New Series: Geometry and Design

My Drafting Table: Geometry and design happen here
My Drafting Table: Geometry and design happen here

Nearly two years ago, after reading “By Hand and Eye” by George Walker and Jim Tolpin, I had the opportunity to take a course by the same name (taught by Jim at the Port Townsend School of Woodworking… if you are a woodworker and have never been there, I highly recommend signing up for one of the classes they offer.  They’re first rate).  It is almost cliche to say this, but the experience was transformative.  My designs have evolved more quickly.  They are better both aesthetically and structurally.  As a result, I believe my woodworking has improved as well.

Still, throughout this experience, I’ve often found myself thinking that there are things I should contribute to what can only be described as a revival of interest in design.  I make my living as a mathematician but keep some measure of daily composure by retreating into my shop in order to make things from wood.  As such, I know an awful lot about geometry.  I’d like to share some of what I know (and some of what I can teach myself on the fly) through an ongoing series of articles on this blog.

I hope to keep the articles somewhat grounded somewhere more concrete than the Euclidean plane by relating many of them to the drafting table pictured above.  I designed and built it in the summer of 2015.  Since I immersed myself in the design techniques described in “By Hand and Eye,” the table seems like a good device for illustrating some of the geometric ideas I hope to explore.  At the moment (and this could very well evolve as I go), I envision organizing this sequence of articles into at least three distinct chapters:

Chapter 1 will probably be the most familiar.  I’ll walk through the design process I followed in order to create the plan and elevation drawings (and a story stick) I used while building the table.  However, I will give a fair amount of attention to the fundamental concepts of geometry in the Euclidean plane that make the techniques of proportion based design work so reliably.  There will be math and logic involved here, but it will be math and logic with a context that should be meaningful to woodworkers and designers.  In addition to laying a reasonable (but not overbearing) geometric foundation for the techniques of design I used on my table, I plan to explore the difference between exact and approximate layout methods.  As an example, consider the common design task of dividing a distance into some number of equally sized pieces.  There is a well known Euclidean construction that will perform this division exacly (at least in principle). However, many designers will frequently use a set of dividers or a sector (or both) to accomplish this task.  The latter approach is by nature an approximation.  Which approach is better?  Which is more efficient?  Are the answers to these questions the same in all circumstances? Some?  Are they the same for all designers? Also, I haven’t decide upon this yet completely, but I suspect I will put some time into the topic of creating isometric drawings in this chapter.

Chapter 2 will be a first step into uncharted territory for me from the perspective of design (less uncharted from the perspective of geometry).  As it turns out, I’m pretty proud of my drafting table.  I’m rather happy with the way it turned out.  I’ve thought about the idea of writing an article about its construction.  In addition to my belief that it is a neat table, there are some nonstandard design features hidden in it that I think others would find useful.  Unfortunately, I did not take enough “in process” photos of the table while I was building it.  I was rather focused on the building and less focused on the documenting.  In addition, I suffer from a measure of ineptitude when it comes to using a camera.  Even the pictures of the finished table end up looking less like what I’d like to see if I were going to read an article about it in a magazine or online. However, something I can do is draw in proper perspective.  There is a rather rich field of geometry that forms the basis for doing this (projective geometry), but depending on your purpose for constructing a perspective drawing, you might need to know anything ranging from none of its fundamentals to a great deal of them.  Chapter 2 will begin at the simple end and explore drawing in proper perspective the way artists do:  I will walk through the process of drawing an object (such as my table) in perspective when I have the object in front of me (ugh… I’m going to need another table for that).  This artists’ approach is very intuitive and can be successfully undertaken without much knowledge of geometry at all (certainly without any real projective geometry).  It is an approach that is particularly useful when you want to create a hand drawn representation of an object that already exists.  To me, an obvious reason to do this is if you want to publish a representative illustration of your work in place of a photograph for a book or article.  Perspective drawings look more natural than our plan and elevation view shop drawings.  They provide the viewer with a better understanding of how our work looks when a high quality photograph is not available.

Chapter 3 will dive much deeper into projective geometry and perspective drawing.  Imagine you have produced a set of plan and elevation view shop drawings for a proposed piece of furniture. Before building it for your client (or even for yourself), you might be wise to get a more accurate representation of how it will look after construction.  The artists’ approach will not work here.  You do not have the object itself to base your perspective drawing upon.  There is no physical object for you to view in perspective yet!  All you have is a geometric model for the object (in the form of your plan and elevation view drawings).  The problem before you is this: How can you transform a set of elevation and plan drawings into a single image that is drawn in proper perspective?  We’ll need at least a rudimentary understanding of projective geometry (and the projective plane) in order to solve this problem.  Projective geometry is a rather vast subject, so I’ll provide the focus upon the parts we need.

Chapter 4, if I ever get there, will address what I would call an inverse problem.  Suppose you are interested  in building an accurate reproduction of an antique piece of furniture, but the curator of the collection in which it resides forbids you from approaching it or touching it with your measurement devices.  You can take pictures from various vantages, measure the distance from your camera to the piece, etc. but that’s about it.  Your photographs can be thought of as perspective images of the piece of furniture.  Do you have enough information to deduce a set of plan and elevation views of the piece?  If so, how?  If not, how much more information do you need? What are some strategies for obtaining it without running the risk of enraging the curator?  We’ll need an even deeper understanding of projective geometry and the projective plane in order answer these questions, but the effort could be worth it in orrder to gain an accurately proportioned set of drawings of the piece you want to reproduce

As I’ve said, I have a day job.  Because of this, I suspect what I’ve described here will be a long term project.  At first, I’d like to try to stick to a schedule of an article a week (perhaps releasing them each weekend), but I can almost guarantee I will deviate from that schedule from time to time.  I’ll make a serious effort to publish them frequently enough that anyone interested in reading them will be able to maintain focus/interest.  I’ll file all of the articles under the Geometry and Design category so that they can be found easily.  I have also placed a menu item on the top of this blog that will take you directly to the Geometry and Design articles.  Comments and feedback are welcome.  Hopefully we can all learn something from this effort.

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