4-Dimensional Point Groups

This repository contains scripts written in Sage for the article Towards a Geometric Understanding of the 4-Dimensional Point Groups by Laith Rastanawi and Günter Rote.

Overview

• logs/: Contains:

  • Outputs of the scripts classify_AxA.sage and classify_DxD.sage: TxT.log, OxO.log, IxI.log and DxD_{orderbound}.log.
  • Cross-reference between CS names and our names for the toroidal groups: toroidal_chiral_cross_reference.log and toroidal_achiral_cross_reference.log.

• precomputed/: Contains precomputed data such as:

  • Groups catalogs.
  • Order catalogs.

• main_functions.sage: Contains definitions, classes and functions that are used in the other scripts. Most notable ones:

  • Q([x1, x2, x3, x4]): Represents the quaternion $x_1 + x_2 i + x_3 j + x_4 k$.

  • quat_2D(alpha, p): Represents the quaternion $e^{\alpha \pi} j^p$, an element of the (binary) dihedral group $2D_2n$.

  • FDR(l, r, star): Represents the 4-dimensional orthogonal transformation $[l, r]$ if star == False, and $\star[l,r]$ otherwise.

  • Q_group(gens): Represents the quaternion group generated by gens.

  • AxB_group(gens): Represents the 4-dimensional point group generated by gens.

  • classify_group(gr): Returns the fingerprint of the group gr.

  • Hurley_pattern(fingerprint): Converts the fingerprint fingerprint of a point group into Hurley pattern.

  • mk_group(gens): Returns the group generated by gens.

  • toroidal_group(*parameters): Returns the toroidal group defined by the parameters parameters.

  • Functions to generate catalogs of point groups:

    • generate_polyhedral_and_axial
    • generate_chiral_toroidal
    • generate_achiral_toroidal
    • generate_tubular

• generate_toroidal_CS.sage: Contains functions to generate catalogs for the toroidal groups using CS classification.

• generate_catalog.sage: Generates groups catalogs and stores them in precomputed/ as json files. The keys are fingerprints of the groups. The values are the names of the groups.

• counting_groups.sage: Generates order catalogs and stores them in precomputed/ as json files. The keys are orders of the groups. The values are the names of the groups.

• generate_oeis_sequences.sage: Generates the following two sequences to be used in OEIS:

  • number of 4-dimensional point groups of order n, and
  • number of 4-dimensional chiral point groups of order n.

  The precomputed catalogs are generated using the main function in counting_groups.sage with orderbound = 10000.

• classify_AxA.sage: Classifies all subgroups of $\pm[A\times A]$, and their achiral extensions. Outputs the results in logs/.

• classify_DxD.sage: Classifies all chiral toroidal groups up to order orderbound//2 and the achiral toroidal groups up to order orderbound. Outputs the results in logs/.

• conjugacy_Dn.sage: Generates dictionaries transform_D4 and transform_D4xD4 to compute conjugations for elements of $2D_{2n}$ by non-elements of $2D_{2n}$.

• table_toroidal_duplications.sage: Checks if we have the correct conjugations for toroidal groups and generates a LaTeX table tables/toroidal_duplications.tex of them.

• table_polyhedral_groups.sage: This script has two parts:

  • Checks that our generators and CS generators for the polyhedral and the axial groups generate geometrically equivalent (but not necessarily equal) groups. The list of generators is given in polyhedral_and_axial_lists.sage.
  • Generates 3 LaTeX tables:
    • tab_polyhedral_and_axial_check.tex: For internal use, with the full information.
    • tab_polyhedral_and_axial_appendix.tex: All polyhedral and axial groups, with generators.
    • tab_polyhedral.tex: An overview of the polyhedral groups.

• table_axial_groups.sage: This script generates 4 LaTeX tables:

  • tab_axial_combined.tex: For internal use, with the full information.
  • tab_axial_full.tex: All axial groups with cross-references.
  • tab_axial_pap.tex: Pyramidal and prismatic.
  • tab_axial_hybrid.tex: Hybrid groups with methods.

  This files loads table_polyhedral_groups.sage to generate a catalog.

• polyhedral_and_axial_lists.sage: Contains two lists of polyhedral and axial groups:

  • polyhedral_list_Cox whose entries consist of [DuVal number, DuVal name, CS generators, our genreators, Coxeter name, method]
  • axial_list whose entries consist of [DuVal number, DuVal name, CS generators, our genreators]

• axial_data.sage: Contains data used in table_axial_groups.sage. The Coxeter names in this files are also used in table_polyhedral_groups.sage.

• BBNWZ_cat.sage: Contains the dictionary BBNWZ_cat which is used to convert to numbering of crystallographic point groups.

• generate_tubes.sage: Generates decompositions of polar orbit polytopes of cyclic-type tubical groups $G$ when the image of the starting point is a rotation center of $G^h$. Outputs are json files which are stored in data/. These files are used in the online gallery.