The original inspiration for the knapsack problem is a puzzle that we are all familiar with. It is one that a vegetable vendor has to solve every day: the vendor goes to the wholesale market to pick up vegetables to sell on a particular day. Of course she cannot carry more than a certain weight, and the total volume cannot exceed the volume of the basket. There are several, at least 10-20, different vegetables she may choose. She knows the profit she can make per kilo of each of the vegetables, and the volume per kilo for each of these as well. Given all this data, she has to choose vegetables that will fit in her basket and are within the weight she can carry, but at the same time will give her maximum profit. This is indeed a complex mathematical problem, but most vendors find a solution for such a problem every day!

This kind of problem is known as “constrained optimization:” the limits on weight and volume are the “constraints” that cannot be exceeded, whereas “optimization” refers to the aim of trying to maximize the profit. “Knapsack problem” is a colloquial name for such problem referring to a problem that a thief may face, very similar that of the vegetable vendor.

A slightly simpler version of this problem is illustrated in this exhibition..

Soap bubbles are formed by the simple physical principle of surface tension, whose mathematical manifestation is the concept of “minimal surfaces” – the idea that soap films always minimize their surface area.

This seemingly simple principle leads to visually appealing and mathematically interesting shapes: individual bubbles are spherical in shape maximizing the volume enclosed, the bubble clusters set- tled on a plane surface display a hexagonal tessellation, the soap films formed by irregular shapes can show a dazzling variety.

Mathematically, some of the soap bubbles are classic examples of surfaces with negative curva- ture, like a saddle, while others are classic examples of surfaces with constant positive curvature, like the spherical bubbles. Soap bubbles have lead to some intriguing mathematical discoveries as well, as we will see below.

In a state of equilibrium, the surface tension on a soap film is the same at all points, and its surface area will have a minimal value; this minimum area property of soap films can be used to solve some mathematical minimization problems.

Imagine having to move a ladder, or a large stick, from one room to another or suppose one is trying to park a car in a tight spot. Most often in such a situation one has to navigate through a rather cramped space. Both tasks require a great deal of ingenuity and dexterity

In the year 1917, the japanese mathematician S. Kakeya considered the problem of finding the area of the smallest convex set in the plane inside which a needle of unit length can be reversed, i.e., turned around through 180◦ . (A “convex set” is nothing but an area which has no “spikes” e.g. a circle or a triangle is convex whereas a star in not convex.)

Indeed, Kakeya conjectured that an equilateral triangle of unit height is the smallest such set; he also observed that if the convexity condition is dropped, that is, if we allow for shapes such as stars or octopus, with “spikes,” then a smaller set is possible. The more interesting question of finding the smallest set (without the convexity condition) in which to turn the needle came to be known as the Kakeya needle problem.