kedlaya / Tetrahedra

This repository consists of code and data associated to the paper "Space vectors forming rational angles" by Kiran S. Kedlaya, Alexander Kolpakov, Bjorn Poonen, and Michael Rubinstein (arXiv:2011.14232).

The code uses a combination of platforms: C++, SageMath version 9.2, and Magma version 2.25-6. (SageMath also uses Singular as an embedded component.) The C++ code depends on Bailey's quad double library QD version 2.3.22 for floating-point computations in quad-double precision. Some of the SageMath code is embedded in Jupyter notebooks; these were originally run on CoCalc.

• Numerical computations/:

  • README.md: C++ compilation notes.
  • low_order_solutions.cc: C++ code to enumerate configurations of 4 lines whose angles are multiples of $\pi/N$ for a fixed value of $N$ (Section 8, Proposition 8.1). The rigorous analysis of the paper shows that this code produces no false negatives.
  • DATA/: output of the C++ code.
  • algebraic_verification.ipynb: Jupyter/SageMath notebook to rule out false positives in the C++ output using algebraic computations (Section 8, Proposition 8.1).
  • group-sporadics.ipynb: Jupyter/SageMath notebook to generate the table of sporadic tetrahedra from the output of the C++ code (Section 11, Table 3).

• Algebraic computations/:

  • torsion_closure.sage: SageMath code to compute torsion closures of general ideals in Laurent polynomial rings (Section 7; see also below).
  • tetrahedra_degenerate.ipynb: Jupyter/SageMath notebook to compute solutions of the Gram determinant equation corresponding to degenerate tetrahedra (Section 9, Lemma 9.3).
  • tetrahedra.ipynb: Jupyter/SageMath notebook to compute solutions of the Gram determinant equation corresponding to nondegenerate tetrahedra (Section 9, Lemma 9.5). This depends on torsion_closure.sage.

• Conversion to angles/:

  • 4-vector-configurations-1-parameter-families.ipynb, 4-vector-configurations-2-parameter-families.ipynb: Jupyter/SageMath notebooks to convert parametric solutions of the Gram determinant equation into families of rational-angle line configurations (Section 1, Theorem 1.2, as well as Section 9, Lemma 9.3 and Lemma 9.5).
  • 4-vector-configurations-tetrahedra.ipynb: Jupyter/SageMath notebook to run consistency checks on the output stored in the DATA/ folder and to interpret it in human-readable format (Section 1, Theorem 1.8, as well as Section 9, Lemma 9.6).

• Larger configurations/:

  • tetrahedra.m, tetrahedra-compute.m: Magma code to assemble rational-angle line configurations with more than 4 lines (Section 10), and to generate the tables of these (Section 11).
  • Maximal_Configurations.txt: output of the previous code.
  • all-2.24-5.save, all-2.25-6: Magma saved workspaces with the tables generated (for the indicated versions of Magma).