---
title: Spherical Grasshopper Problem
url: https://www.emergentmind.com/topics/spherical-grasshopper-problem
type: topic
---

# Spherical Grasshopper Problem

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The spherical grasshopper problem is a geometric optimization problem on the unit sphere \(S^2\subset \mathbb{R}^3\) in which one seeks a measurable colouring, or equivalently a measurable “lawn,” of area \(2\pi\) that maximizes the probability that a grasshopper remains on the same colour after making a geodesic jump of fixed central angle \(\theta\) in a uniformly random direction. In its most studied form, the colouring satisfies an antipodality constraint: exactly one point in each antipodal pair is painted black, so \(\mu(r)+\mu(-r)=1\) for all \(r\in S^2\). The problem was developed in a 2023 essay by van Breugel and was later embedded in a broader computational and harmonic framework that also treats several non-equivalent spherical variants and relates the optimization to Bell inequalities and local hidden-variable approximations to singlet correlations [2307.05359] [2603.08579].

## 1. Mathematical formulation

On \(S^2\), with surface element \(d^2r\) or \(d\Omega(x)\), an antipodal complementary colouring is a measurable function
\[
\mu:S^2\to\{0,1\}
\]
satisfying
\[
\mu(r)+\mu(-r)=1,\qquad \forall r\in S^2.
\]
Equivalently, one may work with a measurable subset \(L\subset S^2\) of area \(|L|=2\pi\), or with an indicator function \(f:S^2\to\{0,1\}\) obeying
\[
\int_{S^2} f(x)\,d\Omega(x)=2\pi.
\]

A grasshopper lands at a uniformly random point \(r_1\in S^2\) and then jumps a fixed geodesic distance \(\theta\) in a uniformly random direction, landing at \(r_2\). The optimization target is
\[
P_{\mathrm{same}}(\theta)=\Pr[\mu(r_1)=\mu(r_2)].
\]
In the one-lawn formulation used in the later framework, this becomes
\[
P(\theta)=\frac{1}{2\pi}\int_{S^2} f(x)\,(K_\theta f)(x)\,d\Omega(x),
\]
where \(K_\theta\) is the normalized circle-averaging operator
\[
(K_\theta g)(x)=\frac{1}{2\pi\sin\theta}\int_{S^2}\delta(\arccos(x\cdot y)-\theta)\,g(y)\,d\Omega(y).
\]
The operator preserves total mass:
\[
\int (K_\theta g)(x)\,d\Omega(x)=\int g(x)\,d\Omega(x).
\]

The 2026 study distinguishes three variants. In the **antipodal complementary** problem, one lawn \(f\) satisfies \(f(x)+f(-x)=1\). In the **antipodal independent** problem, two lawns \(f_1,f_2\) each satisfy \(f_k(x)+f_k(-x)=1\), and success means starting on lawn 1 and landing outside lawn 2. In the **non-antipodal complementary** problem, the antipodal constraint is dropped while retaining one lawn of area \(2\pi\) [2603.08579].

## 2. Integral representations and exact cases

For an antipodal colouring \(\mu\), van Breugel writes
\[
P_{\mu}(\theta)
=\int_{S^2}\!\!\int_{S^2} \mu(r_1)\,\mu(r_2)\,\delta\!\big(\Delta\sigma(r_1,r_2)-\theta\big)\,
d^2r_1\,d^2r_2,
\]
where \(\Delta\sigma(r_1,r_2)\) is the central angle between \(r_1\) and \(r_2\). Introducing the spin field
\[
f(r)=2\mu(r)-1\in\{\pm1\},
\]
the same quantity can be expressed as
\[
P_{\mu}(\theta)
=\tfrac12+\tfrac14\int_{S^2}\!\!\int_{S^2}f(r_1)\,f(r_2)\,
\delta\!\big(\Delta\sigma(r_1,r_2)-\theta\big)\,d^2r_1\,d^2r_2
\]
[2307.05359].

The later framework also gives a double-integral form for two lawns \(L_1,L_2\):
\[
P(\theta)=\frac{1}{4\pi^2\sin\theta}\int_{L_1}d\Omega(x)\int_{L_2^c}d\Omega(y)\,
\delta(\arccos(x\cdot y)-\theta)
\]
[2603.08579].

Several exact cases organize the problem. For a hemispherical lawn, any great-circle hemisphere yields
\[
P(\theta)=1-\theta/\pi.
\]
This quantity serves as the hemisphere colouring bound in the antipodal complementary setting. At \(\theta=\pi/2\), every antipodal one-lawn satisfies
\[
P=1/2.
\]
At \(\theta=\pi\), one-lawn complementary gives \(P=0\), whereas two-lawn complementary gives \(P=1\) [2603.08579]. These exact identities explain why \(\theta=\pi/2\) is a singular point in the antipodal problem and why the large-\(\theta\) behavior differs sharply between one-lawn and two-lawn variants.

## 3. Discretization and numerical optimization

Van Breugel discretized the antipodal problem on a geodesic-icosahedral mesh. Subdividing an icosahedron to depth \(k\) produces \(2N\) points with
\[
2N=2+10\cdot 4^k,
\]
and mesh-spacing parameter
\[
h\approx \sqrt{2\pi/N}.
\]
The projected vertices form an almost-uniform antipodal hexagonal mesh. The Dirac \(\delta\)-function is replaced by a smooth compact-support kernel
\[
\phi(x)=\tfrac14[1+\cos(\pi x/2)]\quad \text{for } |x|\le 2,\qquad 0\ \text{else},
\]
and the unknown colouring is a binary vector \(s\in\{0,1\}^N\) on one hemisphere, with antipodes fixed by \(s_i\leftrightarrow 1-s_i\). The discrete success probability is
\[
P_s(\theta)
=\frac1N\sum_{i=1}^{2N}\frac{\sum_{j=1}^{2N}s_i\,s_j\,\phi\!\big(\tfrac{\Delta\sigma(v_i,v_j)-\theta}{h}\big)}
{\sum_{j=1}^{2N}\phi\!\big(\tfrac{\Delta\sigma(v_i,v_j)-\theta}{h}\big)}.
\]
Optimization is performed by simulated annealing with
\[
T_{n+1}=\alpha T_n,\qquad T_0\approx 0.4,\qquad \alpha\approx 0.99999,
\]
using random antipodal-pair flips accepted with probability \(\min[1,\exp(\Delta P/T)]\), and runs of \(\simeq 150N\) flips until no significant increase in \(P\) is observed [2307.05359].

The later study broadened the discretization toolkit. It considers symmetric spherical \(t\)-designs, with \(N\) up to \(\sim 52\,978\), HEALPix, with \(N\) up to \(\sim 303\,372\), as well as Goldberg-icosahedron subdivisions and custom “Coulomb” point sets for comparison. The area element is approximated by \(h^2\) with
\[
h=\sqrt{4\pi/N},
\]
and the same cosine kernel is used through
\[
\delta(\arccos(x_i\cdot x_j)-\theta)\approx \frac{1}{h}\phi((\theta_{ij}-\theta)/h).
\]
For one lawn,
\[
P_s(\theta)=\frac{4}{\sin\theta\,N^2\,h}\sum_{i,j}s_i\,s_j\,\phi((\theta_{ij}-\theta)/h),
\]
while in the two-lawn problem one replaces \(s_i s_j\) by \(s_{1,i}(1-s_{2,j})\). The optimization combines simulated annealing in a conserved-magnetization Ising model, boundary annealing, and final greedy sweeps. Grid quality is monitored through the “potential energy”
\[
E_i=\sum_j \phi((\theta_{ij}-\theta)/h),
\]
with good grids characterized by small \(\mathrm{Var}(E_i)\) [2603.08579].

## 4. Numerically observed optimal configurations

In van Breugel’s antipodal complementary study, three \(\theta\)-regimes emerge. For
\[
0.10\pi\lesssim \theta \lesssim 0.44\pi,
\]
the numerically optimal colourings are **cogwheels**: the lawn is concentrated in an odd number \(n_0\) of cogs about the equator, symmetric under rotation by \(2\pi/n_0\), with
\[
n_0\approx \lfloor 2\pi/\theta \rceil,\qquad n_0\ \text{odd}.
\]
Each cog has longitudinal width
\[
\Delta\phi\approx \theta,
\]
and latitudinal half-width much smaller than \(\theta\). In this regime,
\[
P_{\mathrm{same}}(\theta)
\]
exceeds the hemisphere bound \(P_{\mathrm{hem}}(\theta)=1-\theta/\pi\), with peaks whenever \(2\pi/\theta\) is near an odd integer [2307.05359].

For
\[
0.45\pi\lesssim \theta \lesssim 0.55\pi,
\]
van Breugel reports **critical solutions** in which same-colour domains break into many small antipodal patches. Their typical diameter \(\xi(\theta)\) shrinks as \(\theta\to \pi/2\), with the numerical scaling
\[
\xi(\theta)\sim |\pi-2\theta|^\alpha,\qquad \alpha\approx 1.
\]
The arrangement has no long-range periodicity and is described as reminiscent of critical Ising clusters. The performance curve is symmetric about \(\theta=\pi/2\), with a smooth dip to \(1/2\) at \(\theta=\pi/2\), as forced by antipodality [2307.05359].

For
\[
0.55\pi\lesssim \theta < \pi,
\]
the numerics yield **stripes with cogs**. Near \(0.55\pi\lesssim \theta \lesssim 0.57\pi\), one sees holed-out cogwheels, such as \(n=7\), whose teeth anticorrelate with neighbours two steps away. For larger \(\theta\), the solutions become three “fan-blades” or antipodal stripes that gradually deform into latitude bands. As \(\theta\to\pi\), the bands proliferate and narrow, with stripe half-width
\[
w\sim \pi-\theta.
\]
In van Breugel’s formulation,
\[
P_{\mathrm{same}}(\theta)\to 0\qquad \text{as } \theta\to\pi,
\]
and the curve exhibits local minima when the number of stripes changes by one [2307.05359].

The later classification recasts the same antipodal complementary phenomenon into **cogwheel**, **labyrinth**, and **stripe** regimes, with numerical boundaries summarized as \(\theta\lesssim 0.41\pi\), \(\theta\approx \pi/2\pm\), and \(\theta\gtrsim 0.57\pi\), respectively. In the stripe regime, the number of stripes of each colour is
\[
n_s\simeq \frac{\pi}{\sqrt{3}\,(\pi-\theta)},
\]
and the stripe-edge modulations have amplitude tending to zero as \(\theta\to\pi\). The same study reports
\[
P(\theta)\to 2/3\qquad \text{as } \theta\to\pi,
\]
while still maintaining the exact identity \(P(\pi)=0\) for the antipodal one-lawn problem [2603.08579]. A plausible implication is that the numerical asymptotics and the exact endpoint behavior must be interpreted with care near \(\theta=\pi\).

## 5. Variants, symmetries, and harmonic interpretation

The three variants studied in the 2026 framework exhibit related but distinct morphologies. In the **antipodal complementary** one-lawn problem, small-\(\theta\) optima are cogwheels with \(k\) odd teeth and
\[
k\simeq \text{nearest odd integer to } 2\pi/\theta.
\]
Higher modes \(m>1\) give
\[
k\simeq \text{nearest odd integer to } 2m\pi/\theta.
\]
At
\[
\theta_q=\pi/q,
\]
the hemisphere and cogwheels with \(k=2q\pm1\) are all near-optimal, with
\[
P=1-1/q
\]
exactly. Around \(\theta\approx \pi/2\), the optima are labyrinthine; for \(\theta\gtrsim 0.57\pi\), they become alternating parallel rings encircling the poles [2603.08579].

In the **antipodal independent** two-lawn problem, the small-\(\theta\) optima are similar cogwheels but with
\[
k\simeq \text{nearest integer to } \pi/\theta,
\]
not necessarily odd, and the two lawns are offset by half a tooth. Mode \(m\) satisfies
\[
k\simeq \text{nearest integer to } m\pi/\theta.
\]
Even \(m\) reproduces complementary one-lawn shapes, whereas odd \(m\) gives offset patterns. Hemispheres are optimal only at \(\theta_q=\pi/q\) for \(q\) even; for \(q\) odd, cogwheels beat the hemisphere. There is no stripe regime, and the symmetry
\[
P(\theta)=P(\pi-\theta)
\]
causes cogwheels to reappear for \(\theta>\pi/2\) [2603.08579].

In the **non-antipodal complementary** one-lawn problem, the cogwheel regime persists with
\[
k\simeq \text{nearest integer to } 2\pi/\theta,
\]
where \(k\) may be even. Intermediate optima include islands, bands, and a cube-symmetric “squircle” pattern at \(\theta=\pi/2\). As \(\theta\to\pi\), the lawn approaches two polar caps of colatitude \(\pi/3\) separated by a belt, and
\[
P(\pi)=1
\]
[2603.08579].

A spectral account is obtained by expanding the indicator in orthonormal spherical harmonics,
\[
\mu(r)=\sum_{\ell=0}^{\infty}\sum_{m=-\ell}^{\ell}\hat{\mu}_{\ell m}Y_{\ell m}(r),
\]
with
\[
\hat{\mu}_{\ell m}=\int_{S^2}\mu(r)\,Y_{\ell m}^*(r)\,d\Omega(r),
\qquad
\sum_{\ell m}|\hat{\mu}_{\ell m}|^2=2\pi,
\qquad
\hat{\mu}_{00}=\sqrt{\pi}.
\]
Under antipodal complementarity,
\[
\mu(r)+\mu(-r)=1\quad \Rightarrow \quad \hat{\mu}_{\ell m}=0 \ \text{for even } \ell\ge 2.
\]
The success probability becomes
\[
P(\theta)=\frac{1}{2\pi}\sum_{\ell m} |\hat{\mu}_{\ell m}|^2\,P_\ell(\cos\theta),
\]
with the upper bound
\[
P(\theta)\le \tfrac12+\tfrac12\max_{\text{allowed }\ell} P_\ell(\cos\theta).
\]
Sectoral harmonics \((|m|=\ell)\) correspond to cogwheel patterns concentrated near the equator, zonal harmonics \((m=0)\) to stripes or rings, and mixed \((\ell,m)\) content to islands and labyrinths. Numerically, the maximizing \(\ell^*\) rises from \(\ell^*=1\) at small \(\theta\), tends to infinity near \(\theta\to \pi/2\), falls to \(\ell^*=3\) at stripe onset, and rises again as \(\theta\to\pi\) [2603.08579].

## 6. Relation to planar results, Bell inequalities, and open questions

The spherical antipodal complementary problem was motivated by Bell inequalities. Quantum singlet measurements on axes separated by \(\theta\) have correlation
\[
E_Q(\theta)=-\cos\theta.
\]
In the one-lawn complementary interpretation, any deterministic local hidden-variable model yields
\[
E_{\mathrm{LHV}}(\theta)=-(2P(\theta)-1).
\]
Hence the maximal grasshopper success probability is exactly the best classical approximation to the singlet correlation at that separation angle. The gap
\[
\Delta(\theta)=|E_Q(\theta)-E_{\mathrm{LHV}}^*(\theta)|
\]
quantifies the Bell-inequality violation achievable with random axes separated by \(\theta\). CHSH at \(\theta=\pi/4\) and Braunstein–Caves at \(\theta=\pi/(2n)\) appear as special cases, and solving the grasshopper problem supplies an infinite family of optimal Bell inequalities [2603.08579].

The numerics also relate the sphere to the planar grasshopper problem studied by Goulko and Kent. Van Breugel reports that spherical cogwheel solutions have the same qualitative shape as the two-dimensional disc-problem cogs, except that on \(S^2\) antipodality forces the number of cogs to be odd and makes cogs and holes identical by symmetry. Stripe patterns and holed-out fans likewise mirror the planar “three-bladed fan” and stripe solutions after “wrapping” onto \(S^2\). By contrast, the critical small-patch regime has no planar analogue because the planar problem lacks the antipodal constraint and therefore does not enforce the \(\theta\leftrightarrow \pi-\theta\) symmetry [2307.05359].

The later framework places the problem near several classical and physical models. The stripe limit as \(\theta\to\pi\) is compared to Buffon-type geometry and the “needle crossing stripes” problem; antipodal lawns are described as closely related to the double-cap conjecture; and the emergent stripe, labyrinth, and droplet patterns are compared to modulated-phase pattern formation in block copolymers, magnetic garnets, ferrofluids, Turing patterns on curved surfaces, and dipolar Bose–Einstein condensate droplet arrays [2603.08579].

Several open questions remain explicit. In the cogwheel regime, when \(2\pi/\theta\) lies halfway between odd integers, one finds multiple nearly degenerate colourings, such as 7- or 9-cog wheels at \(\theta\approx 2\pi/7.9\), and it is open whether a unique global optimum exists. Below \(\theta\approx 0.10\pi\), van Breugel’s numerics revert to hemisphere-like colourings, possibly as a resolution artefact, so analytic or higher-resolution work is needed to determine whether infinitely many cogwheels persist as \(\theta\to 0\). No closed-form approximation of cog shapes is known. Parallel tempering and adaptive Monte Carlo moves that flip multiple well-chosen points at once were proposed as possible algorithmic improvements. Extensions to higher-dimensional spheres or torii, other metrics, variable-jump distributions, and formulations without antipodality were identified as natural directions for further study [2307.05359].

Source: https://www.emergentmind.com/topics/spherical-grasshopper-problem