Matrices and Tilings with Right Trominoes
After the introduction of polyominoes by S. W. Golomb, many questions have arisen concerning tilings of plane regions with these figures. In this article we focus our attention on the study of tilings with right trominoes, also called L-trominoes. We give an explicit, recursive way of obtaining transfer matrices for tilings of this kind which allows us to obtain (via computer calculations) results about enumerating tilings of rectangles and the corresponding generating functions, as well as some statements about the existence of tilings in a class of regions called strips.
The Mystery of Robert Adrain
Frank J. Swetz
Robert Adrain (1775-1843) was a leading mathematician and educator in the early United States. Although self-educated in mathematics, he held professorships in the subject at Columbia, Rutgers, and the University of Pennsylvania. He was an avid problem-solver, researcher, and editor of several mathematical related journals. He has been credited with doing some of the first Â“real mathematicsÂ” in the U.S. Among his accomplishments are a derivation of a probability distribution for errors and successful applications of the Method of Least Squares. Despite his contributions to the growth of American mathematics, Adrain has received little attention. This article calls attention to that fact and suggests further consideration of this man’s accomplishments, particularly as a pioneering mathematics educator.
Touching the Z2 in Three-Dimensional Rotations
Vesna Stojanoska and Orlin Stoytchev
Rotations in three-dimensional space have the following fascinating property: a full 360ã€« rotation of an object is topologically nontrivial, i.e., you cannot deform this motion to the trivial one, but a 720ã€« rotation is trivial. This fact can be demonstrated by attaching three (or more) strands to the object you rotate and fixing their other ends to your desk surface. By rotating the object you produce a kind of a braid. Then, without further rotating the object, you try to unplait the braid by moving the strands around. The braids obtained from an even number of full rotations, no matter how complicated, are trivial, while those coming from an odd number of rotations are nontrivial. This connection between three-dimensional rotations and braids can be made rigorous to prove that the fundamental group of SO(3) is indeed Z2 .
Packing Squares in a Square
Here is a bounty problem from Paul Erdös: Place n nonoverlapping squares (not necessarily of the same size) inside a unit square. What is the largest possible value for the sum of the side lengths of the n squares? Around 1932 Erdös conjectured that when n = k2 + 1 the answer is k. In 1995 he and Soifer provided conjectures for all values of n. This note shows that it suffices to prove the original conjecture: that is the original conjecture actually implies the more general conjecture.
Why is the Sum of Independent Normal Random Variables Normal? "
Bennett Eisenberg and Rosemary Sullivan
The fact that the sum of independent normal random variables is normal is fundamental in probability and statistics. The standard proofs using convolutions and moment generating functions do not give much insight into why this is true. This paper gives two more proofs, one geometric and one algebraic, that provide more insight into why the sum of independent normal random variables must be normal.
A Converse to a Theorem on Linear Fractional Transformations
An interesting geometric fact about a linear fractional transformation (also called Möbius transformation and bilinear transformation) is that it maps circles and lines to circles and lines in a bijective fashion. Naturally we want to ask: what can we say about an arbitrary bijective function that maps circles and lines onto circles and lines? We will show that any such function is either a linear fractional transformation or the complex conjugate of a linear fractional transformation.
Thomas Q. Sibley
The Bolzano-Weierstrass theorem asserts, under appropriate circumstances, the convergence of some subsequence of a sequence. While this famous theorem ignores the actual limit of the subsequence, it is natural to investigate such limits. This note characterizes the set of possible limits of subsequences of a given sequence.
PROOF WITHOUT WORDS
Proof Without Words: Isosceles Dissections
We show that every triangle can be dissected into four isosceles triangles, that every acute triangle can be dissected into three isosceles triangles, and that a triangle can be dissected into two isosceles triangles if and only if one angle is three times another or the triangle is right angled. This item was motivated by an earlier proof without words that dissected an arbitrary triangle into six isosceles triangles.
Proof Without Words: Exponential Inequalities
This paper gives two visual proofs of the following exponential inequalities: