As dimension increases, the ratio of volume between a unit ball and a unit cube goes to zero. Said another way, if you have a high-dimensional ball inside a high-dimensional box, nearly all the volume is in the corners. This is a surprising result when you first see it, but it’s well known among people […]

# Category: Math

## Higher dimensional squircles

The previous post looked at what exponent makes the area of a squircle midway between the area of a square and circle of the same radius. We could ask the analogous question in three dimensions, or in any dimension. (Is a shape between a cube and a sphere a cuere?) In more conventional mathematical terminology, […]

## History of the term “Squircle”

Architect Peter Panholzer coined the term “squircle” in the summer of 1966 while working for Gerald Robinson. Robinson had seen a Scientific American article on the superellipse shape popularized by Piet Hein and suggested Panholzer use the shape in a project. Piet Hein used the term superellipse for a compromise between an ellipse and a […]

## Putting topological data analysis in context

I got a review copy of The Mathematics of Data recently. Five of the six chapters are relatively conventional, a mixture of topics in numerical linear algebra, optimization, and probability. The final chapter, written by Robert Ghrist, is entitled Homological Algebra and Data. Those who grew up with Sesame Street may recall the song “Which […]

## Elementary solutions to differential equations

Differential equations rarely have closed-form solutions. Some do, and these are emphasized in textbooks. For this post we want to look specifically at homogeneous second order linear equations: y ” + a(x) y‘ + b(x) y = 0. If the coefficient functions a and b are constant, then the solution can be written down in terms […]

## Finite rings

It occurred to me recently that I rarely hear about finite rings. I did a Google Ngram search to make sure this isn’t just my experience. Source Why are finite groups and finite fields common while finite rings are not? Finite groups have relatively weak algebraic structure, and demonstrate a lot of variety. Finite fields […]

## Monads and generalized elements

Paolo Perrone gives a nice, succinct motivation for monads in the introduction to his article on probability and monads. … a monad is like a consistent way of extending spaces to include generalized elements of a specific kind. He develops this idea briefly, and links to his dissertation where he gives a longer exposition (pages […]

## Mixing error-correcting codes and cryptography

Secret codes and error-correcting codes have nothing to do with each other. Except when they do! Error-correcting codes Error correcting code make digital communication possible. Without some way to detect and correct errors, the corruption of a single bit could wreak havoc. A simple example of an error-detection code is check sums. A more sophisticated […]

## US Army applying new areas of math

Many times on this blog I’ve argued that the difference between pure and applied math is motivation. As my graduate advisor used to say, “Applied mathematics is not a subject classification. It’s an attitude.” Traditionally there was general agreement regarding what is pure math and what is applied. Number theory and topology, for example, are […]

## Riffing on mistakes

I mentioned on Twitter yesterday that one way to relieve the boredom of grading math papers is to explore mistakes. If a statement is wrong, what would it take to make it right? Is it approximately correct? Is there some different context where it is correct? Several people said they’d like to see examples, so […]

## A genius can admit finding things difficult

Karen Uhlenbeck has just received the Abel Prize. Many say that the Fields Medal is the analog of the Nobel Prize for mathematics, but others say that the Abel Prize is a better analog. The Abel prize is a recognition of achievement over a career whereas the Fields Medal is only awarded for work done […]

## Thermocouple polynomials and other sundries

I was looking up something on the NIST (National Institute of Standards and Technology) web site the other day and ran across thermocouple polynomials. I wondered what that could be, assuming “thermocouple” was a metaphor for some algebraic property. No, it refers to physical thermocouples. The polynomials are functions for computing voltage as a function […]

## Counting irreducible polynomials over finite fields

You can construct a finite field of order pn for any prime p and positive integer n. The elements are polynomials modulo an irreducible polynomial of degree n, with coefficients in the integers mod p. The choice of irreducible polynomial matters, though the fields you get from any two choices will be isomorphic. For example, […]

## Average distance between planets

What is the closest planet to Earth? The planet whose orbit is closest to the orbit of Earth is clearly Venus. But what planet is closest? That changes over time. If Venus is between the Earth and the sun, Venus is the closest planet to Earth. But if Mercury is between the Earth and the […]

## All elliptic curves over fields of order 2 and 3

Introductions to elliptic curves often start by saying that elliptic curves have the form y² = x³ + ax + b. where 4a³ + 27b² ≠ 0. Then later they say “except over fields of characteristic 2 or 3.” What does characteristic 2 or 3 mean? The order of a finite field is the number of […]

## Efficient modular arithmetic technique for Curve25519

Daniel Bernstein’s Curve25519 is the elliptic curve y² = x³ + 486662x² + x over the prime field with order p = 2255 – 19. The curve is a popular choice in elliptic curve cryptography because its design choices are transparently justified [1] and because cryptography over the curve can be implemented very efficiently. This […]

## Chaos + Chaos = Order

If you take these chaotic-looking values for your x-coordinates

and these chaotic-looking values for your y coordinates

you get this image that looks more ordered.

The image above is today’s exponential sum.

## An attack on RSA with exponent 3

As I noted in this post, RSA encryption is often carried out reusing exponents. Sometimes the exponent is exponent 3, which is subject to an attack we’ll describe below [1]. (The most common exponent is 65537.) Suppose the same message m is sent to three recipients and all three use exponent e = 3. Each […]

## Public key encryption based on squares and non squares

The RSA encryption algorithm depends indirectly on the assumption that factoring the product of large primes is hard. The algorithm presented here, invented by Shafi Goldwasser and Silvio Micali, depends on the same assumption but in a different way. The Goldwasser-Micali algorithm is more direct than RSA, thought it is also less efficient. One thing […]

## An infinite product challenge

Gil Kalai wrote a blog post yesterday entitled “Test Your Intuition (or knowledge, or programming skills) 36.” The challenge is to evaluate the infinite product I imagine there’s an elegant analytical solution, but since the title suggested that programming might suffice, I decided to try a little Python. I used primerange from SymPy to generate […]