In modern times, we have all of these neat contraptions - such as rules and protractors - to help us make precise drawings. But what about back in Archimedes' time? The Greeks were making very precise blueprints for immensely complicated pieces of architecture without any of the newfangled gadgets that we have today; they had only a compass (for making circles) and a straightedge (without any distance markings). How did they do it? In this project, we were able to find out firsthand.
Why bother, though? Why would we want to make drawings without a ruler when we have multiple at our beck and call? This is a similar question to asking, "Why must we learn math, when we have calculators?" This project was purely conceptual; it helps to show us why geometrical concepts work the way they do, based on axiomatic logic. These fancy words simply mean that we are starting with an axiom - a fact that everyone can agree on as true - and working upward from there, saying "If this is true, then this must also be true." Even more simply: the Art of Construction project was all about proofs.
Why bother, though? Why would we want to make drawings without a ruler when we have multiple at our beck and call? This is a similar question to asking, "Why must we learn math, when we have calculators?" This project was purely conceptual; it helps to show us why geometrical concepts work the way they do, based on axiomatic logic. These fancy words simply mean that we are starting with an axiom - a fact that everyone can agree on as true - and working upward from there, saying "If this is true, then this must also be true." Even more simply: the Art of Construction project was all about proofs.
I started with Benchmark #1: choosing an image to reconstruct using geometrical constructions; this image is shown above. It depicts the NCC-59804 USS Princeton, a Niagara-class cruiser from Star Trek, with the Star Fleet emblem in the background. Actually, when first asked to come up with an image, my first thought was of the USS Enterprise, because I knew that I had to construct it. The USS Enterprise has a lot of circles and curves that are very easy to construct using a compass, if you know how to use it well. I changed my mind, though, simply because the USS Enterprise had been done to death; it's old, and I wanted something new. So, I went online and looked up the different starship classes from Star Trek, and I found that the Niagara-class fast cruiser was unusual enough to be interesting while still possessing the same circles and curves. The best part: it is identical to the USS Enterprise in every detail except the warp pods. The Niagara-class is identifiable by its third warp pod, located underneath the ship, which meant that a top view meant hardly any extra work.
Next, I remade my image using only a compass and a straight-edge: my Benchmark #2. This was made much easier by the fact that I had an image to go off of, so instead of setting my compass to an arbitrary width, I could make my image more precise. My image required a lot of work and time; my first draft took me four hours because I didn't know what I was doing, but even my second draft (seen above) took me a couple of hours! If you want to make it yourself, I typed up instructions to do so. You can see them in the following post. (Warning: the instructions are very extensive! Follow them only if you have a lot of time on your hands.) I did make one change from the original image: the Star Fleet emblem proved very difficult to construct, and at any rate I needed a square and a hexagon, both of which were missing from the USS Princeton. (The full list of required elements is as follows: an equilateral triangle, a square, a regular polygon (I chose a hexagon), a rosette (also called petal flower), and a circle (I'll bet you can't find one of those on my image).) So, I changed the background to incorporate those constructions. It may be easier to see what the finished construction looked like in the image below, in which I went over the picture's lines with a sharpie.
For Benchmark #3, I simply decorated my geometric construction to make it appear more finished and look more presentable. I decided to construct my image on a canvas, and paint it. You can see my finished art piece above. I decorated the background - the squares and hexagon - to appear as outer space, with the petal flowers painted to be stars. And, of course, the USS Princeton was painted to look like what it is. In Benchmark #3, we were allowed to embellish our image using non-construction techniques. My image needed few embellishments. The only details on my painting that weren't constructed were the yellow and blue details on the USS Princeton, and, of course, the writing: "NCC 59-804".
You can imagine, with the complexity of my construction, that I would have come across several stumbling blocks. The main one was the parabolic shape that makes up the body of the ship (that the huge circle and the warp pods are connected to); I know the equation for a parabola (y = a(x - h)^2 + k, and it expands into a quadratic equation: y = ax^2 + bx + c), but I don't know how to create one using geometric constructions. I still don't. I went to my teacher, who advised that I approximate it with an oval shape and two straight lines. I think that there is a way to do it, but it is extremely complicated. I feel like I very successfully constructed some rather abstract shapes, though, by breaking them down into a few component curves and lines. That's why, in the Art of Construction project, the most useful habit of a mathematician was, for me, "Take Apart and Put Back Together." If I were given the capability to redo this project, I think that I would have constructed the embellishments that I put on in Benchmark #3. The embellishments that I made were minor details, and because my project was complex enough without them I deemed it unnecessary. However, I believe that I could have constructed them, and I think that, now that I know the amount of time the constructions took me, I would probably have had time to construct them. Besides, upon taking a closer look at them, I found that they look more difficult than they actually are.
You can imagine, with the complexity of my construction, that I would have come across several stumbling blocks. The main one was the parabolic shape that makes up the body of the ship (that the huge circle and the warp pods are connected to); I know the equation for a parabola (y = a(x - h)^2 + k, and it expands into a quadratic equation: y = ax^2 + bx + c), but I don't know how to create one using geometric constructions. I still don't. I went to my teacher, who advised that I approximate it with an oval shape and two straight lines. I think that there is a way to do it, but it is extremely complicated. I feel like I very successfully constructed some rather abstract shapes, though, by breaking them down into a few component curves and lines. That's why, in the Art of Construction project, the most useful habit of a mathematician was, for me, "Take Apart and Put Back Together." If I were given the capability to redo this project, I think that I would have constructed the embellishments that I put on in Benchmark #3. The embellishments that I made were minor details, and because my project was complex enough without them I deemed it unnecessary. However, I believe that I could have constructed them, and I think that, now that I know the amount of time the constructions took me, I would probably have had time to construct them. Besides, upon taking a closer look at them, I found that they look more difficult than they actually are.