Spherical Power Cells: Geometry & Energy Applications

• Spherical power cells are weighted Voronoi-type partitions that allow direct control over cell areas for accurate spherical discretization in numerical simulations.
• They are constructed via halfspace intersections and lifting techniques, enabling efficient parallel computations with linear scaling in large-scale applications.
• In addition to geometric applications, spherical power cells serve as a conceptual model for phase transitions in Coulomb-coupled spherical lattices, guiding energy device design.

Searching arXiv for papers on spherical power cells and closely related spherical power-diagram methods. Spherical power cells are weighted Voronoi-type partitions of the sphere in which each site $\vec{p}_i$ is assigned a weight $w_i$, and the corresponding cell consists of those points on $\mathbb{S}^2$ minimizing the modified squared distance $\|\vec{p}_i-\vec{r}\|^2-w_i$. In the available literature, the term appears in at least two distinct but related senses: as a geometric object for spherical discretization and tessellation, and as a speculative device concept tied to energy release in Coulomb-coupled spherical lattices. The geometric sense is formalized through spherical power diagrams and their algorithmic use in numerical methods (Caplan et al., 11 Aug 2025), while the device-oriented sense is introduced as a conceptual application of phase transitions on spherical lattices (Bachmann et al., 2024). Related work on equal-area spherical subdivision, Gaussian spherical tessellations, and spherical shell mechanics provides adjacent mathematical and physical context (Malkin, 2016, Lybrand et al., 2021, Couturier et al., 2021).

1. Definition and core geometric formulation

In the geometric literature, spherical power cells arise from a weighted partition of the sphere. For sites $\{\vec{p}_i\}\subset \mathbb{S}^2$ with weights $w_i$, the spherical power cell is defined by

$$\mathbb{P}_i=\left\{\vec{r}\in\mathbb{S}^2:\ \|\vec{p}_i-\vec{r}\|^2-w_i<\|\vec{p}_j-\vec{r}\|^2-w_j,\ \forall j\neq i\right\},$$

which generalizes the spherical Voronoi construction recovered in the equal-weight case (Caplan et al., 11 Aug 2025). The unweighted cells are

$$\mathbb{V}_i=\left\{\vec{r}\in\mathbb{S}^2:\ \|\vec{p}_i-\vec{r}\|<\|\vec{p}_j-\vec{r}\|,\ \forall j\neq i\right\}.$$

This weighted formulation is significant because it permits direct control over cell measures, especially cell area, which is not generally available in ordinary Voronoi partitions. In the cited shallow-water discretization, this control is described as crucial for conservation, specifically for matching power-cell areas to prescribed target areas through weight adjustment (Caplan et al., 11 Aug 2025).

A second formal viewpoint is given through lifting. Power diagrams in $\mathbb{R}^3$ correspond to ordinary Voronoi diagrams of lifted sites in $\mathbb{R}^4$ under

$w_i$0

and the spherical power cell is then the projection of the lifted Voronoi cell back to the sphere (Caplan et al., 11 Aug 2025). This establishes a direct equivalence between weighted spherical partitioning and standard Voronoi geometry in a higher-dimensional embedding.

A related but less formal connection appears in the study of Gaussian spherical tessellations. There, spherical cells induced by random hyperplanes are described as intersections of the sphere with polyhedral cones and as spherical polytopes. The discussion states that such cells function much like spherical power cells when attention is restricted to the “relevant” faces that actually delimit a cell (Lybrand et al., 2021). This suggests a structural analogy rather than an explicit identification.

2. Construction, representation, and algorithmic properties

A direct construction of spherical power cells is given by halfspace intersection. Each cell is written as

$w_i$1

with

$w_i$2

where

$w_i$3

This representation leads to a halfspace clipping algorithm in which an initial large cell is iteratively clipped against bisector halfspaces associated with neighboring sites (Caplan et al., 11 Aug 2025).

The cited implementation emphasizes that each cell can be computed independently, so the algorithm is naturally parallelizable. To reduce the number of candidate neighbors, a spherical quadtree is built by recursively subdividing the faces of an octahedron circumscribing the sphere. A “Radius of Security Theorem” is used for early termination when more distant bisectors cannot contribute to the final cell (Caplan et al., 11 Aug 2025).

Empirical performance claims are specific. The reported scaling is linear with the number of sites, $w_i$4, and spherical Voronoi diagrams of 100 million sites can be computed in under 2 minutes on a single machine (Caplan et al., 11 Aug 2025). At simulation scales between 10,000 and 200,000 particles, power-diagram calculation is reported to comprise about 12% of runtime per timestep, while iterative linear solves account for 70% (Caplan et al., 11 Aug 2025). These figures pertain to the Voronoi/power-diagram computation framework used in the spherical shallow-water solver.

The broader tessellation literature provides complementary geometric insight. In Gaussian spherical tessellations in $w_i$5, a cell can be described by only $w_i$6 relevant faces even when $w_i$7 hyperplanes are present, and the resulting cell radius is bounded above, up to constants, by $w_i$8 with high probability (Lybrand et al., 2021). This does not define power cells directly, but it shows that spherical cells induced by many constraints often admit a compressed face description.

3. Area control, equal-area subdivision, and relation to other spherical grids

The distinctive feature of spherical power cells, relative to ordinary spherical Voronoi cells, is weighted control of geometry. In the numerical framework of spherical shallow water, weights are adjusted so that cell areas match prescribed targets exactly up to numerical tolerance (Caplan et al., 11 Aug 2025). This places spherical power cells in a wider family of spherical partitioning methods aimed at controlling area distribution.

A separate approach to equal-area partitioning is the isolatitudinal band construction that divides the sphere into latitudinal bands of near-constant span and then subdivides each band longitudinally into equal-area cells (Malkin, 2016). The area between latitudes is

$w_i$9

and the number of longitudinal cells in a band is chosen from the target cell area $\mathbb{S}^2$0 using formulas of the form

$\mathbb{S}^2$1

or

$\mathbb{S}^2$2

The method achieves equal-area cells with relative accuracy better than $\mathbb{S}^2$3 and maintains nearly uniform latitude steps, with residuals typically less than one degree for fine grids (Malkin, 2016).

That equal-area method is explicitly described as relevant to constructing spherical power cells (Malkin, 2016). The connection is not that the banded grid is itself a power diagram, but that it provides a practical, uniform basis for defining spherical power diagrams or cells. A plausible implication is that equal-area band constructions and weighted power diagrams address overlapping design requirements—uniform coverage, controllable cell area, and efficient point location—through different geometric mechanisms.

The contrast with other spherical discretizations is also explicit. HEALPix and Lambert-type isolatitudinal equal-area tessellations preserve equal area but exhibit non-uniform latitude steps, especially near the poles or equator, whereas the banded equal-area method prioritizes near-constant latitudinal span (Malkin, 2016). Spherical power cells, by contrast, derive their flexibility from weight assignment rather than from an a priori band structure (Caplan et al., 11 Aug 2025).

4. Role in Lagrangian discretization of the spherical shallow water equations

The most developed operational use of spherical power cells in the provided literature is a Lagrangian method for the spherical shallow water equations (Caplan et al., 11 Aug 2025). In this framework, the fluid is represented by particles, each associated with a spherical power cell carrying local values $\mathbb{S}^2$4 and $\mathbb{S}^2$5. The governing equations are given as

$\mathbb{S}^2$6

$\mathbb{S}^2$7

with the material derivative

$\mathbb{S}^2$8

Discrete operators are constructed from cell geometry. The cited first-order discrete gradient and divergence formulas use cell areas, edge lengths, centroids, and tangent-space projection: $\mathbb{S}^2$9

$\|\vec{p}_i-\vec{r}\|^2-w_i$0

where $\|\vec{p}_i-\vec{r}\|^2-w_i$1 projects to the tangent space at $\|\vec{p}_i-\vec{r}\|^2-w_i$2 (Caplan et al., 11 Aug 2025).

Time advancement is semi-implicit. After particle advection, heights are updated through a linear system

$\|\vec{p}_i-\vec{r}\|^2-w_i$3

followed by a velocity update using the new heights (Caplan et al., 11 Aug 2025). The paper notes that artificial viscosity is not needed to stabilize the simulation (Caplan et al., 11 Aug 2025).

Mass conservation is enforced by solving a semi-discrete optimal transport problem. If each particle mass is constant,

$\|\vec{p}_i-\vec{r}\|^2-w_i$4

then after updating $\|\vec{p}_i-\vec{r}\|^2-w_i$5, the target cell area becomes

$\|\vec{p}_i-\vec{r}\|^2-w_i$6

Weights are then adjusted by Newton’s method to match actual power-cell areas to these target areas via the energy

$\|\vec{p}_i-\vec{r}\|^2-w_i$7

with Newton step

$\|\vec{p}_i-\vec{r}\|^2-w_i$8

The partition remains exact up to numerical tolerance, and mass is therefore perfectly conserved modulo solver tolerance (Caplan et al., 11 Aug 2025).

The reported conservation properties are restricted but clear: momentum and energy conservation are described as comparable to the latest Lagrangian approach for simulating the spherical shallow water equations (Caplan et al., 11 Aug 2025). The main methodological significance is that spherical power cells give a moving-control-volume formulation: Lagrangian in particle motion, but geometrically analogous to finite-volume discretization.

5. “Spherical Power Cell” as an energy-device concept

A distinct usage appears in the “Crystal Ball” model of phase transitions on a classical spherical lattice (Bachmann et al., 2024). There, a Spherical Power Cell is conceptualized as a device in which a lattice of Coulomb-coupled ions on a sphere is triggered to undergo a phase transition, here by controlled removal of ions, releasing stored electrostatic potential energy as directed kinetic energy in some of the remaining ions (Bachmann et al., 2024).

The underlying model treats $\|\vec{p}_i-\vec{r}\|^2-w_i$9 point particles, typically Boron nuclei with charge $\{\vec{p}_i\}\subset \mathbb{S}^2$0 and mass $\{\vec{p}_i\}\subset \mathbb{S}^2$1, constrained to a sphere of radius $\{\vec{p}_i\}\subset \mathbb{S}^2$2. The Coulomb force on particle $\{\vec{p}_i\}\subset \mathbb{S}^2$3 is

$\{\vec{p}_i\}\subset \mathbb{S}^2$4

with drag

$\{\vec{p}_i\}\subset \mathbb{S}^2$5

and the total force is projected to the tangent plane before the particle positions are projected back to the sphere after each update (Bachmann et al., 2024).

The induced “phase transition” is the removal of $\{\vec{p}_i\}\subset \mathbb{S}^2$6 particles, creating a hole that forces re-equilibration. The central quantity is the average peak kinetic energy of a single remaining particle: $\{\vec{p}_i\}\subset \mathbb{S}^2$7 with fitted constant $\{\vec{p}_i\}\subset \mathbb{S}^2$8 from simulations, valid in the regime $\{\vec{p}_i\}\subset \mathbb{S}^2$9 (Bachmann et al., 2024). An alternative binary-interaction interpretation is

$w_i$0

with $w_i$1 expressed in terms of $w_i$2, $w_i$3, and mean lattice spacing $w_i$4 (Bachmann et al., 2024).

The reported numerical example is specific: for $w_i$5, increasing $w_i$6 from 1 to 6 can boost the maximum kinetic energy experienced by any particle by an order of magnitude, from roughly $w_i$7 to $w_i$8 (Bachmann et al., 2024). The energy is stated to be localized primarily in particles adjacent to the created hole. The paper further states that the scaling law provides a first-principles guide for predicting and optimizing energy output as a function of size, coverage, and trigger action (Bachmann et al., 2024).

This is a conceptual rather than experimentally established device notion. The same source explicitly notes that quantum mechanical extensions such as R-matrix or Density Functional Theory would be needed to capture effects relevant for fusion or lower-temperature regimes (Bachmann et al., 2024). Accordingly, the term “Spherical Power Cell” here denotes a proposed energy-harvesting architecture, not a demonstrated technology.

6. Adjacent physical and geometric contexts

The phrase “spherical cell” also occurs in other technical literatures that are not about spherical power cells in the weighted Voronoi sense, but which illuminate the broader semantics of the term.

In membrane and porous-media theory, a spherical cell may denote a concentric three-region model consisting of a solid core, porous layer, and liquid envelope, used for analytical study of micropolar flow through globular membranes (Khanukaeva, 2019). In that setting, the geometry is radial rather than tessellational, and the key quantity is hydrodynamic permeability $w_i$9, computed from matched solutions of micropolar and Brinkman-type equations. Increasing the micropolarity number $$\mathbb{P}_i=\left\{\vec{r}\in\mathbb{S}^2:\ \|\vec{p}_i-\vec{r}\|^2-w_i<\|\vec{p}_j-\vec{r}\|^2-w_j,\ \forall j\neq i\right\},$$0 and the scale factor $$\mathbb{P}_i=\left\{\vec{r}\in\mathbb{S}^2:\ \|\vec{p}_i-\vec{r}\|^2-w_i<\|\vec{p}_j-\vec{r}\|^2-w_j,\ \forall j\neq i\right\},$$1 decreases permeability, while increasing porous-layer porosity $$\mathbb{P}_i=\left\{\vec{r}\in\mathbb{S}^2:\ \|\vec{p}_i-\vec{r}\|^2-w_i<\|\vec{p}_j-\vec{r}\|^2-w_j,\ \forall j\neq i\right\},$$2 increases it (Khanukaeva, 2019). This usage is terminologically adjacent but conceptually distinct.

In biomechanics, spherical shell models describe pressurized biological cells probed by spherical indenters or compressed between plates (Couturier et al., 2021). The shell has radius $$\mathbb{P}_i=\left\{\vec{r}\in\mathbb{S}^2:\ \|\vec{p}_i-\vec{r}\|^2-w_i<\|\vec{p}_j-\vec{r}\|^2-w_j,\ \forall j\neq i\right\},$$3, thickness $$\mathbb{P}_i=\left\{\vec{r}\in\mathbb{S}^2:\ \|\vec{p}_i-\vec{r}\|^2-w_i<\|\vec{p}_j-\vec{r}\|^2-w_j,\ \forall j\neq i\right\},$$4, Young’s modulus $$\mathbb{P}_i=\left\{\vec{r}\in\mathbb{S}^2:\ \|\vec{p}_i-\vec{r}\|^2-w_i<\|\vec{p}_j-\vec{r}\|^2-w_j,\ \forall j\neq i\right\},$$5, Poisson’s ratio $$\mathbb{P}_i=\left\{\vec{r}\in\mathbb{S}^2:\ \|\vec{p}_i-\vec{r}\|^2-w_i<\|\vec{p}_j-\vec{r}\|^2-w_j,\ \forall j\neq i\right\},$$6, and internal pressure $$\mathbb{P}_i=\left\{\vec{r}\in\mathbb{S}^2:\ \|\vec{p}_i-\vec{r}\|^2-w_i<\|\vec{p}_j-\vec{r}\|^2-w_j,\ \forall j\neq i\right\},$$7, with bending stiffness

$$\mathbb{P}_i=\left\{\vec{r}\in\mathbb{S}^2:\ \|\vec{p}_i-\vec{r}\|^2-w_i<\|\vec{p}_j-\vec{r}\|^2-w_j,\ \forall j\neq i\right\},$$8

stretching stiffness $$\mathbb{P}_i=\left\{\vec{r}\in\mathbb{S}^2:\ \|\vec{p}_i-\vec{r}\|^2-w_i<\|\vec{p}_j-\vec{r}\|^2-w_j,\ \forall j\neq i\right\},$$9, thinness parameter

$$\mathbb{V}_i=\left\{\vec{r}\in\mathbb{S}^2:\ \|\vec{p}_i-\vec{r}\|<\|\vec{p}_j-\vec{r}\|,\ \forall j\neq i\right\}.$$0

and dimensionless pressure

$$\mathbb{V}_i=\left\{\vec{r}\in\mathbb{S}^2:\ \|\vec{p}_i-\vec{r}\|<\|\vec{p}_j-\vec{r}\|,\ \forall j\neq i\right\}.$$1

For small deformation, the leading-order force–displacement law is

$$\mathbb{V}_i=\left\{\vec{r}\in\mathbb{S}^2:\ \|\vec{p}_i-\vec{r}\|<\|\vec{p}_j-\vec{r}\|,\ \forall j\neq i\right\}.$$2

where $$\mathbb{V}_i=\left\{\vec{r}\in\mathbb{S}^2:\ \|\vec{p}_i-\vec{r}\|<\|\vec{p}_j-\vec{r}\|,\ \forall j\neq i\right\}.$$3 depends on pressure (Couturier et al., 2021). This work concerns pressurized spherical shells rather than spherical power diagrams or energy devices, but it reinforces that spherical cells in applied mechanics often denote bounded spherical objects whose material parameters are inferred from controlled loading.

A final neighboring usage is “power spectrum” in the spherical Fourier-Bessel basis (Wen et al., 2024). Despite the shared word “power,” this is unrelated to spherical power cells. The SFB basis preserves curved-sky geometry and fully accounts for wide-angle effects in three-dimensional galaxy clustering, with discrete radial eigenmodes and mappings to angular power spectra (Wen et al., 2024). The overlap is purely terminological.

7. Conceptual scope, misconceptions, and significance

The literature supports at least two non-equivalent meanings of “Spherical Power Cells.” The first is rigorous and geometric: weighted spherical cells in a power diagram, used for area-controlled partitioning and Lagrangian numerical discretization (Caplan et al., 11 Aug 2025). The second is speculative and device-oriented: a proposed energy-harvesting architecture based on defect-induced reconfiguration of a Coulomb lattice on a sphere (Bachmann et al., 2024).

A common misconception would be to treat these as the same concept. The data do not support that identification. In the numerical-analysis literature, “power” refers to power-diagram geometry, i.e., weighted distance and weighted Voronoi partitions (Caplan et al., 11 Aug 2025). In the lattice-dynamics literature, “power” refers to energy output or harvesting potential from phase transitions (Bachmann et al., 2024). The shared phrase therefore spans geometry and energetics but does not denote a unified standard term.

Another possible misconception is that all spherical tessellations are power diagrams. The available sources instead show a hierarchy: equal-area banded grids (Malkin, 2016), Gaussian spherical hyperplane tessellations (Lybrand et al., 2021), spherical Voronoi diagrams, and spherical power diagrams (Caplan et al., 11 Aug 2025) are related but distinct constructions. The random tessellation results show that cells can be small and describable by relatively few relevant faces (Lybrand et al., 2021); the equal-area band method shows that near-constant latitude span can coexist with equal area (Malkin, 2016); the power-diagram framework adds explicit weight control and direct enforcement of target areas (Caplan et al., 11 Aug 2025).

The strongest established significance of spherical power cells is therefore methodological. They provide exact geometric control over moving spherical control volumes, enable semi-discrete optimal transport for mass conservation, and support scalable computation at very large site counts (Caplan et al., 11 Aug 2025). The device interpretation remains a proposal derived from a classical $$\mathbb{V}_i=\left\{\vec{r}\in\mathbb{S}^2:\ \|\vec{p}_i-\vec{r}\|<\|\vec{p}_j-\vec{r}\|,\ \forall j\neq i\right\}.$$4-body model with explicit scaling laws for peak kinetic energy after particle removal (Bachmann et al., 2024). This suggests two research directions under the same label: one mature and computational, the other exploratory and physical.