The Giza Pyramids: Geometry and Physical Behavior

Abstract

This study tested whether an idealized Great Pyramid reference geometry produced steady-state thermal behavior that differed systematically from controlled alternatives. Twenty-one limestone geometries were evaluated in two angle-controlled families, five cases were repeated in granite, and fifteen additional grid-refinement runs were completed. All 41 solver executions met the iterative stopping tolerance. Thermal response changed monotonically with face angle in both families, while geometry-normalized measures nearly collapsed at equal angles. The reference geometry fell between its immediate neighbors and ranked eleventh of twenty-one for the principal thermal metrics. Its preregistered classification was therefore smooth trend, not local extreme. Spatial convergence was not established, so the findings support controlled comparison rather than high-fidelity prediction of the real pyramid.

Keywords

Great Pyramid; parametric geometry; steady-state conduction; finite difference; thermal response; numerical refinement; ASRE-Lab

1. Introduction

The external geometry of the Great Pyramid can be studied as a physical form without assuming a historical function. This work isolates geometry through controlled computational comparison. It does not determine builder intent and does not establish that the pyramid served a thermal or energy-related purpose.

1.1 Research question

Does the geometry of the Great Pyramid produce measurable steady-state thermal behavior that differs systematically from controlled alternative geometries under identical physical conditions?

2. Methods

The study was conducted in July 2026 using a frozen comparison protocol and ASRE-Lab’s declared solver workflow.

2.1 Study design and protocol freeze

The protocol defined 21 primary geometries: R0, ten Family A fixed-base variants, and ten Family B fixed-volume variants. The angle offsets were −10°, −8°, −6°, −4°, −2°, +2°, +4°, +6°, +8°, and +10°. Twenty-one limestone runs, five granite sensitivity runs, and fifteen additional refinement runs produced 41 solver executions.

2.2 Reference geometry

R0 used a square base of 230.33 m, a face angle of 51.846°, a derived height of 146.590628 m, and an analytical volume of approximately 2.5923×10⁶ m³.

2.3 Parametric geometry families

Family A held the 230.33 m base constant. Family B recalculated base and height to preserve the reference volume. For base length B, height H, and face angle θ:

2.4 Materials

The primary material was homogeneous limestone with k = 1.3 W/(m·K). Granite sensitivity cases used k = 2.5 W/(m·K).

2.5 Governing equation and physical assumptions

The solver represented a solid square pyramid with constant isotropic conductivity, a uniform volumetric heat source, and steady-state conduction:

The internal chambers and real heterogeneous material distribution were outside this first controlled comparison.

2.6 Boundary conditions and numerical settings

2.7 Numerical implementation

pyramid_thermal_conduction_v1 used a masked Cartesian finite-difference domain. The displayed CAD geometry was not used as a finite-element mesh. Production execution used backend revision 9f3c240893738b125b7dd6997a2068228b8385b9 and solver version 1.0.0.

2.8 Solver benchmark and evidence

ASRE-Lab retained run status, inputs, outputs, convergence evidence, and solver identity within the study workflow. Across the completed study, iteration counts ranged from 132 to 790 and final residuals ranged from 9.063×10⁻⁷ to 9.984×10⁻⁷.

2.9 Spatial-refinement study

R0, A−10, A+10, B−10, and B+10 were rerun at n = 17, 25, and 33, in addition to their n = 41 primary cases. These fifteen runs tested sensitivity to grid spacing; they did not establish spatial convergence.

2.10 Outcomes

Additional outcomes included maximum temperature gradient, analytical and masked numerical volume, active-cell count, iteration count, final residual, and convergence status.

2.11 Geometry-normalized response measures

2.12 Pre-specified analysis plan

The complete dataset was analyzed across angle, height, base size, volume, material, and resolution. Individual interesting cases were not selected in place of the full controlled series.

2.13 Pre-specified interpretation of the reference geometry

R0 could be classified as a local extreme, smooth trend, transition, or ordinary interior point. A local extreme required R0 to lie above or below both immediate neighbors in a controlled family.

2.14 Material-sensitivity analysis

R0, A−10, A+10, B−10, and B+10 were repeated in granite. The test separated conductivity effects from geometry-normalized response.

3. Results

The geometry mattered. The Great Pyramid's reference geometry was not anomalous.

All 41 runs completed; no failed, queued, running, or partial runs remained. All runs satisfied the iterative stopping tolerance.

3.1 Primary limestone response across the 21 geometries

Family A ΔTmax increased monotonically from 4.111895 °C at 41.846° to 8.840955 °C at 61.846°. Family B increased monotonically from 5.198058 °C to 6.844561 °C across the same angles.

3.2 Geometry-normalized response

At R0, Gmax was 0.042392 and Gmean was 0.011999. At equal angles, normalized G metrics nearly collapsed across the two families, indicating that much of the absolute difference was related to scale or volume while normalized response tracked geometry.

3.3 Immediate-neighbor comparison and R0 classification

R0 lay between both immediate neighbors in both families and therefore did not meet the preregistered local-extreme criterion. Its final classification was Smooth trend. R0 ranked 11th of 21 limestone geometries for each principal thermal metric.

3.4 Material sensitivity

R0 limestone ΔTmax was 6.153686 °C and granite ΔTmax was 3.199886 °C, giving a granite/limestone ratio of approximately 0.519995. Across the five repeated geometries, granite produced about 0.52 times the limestone temperature rise, consistent with the conductivity ratio. G values remained nearly unchanged.

3.5 Spatial refinement

R0 ΔTmax increased from 5.771502 °C at n = 17 to 5.983398 °C at n = 25, 6.083625 °C at n = 33, and 6.153686 °C at n = 41.

At R0, the n = 33→41 changes were +1.151626% for ΔTmax, +4.760920% for ΔTmean, +1.854208% for maximum gradient, and −1.870577% for numerical volume.

3.6 Benchmark, convergence, evidence, and Scientific Trust

All runs met iterative tolerance, but iterative convergence is not spatial convergence. The R0 masked numerical volume at n = 41 was 7.817187% above analytical volume. The results are therefore strongest as controlled comparisons within one declared numerical method, not as high-fidelity predictions of the real pyramid.

4. Discussion

Angle produced systematic changes under the controlled thermal model. The two geometry families differed in absolute response, while normalization substantially reduced their separation.

4.1 Main geometric behavior

Both limestone families showed monotonic ΔTmax increases as face angle increased. Family A changed more strongly because its fixed base allowed height and volume to change together.

4.2 Position of the Great Pyramid reference geometry

R0 was a smooth interior point in both controlled angle series, not a local thermal extreme. This result does not support an anomalous reference geometry under the tested model.

4.3 Material sensitivity

The approximately 0.52 granite-to-limestone temperature-rise ratio followed the conductivity change, while normalized G measures remained nearly unchanged.

4.4 Numerical evidence and interpretation limits

Iterative stopping was achieved for every run, but the refinement changes and masked-volume discrepancy prevent a claim of spatial convergence or real-pyramid predictive fidelity.

5. Limitations

The model used homogeneous solids, omitted chambers and heterogeneous construction, imposed idealized boundaries and a uniform numerical probe, and used a masked Cartesian grid with measurable volume error. It does not reproduce the complete Great Pyramid or its environment.

6. Conclusion

Under this idealized steady-state model, geometry produced systematic thermal differences. R0 was a smooth interior point in both controlled angle series, not a local thermal extreme. The study makes no claim about historical purpose or builder intent.

7. Data and Reproducibility Statement

The study definition, run identities, solver settings, evidence status, and reported outputs were retained through ASRE-Lab. The complete geometry, primary-result, and spatial-refinement records are reported in the appendices below.

8. Software Availability

ASRE-Lab is available at github.com/eslammohamed2009b-a11y/ASRE-LAB. The production execution backend revision was 9f3c240893738b125b7dd6997a2068228b8385b9; solver version 1.0.0.

9. Author Contributions

Eslam M. Badawi conceived the study, designed the protocol, developed the software workflow, executed the study, analyzed the evidence, and wrote the manuscript.

10. Funding

This independent research received no declared external funding.

11. Competing Interests

The author declares no competing interests.

12. Ethics Statement

This computational study involved no human participants, animals, or personal data.

13. Acknowledgments and AI-Assistance Disclosure

Generative AI tools assisted software development, drafting, language editing, and workflow organization. Numerical evidence came from the declared solver workflow; final interpretation remains the author's responsibility.

References

1. W. M. F. Petrie, The Pyramids and Temples of Gizeh. London: Field & Tuer, 1883.
2. M. Balezin et al., “Electromagnetic properties of the Great Pyramid: First multipole resonances and energy concentration,” Journal of Applied Physics 124, 034903 (2018), doi:10.1063/1.5026556.
3. E. C. Robertson, Thermal Properties of Rocks, USGS Open-File Report 88-441, 1988, doi:10.3133/ofr88441.
4. S. V. Patankar, Numerical Heat Transfer and Fluid Flow. Hemisphere, 1980.
5. P. J. Roache, Verification and Validation in Computational Science and Engineering. Hermosa, 1998.
6. E. M. Badawi, ASRE-Lab, GitHub repository, 2026.

Appendix A. Pre-registered geometry table

Appendix B. Primary results

Appendix C. Spatial-refinement results

Appendix D. Results-handoff checklist

• 41 of 41 solver executions completed.
• No failed, queued, running, or partial runs remained.
• All runs met the iterative stopping tolerance.
• Spatial convergence was not established.
• R0 classification recorded as Smooth trend.
• No historical-purpose or builder-intent claim was made.