Arbelos
Arbelos |
---|
Arbelos
- This modern knife in the shape of an arbelos is used to make shoes.
Contents
Basic Description
The word "arbelos" means shoemaker’s knife in Greek. The first mathematician to study mathematical properties of this plane region is Archimedes, who wrote all his thoughts about the arbelos in his Book of Lemmas.
In geometry, an arbelos is a figure bounded by three semicircles, tangent in pairs, and with diameters lying on the same line. Graphically, an arbelos is the green region in the picture above. The position of central point B is arbitrary and can be anywhere along the diameter. Here we assume that the three diameters are as the image shows.
From now on, let's think in a semicircle way!
A More Mathematical Explanation
- Note: understanding of this explanation requires: *geometry, a little bit of algebra
Properties
Arbelos has lots of unexpected but interesting properties.
Arc Length
Area
Rectangles
Tangents
Those are the four main properties of the arbelos. But Archimedes did not stop exploring this amazing figure; he found more fascinating things inside the arbelos.
Archimedes' Twin Circles
If two circles are inscribed in an arbelos tangent to the line segment , one on each side, then the two circles are congruent and have the same diameter . Because the two circles were first found by Archimedes, they are called Archimedes' Twin Circles. (Figure 3)
Proving The Twin Circles Congruent
Archimedes’ Circles and the Problem of Apollonius
Bankoff Circle
In addition to the Archimedes' twin circles, there is another circle inside the arbelos that is congruent to the twin circles. It is called the Bankoff Circle, the black circle in the figure above. Leon Bankoff, a dentist in Beverly Hills and also a mathematician, found the third circle and wrote it down in his article named Are The Twin Circles of Archimedes Really Twins?
The Bankoff circle and the twin circles are all identical. The diameter of the Bankoff Circle is also .
Let x be the radius of the Bankoff Circle and let A be the area of .
Let be the radius of the Bankoff Circle and let be the area of .
- (half of the diameter of the enclosing circle)
- (the sum of the radius of circle and the radius of circle )
- (the sum of the radius of circle and the radius of circle )
According to a property found by Pappus, the altitude to the base of is two times the radius of circle .
Therefore,
- The area of :
According to Heron's Formula, it states that the area of a triangle with sides a, b, and c is where s is the semiperimeter of the triangle.
- Let
- Then
Because the area of is the sum of the areas of , , and , therefore Recall that is the radius of the Bankoff Circle.
We have three equations all representing the area of and we know that . <template>AlignEquals
Pappus Chain
Definition
A chain of inscribed circles is called a Pappus Chain when its first circle is tangent to the three semicircles forming the arbelos, and all the subsequent circles are tangent to one another and to the boundaries of the arbelos. The Pappus Chain is named after Pappus of Alexandria, a great Greek mathematician who studied and wrote about it in the 4th century A.D. The figure above shows all of the three variations of the chain: leftward, rightward, and downward. The default position is the one that extends to the right.
Properties
- Height
- Ellipse
Pappus Chain and Steiner Chain
Why It's Interesting
The arbelos has attracted lots of professional mathematicians as well as many interested amateurs, and it still inspires people in many ways. Today, you can see the arbelos in areas ranging from shoemaking to art design.
Applications
Leather cutting
A modern head knife, used for leather cutting and shoemaking, resembles the shape of an arbelos as the main image shows. This kind of knife has a curved blade, which mathematically is the circumference of the largest semicircle; leather-crafters use the blade to cut, skive, or trim leather. Its two sharp cusp points on each side are perfect for cutting right angles. Because of the shape of an arbelos, the head knife is an important, easy, and very useful tool for cutting leather and making shoes.
Art & Design
Put the arbelos in a fractal pattern and you will get figures like the one above. A fractal pattern is a fragmented geometric shape, which consists of lots of reduced-size copies of the whole graph. The main shape of this fractal is a little bit different from the classic arbelos because the two smaller semicircles become two whole circles, making the entire figure more heart-shaped. Yet, it is a pretty design and it comes from a mathematics website that is actually called “Arbelos.”
You can also find an abstract public sculpture in the shape of arbelos located in the Netherlands.
Mechanics
The arbelos in mathematics is similar to the 3-D Mohr’s circle in mechanics. Mohr’s circle is a geometric representation of the state of stress at a point: the normal force and shear force. It is very useful to perform quick and efficient estimations. To understand how the diagram is drawn and how it is in the shape of the arbelos, we need to know the mathematical problems Mohr has studied. (This section is hard to understand; it requires knowledge about matrices, vectors, solid geometry, and algebraic calculation.)
First, let be a unit vector in
Let be a symmetric 3 by 3 matrix. For convenience, let the eigenvalues of the matrix M be and other numbers 0. (To produce the arbelos, assume that are different numbers and .)
Next, use a pair of real numbers to denote the normal and tangential components of vector .
Because is a unit vector, so
Therefore, is the length of the normal component of . Thus, is the normal vector of . See the left image.
- According to the left image, the tangential vector of is .
Remember that is the length of the tangential component of , .
Because represents any unit vector over a unit sphere in , there is a range for (x, y) in as
varies in .
Now let . Then
Because is a unit vector and
Therefore . This equations represents a triangular plane in . See the right figure.
Then draw a line with three points as shown in Figure 7. Let the abscissae of be in an increasing order.
In order to prove that the range is an arbelos, we need to prove that the three vertices of this triangle in the right figure, map to the three points of the arbelos as Figure 7 shows, and the three sides of this triangle are arc , arc , and arc in the arbelos. We take one side of the triangle as an example and it can be proved that the other two sides are mapped to the other arcs in the same way. Assume we want to prove the arc . Let , then .
Substitute and with and in the expression y^2 + x^2 - (\lambda_1 + \lambda_2) x, using .
So
Because ,
Recall that
Because
For the reason that
Therefore, we get
<template>AlignEquals
Archimedean Circles
Any circle defined in the arbelos that has radius , the same as the radii of Archimedes’ Twin Circles, is called an Archimedean Circle.
There are three famous Archimedean Circles: the Bankoff Circle, the Schoch Circle, and the Woo Circle. We discuss the Bankoff Circle above. To learn more about the other two circles, see
the Woo Circles and The Archimedean Circles of Schoch and Woo .
Teaching Materials
- There are currently no teaching materials for this page. Add teaching materials.
References
[1] Boas, Harold P. Reflections on the Arbelos. Retrieved from http://www.math.tamu.edu/~harold.boas/preprints/arbelos.pdf.
[2] Nelsen, Roger B. Proof Without Words: The Area of an Arbelos. Retrieved from http://legacy.lclark.edu/~mathsci/arbelos.pdf.
[3] Wikipedia (Arbelos). . Arbelos. Retrieved from http://en.wikipedia.org/wiki/Arbelos.
[4] Radius of the Twin Circles. Retrieved from http://www.geogebra.org/en/examples/frisbee/worksheets/pappus_chain.html.
[5] Weisstein, Eric W. Arbelos. From MathWorld--A Wolfram Web Resource. Retrieved from http://mathworld.wolfram.com/Arbelos.html.
[6] Bankoff, L. Are the Twin Circles of Archimedes Really Twins? Math. Mag. 47, 214-218, 1974.
[7] van Lamoen, Floor and Weisstein, Eric W. Pappus Chain. From MathWorld--A Wolfram Web Resource. Retrieved from http://mathworld.wolfram.com/PappusChain.html.
Future Directions for this Page
- An applet or applets to demonstrate different kinds of arbelos and Archemedes' circles with a moveable point B.
- More interesting and important applications of arbelos.
- Or anything you think would be necessary for this page. You are welcome to edit it!
Leave a message on the discussion page by clicking the 'discussion' tab at the top of this image page.