Appendix E: Thesis Research Reproduction & Setup

Each script uses the PyEQSP Python library (this repository) to reproduce results originally computed in MATLAB using the EQ Sphere Partitioning Toolbox.

Paul Leopardi, “Distributing points on the sphere: Partitions, separation, quadrature and energy”, UNSW, 2007.

Note

Beta Feedback Wanted If you are running these thesis reproduction scripts on your local system, we would love to know if the results are consistent with your hardware. Please share a screenshot or report any numerical variances in our Feedback Hub.

Script Inventory

The following table maps the scripts in examples/phd-thesis/src/ to the original thesis figures.

Script	Thesis Figure	Backend	Description
fig_3_1_partition_s2_33.py	Fig 3.1	Mayavi	3D partition EQ(2, 33) on S²
fig_3_3_algorithm_s3_99.py	Fig 3.3	Matplotlib (Agg)	Algorithm illustration for EQ(3, 99)
fig_3_4_max_diam_s2.py	Fig 3.4	Matplotlib (Agg)	Max diameter coefficients of EQ(2, N), N=1..100,000
fig_3_5_max_diam_s3.py	Fig 3.5	Matplotlib (Agg)	Max diameter coefficients of EQ(3, N), N=1..100,000
fig_3_6_max_diam_s4.py	Fig 3.6	Matplotlib (Agg)	Max diameter coefficients of EQ(4, N), N=1..100,000
fig_3_7_max_diam_multi_dim.py	Fig 3.7	Matplotlib (Agg)	Max diameters of EQ(d, 2^k), d=2..8, k=1..20 (parallelized)
fig_4_1_eqp_s2_33.py	Fig 4.1	Mayavi	3D EQ code EQP(2, 33) with partition EQ(2, 33)
fig_4_2_min_dist_s2.py	Fig 4.2	Matplotlib (Agg)	Min distance coefficient of EQP(2, N), N=1..20,000
fig_4_3_min_dist_s3.py	Fig 4.3	Matplotlib (Agg)	Min distance coefficient of EQP(3, N), N=1..20,000
fig_4_4_min_dist_s4.py	Fig 4.4	Matplotlib (Agg)	Min distance coefficient of EQP(4, N), N=1..20,000
fig_4_5_packing_s2.py	Fig 4.5	Matplotlib (Agg)	Packing density of EQP(2, N), N=1..20,000
fig_4_6_packing_s3.py	Fig 4.6	Matplotlib (Agg)	Packing density of EQP(3, N), N=1..20,000
fig_4_7_packing_s4.py	Fig 4.7	Matplotlib (Agg)	Packing density of EQP(4, N), N=1..20,000
fig_4_8_wyner_s2_s3_s4.py	Fig 4.8	Matplotlib (Agg)	Wyner ratios for EQP(2), EQP(3), EQP(4), N=2..20,000
fig_4_9_wyner_s5_s6_s7.py	Fig 4.9	Matplotlib (Agg)	Wyner ratios for EQP(5), EQP(6), EQP(7), N=2..20,000
fig_4_10_eqp_voronoi_s2_33.py	Fig 4.10	Mayavi	EQP(2, 33) Voronoi edges on S² in 3D
fig_5_1_normalized_energy.py	Fig 5.1	Matplotlib (Agg)	Normalized energy of EQP(d, N), d=2,3,4
fig_5_2_diff_normalized_energy.py	Fig 5.2	Matplotlib (Agg)	Convergence of normalized energy
fig_5_3_energy_coeff_s2.py	Fig 5.3	Matplotlib (Agg)	Energy coefficient of EQP(2, N), N=2..20,000
fig_5_4_energy_coeff_s3.py	Fig 5.4	Matplotlib (Agg)	Energy coefficient of EQP(3, N), N=2..20,000
fig_5_5_energy_coeff_s4.py	Fig 5.5	Matplotlib (Agg)	Energy coefficient of EQP(4, N), N=2..20,000

All scripts save a PNG output file to the current directory and print a confirmation message on completion.

Reproduction Notes

• 3D Figures: Visualizations using Mayavi are not always bitwise identical across hardware due to non-deterministic GPU rasterization, even when the underlying mesh is identical.

• Min-distance: Unified search using KDTrees ensures \(O(N \log N)\) scaling, allowing \(N=20,000\) to complete in seconds.

• Riesz Energy: Exact summations for \(s > 0\) use memory-efficient block-tiling to ensure \(O(N)\) peak RAM.

Running Time Benchmark (Section 3.10.2)

Beyond figure reproduction, the project includes a formal replication of the performance benchmark described in Section 3.10.2: Running time. It verifies that the recursive zonal partitioning algorithm scales as \(O(N^{0.6})\).

To execute:

python3 benchmarks/src/benchmark_eq_regions.py --max-d 11 --max-k 22

Technical Setup & Troubleshooting

Reproduction scripts for 3D figures require the Mayavi engine. We strongly recommend using the venv_sys environment for this work. See the 3D Plotting & System-Integrated Setup (venv_sys) section in Appendix B for more.

Backend Configuration

Rendering 3D Great Circle arcs and Voronoi cells on the sphere requires specific VTK/Mayavi backends. On Linux systems, calibrate your environment as follows:

export QT_API="pyqt5"
export QT_QPA_PLATFORM="xcb"

Warning

Without these exports, Mayavi may fail to initialize a window or crash with a Segmentation Fault when attempting to rasterize GREAT CIRCLE edges.

Bitwise Reproducibility

PyEQSP aims for identical numerical results to the original thesis. But users should be aware of two potential sources of variance:

1. Hardware Rasterization: 3D plots generated via Mayavi/VTK use GPU hardware. Minor variances (e.g., ~2 pixel differences) can occur between different GPUs or drivers due to non-deterministic anti-aliasing.

2. Floating-Point Drift: While we use numpy.longdouble for critical recursions, extreme depths in high dimensions (\(d > 8\)) may show sub-microscopic differences across different CPU architectures.

Headless Execution

To run reproduction scripts on a server or in CI without a display:

export HEADLESS=1
python3 examples/phd-thesis/src/fig_3_4_max_diam_s2.py

Numerical scripts (Matplotlib Agg backend) will save PNGs directly. 3D scripts (Mayavi) will attempt to use an offscreen buffer if xvfb is available.