One of the most remarkable natural talents of human beings is their ability to navigate, in a meaningful way, through enormous problem spaces. What do I mean by an “enormous problem space?” Consider a 21-city Traveling Salesperson Problem (TSP), where the task is to find the shortest, or, at least, a very short tour of all of the cities. The number of all possible tours, which means the number of all different ways to visit the cities, is 20!/2 (twenty factorial divided by two). This is approximately equal to 1.2×1018. This is a 1 followed by 18 zeros. This is a very big number. If you examined three tours per second and started at the Big Bang, which was almost 14 billion years ago, you would have just finished checking all tours in a 21-city TSP. This is an insanely large search space! But, we should reserve the word “insane” for something even more remarkable. Here it is: A subject (a psychology graduate student) is able to produce the shortest tour in a 21-city TSP almost half of the time, and each tour is produced in just over one minute. And when their tour was not actually the shortest, it was, on average, longer than the shortest one by only 2%. Humans are able to produce a TSP tour without considering any alternatives and they do this without any special training or experience with the TSP. The only way to accomplish what appears to be an insanely creative act is to identify and use all of the relevant symmetries of the problem’s representation. It is symmetries that allow humans to solve this kind of difficult optimization problems.
Symmetries also show up in what had been considered for years to be a very different type of problems called insight problems. An example of an insight problem is being asked to construct 4 identical triangles using 6 matchsticks. Arranging the matchsticks on the surface of your desk will not produce the solution. There is simply not enough symmetry in such two-dimensional (2D) geometrical shapes. The insight that is needed here is to realize that it is a 3D geometrical shape that will maximize the number of symmetries resulting in the solution. This connection between symmetry and geometry is not coincidental. Mathematicians know that symmetry is the sine qua non of geometry. Physicists are also hooked on symmetry. They claim that most, if not all breakthroughs in their science, were accomplished by adding a missing symmetry to the existing systems. And once symmetry is present, nature solves a constrained optimization problem that is called a least-action principle. So, again, symmetry and optimization are closely related.
Polya, a mathematics educator at Stanford, said 70 years ago, that “problems concerned with greatest and least values, or maximum and minimum problems, are more attractive, than other mathematical problems of comparable difficulty. We wish to obtain a certain object at the lowest possible price, or the greatest possible effect with a certain effort, or the maximum work done within a given time and, of course, we wish to run the minimum risk. Mathematical problems on maxima and minima appeal to us.” If symmetry and optimization are so ubiquitous, so fundamental in math and natural science, so intuitive to all of us, and so much fun, why aren’t we teaching these concepts to our kids starting in kindergarten? Don’t blame our educational system. No educational system in recorded history appreciated the importance of symmetry and optimization. These concepts have long been known to scientists, but have never been used to inspire young, inquisitive minds.