{"id":1216,"date":"2015-03-01T01:00:39","date_gmt":"2015-03-01T01:00:39","guid":{"rendered":"http:\/\/blogs.ams.org\/visualinsight\/?p=1216"},"modified":"2015-07-29T00:47:16","modified_gmt":"2015-07-29T00:47:16","slug":"schmidt-arrangement","status":"publish","type":"post","link":"https:\/\/blogs.ams.org\/visualinsight\/2015\/03\/01\/schmidt-arrangement\/","title":{"rendered":"Schmidt Arrangement"},"content":{"rendered":"
\"Schmidt<\/a>

Schmidt Arrangement of the Eisenstein Integers – Katherine Stange<\/p><\/div><\/div>\n

This picture drawn by Katherine Stange<\/a> shows what happens when we apply fractional linear transformations<\/p>\n

$$ z \\mapsto \\frac{a z + b}{c z + d} $$<\/p>\n

to the real line sitting in the complex plane, where $a,b,c,d$ are Eisenstein integers<\/a><\/b>: that is, complex numbers of the form <\/p>\n

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

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

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.<\/p>\n

The Eisenstein integers are the algebraic integers<\/b><\/a> 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<\/b><\/a>. 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<\/b> by taking the real line together with the point at infinity and acting on it in all possible ways by transformations <\/p>\n

$$ z \\mapsto \\frac{a z + b}{c z + d}, \\quad a,b,c,d \\in \\mathcal{O}_K $$<\/p>\n

These transformations form a group called the Bianchi group<\/b><\/a> $\\mathrm{PSL}_2(\\mathcal{O}_K)$. <\/p>\n

Stange has more pictures of Schmidt arrangements here: <\/p>\n

• Katherine E. Stange, Schmidt arrangements<\/a>.<\/p>\n

She drew them using Sage. She explains the underlying mathematics here:<\/p>\n

• Katherine E. Stange, Visualising the arithmetic of imaginary quadratic fields<\/a>.<\/p>\n

As she describes, Schmidt arrangements have a rich geometrical structure that reflects the arithmetic of their number fields:<\/p>\n

\nWe study the orbit of $\\mathbb{R} \\cup \\{\\infty\\}$ under the Bianchi group<\/a> $\\mathrm{PSL}_2(\\mathcal{O}_K)$, where $K$ is an imaginary quadratic field<\/a>. 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<\/a> form of $\\mathcal{O}_K$. Second, we show that the circles themselves are in bijection with certain ideal classes<\/a> in orders<\/a> of $\\mathcal{O}_K$, the conductor<\/a> being a certain multiple of the curvature. This allows us to count circles with class numbers<\/a>. Third, we show that the arrangement of circles is connected if and only if $\\mathcal{O}_K$ is Euclidean<\/a>. These results are meant as foundational for a study of a new class of thin groups generalising Apollonian groups, in a companion paper.\n<\/p><\/blockquote>\n

For more, see:<\/p>\n

• Katherine E. Stange, The Apollonian structure of Bianchi groups<\/a>.<\/p>\n

\n

Visual Insight<\/i> is a place to share striking images that help explain advanced topics in mathematics. I\u2019m always looking for truly beautiful images, so if you know about one, please drop a comment here<\/a> and let me know!<\/p>\n

<\/div>","protected":false},"excerpt":{"rendered":"

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<\/a><\/b> that is, complex numbers of the form $ m + n \\exp(2 \\pi i\/3) $ with $m,n$ being integers. <\/p>\n

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