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Chain Reaction: Inversion and the Pappus Chain Theorem

Difficulty
Time Required Average (6-10 days)
Prerequisites Good grasp of Euclidean geometry, a firm understanding of how to construct a mathematical proof, determination
Material Availability Readily available
Cost Very Low (under $20)
Safety No issues

Abstract

Here is a challenging problem for anyone with an interest in geometry. This project requires background research to solve it, but it is an excellent illustration of visual thinking in mathematics.

Objective

The objective is to prove that the height, hn, of the center of the nth inscribed circle, iCn, above the line segment AC is equal to n times the diameter of iCn.

Math Chain of Circles

Credits

Andrew Olson, Science Buddies, author,
Prof. Harold P. Boas and Alex Bogomolny, for their inspiring and insightful publications on the subject, and
Prof. David Joyce, for the Geometry Applet.

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Last edit date: 2013-05-31

Introduction

Important: You will need the current version of Java installed on your computer for this project. If you do not have Java, some figures and images in this project may not display properly in your browser. To check for the current version of Java on your machine, visit Java.

Figure 1 below shows a series of circles (iC1, iC2, iC3, …, iC30), inscribed inside 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, AC, 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.

The chain of inscribed circles is sometimes called a Pappus Chain, for Pappus of Alexandria, who studied and wrote about it in the 4th century A.D. The inscribed circles are tangent to one another, and to the boundaries of the arbelos. That is, iC1 is tangent to each of the three semicircles that form the boundary of the arbelos, while each successive circle is tangent to the preceding one and to two of the semicircles that bound the arbelos (note that, in its default position, the figure illustrates just one of three possible configurations of the chain). Pappus proved a theorem (which he called "ancient"), which states that the height, hn, of the center of the nth inscribed circle, iCn, above the line segment AC is equal to n times the diameter of iCn.


Figure 1. Chain of circles inscribed in the arbelos.

Pappus' proof, relying solely on Euclidean geometry, ran over many pages. The modern proof is much simpler and uses the powerful method of circle inversion, invented in the 1820's by Jacob Steiner. Try manipulating the figure by clicking and dragging one of the orange points, A, or B. Note that as you do this, not only do you re-size the arbelos and the chain of inscribed circles, you also cause the corresponding black point, A' or B', to move as well. Point A' is the inverse of point A, and point B' is the inverse of point B, both points having been inverted through a circle whose center is at point C.

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 an image may display stating "This plugin is disabled" or the diagram may 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.

Now, back to Pappus' chain of circles
Figure 1, above, showed the result of inverting points A and B. Things get really interesting when we invert other features of the diagram, again using for the inversion a circle whose center is at point C. For instance, what is the inverse of semicircle AC, or semicircle BC, or one of the inscribed circles, iCn? It turns out that they are all related in a very interesting way. You'll need to do further research on circle inversion (see the Bibliography to get started) so that you can prove the following facts to yourself, but for now, take them as given:

  • The inverse of the center point of the circle of inversion is a point at infinity.
  • The inverse of a circle whose circumference passes through the center of the circle of inversion is a line.
  • The inverse of a circle whose circumference does not pass through the center of the circle of inversion is a circle.
  • Inversion preserves angles, but with reversed sign (i.e., inversion is an anti-conformal mapping).

What is the inverse of a circle whose center lies on the circumference of the circle of inversion?

Figure 2, below, again illustrates the Pappus Chain, this time with three additional features. The circle of inversion (in dark green), has its center at C and radius CI. The semicircles AC and BC are inverted to two parallel lines, perpendicular to line AC and, of course, passing through points A' and B', respectively. What do you think would be the result of inverting the inscribed circles, iC1iCn?


Figure 2. Chain of Inscribed Circles and the Circle of Inversion.

Your goal is to gain a sufficient understanding of the principles of circle inversion and their application to the arbelos so that you can demonstrate to yourself and others that the statements in the preceding two paragraphs are indeed true. Then, using these facts and an additional insight or two, prove Pappus' Theorem:

hn = n × diameter(iCn).

Terms and Concepts

To do a project on Pappus' Theorem, you should do research that enables you to understand the following terms and concepts:

  • Circle inversion
  • Inversive geometry
  • Conformal mapping
  • The arbelos

Bibliography

Circle inversion. This is the heart of the matter. Here are a couple of links to get you started on your research.

Here is an excellent introduction to the concept and method of circle inversion (written and illustrated by the author the Geometry Applet, Prof. David Joyce):

Another reference is: Eric W. Weisstein. Inversion. From MathWorld--A Wolfram Web Resource.

The arbelos. There are several websites with more information on the fascinating properties of the arbelos.

Alex Bogomolny has a wide-ranging mathematics website. Here is a link to his page on the arbelos:

Geometry Applet. The Geometry Applet was kindly provided to Science Buddies by its author, Professor David Joyce, who wrote it for illustrating an amazing online edition of Euclid's Elements:

Pappus of Alexandria. You'll find a brief biography of Pappus here:

Materials and Equipment

  • Computer with Internet access for research (and possibly creation of figures)
  • Paper
  • Pencil
  • Compass
  • Straightedge

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Experimental Procedure

  1. Use the resources in the Bibliography section and your own research to gain an understanding of the principles of circle inversion.
  2. Apply your new knowledge to prove the assertions regarding Figure 2 in the Introduction.
  3. Experiment with Figure 2, above, to gain insight into how the method of inversion can be elegantly applied to prove Pappus' Theorem.
  4. Complete your proof.

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Variations

For a more basic geometry project, try:

If you'd like to try making your own dynamic diagram with the Geometry Applet, check out these links:

  • Geometry Applet How-to Pages
  • Combining Computer Science and Math: Inscribing a Circle in a Triangle Using the Geometry Applet
  • The Introduction notes that "in its default position, the figure illustrates just one of three possible configurations of the chain." A second configuration can be obtained by sliding point B closer to point C, which has the effect of "flipping" the arbelos. It is as if the chain of circles in the original figure were descending toward point A. The theorem also holds for a chain of inscribed circles descending toward point B. Use the Geometry Applet to construct a figure which illustrates this configuration.
  • The arbelos has many other fascinating features. Perhaps your research will turn up some other feature which you wish to explore for your science fair project.

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