kedlaya / tetrahedra Goto Github PK

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).