Upon Reflection talks about the relationship between the geometry of reflection in a line, and specular reflection in a mirror. It also shows how to use specular reflection to compute a light triangle, which is the smallest-perimeter triangle that can be inscribed in another triangle.

Circular Reasoning first shows an interesting property of circles: if you draw a line from a point P and it cuts a circle in points Q and R, the product of the distances PQ and PR is equal to the value of the point with respect to the equation of the circle. We then continue the discussion from January by talking about Ptolemy's Theorem, and showing how it can be used to link Snell's Law and Fermat's Principle of Least Time for refraction.

Marco Corvi has observed that this linkage seems to have a flaw. The basic problem is that when we choose a new point on the surface that is to the left or right of the point dictated by Snell's Law, the condition that four of the points remain on a circle can be violated. This is correct, but not a fatal flaw in the argument. Rather, it was just a flaw in my presentation. Here's the way to fix things up.

Ptolemy's Theorem can be considered a generalization of the triangle inequality. This states that for any triangle ABC, |AC|>=|AB|+|BC|. The two sides are equal if the triangle is degenerate (that is, B lies on the line segment AC). Otherwise the triangle actually encloses some area, and the relation is strictly greater-than.

Similarly, Ptolemy's Theorem states (take a deep breath) that the product of the lengths of the diagonals of a quadrilateral is greater than or equal to the sum of the products of the lengths of opposite sides.

If the four points lie on a circle, then equality holds. If it's a more general quadrilateral, then like the triangle inequality, the relation is strictly greater-than.

In the final step of the column, I use Ptolemy's Theorem to substitute for four points on a circle, and four points off the circle, to show that the distance covered by the four non-cyclic points is longer. In this step, when I substitute for the cyclic points the replacement is exact. When I substitute for the non-cyclic points, the value given by Ptolemy's Theorem is larger than the value we're replacing. But since we're only showing that this value is bigger than the one for the cyclic points, it's okay.

In other words, if lengths of the non-cyclic points is N and the length of the cyclic points is C, and their substituted values from Ptolemy's Theorem are N' and C', then N'>N and C'=C. We start with N>C, so the new version N'>C' still holds, since N'>N, and we've returned to Snell's Law.

Aperiodic Tiling discusses the world of creating non-repeating patterns that fill the plane. Although many tiles can be used to create non-periodic patterns, there are some that cannot tile the plane periodically at all, no matter how they are used, but can only do so without creating a pattern that tiles by translation. Such tiles are called aperiodic.

Two sets of aperiodic tiles are particularly useful for computer graphics, since they're based on squares. You can download Adobe Acrobat (or pdf) format templates for the Robinson Tiles and the Ammann Tiles by selecting these links. I recommend gluing the front patterns to a piece of cardboard, cutting them out, then cutting out the back patterns and gluing them individually to the tiles. This is a lot easier than trying to get the fronts and backs to line up on opposite sides of the cardboard before cutting them out.

Aperiodic Tiles, Part 2 discussed the two different kinds of Penrose tiles: the kite/dart pair, and the thin/thick pair. I've saved links here for prototype tiles for both kinds in Adobe Acrobat (or pdf) format.

You can get files for the geometry of kites and darts for designing your own tiles, or download images of kites and darts using Conway's decorations. Similarly, you can get files for the geometry of thin and thick rhombs for designing your own tiles, or download images of thin and thick rhombs using Conway's decorations. Finally, you can get a copy of the Gummelt decagon for building up your own overlapped tilings.

When To Fold discusses the ubiquitous corrugated cardboard box, and some of the mechanics behind how they're designed and made.

A couple of people have asked about the photos. I shot all the pictures using a Kodak DC120 hand-held digital camera in one of our conference rooms; I just turned on the overhead lights (fluorescents, unfortunately) and played with the shutter speed until things look good. I imported the files into Photoshop and noticed that the brown boxes didn't look so great against the brown wooden tabletop. So I used the magnetic and freehand lasso tools to selectKnow the boxes, inverted the selection, and applied Desaturate. The result is that the background went gray while the box stayed nicely in color. I then used Levels to adjust the gray background to a pleasing level against the brown cardboard box.

The Triangular Manuscripts offer a tentative translation of some strange manuscripts discovered in the back of a dresser by a former student of my friend Dr. Stan Conversion.