Jung’s Theorem: Enclosing a Set of Points

Let us imagine that we have a finite set $latex {P}&fg=000000$ of points in the plane $latex {\mathbb{R}^2}&fg=000000$ (Fig. 1a). How large a circle is required to enclose them. More specifically, what is the minimum radius of such a bounding circle? The answer is given by Jung's Theorem. Left: a set P of points in … Continue reading Jung’s Theorem: Enclosing a Set of Points →

Is There Anyone Out There? The Drake Equation gives a Clue

The Drake Equation is a formula for the number of developed civilizations in our galaxy, the Milky Way. This number is determined by seven factors. Some are known with good accuracy but the values of most are quite uncertain. It is a simple equation comprising seven terms multiplied together [TM193 or search for “thatsmaths” at … Continue reading Is There Anyone Out There? The Drake Equation gives a Clue →

Think of a Number: What are the Odds that it is Even?

Pick a positive integer at random. What is the chance of it being 100? What or the odds that it is even? What is the likelihood that it is prime? Since the set $latex {\mathbb{N}}&fg=000000$ of natural numbers is infinite, there are difficulties in assigning probabilities to subsets of $latex {\mathbb{N}}&fg=000000$. We require the probability … Continue reading Think of a Number: What are the Odds that it is Even? →

Resolution of Paradox: a Gateway to Mathematical Progress

A paradox is a statement that appears to contradict itself, or that is counter-intuitive. The analysis of paradoxes has led to profound developments in mathematics and logic. One of the richest sources of paradox is the concept of infinity. Hermann Weyl, one of the most brilliant mathematicians of the twentieth century, defined mathematics as “the … Continue reading Resolution of Paradox: a Gateway to Mathematical Progress →