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The number $π$ and summation by $SL(2,\mathbb Z)$

Nikita Kalinin, Mikhail Shkolnikov

TL;DR

The paper derives a new formula for $\pi$ through an $SL(2,\mathbb Z)$ lattice-cropping construction on unimodular polygons. By defining $f(a,b,c,d)=\sqrt{a^2+b^2}+\sqrt{c^2+d^2}-\sqrt{(a+c)^2+(b+d)^2}$ for all nonnegative integers with $ad-bc=1$, and associating the cropped area to $\tfrac{1}{2}f(a,b,c,d)^2$, the author proves $\displaystyle \sum f(a,b,c,d)=2$ and $\displaystyle \sum f(a,b,c,d)^2=2-\frac{\pi}{2}$, leveraging limits ${\rm Area}(P_n)\to {\rm Area}(D^2)$ and ${\rm Perimeter}(P_n)\to 2\pi$ for the evolving unimodular polygons. The approach links discrete lattice geometry to the circle, yielding a novel $\pi$-formula and motivating broader questions, including higher dimensions, zeta-type sums, and potential modular-analytic connections. The work also outlines future directions, such as extensions to other powers of $f$, tropical geometry interpretations, and relations to convex-domain coordinates. Overall, it provides a bridge between lattice combinatorics, tropical geometry, and classical constants, with potential for further exact and analytic developments.

Abstract

We obtained a new formula for $π$.

The number $π$ and summation by $SL(2,\mathbb Z)$

TL;DR

The paper derives a new formula for through an lattice-cropping construction on unimodular polygons. By defining for all nonnegative integers with , and associating the cropped area to , the author proves and , leveraging limits and for the evolving unimodular polygons. The approach links discrete lattice geometry to the circle, yielding a novel -formula and motivating broader questions, including higher dimensions, zeta-type sums, and potential modular-analytic connections. The work also outlines future directions, such as extensions to other powers of , tropical geometry interpretations, and relations to convex-domain coordinates. Overall, it provides a bridge between lattice combinatorics, tropical geometry, and classical constants, with potential for further exact and analytic developments.

Abstract

We obtained a new formula for .

Paper Structure

This paper contains 5 sections, 5 theorems, 9 equations, 2 figures.

Table of Contents

  1. History: geometric approach to $\pi$
  2. $SL(2,\mathbb Z)$-way to cut corners
  3. Proofs
  4. Questions
  5. Aknowledgement

Key Result

Lemma 1

For all $a,b,c,d\in \mathbb Z_{\geq 0}$ with $ad-bc=1,$ such that $(a,b),(c,d)$ belong to the same quadrant, there is a corner of $P_n$ for some $n\geq 0$ supported by the primitive vectors $(a,b)$ and $(c,d).$ In $P_{n+1}$ this corner is cropped by the line orthogonal to $(a+c,b+d)$ and tangent to

Figures (2)

  • Figure 1: The disc is inscribed in the square $P_0$. Then, $P_1$ is the only unimodular octagon circumscribing $D^2$ which can be obtained by corner cuts of $P_0.$
  • Figure 2: The plot of $F$ and its corner locus (tropical analytic curve) $C$ for a disc.

Theorems & Definitions (12)

  • Definition 1
  • Example 1
  • Remark 1
  • Definition 2
  • Example 2
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Definition 3
  • Lemma 4
  • ...and 2 more