polymake exhibits a few particularities, making it special to work with.
Firstly, polymake can be used within a Perl script. Moreover, users can extend polymake and define new objects, properties, rules for computing properties, and algorithms.[5]
Secondly, it exhibits an internal client-server scheme to accommodate the usage of Perl for object management and interfaces as well as C++ for mathematical algorithms.[6] The server holds information about each object (e.g., a polytope), and the client sends requests to compute properties. The server has the job of determining how to complete each request from information already known about each object using a rule-based system. For example, there are many rules on how to compute the facets of a polytope. Facets can be computed from a vertex description of the polytope, and from a (possibly redundant) inequality description. polymake builds a dependency graph outlining the steps to process each request and selects the best path via a Dijkstra-type algorithm.[6]
polymake divides its collection of functions and objects into 10 different groups called applications. They behave like C++ namespaces. The polytope application was the first one developed and it is the largest.[7]
Common: "helper" functions used in other applications.[8]
Fan: functions for polyhedral complexes (which generalize simplicial complexes), planar drawings of 3-polytopes, polyhedral fans, and subdivisions of points or vectors.[9]
Matroid: computation of standard properties of a matroid, like bases and circuits. This application can also compute more advanced properties like the Tutte polynomial of a matroid and realizing the matroid with a polytope.[14]
Polytope: over 230 functions or calculations that can be done with a polytope. These functions range in complexity from simply calculating basic information about a polytope (e.g., number of vertices, number of facets, tests for simplicial polytopes, and converting a vertex description to an inequality description) to combinatorial or algebraic properties (e.g., H-vector, Ehrhart polynomial, Hilbert basis, and Schlegel diagrams).[7] There are also many visualization options.
polymake version 1.0 first appeared in the proceedings of DMV-Seminar "Polytopes and Optimization" held in Oberwolfach, November 1997.[2] Version 1.0 only contained the polytope application, but the system of "applications" was not yet developed. Version 2.0 was in July 2003, [17] and version 3.0 was released in 2016.[18] The last big revision, version 4.0, was released in January 2020.[19]
polymake is highly modularly built and, therefore, displays great interaction with third party software packages for specialized computations, thereby providing a common interface and bridge between different tools. A user can easily (and unknowingly) switch between using different software packages in the process of computing properties of a polytope.[20]
Below is a list of third-party software packages that polymake can interface with as of version 4.0. Users are also able to write new rule files for interfacing with any software package. Note that there is some redundancy in this list (e.g., a few different packages can be used for finding the convex hull of a polytope). Because polymake uses rule files and a dependency graph for computing properties,[5] most of these software packages are optional. However, some become necessary for specialized computations.
4ti2: software package for algebraic, geometric and combinatorial problems on linear spaces
singular: computer algebra system for polynomial computations, with special emphasis on commutative and non-commutative algebra, algebraic geometry, and singularity theory
sketch: for making line drawings of two- or three-dimensional solid objects
^ abGawrilow, Ewgenij; Joswig, Michael (2000-01-01). Kalai, Gil; Ziegler, Günter M. (eds.). polymake: a Framework for Analyzing Convex Polytopes. Polytopes—combinatorics and computation, DMV Seminar. Birkhäuser Basel. pp. 43–73. doi:10.1007/978-3-0348-8438-9_2. ISBN9783764363512.