CTK Insights

Parabola has an easily verifiable property. The segment joining points and crosses -axis in point . The equation of the segment is , from which . This may be a curious fact in its own right. What does it say? Taken at a face value, it simply shows a way to obtain the product of [...]

In a recent post, I have implied that Socrates new how to dissect a 2×1 rectangle into a square. There is actually no evidence that he did. However, he certainly knew how to produce a square half the area of a given one. How would he relate the two problems? A sangaku tablet has preserved [...]

Nowadays, finding the area of curvilinear shapes falls in the purview of calculus. But the problem of finding areas draw much interest in antiquity and preoccupied mathematicians ever since. One of the acknowledged results by Hippocrates of Chios (470-410 B.C.) is the Squaring of a Lune. The problem of squaring a shape refers to a [...]

One of the first challenge problems Paul H. Nahin offers in his new book comes from his experience as a freshman at Stanford. This is a nice yarn. When I was a freshman at Stanford I did well enough during the first two terms of calculus to be allowed to transfer into the honors section [...]

A frequently cited math curiosity relates the relative increase in the rope length between the rope laid on the Earth's equator and that around an average size watermelon. In both case the sought increase in length is due to the uniform expansion of the rope to, say, 1 ft away from the surface. It often [...]

A hyperbola has two axes of symmetry: one that crosses the hyperbola while the other does not. Two different 3D shapes are obtained when a hyperbola is made to rotate around its axes. Two sheet hyperboloid One sheet hyperboloid The equation of one is and that of the other is The one sheet hyperboloid has [...]

I just received a review copy of Fascinating Mathematical People by Donald Albers and Gerald Alexanderson (Princeton University Press, 2011). Right now I am into something else, but could not forego getting a quick first impression. Looks like I am going to enjoy reading the book. Here's something that caused me a healthy chuckle. I [...]

What's the task? The task is to combine several 3- and 4-pyramids into larger 3- and 4-pyramids. What's the setup? You'll need 4 tetrahedra and 6 square pyramids. Having 8 tetrahedra and 8 square pyramids will allow to complete 3- and 4-pyramids simultaneously. For your convenience, here are the maps of the pyramids. Just cut, [...]

The setup Draw 6 dots more or less evenly distributed over a circle: The activity Join the dots by lines of two colors, say, red and blue. Join all pairs of dots. See how many lines of the two colors emanate from each dot. What's to observe At every dot there are at least three [...]

The setup A cube whose faces have been partitioned by the midlines into four squares. The activity Color the squares in as many colors as you wish, with a single caveat: no two adjacent squares, i.e., squares that share an edge, may be of the same color. The task Determine the maximum possible number of [...]