On a piece of paper, design a table with four legs. This is really a very simple exercise that illustrates several practical applications of Euclidean geometry. Make the tabletop square. (Actually, this is not mandatory, but most of us today are used to square or rectangular-shaped tabletops, windows, picture frames or sheets of paper.) To make certain that a particular area is square, all one has to do is make sure each corner registers at exactly 90 degrees. However, with an understanding of Euclidean Geometry and the properties of right triangles, it is possible to determine if our four-sided shape is square without measuring each and every corner. If two corners are square, then the whole figure is square. Also, it is true that if both diagonal measurements are square, then the whole object is also square. This last geometric property is an invaluable way for builders and home-improvement workers to doublecheck their handiwork. Just by building a table, we can see this aspect of Euclidean geometry applied to our daily lives.
Design a two-story house. When you do this, you will first of all discover the properties of the right triangle as described in step 1. But as you design the second story of the house, you will discover the world of parallel lines, as described by the Greek mathematician and philosopher over 20 centuries ago. To concur with Western architectural standards, your house will have to have walls that rise at perpendicular angles to the ground and thus for all practical purposes act as parallel lines. This means that if all walls are continued straight up from the ground, then the second floor will be the exact size as the first floor.
Draw and design the layout for an oval-shaped racetrack that is to be used in track and field events at your local high school. If you study the shape closely, you might notice that your ellipse or oval is really a circle divided in two connected by two parallel and perfectly straight segments of gravel runway. This design exercise underscores Euclid’s understanding of the nature of a circle and also his most famous axiom, which says that any two points can be joined by a straight line. Euclid defined the creation of a circle from a center point, where a straight line of one constant length, the radius, determines the overall shape. If the two ends of your racetrack are circular in nature, then every spot on the circumference of the curve will be equidistant from one point that is located at the center of that same curve. Also note the obvious reference to Euclid’s theory about two points being joined by a straight line. This idea is easily illustrated by emphasizing the sprinting events that take place at a track and field event. In this event contestants run along a straight section of track to see who can reach the finish line first. In reality, the path of the runners is the same as connecting two dots on a piece of paper with a pencil and a ruler. As you can see, a verbal discussion of Euclidean Geometry can easily become very wordy and complex, but essentially most designers today live in the world of Euclidean Geometry. By better understanding this textbook view of squares, circles and triangles, one can become a better designer.