Dynamical systems offer a rich field for student experimentation. A standard picture (the orbit diagram) that shows the location of attractive periodic points is often called the bifurcation diagram. The authors say that the real bifurcation diagram should show both attractive _and_ repelling periodic points, provide one, and much else on f(f(f( ... (x) ... ))).is the Minimum Value of Pi
If we define a circle to be the set of points (x, y) such that xp + yp = 1, p >= 1, then the ratio of the circle's circumference to its diameter is a function of p. When p = 2 the ratio takes its smallest value, showing that our familiar circles are more circular than any other circles.Optimal Card-Collecting Strategies for Magic: The Gathering
Games that involve the use of cards--not the usual playing cards with pictures of kings and queens, but those with pictures of monsters and alien landscapes--are very popular with a segment of the population. How shall cards be purchased (they come in packages of different sizes) to best collect a complete set? Most of the time, but not always, by buying big packages.Artemus Martin: An Amateur Mathematician of the Nineteenth Century and His Contribution to Mathematics
Artemus Martin (1835-1918) lived in a mathematical world very different from ours, and very different from the European mathematical world of his time. In telling us what he did and how he did it, the authors give us a valuable glimpse into the past and keep us from making (usually unconsciously) the common and wrong assumption that everything has always been much as it is now.Contumacious Spheres
Let us put n-dimensional unit spheres in an n-dimensional box, packing them regularly with centers at (+-1, +-1, ... +-1). In the center there is a sphere tangent to the spheres around it. For n = 2, think of four fifty-cent pieces surrounding a dime and for n = 3 eight tennis balls surrounding a golf ball. When n = 4, the inner sphere is also a unit sphere. When n = 9, it is tangent to the box. When n >= 10, it extends outside the box. The volume of the box, however remains greater than the volume of the sphere until dimension 1206. Strange things also happen at dimension 264.Partially Differentiable, Yes; Continuous, No
You may know the function that has partial derivatives everywhere but fails to be continuous at the origin, but you probably do not know when the example first appeared in print (1884). How about the function that tends to zero along every line through the origin but which is discontinuous at (0, 0)?
For a subgroup H of G, it is possible to pick an operation table for G showing properties of the normalizer, the largest subgroup of G having H as a normal subgroup.Getting Normal Probability Approximations Without Using Normal Tables
What to do if stranded on a desert island without normal tables? Actually, the question is, how to make students understand better the normal density?Normal Lines and Curvature
Take normals to a curve at x and x + h and let h approach zero. What happens to their point of intersection? It _doesn't_ go off to the point at infinity.A Picture of Real Arithmetic
Stereographic projection enables us to add and multiply numbers geometrically, both on the same diagram.Integrals of cos2n x and sin2n x
Complex numbers let the integrals of the title be evaluated fairly quickly.
A proof that there are no contradictions.