements in a wall.
      George

    • jlsmith says:

      Joe

      If I could jump in I would offer this advice:
      Since you have confidence in your ‘eye’ you could start the process by laying out what your ‘eye’ sees a correct configuration using painters tape. This will allow you to view the layout in space and time. The tape will allow you to make multiple adjustments based on your direct experience. Once you are satisfied with the layout you can then ‘reverse engineer’ it by constructing a series of drawings using the measurements of the tape layout. These drawings can then be analyzed/deconstructed searching for a pattern. They probably won’t reveal an exact proportioning system (but they might if your eye is really good) but you should be able to adjust the layout in such a way to bring the entire layout into a proportioning system (it should be noted that this can be a time consuming process; being able to resolve all the existing conditions with a new proportioning system is typically very difficult, and thus compromises while probably be required). Once the drawing has been adjusted you can then tape the walls one more time based on the adjusted drawings which will allow you to test the layout before you cut any wood. It may sound like a lot of work but given your concern that “If they are wrong it will bother me every time I see it”, it should be well worth the effort.

  3. Frank Vucolo says:

    This is a really, really, really good blog.
    I appreciate it.

    Thank you,
    Frank

  4. jlsmith says:

    George

    Did you happen to see this article:
    SYMMETRY IN THE SUBATOMIC WORLD?
    (the Dan Brown reference is gratuitous but the substance of the article is interesting)

    http://news.discovery.com/space/symmetry-in-the-subatomic-world.html

    • walkerg says:

      Jlsmith,
      I had not read this before. I was planning on addressing the golden rectangle and phi in an upcoming post. You may find my views on this take an unexpected tack.

      George

      • jlsmith says:

        A general comment regarding the idea of how does one start a design (ie Terry’s comment or Joe’s desire for a ‘running start’). The idea of how do you start is not complicated it can be difficult (when I use to teach architectural design I referred to this as ‘the fear of the blank page’). You just need to start, do something, anything to get yourself started (brainstorming). Often times a start can be found in some sort of limiting factor or restriction. For example in Joe’s case it could be something in the existing conditions perhaps the length of the shorter wall is a starting point given its length limits the number of panels it could be divided into and still look ‘OK’. There it is, a start. Now the distance between where you start and where you end up can be long, but you have to start in order to get to the end. There is no proportioning system that will generate an idea (a common mistake made by those who are just beginning to use proportioning systems). A proportioning system is not an ‘idea’ tool. It a harmonizing system. By applying a proportioning system one is able decide between the infinite number of relationships between the elements.

  5. Terry OD says:

    I have been trying to understand this — for example, how is the square determined? I notice that Wittkower wrote that Serlio’s geometry was done after the door dimensions were chosen, so it seems to not be used as a method to initially establish the dimensions.

    This seems similar to the medieval and Renaissance canons for laying out text pages in a book. The goal seems to have been to establish formal relationships between the edges and diagonals of the page and the rectangles containing text, to establish a degree of order, balance, and proportionality. When those relationships were neglected, the standard of book design plummeted. While the means may have been a bit different, the resulting harmony created by the use of ‘regulating lines’ is very similar to what architects were striving for.

    -Terry

    • joe sullivan says:

      Friends:

      Thank you for the advice on using tape and such for layout. I do plan to do just that, but wouldlike a bit of a running start.

      Back to the idea of relationships — it seems there has to be an elusive balance between the math and the eye, if you will. For example, you can study Fibonacci sequences and the related Golden Proportions, and sure enough, they apply or ALMOST apply to many great works of art and architecture. but they can also fall flat. Seems the difference between mathematics and aesthetic genius is somewhere in the ALMOST.

      Cheers!

    • walkerg says:

      Terry,

      I try not to read too much into this. I simply take this as a quick nonmathematical way to divide up a space into proportional units ie thirds, fourths, etc. Wittkower does point out that this drawing by Serlio did produce a door opening that was two squares high. Again using this simple geometrical method resulted in simple proportions 1:2, 1:3.
      The information about medeival manuscripts is interesting. Had not heard that before.

      George

  6. Jeff Burks says:

    For further reading on the proportions of page layout and how this applies to this discussion read:

    http://en.wikipedia.org/wiki/Canons_of_page_construction

    • Jeff Burks says:

      This discussion just reminded me of my favorite quote from Frank Chouteau Brown &ect from the Study of the Orders Vol II. (1906)

      “But it must always be remembered that the “rules” to which we have now reduced the Classic Orders, are not to be considered as the principles upon which they were first designed. Rather, these rules and systems of proportioning the details of the Classic Orders of architecture have been invented by enthusiastic theorists and student of later times to fit the old examples. The people who erected these ancient monuments understood no such rules, but rather created their work under the direct influence of a vital artistic instinct and life of which to-day we are imitating the mere empty forms….

      Rather than quote to whole passage I would just refer others to page 154 under the heading “Rules of Classic Architecture, Their Use and Misuse.”, which can be read in full at Google Books.

      The reason I bring this up is to remind people that many times, when others have bandied about such rigid ideas as rules and systems for design, they are overlooking the fact that the majority of these so called rules are the modern inventions of scholars attempting to grapple with the past.

      For example, one could take a rulebook for the Doric Order (where everything is broken down by column diameters and such), and attempt to apply them to existing ancient structures of that supposed Order…. and be sorely disappointed to find that they don’t fit because the rule book was based on averages, or prominent examples.

      • jlsmith says:

        Jeff

        While of course Frank Chouteau Brown is correct in broadest of terms (even the Renaissance masters couldn’t agree on the exact proportions of The Orders, (see ‘The Classical Orders of Architecture’ by Robert Chitham for a side by side comparison) I don’t agree with the following: “The people who erected these ancient monuments understood no such rules, but rather created their work under the direct influence of a vital artistic instinct and life…”

        One can travel (today) to any of the ancient Greek or Roman sites and see remarkably similar vocabulary being used in the architectural remains of civic buildings. However, the same is not true when it comes to common private housing (Roman vs Greek for example). So clearly cultural differences did manifest themselves in architectural forms. So I find it very difficult to understand how the similarity in civic architecture could have taken place without the use of common knowledge of aesthetic principles that goes well beyond “vital artistic instinct”.

      • walkerg says:

        Jeff,

        My intent is not to share information about any proportional system as a paint by number or wooden aproach. The goal is to help woodworkers grasp some of the basics in recognizing and using proportions in a practical way. I will comment though that much of the finest American work to come out of cabinet shops from the 18th and early 19th century relied heavily on simple proportions and the orders as laid down in vest pocket design books written for the common workman. These simple lessons still can provide a basis for solid design.

        George

  7. joe sullivan says:

    Jeff:

    Interesting comments, and at least partly true. While it is almost certainly the case that the “orders” were not rigorously applied the way a computer would do it, it is also true that various classical buildings with similar characteristics do follow the “orders” to a large degree. Doubtless, this was partly done by eye and rule of thumb. However, the Greeks and Romans spent a great deal of time on math and engineering and understood both quite well. Temples were both sacred and objects of civic (national) pride. As such, they got the best efforts of very sophisticated craftsmen and designers who understood very well what they were about. For example the curving tapers of the columns of the Parthenon are so subtle that lines from opposite sides intersect at a significant distance from the column, and they all do it at the same spot. This required precise calculation. Of course everything else in that superb temple approaches perfection, too. If you visit the British Museum and walk from a room containing a very fine Greek temple from Asia Minor, and pass into the room with the Elgin Marbles, you are struck by the difference.

    Similarly, in a wholly different aesthetic, the Taj Mahal, a massive marble building, seems to almost float. I spent much time staring at it and photographing it, trying to understand this phenomenon. Only later, while looking at a near exact scale model in a shop in Agra did it strike me. The famously out-of-plumb minarets create a visual tension that lightens the whole building. The model lacked the few degrees of out-of-plumbness, and also lacked the lightness. This could not have been done on the fly or after the fact. It is a remarkably subtle act of genius.

    Book design is another matter. I know typographers and layout people who do indeed study the cannons and rules. However, while they use rulers and such, most of the balance is done by eye. In typography and layout, we come much closer to the sense of the very interesting comments you posted.

    Excellent posts, by the way.

    Cheers!

    Joe

  8. joe sullivan says:

    Mr. Smith:

    Quite right — the differences in the lengths of the walls is one of the challenges in my very simple problem. Panels of a size that will “break even,” feel too narrow. Also, there is a wide door frame on one side and a stair with no frame on the other. And, though I can generally visualize the end result, there is the question of how to deal with the spaces over the door frame, and the cornice molding. Finally, and importantly, there is the question of the widths of the styles and rails as they relate to the width of the raised panels.

    Joe

    • jlsmith says:

      Joe

      Sometimes the simple problems prove to be quite a challenge to solve. Being an architect that does a fair amount of residential work I am familiar with these sort of ‘simple’ problems. However, I do not want to high jack Mr. Walker’s blog so if you want to continue this conversation feel free to contact me directly: jlsmith-shop@comcast.net

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