Abstract
This is an interesting geometry project that goes back to the time of Archimedes, the famous Greek mathematician. You can combine this mathematical project with computer science and take this ancient problem into the twenty-first century with a dynamic diagram using the geometry applet.Objective
Introduction
Figure 1 below shows an arbelos. What is an arbelos? The arbelos is the white region in the figure, bounded by three semicircles. The diameters of the three semicircles are all on the same line segment, AB, and each semicircle is tangent to the other two. The arbelos has been studied by mathematicians since ancient times, and was named, apparently, for its resemblance to the shape of a round knife (called an arbelos) used by leatherworkers in ancient times.
Notes on How to Manipulate the Diagram
The diagram is illustrated using the Geometry Applet (by kind permission of the author, see Bibliography). If you have any questions about the applet, send us an email at: scibuddy@sciencebuddies.org. With the help of the applet, you can manipulate the diagram by dragging points.
In order to take advantage of this applet, be sure that you have enabled Java on your browser. If you disable Java, or if your browser is not Java-capable, then the diagram will still appear, but as a plain, still image.
If you click on a point in the diagram, you can usually move it in some way. The free points, usually colored red, can be freely dragged about, and as they move, the rest of the diagram (except the other free points) will adjust appropriately. Sliding points, usually colored orange, can be dragged about like the free points, except their motion is limited to either a straight line, a circle, a plane, or a sphere, depending on the point. Other points can be dragged to translate the entire diagram. If a pivot point appears, usually colored green, then the diagram will be rotated and scaled around that pivot point. (Note that diagrams will often use only one or two of the above types of points.)
You can't drag a point off the diagram, but frequently parts of the diagram will be moved off as you drag other points around. If you type r or the space key while the cursor is over the diagram, then the diagram will be reset to its original configuration.
You can also lift the diagram off the page into a separate window. When you type u or return the diagram is moved to its own window. Typing d or return while the cursor is over the original window will return the diagram to the page. Note that you can resize the floating window to make the diagram larger.
An interesting property of the arbelos is the "twin circles" discovered by Archimedes (see Figure 2, another dynamic diagram). If you draw circles tangent to the line CD and inscribed within the arbelos, the circles will be congruent. Archimides proved this in his Book of Lemmas (proposition 5).
This project centers around a curious property of the arbelos that involves the twin circles. The line segment EF in Figure 3 passes through the center of the right-hand twin circle. Its endpoint F is the point of tangency where the right-hand twin circle touches the smaller semicircle of the arbelos. The endpoint E is the intersection of the extension of CD and the line from F through the center of the right-hand twin circle. Curiously enough, line segment EF is congruent with line segment AC, the diameter of the other semicircle inside the largest semicircle of the arbelos. Can you prove that EF = AC?
Terms, Concepts and Questions to Start Background Research
To do this project, you should do research that enables you to understand the following terms and concepts:
Bibliography
Materials and Equipment
To do this experiment you will need the following materials and equipment:
Experimental Procedure
Variations
Credits
Last edit date: 2006-04-19 23:59:59
If you like this project, you might enjoy exploring careers in Pure Mathematics.
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