French mathematician Joseph Bertrand offered this observation in his Calcul des probabilités (1889). Inscribe an equilateral triangle in a circle, and then choose a chord of the circle at random. What is the probability that this chord is longer than a side of the triangle? There seem to be three different answers:

1. Choose two random points on the circle and join them, then rotate the triangle until one of its vertices coincides with one of these points. Now the chord is longer than a side of the triangle when its farther end falls on the arc between the other two vertices of the triangle. That arc is one third of the total circumference of the circle, so by this argument the probability is 1/3.

2. Choose a radius of the circle, choose a point on that radius, and draw a chord through that point that’s perpendicular to the radius. Now imagine rotating the triangle so that one of its sides also intersects the radius perpendicularly. Our chord will be longer than a side of the triangle if the point we chose is closer to the circle’s center than the point where the triangle’s side intersects the radius. The triangle’s side bisects the radius, so by this argument the probability is 1/2.

3. Choose a point anywhere in the circle and draw the chord for which this is the midpoint. This chord will be longer than a side of the triangle if the point we chose falls within a concentric circle whose radius is half the radius of the larger circle. That smaller circle has one-fourth the area of the larger circle, so by this argument the probability is 1/4.

Further methods yield still further solutions. After more than a century, the implications of Bertrand’s conundrum are still being discussed.

What is the shape of the moon’s path around the sun? The moon orbits the earth, and the earth orbits the sun, so many of us imagine it looks something like the image on the left, a looping motion in which the moon periodically slides “backward” during its progress around the larger body.

But it’s not! The shape is closer to a 13-gon with rounded corners; there are no loops. Helmer Aslaksen, a mathematician at the National University of Singapore, writes, “I like to visualize this as follows. Imagine you’re driving on a circular race track. You overtake a car on the right, and immediately slow down and go into the left lane. When the other car passes you, you speed up and overtake on the right again. You will then be making circles around the other car, but when seen from above, both of you are driving forward all the time and your path will be convex.”

More at his page.

(Thanks, Drake.)

From the Royal Society of Chemistry’s Chemistry World blog: In 1955, when impish graduate student A.T. Wilson published a paper with his humorless but brilliant supervisor, Melvin Calvin, Wilson made a wager with a department secretary that he could sneak a picture of a man fishing into one of the paper’s diagrams. He won the wager — can you find the fisherman?

In 1958 psychologist Robert Plutchik suggested that there are eight primary emotions: joy, sadness, anger, fear, trust, disgust, surprise, and anticipation. Each of the eight exists because it serves an adaptive role that gives it survival value — for example, fear inspires the fight-or-flight response.

He arranged them on a wheel to show their relationships, with similar emotions close together and opposites 180 degrees apart. Like colors, emotions can vary in intensity (joy might vary from serenity to ecstasy), and they can mix to form secondary emotions (submission is fear combined with trust, and awe is fear combined with surprise).

When all these combinations are included, the system catalogs 56 emotions at 1 intensity level. And in his final “structural model” of emotions, the petals are folded up in a third dimension to form a cone.