Tag Archives: trigonometry

Parallelogram Cosine Problem

Another challenging problem from Presh Talwalkar. I certainly could not have solved it on a timed test at the age of 16.

One Of The Hardest GCSE Test Questions – How To Solve The Cosine Problem

Construct a hexagon from two congruent parallelograms as shown. Given BP = BQ = 10, solve for the cosine of PBQ in terms of x.

This comes from the 2017 GCSE exam, and it confused many people. I received many requests to solve this problem, and I thank Tom, Ben, and James for suggesting it to me.”

See the Parallelogram Cosine Problem

Square Wheels

I came across the following problem from an Italian high school exam on the British Aperiodical website presented by Adam Atkinson:

“There have been various stories in the Italian press and discussion on a Physics teaching mailing list I’m accidentally on about a question in the maths exam for science high schools in Italy last week. The question asks students to confirm that a given formula is the shape of the surface needed for a comfortable ride on a bike with square wheels.

What do people think? Would this be a surprising question at A-level in the UK or in the final year of high school in the US or elsewhere?”

I had seen videos of riding a square-wheeled bicycle over a corrugated surface before, but I had never inquired about the nature of the surface. So I thought it would be a good time to see if I could prove the surface (cross-section) shown would do the job. See Square Wheels.

Complex Numbers – Geometric Viewpoint

This may be a futile attempt at an elementary introduction to complex variables by emphasizing their geometric properties. The elementary part is probably undermined by an initial discussion of field extensions and a necessary reference to trigonometry. Hopefully, the suppression of the explicit use of complex powers of Euler’s constant e until the very end will allow the geometric ideas to have center stage. A primary goal of the essay is to realize that complex polynomials involve sums of circles in the plane. The image of real polynomials as wavy curves in the plane is misleading for an understanding of complex behavior. See Complex Numbers – Geometric Viewpoint.

Geometric Puzzle Medley

This is a collection of simple but elegant puzzles, mostly from a British high school math teacher Catriona Shearer, for which I thought I would show solutions (solutions for a number of them had not been posted yet on Twitter at the time of writing).  See the Geometric Puzzle Medley.

Apparently Catriona Shearer creates these problems herself, which shows an especially gifted talent. Ben Olin, of Math with Bad Drawings fame, had an interesting interview with Ms. Shearer. The reason for the interest in her work becomes evident the more of her geometry problems one sees. They are especially elegant and minimalist, and often have simple solutions, as exemplified by the “5 Problem” or “Shear Beauty” problem illustrated here. Words, such as “beauty” and “elegance”, are often bandied about concerning various mathematical subjects, but as with any discussion of esthetics, the efforts at explanation usually fall flat. Shearer’s problems are one of the best examples of these ideas I have ever seen. If you contemplate her problems and even solve them, you will understand the meaning of these descriptions.

One of the key aspects of mathematics is often its “hidden-ness” (some would say “opacity” or “incomprehension”). Her problems appear to have insufficient information to solve. But as you look at the usually regular figures, you see that there are inherent rigid constraints that soon yield specific information that leads to a solution. This discovery is akin to the sensation of discovering Newton’s mathematical laws underlying physical reality. It is the essence of one of the joys of mathematics.

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Perspective Map

A number of recent puzzles have involved perspective views of objects. I had never really explored the idea of a perspective map in detail. So some of the properties associated with it always seemed a bit vague to me. I decided I would derive the mathematical equations for the perspective or projective map and see how its properties fell out from the equations. With this information in hand I then addressed some questions I had about the article “Dürer: Disguise, Distance, Disagreements, and Diagonals!” by Annalisa Crannell, Marc Frantz, and Fumiko Futamura concerning a controversy over Albrecht Dürer’s woodcut St. Jerome in His Study (1514). And finally, I read somewhere that a parabola under a perspective map becomes an ellipse, so I was able to show that as well. See the Perspective Map.