Schmidt Arrangement, Algebra Integer, Gaussian Integer, Eisenstein Integer, Modular Group

By Xah Lee. Date: . Last updated: .

Schmidt Arrangement

math schmidt arrangement Katherine Stange 2015-03-01
Schmidt Arrangement by Katherine Stange. (more pictures here

in a glance, it looks like geometric inversion on parts of hexagonal grid… See Circle Inversion Gallery

Here is what mathematician [John Baez] says:

That's close. You build it by taking the real line in the complex plane and applying all possible transformations

f[z] ↦ (a*z + b)/(c*z + d)

where a,b,c,d are Eisenstein integers, meaning numbers of the form

m + n Sqrt[-3]

The Eisenstein integers lie on a hexagonal grid!

for more math details, see [Schmidt Arrangement By John Baez. At , accessed on 2015-03-01 ]

Here is more in the conversation:

John Baez wrote:

+Xah Lee - you will love the Eisenstein integers as soon as you know what they are: they are like integers, but they're complex numbers instead of real numbers. You can define prime numbers for these other integers, you can prove unique factorization into primes, and so on.

And the Eisenstein integers are just one of many kinds of generalized integers, called algebraic integers. This puts the usual theory of prime numbers in a bigger context.

The group of fractional linear transformations

z ↦ (az + b)/(cz + d)

where a,b,c,d are all integers is incredibly important and beautiful. It's called the modular group and I could talk about it for days - it's one of the stars of modern number theory! For example, Fermat's last theorem was proved with the help of this group and its relatives.

Using Eisenstein integers or some other kind of integers for a,b,c,d gives variations on this game.

John Baez wrote:

+Xah Lee wrote: “is there some interesting thing about the term fractional linear transformation? I'm guessing that term came from the form of the formula (az + b)/(cz + d)…”


“do you prefer this term than Möbius transformation?”

No big deal - I just think it's a bit more self-explanatory, and I think I've seen it in more places.

Harald Hanche-Olsen wrote:

+Xah Lee The term fractional linear transformation is, I think, related to the fact that the complex plane with a point at infinity added to a.k.a. the Riemann sphere to is also the complex projective line, defined as the set of all one-dimensional complex subspaces of the standard two-dimensional complex vector space. Just associate the linear span of (z,1) with z, more generally the span of (z,w) with z/w, and the span of (z,0) with ∞ (when z≠0).

A linear transformation of two-dimensional space will typically have the form (z,w)↦(az+bw,cz+dw). Now apply that to (z,1) [representing z] and reinterpret the result as a complex number (or ∞), and you have the map z↦(az+b)/(cz+d). So it arises from a linear map, and there is a fraction involved, so calling it “fractional linear” isn't totally crazy. (But I do prefer “Möbius transformation” personally.)

Michael Nelson wrote:

+Xah Lee The main point is that the rational numbers are difficult to work with. This is because the rational numbers lack “completeness” type properties. Quite often, algebraic structures become simplified when we view them over a field extension. For example, the set of solutions to an equation like:

y^2 = x^3 - x + 1

looks like a torus when you allow x and y to be complex numbers. It's not so obvious what they look like over the rational numbers though (it's still a torus). The eisenstein integers are the ring of integers for a certain field extension of Q; so it represents a part of this field extension simplication process. By the way, once we understand what the algebraic structure "looks like" over a bigger field, a powerful method to bring that algebraic structure back down to the rationals is via Galois Theory.

Also, I highly recommend visiting this website for more complex beauties.

He has many different phase portraits on his website, along with the code to make them. And he also has a very beautiful book that goes into detail about them. There's definitely a lot of insight to be gained by understanding these visualization methods.

Algebraic Integer

In number theory, an algebraic integer is a complex number that is a root of some monic polynomial (a polynomial whose leading coefficient is 1) with coefficients in ℤ (the set of integers). The set of all algebraic integers is closed under addition and multiplication and therefore is a subring of complex numbers denoted by A. The ring A is the integral closure of regular integers ℤ in complex numbers.

(a monic polynomial is a univariate polynomial in which the leading coefficient (the nonzero coefficient of highest degree) is equal to 1.)

Algebraic integer

Gaussian integer

In number theory, a Gaussian integer is a complex number whose real and imaginary parts are both integers. The Gaussian integers, with ordinary addition and multiplication of complex numbers, form an integral domain, usually written as Z[i].

Gaussian integer

In mathematics, Eisenstein integers (named after Gotthold Eisenstein), also known as Eulerian integers (after Leonhard Euler), are complex numbers of the form

z = a + b * ω

where a and b are integers and

ω = 1/2 * (-1 + ⅈ * Sqrt[3]) = ⅇ^(2*π*ⅈ/3)

is a primitive (non-real) cube root of unity. The Eisenstein integers form a triangular lattice in the complex plane, in contrast with the Gaussian integers, which form a square lattice in the complex plane.

Eisenstein integer lattice
Eisenstein integer lattice. ([image source 2015-03-03 ])

Eisenstein integer

Modular group

In mathematics, the modular group Γ is a fundamental object of study in number theory, geometry, algebra, and many other areas of advanced mathematics. The modular group can be represented as a group of geometric transformations or as a group of matrices.

The modular group Γ is the group of linear fractional transformations of the upper half of the complex plane which have the form

f[z] := ( a * z + b ) / ( c * z + d )

where a, b, c, and d are integers, and a * d − b * c = 1. The group operation is function composition.

Modular group