People have looked into circle packings with two kinds of circles. Of course they did. Are they fun to look at? Indeed they are:
https://en.wikipedia.org/wiki/File%3A2-d_dense_packing_r1.svg
Just scroll down to find some more! The article is a bit short, but here it is:
https://en.wikipedia.org/wiki/Circle_packing#Unequal_circles
Half a year ago, I filled in some sorry's for the massive project [1] to formalize the Fields-medal winning proof that sphere packing in dimension 8 is optimized by the E8-lattice. Last week it was announced that all remaining sorry's were filled by Gauss, an autoformalization agent. Gauss was able to build on the blueprint and other scaffolding built by the community. A few days later, Gauss also formalized the proof in dimension 24, this time working directly from the published paper, without mayor community input [3].
Since Lean verifies the generated proofs, hallucinations are not a problem.
The community now processes the generated proofs to make sure it satisfies the community standards and remains usable in the future [2].
[1] https://thefundamentaltheor3m.github.io/Sphere-Packing-Lean/
I derived Planck mass from sphere packing – 0.574 ppm, zero free param
https://zenodo.org/records/18089296
#HackerNews #PlanckMass #SpherePacking #Physics #Research #ScienceInnovation
We derive M_Pl/m_e = π⁴⁵ × (1 + 2α + α/13 − (8/9)α²) from pure geometry. ZERO free parameters. ALL coefficients from sphere packing:- 45 = (K₃² − K₃ − 2τD)/2 where K₃ = 12 (3D kissing number)- 13 = K₃ + 1 (same factor as Weinberg angle sin²θ_W = 3/13)- −8/9 = −(K₃−4)/(K₃−3) from tetrahedral cluster geometry- 2 = K₄/K₃ = 24/12 (Dirac g-factor) RESULT: 5.74×10⁻⁷ relative error (0.574 ppm) — 38× better than direct G measurements (22 ppm). CROSS-VALIDATION: sin²θ₁₃ = 1/45 (neutrino mixing angle) independently confirms the exponent, suggesting unified geometric origin for Planck scale and Standard Model. The formula emerges from the sedenionic dimensional cascade: physical reality projects from 16D sedenion algebra 𝕊 through octonions 𝕆 (8D) and quaternions ℍ (4D) to observable ℝ³. Kissing numbers K(16)=4320, K(8)=240, K(4)=24, K(3)=12 quantify information loss at each stage. Key identity: e^π − π = 19.999 ≈ 20 = K(8)/K(3) connects transcendental constants to lattice geometry (error 0.0045%). Package includes:- UCT_Planck_Mass_v5_1.pdf (docx) - Complete derivation (4 pages, 3 figures)- planck_mass_validation.py (ipynb) - Reproducible Python code (seed=42)- Data with results - Bootstrap (N=10,000) and Monte Carlo (N=100,000) resultshttps://colab.research.google.com/drive/1UtKmlkaYFHegx4S98Vv_B18FuylP3hlm?usp=sharing All numerical verification uses CODATA 2022 constants.
"In higher dimensions, mathematicians still don’t know the answer [to the sphere packing problem.] [...] Now, in a short manuscript posted online in April, the mathematician Boaz Klartag has bested these previous records by a significant margin. Some researchers even believe his result might be close to optimal."
https://www.quantamagazine.org/new-sphere-packing-record-stems-from-an-unexpected-source-20250707/
New Sphere-Packing Record Stems from an Unexpected Source
https://www.quantamagazine.org/new-sphere-packing-record-stems-from-an-unexpected-source-20250707/
#HackerNews #SpherePacking #SpherePackingRecord #Mathematics #UnexpectedSource #QuantumPhysics
Bo’az Klartag: Striking new Lower Bounds for Sphere Packing in High Dimensions
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