With thanks to Ray Cross, for pointing me in this direction
Arbelos is an old Greek name for a shoemaker's knife. The curved blade with points at each end is an ideal shape for cutting leather precisely, and so has been a shoemakers' tool for centuries.
Mathematically, the Arbelos has some lovely properties that have made it worthy of study for millenia. It is often referred to as the Arbelos of Archimedes.
The Arbelos is made up of three semi-circles with centres on the same interval. Two of these are contained exactly within the third larger circle, and so all three 'kiss', just like the Farey Numbers and Ford Circles and the Apollonian Gasket!
In the task, we will explore some of these properties using a dynamic GXWeb model, built upon a half-unit circle. Feel free to explore further...
Giant Arbelos sculpture in the Netherlands
Try building your own Arbelos model using GXweb and use it to explore both the geometry and the algebra involved.
The Pappus chain is a ring of circles between two tangent circles (in this case, the Arbelos) investigated by Pappus of Alexandria in the 3rd century AD.
To be more precise, here we are using what might be termed a double arbelos made up of full circles rather than the upper half-circles which are commonly used.
The (outer) circle with centre O has radius 12. Circle centre C1 has radius r. Circle C2 forms both the second blade of the Arbelos and the first link in the Pappus Chain.
The grey circle C3, standing on the tangent point of the shoemaker blade, has an important relationship with the Arbelos - can you guess what this might be?
Next, consider the circle with centre P1: Can you find the radius and area of this circle (and subsequent circles) in terms of r?
Finally, you will see that the centres of the circles forming the Pappus Chain all lie on an ellipse - can you determine the parametric and cartesian equations of this curve in terms of r?
Drag the red circle or use the input box or slider to explore. Use the button below the GXWeb model to switch between Arbelos and Pappus views!
Arbelos HINT: The height of the circle standing at the tangent point of circles C1 and C2 is the geometric mean of the diameters of the two circles: height = √(2⋅r⋅(1−2⋅r)).
Pappus Chain HINT: The height of the centre of the circle P1 lies 2⋅radius of circle P1 above the x-axis.
The height of the centre of the next circle (P2) lies 4⋅radius of that circle above the x-axis.
Pappus Ellipse HINT: The two focal points of the Pappus ellipse lie at the centres of the two circles which define the chain: the circle with centre 0 and circle C1. The centre of the ellipse lies halfway between these points.
Use the CAS Evaluate option from the dropdown menu above to solve an expression or equation in the f(x) MathBox. Click on the following to try:
Solve(x^2=x+1,x)
Solve(x^2=x+1,x),x
Notice the difference between these two results? Adding an extra ,x at the end forces an exact rather than a numeric result!
And some more to try (just click on each line):
pFactor(320) (Prime factors)
sqcomp(x^2-4*x+3,x),x (Complete the Square)
nthroot(10,3) (Enter into the MathBox as \ nthroot [space] 3 [space] 10 [space])
Use the Solve option from the dropdown menu above to solve an expression or equation in the f(x) MathBox, or enter solve(an equation) in f(x),g(x) or h(x).
Use the Derivative option from the dropdown menu above to differentiate a function in the f(x) MathBox, or enter d/dx(a function) in f(x),g(x) or h(x).
Use the Integral option from the dropdown menu above to integrate a function in the f(x) MathBox, or enter int(a function) in f(x).
For definite integrals, perform the integration as described using the dropdown, and enter end values when requested. Try ∫(x2−x−1,0,x) for example. Use the x box or slider to vary this.
Display the tangent (or Normal) of f(x) for the current value of x using the Tangent/Normal button, or enter tangent (or tngnt) or normal directly into the graph mathBoxes - set the desired x-value first.
Continued fractions are awesome. Every real number, rational and irrational, can be represented by a continued fraction - the rational ones are finite, of course, but both finite and infinite offer some wonderful patterns and opportunities to explore!
Use the MathBox above to enter a value to express as a continued fraction.
The powerful Wolfram Alpha online CAS engine will answer almost anything you care to ask - within reason! From the continued fraction of pi to Solve x^2=x+1 to the population of Australia!
QR Codes are a great way to share data and information with others, even when no Internet connection is available. Most modern devices either come equipped with QR readers in-built, or freely available.
The default link here is the Geometry Expressions Showcase, but you can use it as an alternative to sending your assessment data via email, web or Google Cloud. You can even use it to send your own messages!
