It is likely that discussing Math may emote a visceral reaction from people of all shapes and types. While there are some out there that adore numbers, there are probably many more that physically recoil at the sight of math problems. For those who are fascinated by how math works or are just interested in digging into how math can literally describe and impact every single thing within the universe, the good folks over at Quanta Magazine have developed a Map of Mathematics. It is a really neat and intuitively interactive map that breaks down the dependencies and interdependencies of different levels of Mathematics, going from the basics of Numbers, Shapes, and Change, to things like Prime Reciprocity, Continuous Symmetries, and Einstein’s Equations.
The discussion, analysis and obsession with Prime Numbers is one that goes back millennia and has spawned all sorts of theories:
Prime numbers are whole numbers larger than 1 that are not divisible by any whole number apart from 1 and themselves. They’re like the atoms of number theory — you can use primes to make any other number.
In the third century B.C., Euclid proved that there are infinitely many primes. He argued that if we multiply all known primes together and add 1, then either this new number N is prime, or N can be divided by a number that’s not on our original list of primes — a new prime. This proves infinitude, but it’s stillborn as a technique: It tells us nothing about the distribution of primes and provides no way to investigate further questions about them.
Today, mathematicians are interested in understanding how often primes occur.
Number theorists create functions of real or complex variables, called analytic functions, that let them study questions about prime numbers. For example, they might ask: Approximately how many primes exist in a short interval? Or in how many ways can a natural number be expressed as a sum of three squares? Analytic functions have properties that address these questions.
The field dates back to the work of Peter Gustav Lejeune Dirichlet in the 19th century. Dirichlet studied “arithmetic progressions,” the list of numbers we get by starting with a natural number A and adding to it multiples of a natural number B. For example, with A = 4 and B = 7 we get: 4, 11, 18, 25, and so on. Dirichlet used analytic functions to prove that as long as A and B don’t have any common prime factor (as in our example), such an arithmetic progression must contain infinitely many primes.
Shapes have long been able to be described and defined using simple to complex math formulas. Interestingly, some of the hardest shapes to define via math are those that have 3 or 4 dimensions.
Mathematicians have understood one- and two-dimensional manifolds since the 19th century. A surprising discovery in the mid-20th century was that shapes with five or more dimensions are also relatively easy to analyze — those extra dimensions provide mathematicians with more room to maneuver, which allows them to bring more techniques to bear. Many of the hardest open problems in topology are in dimensions three and four, where mathematicians still search for a better understanding of how to tell manifolds apart and how to understand the characteristics that distinguish them.
You could get lost in this map of Math for hours. Some of the theories are mind bending on the highest order, yet this interactive map does an exceptional job of making the complex read clearly and succinctly. In today’s data driven world, getting a better grasp of Math concepts and theory may be a good career move.
