This picture drawn by Katherine Stange shows what happens when we apply fractional linear transformations

$$ z \mapsto \frac{a z + b}{c z + d} $$

to the real line sitting in the complex plane, where $a,b,c,d$ are **Eisenstein integers**: that is, complex numbers of the form

$$ m + n e^{2 \pi i/3} $$

where $m,n$ are integers. The result is a complicated set of circles and lines called the ‘Schmidt arrangement’ of the Eisenstein integers.

There are infinitely many circles in the Schmidt arrangement, but this picture shows only those with radius $\gt 1/20$ that pass through a certain parallelogram. The curvature of each circle—that is, the reciprocal of its radius—is always an integer multiple of $\sqrt{3}$. Each circle is colored according to whether this integer is even or odd.

The Eisenstein integers are the **algebraic integers** of the field $\mathbb{Q}[\sqrt{-3}]$, meaning those that are roots of polynomials with leading coefficient 1. More generally, for any square-free integer $N$ we can form a field $K = \mathbb{Q}[\sqrt{-N}]$, called a **imaginary quadratic number field**. The algebraic integers in such a field form a ring known as $\mathcal{O}_K$, and we can use these to define a **Schmidt arrangement** by taking the real line together with the point at infinity and acting on it in all possible ways by transformations

$$ z \mapsto \frac{a z + b}{c z + d}, \quad a,b,c,d \in \mathcal{O}_K $$

These transformations form a group called the **Bianchi group** $\mathrm{PSL}_2(\mathcal{O}_K)$.

Stange has more pictures of Schmidt arrangements here:

• Katherine E. Stange, Schmidt arrangements.

She drew them using Sage. She explains the underlying mathematics here:

• Katherine E. Stange, Visualising the arithmetic of imaginary quadratic fields.

As she describes, Schmidt arrangements have a rich geometrical structure that reflects the arithmetic of their number fields:

We study the orbit of $\mathbb{R} \cup \{\infty\}$ under the Bianchi group $\mathrm{PSL}_2(\mathcal{O}_K)$, where $K$ is an imaginary quadratic field. The orbit, called a Schmidt arrangement $S_K$, is a geometric realisation, as an intricate circle packing, of the arithmetic of $K$. This paper presents several examples of this phenomenon. First, we show that the curvatures of the circles are integer multiples of $\sqrt{-\Delta}$ and describe the curvatures of tangent circles in terms of the norm form of $\mathcal{O}_K$. Second, we show that the circles themselves are in bijection with certain ideal classes in orders of $\mathcal{O}_K$, the conductor being a certain multiple of the curvature. This allows us to count circles with class numbers. Third, we show that the arrangement of circles is connected if and only if $\mathcal{O}_K$ is Euclidean. These results are meant as foundational for a study of a new class of thin groups generalising Apollonian groups, in a companion paper.

For more, see:

• Katherine E. Stange, The Apollonian structure of Bianchi groups.

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