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American Mathematical Monthly -February 2007

February 2007

The Lighthouse Theorem, Morley & Malfatti — A Budget of Paradoxes
By Richard K. Guy
rkg@cpsc.ucalgary.ca
The Lighthouse Theorem is a simple but striking corollary to theorems of Euclid, but does not appear in geometry texts. It states that n equally inclined beams from each of two lighthouses intersect in n2 points that are the vertices of n regular n-gons. It has several ramifications and may be applied to Morley's theorem, to the construction of Malfatti circles, and to other configurations that remain to be investigated.

Stepping to Infinity Along Gaussian Primes
By Po-Ru Loh
ploh@caltech.edu
A problem posed by Basil Gordon, sometimes misattributed to Erdös, asks whether one can walk to infinity along Gaussian primes with steps of bounded size. We consider walks restricted to angular sectors of the complex plane, and we show that for a given step size and starting point (i) there is a lower bound on the angle that must be swept out by a prime walk and (ii) by taking the union of angular sectors, we can create a region on which no prime walk exists that comes arbitrarily close to covering the complex plane.

Notes

A Simple Proof of Sharkovsky's Theorem Revisited
By Bau-Sen Du
dubs@math.sinica.edu.tw

The LSB Theorem Implies the KKM Lemma
By Gwen Spencer and Francis Edward Su
gspencer@hmc.edu, su@math.hmc.edu

Pointwise Products of Uniformly Continuous Functions on Sets in the Real Line
By Sam B. Nadler, Jr. and Donna M. Zitney
nadler@math.wvu.edu, dzitney@math.wvu.edu

Problems and Solutions

Reviews

Analysis I: Convergence, Elementary Functions. Analysis II: Differential and Integral Calculus, Fourier Series, Holomorphic Functions.
By Roger Godement
Reviewed by Gerald B. Folland
folland@math.washington.edu

 

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