"The Arbelos" University of Kansas.

Let be the length of the gap between the bases (so that the diameter of the top arc is ) and let be the abscissa of the intersection of the radical axis with the axis, assuming the origin is at the leftmost point of the arbelos [].

This function displays the arbelos.

Welch, The Arbelos, Master’s thesis, University of Kansas, 1949

The Arbelos - University of Kansas

Motivated by the computational advantages offered by Mathematica, I decided some time ago to embark on collecting and implementing properties of the fascinating geometric figure called the arbelos. I have since been impressed by the large number of surprising discoveries and computational challenges that have sprung out of the growing literature concerning this remarkable object. I recall its resemblance to the lower part of the iconic canopied penny-farthing bicycle of the 1960s TV series The Prisoner, Punch’s jester cap (of Punch and Judy fame), and a yin-yang symbol with one arc inverted; see Figure 1. There is now an online specialized catalog of Archimedean circles (circles contained in the arbelos) [] and important applications outside the realm of mathematics and computer science [] of arbelos-related properties.

Reflections on the ArbelosHarold P

Leon Bankoff was the person who stimulated the extraordinary attention on the arbelos over the last 30 years. Schoch drew Bankoff’s attention to the arbelos in 1979 by discovering several new Archimedean circles. He sent a 20-page handwritten note to Martin Gardner, who forwarded it to Bankoff, who then gave a 10-chapter manuscript copy to Dodge in 1996. Due to Bankoff’s death, a planned joint work was interrupted until Dodge reported some discoveries []. In 1999 Dodge said that it would take him five to ten years to sort all the material in his possession, then filling three suitcases. Currently this work is still forthcoming. Not surprisingly, like Volume 4 of The Art of Computer Programming, it appears that important work needs a substantial time to be developed.


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We have seen that there are some Archimedean circles other than the twins, namely the Bankoff circle and those mentioned in properties 21 through 27. There are also non-Archimedean twins, that is, pairs of circles of the same radius, different than that of the twins, appearing at significant places within the arbelos.

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In this section we generalize the shape of an arbelos by allowing the arcs to cross and by considering a 3D version. To set the context of the first of those generalizations, we need the concept of the radical axis of two circles.

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For fifteen years, competitors and mathematicians interested in high-level compteitions found it near-impossible to think of the prestigious USA Mathematical Olympiad without having in mind Dr Samuel L Greitzer, founding chairman of the committee in charge of the USAMO. Even the abbreviation USAMO centred on his name Sam, as his friends and admirers called him. Throughout his tenure and for years afterwards (through the journal Arbelos, which he prepared more or less single-handedly). Sam’s concern with the mathematical well-being of highly talented students was similarly centralized. Only death could terminate his devotion, on February 22, 1988, at age 82.

arbelos | Circle | Eigenvalues And Eigenvectors

The circle in Figure 3 is called the radical circle of the arbelos and the line is its radical axis (this terminology will be clarified in ). To illustrate properties 3-11 and 25, 26, we draw and label points and show some coordinates, lines, and circles in Figure 4.

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The coordinates of the tangent points , , and are obtained as the intersections of the lines joining the centers of the three arcs of the arbelos and the incircle.

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These four solutions give the centers in pairs: , , , , where and are the reflections of and in the diameter of the arbelos; only the last expression is valid. The result also shows that the twins are indeed of the same radius . Any circle with radius equal to the twins’ radius is called Archimedean. A nice interpretation of arises when considering and as resistances: then is the resistance resulting from connecting and in parallel; that is, . The function computes the value of the centers and the common value of the radius of the twins.