Gravity in Hades
Contrary to public opinion, even among those of us who should know better, gravity does not necessarily decrease with depth from a planet's surface. Gravity within a heavenly body depends upon the density function of the body. We consider various historical models of the Earth, and generate their corresponding gravitational accelerations as functions in terms of distance from the center of the Earth. We conclude with an analysis of a body falling down a hole through the Earth, and show that a rock actually falls from the Earth to its center in about 19.2 minutes, rather than the 21.2 minutes as given by the standard homogeneous Earth model.
Beyond the Celestrial Sphere: Oriented Projective Geometry and Computer Graphics
Kevin G. Kirby
Software for computer graphics represents 3D space a little differently than one might expect. Euclidean geometry is not quite right. This paper introduces an alternative, called oriented projective geometry. In this geometry our familiar 3D space is augmented by a sphere of points at infinity and, beyond that, another copy of Euclidean space. We explain the mathematics of points, lines, planes, and transformations in this framework, and illustrate how it is particularly well-suited for computer graphics computations.
At the Flat Earth Society
"I have a differential geometer on the line. He wants to know if he can join the local branch without joining the global organization."
Areas and centroids for triangles within triangles
Given triangle ABC, let A', B', C' be points on the sides CB, AC, BA respectively. We then have a second triangle A'B'C'. A third triangle is determined by the vertices P, Q, R, the intersection points of the segment pairs AA' and BB', BB' and CC', CC' and AA' respectively, as shown in the figure. In this note we use vector algebra to find known results about ratios of the areas of these three triangles. We also obtain some new results concerning the relative positions of the centroids of the triangles, including necessary and sufficient conditions for the three centroids to be collinear.
Tiling Large Rectangles
Darren A. Narayan and Allen J. Schwenk
Search through your attic or basement and you will likely find several 4x6 and 5x7 photographs. Select some of these photos and put them together to form a rectangle. Which ones can you build?
Unique Rook Circuits
Richard K. Guy and Marc M. Paulhus
A chess rook is to visit each square of a rectangular board just once and return to its original square. We are concerned with arranging a minimal number of forbidden squares so that there is a unique circuit of the remaining squares.
Proof Without Words: Nonnegative Integer Solutions and Triangular Numbers
Matthew J. Haines and Michael A. Jones
Lattice points are used to equate the number of nonnegative integer solutions of a linear equation in three variables with three constraint inequalities to a linear combination of triangular numbers.
An Encryption Algorithm using a Variation of the Turning-Grille Method
The concept of placing a square grid (or grille) with openings over a selection of text as a method of encryption was conceived by the 16th century mathematician Girolamo Cardano in his De Subtilitate. Variations of this coding technique were used as late as the mid-twentieth century. In this note, we provide an introduction to grids, introduce a linear variation of the grid concept, describe a procedure for using grids to encode messages, and demonstrate how to use modern encryption techniques in conjunction with grids. We conclude the paper by inviting the reader to decode a passage of encrypted text.
Axial View of Trigonometric Functions
M. Vali Siadat
In terms of modern pedagogy, having visual interpretations of trigonometric functions is useful and quite helpful. This axial view of trigonometric functions can help develop a better sense of their magnitudes, as well as providing for a quick determination of their signs.
PROOFS WITHOUT WORDS
The Cosine of a Difference
William T. Webber and Matthew Bode
The formula for the cosine of a difference is demonstrated using technique similar to that used by Priebe and Ramos (December 2000) for the sine of a sum.