Do you like the summer heat? Our lead article makes August/September even hotter as Erwan Brugallé and Kristin Shaw introduce us to a new but growing area of geometry in their paper "A Bit of Tropical Geometry." Ever wonder what angles in a right triangle can be trisected with a ruler and compass? If the sides have integer length, then you will find the answer in Wen D. Chang and Russell A. Gordon's note "Trisecting Angles in Pythagorean Triangles." If you like our Problem Section, then you will love this month's longer than usual version. We close with Michael Barr's review of Noson Yanoksy's The Outer Limits of Reason. Stay tuned for the October issue when we get a lesson in additive combinatorics from Persi Diaconis, Xuancheng Shao, and Kannan Soundararajan.—Scott Chapman
Vol. 121, No. 7, pp.563-660.
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ARTICLES
A Bit of Tropical Geometry
Erwan Brugallé and Kristin Shaw
This friendly introduction to tropical geometry is meant to be accessible to first year students in mathematics. The topics discussed here are basic tropical algebra, tropical plane curves, some tropical intersections, and Viro’s patchworking. Each definition is explained with concrete examples and illustrations. The text is a modification of a translation from a French text by the first author. There is also a newly-added section highlighting new developments and perspectives on tropical geometry. In addition, the final section provides an extensive list of references on the subject.
To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.121.07.563
Mathbit: Another Algebraic Pythagorean Proof
Marcelo Brafman
To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.121.07.589
Four Quotient Set Gems
Bryan Brown, Michael Dairyko, Stephan Ramon Garcia, Bob Lutz, and Michael Someck
Our aim in this note is to present four remarkable facts about quotient sets. These observations seem to have been overlooked by the Monthly, despite its intense coverage of quotient sets over the years.
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Blind-friendly von Neumann’s Heads or Tails
Vinícius G. Pereira de Sá and Celina M. H. de Figueiredo
The toss of a coin is usually regarded as the epitome of randomness, and has been used for ages as a means to resolve disputes in a simple, fair way. Perhaps as ancient as consulting objects such as coins and dice is the art of maliciously biasing them in order to unbalance their outcomes. However, it is possible to employ a biased device to produce equiprobable results in a number of ways, the most famous of which is the method suggested by von Neumann back in 1951. This paper addresses how to extract uniformly distributed bits of information from a nonuniform source. We study some probabilities related to biased dice and coins, culminating in an interesting variation of von Neumann’s mechanism that can be employed in a more restricted setting where the actual results of the coin tosses are not known to the contestants.
To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.121.07.600
Zero Sums on Unit Square Vertex Sets and Plane Colorings
Richard Katz, Mike Krebs, and Anthony Shaheen
We prove that if a real-valued function of the plane sums to zero on the four vertices of every unit square, then it must be the zero function. This fact implies a lower bound in a “coloring of the plane” problem similar to the famous Hadwiger–Nelson problem, which asks for the smallest number of colors needed to assign every point in the plane a color so that no two points of unit distance apart have the same color.
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Mathbit: A Refinement of a Theorem of J. E. Littlewood
D. Zagier
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Positive Linear Maps and Spreads of Matrices
Rajendra Bhatia and Rajesh Sharma
The farther a normal matrix is from being a scalar, the more dispersed its eigenvalues should be. There are several inequalities in matrix analysis that render this principle more precise. Here it is shown how positive unital linear maps can be used to derive many of these inequalities.
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NOTES
Trisecting Angles in Pythagorean Triangles
Wen D. Chang and Russell A. Gordon
Suppose that a2 + b2 = c2, where a, b, and c are relatively prime positive integers, and consider the right triangle T with sides a, b, and c. We prove that both of the acute angles in T can be trisected with a compass and unmarked straightedge if and only if c is a perfect cube.
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Mathbit: Yet Another Direct Proof of the Uncountability of the Transcendental Numbers
Diego Marques
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Points Covered an Odd Number of Times by Translates
Rom Pinchasi
Given an odd number of axis-aligned unit squares in the plane, it is known that the area of the set whose points in the plane that belong to an odd number of unit squares cannot exceed the area of one unit square, that is, 1. In this paper, we consider the same problem for other shapes. Let T be a fixed triangle and consider an odd number of translated copies of T in the plane. We show that the set of points in the plane that belong to an odd number of triangles has an area of at least half of the area of T . This result is best possible. We resolve also the more general case of a trapezoid and discuss related problems.
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On the Characterization of Galois Extensions
Meinolf Geck
We present a shortcut to the familiar characterizations of finite Galois extensions, based on an idea from an earlier MONTHLY note by Sonn and Zassenhaus.
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Mathbit: Simpson’s Rule and Sums of Powers
Samuel G. Moreno and Esther M. García-Caballero
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Continuous Images of Cantor’s Ternary Set
F. Dreher and T. Samuel
The Hausdorff–Alexandroff Theorem states that any compact metric space is the continuous image of Cantor’s ternary set C. It is well known that there are compact Hausdorff spaces of cardinality equal to that of C that are not continuous images of Cantor’s ternary set. On the other hand, every compact countably infinite Hausdorff space is a continuous image of C. Here, we present a compact countably infinite non-Hausdorff space that is not the continuous image of Cantor’s ternary set.
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Extended Echelon Form and Four Subspaces
Robert A. Beezer
Associated with any matrix, there are four fundamental subspaces: the column space, row space, (right) null space, and left null space. We describe a single computation that makes readily apparent bases for all four of these subspaces. Proofs of these results rely only on matrix algebra, not properties of dimension. A corollary is the equality of column rank and row rank.
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PROBLEMS AND SOLUTIONS
Problems 11789-117895
Solutions 11655, 11656, 11657, 11659, 11660, 11664, 11667
To purchase from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.121.07.648
REVIEWS
The Outer Limits of Reason By Noson Yanofsky
Reviewed by Michael Barr
To purchase from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.121.07.658