## polymake: a Framework for Analyzing Convex Polytopes (1999)

Citations: | 94 - 15 self |

### BibTeX

@MISC{Gawrilow99polymake:a,

author = {Ewgenij Gawrilow and Michael Joswig},

title = {polymake: a Framework for Analyzing Convex Polytopes},

year = {1999}

}

### Years of Citing Articles

### OpenURL

### Abstract

polymake is a software tool designed for the algorithmic treatment of polytopes and polyhedra. We give an overview of the functionality as well as of the structure. This paper can be seen as a first approximation to a polymake handbook. The tutorial starts with the very basics and ends up with a few polymake applications to research problems. Then we present the main features of the system including the interfaces to other software products. polymake is free software; it is available on the Internet at http://www.math.tu-berlin.de/diskregeom/polymake/.

### Citations

8511 | Introduction to Algorithms - Cormen, Leiserson, et al. - 1990 |

1494 | The C++ Programming Language
- Stroustrup
- 1997
(Show Context)
Citation Context ...s highly customizable and extendible. However, it requires a bit of programming to extend the system. 28 7.1 The C/C++ Interface All polymake generators and clients are written in C++, cf. Stroustrup =-=[32]-=-. They use an interface to the polymake system which is implemented in a special class Poly derived from the usual iostream class. Since version 1.3 the code is designed for compilation with the GNU C... |

688 | Algorithms in Combinatorial Geometry - Edelsbrunner - 1987 |

457 |
Reflection groups and Coxeter groups
- Humphreys
- 1990
(Show Context)
Citation Context ...ly 3-dimensional regular polytopes, i.e. they have pairwise congruent facets as well as pairwise congruent vertexsgures. They are closely related tosnite re ection groups, cf. Coxeter [13], Humphreys =-=[21-=-]. There are three innite families of regular d-polytopes: the simplices, the cubes and the cross-polytopes. For each of these classes there is a corresponding polymake generator: simplex, cube, cross... |

357 |
Lectures on Polytopes
- Ziegler
- 2007
(Show Context)
Citation Context ...Polytopes The purpose of this section is to outline the mathematical foundations of polymake. We recall the denition and the basic properties of polytopes and polyhedra. For more details see Ziegler [=-=-=-34]. Take any set of points P R d . Let conv(P ) = ( X p2P c p p c p 0; X p2P c p = 1 ) and cone(P ) = ( X p2P c p p c p 0 ) denote the convex hull and the cone of P , respectively. The convex hull... |

223 |
Regular Polytopes
- Coxeter
- 1973
(Show Context)
Citation Context ... They are the only 3-dimensional regular polytopes, i.e. they have pairwise congruent facets as well as pairwise congruent vertexsgures. They are closely related tosnite re ection groups, cf. Coxeter =-=[13-=-], Humphreys [21]. There are three innite families of regular d-polytopes: the simplices, the cubes and the cross-polytopes. For each of these classes there is a corresponding polymake generator: simp... |

183 | A pivoting algorithm for convex hulls and vertex enumeration of arrangements and polyhedra
- Avis, Fukuda
- 1991
(Show Context)
Citation Context ... matters of arithmetic in polymake. For details see Fukuda's CDD homepage [17] as well as Fukuda and Prodon [18]. 4.1.2 LRS The algorithm behind LRS is the reverse search algorithm of Avis and Fukuda =-=[6]-=-, implemented by Avis [4, 3]. Like CDD it is also restricted to full-dimensional input. 4.1.3 PORTA This is Christof's and Loebel's implementation of the convex hull computation by FourierMotzkin elim... |

180 |
The Art of Computer Programming: Fundamental Algorithms, volume 1. Addison-Wesley, 3rd edition
- Knuth
- 1997
(Show Context)
Citation Context ...stency checking does not take place and there is no penalty with respect to eciency during run-time. Overhead from unnecessary copying of data is almost avoided by using reference counting, cf. Knuth =-=[26]-=-. There are template classes Vector and Matrix which allow standard Linear Algebra computations such as solving systems of linear equations, inverting matrices, computing scalar products, and so on. A... |

85 |
STL Tutorial and Reference Guide
- Musser, Derge, et al.
- 2001
(Show Context)
Citation Context ...ritten in C++ using the class library which is described below. 25 5.6 The polymake Template Library The most recent release of the ANSI C++ Standard includes the Standard Template Library (STL), cf. =-=[29]-=-. The polymake Template Library (PTL) is built on top of STL and designed in a similar spirit. It features a variety of C++ template classes which are used by the polymake clients. Most importantly, t... |

83 | How good are convex hull algorithms
- Avis, Bremner, et al.
- 1997
(Show Context)
Citation Context ...f polymake oers interfaces to the programs listed below. See Ziegler [34, Section 1.2] for an introduction to convex hull computation by FourierMotzkin elimination. See also Avis, Bremner and Seidel [=-=5]-=- as well as Bremner [10] for a thorough discussion of various convex hull computation methods and their complexity. 4.1.1 CDD This package has been developed by Fukuda. It implements the double descri... |

76 | Double description method revisited
- Fukuda, Prodon
- 1995
(Show Context)
Citation Context ...he GNU Multiprecision Library (GMP) via polymake's C++-wrappers. See also the Section 5.7 on matters of arithmetic in polymake. For details see Fukuda's CDD homepage [17] as well as Fukuda and Prodon =-=[18]-=-. 4.1.2 LRS The algorithm behind LRS is the reverse search algorithm of Avis and Fukuda [6], implemented by Avis [4, 3]. Like CDD it is also restricted to full-dimensional input. 4.1.3 PORTA This is C... |

57 | lrs: A Revised Implementation of the Reverse Search Vertex Enumeration Algorithm
- Avis
- 2000
(Show Context)
Citation Context ... polymake. For details see Fukuda's CDD homepage [17] as well as Fukuda and Prodon [18]. 4.1.2 LRS The algorithm behind LRS is the reverse search algorithm of Avis and Fukuda [6], implemented by Avis =-=[4, 3]-=-. Like CDD it is also restricted to full-dimensional input. 4.1.3 PORTA This is Christof's and Loebel's implementation of the convex hull computation by FourierMotzkin elimination. PORTA [11] does not... |

43 |
PORTA: POlyhedron Representation Transformation Algorithm
- Christof, Löbel
- 1997
(Show Context)
Citation Context ...oordinate. There are very many tools available which allow to deal with various aspects of polytope theory on a computer. Among these are Avis' LRS [4], Fukuda's CDD [17], Christof and Loebel's PORTA =-=[11]-=-, The Geometry Center's GEOMVIEW [28], and many more. The purpose of polymake is to combine the features of these (and other) programs and to go beyond. Many features of the polymake system are realiz... |

25 | Combinatorial aspects of convex polytopes - Bayer, Lee - 1993 |

23 |
The interactive geometry software Cinderella
- Richter-Gebert, Kortenkamp
- 1999
(Show Context)
Citation Context ... 6: A 2-polytope in the spherical model, that is, a hyperplane arrangement on the 2-sphere with one chamber (and its opposite) shaded. The far hyperplane is the boundary of the projection. Cinderella =-=[31]-=- postscript output. Suppose you are given the set of facets of a full-dimensional polyhedron together with the far hyperplane in the spherical model. Then the property of being pointed is equivalent t... |

22 | Linear programming, the simplex algorithm and simple polytopes
- Kalai
- 1997
(Show Context)
Citation Context ...ous pivot strategies. The algorithm indicated clearly does not allow for ecient linear programming but it has the advantage that it can also deal with abstract objective functions as studied by Kalai =-=[24]-=- and others. 4.4 Visualization Up to now polymake does not provide many tools for visualization on its own. Instead, once again, we usually rely on packages developed by other groups. 15 4.4.1 GEOMVIE... |

15 |
Neighborly cubical polytopes, Discrete Comput. Geometry 24 (2000), 325–344, The Branko Grünbaum birthday issue
- Joswig, Ziegler
(Show Context)
Citation Context ...PLICIAL(true) Recently, there seems to be a growing interest in cubical polytopes, i.e. polytopes whose proper faces are (combinatorial) cubes, cf. Blind and Blind [8] [9], see also the references in =-=[23]-=-. Given that the graph of a cubical polytope does not contain a triangle [22, Proposition 3.2], the following characterization follows from an easy induction on the dimension. Lemma 4.2 The d-polytope... |

10 |
Convex polytopes without triangular faces
- Blind, Blind
- 1990
(Show Context)
Citation Context ... polymake random.poly SIMPLICIAL SIMPLICIAL(true) Recently, there seems to be a growing interest in cubical polytopes, i.e. polytopes whose proper faces are (combinatorial) cubes, cf. Blind and Blind =-=[8]-=- [9], see also the references in [23]. Given that the graph of a cubical polytope does not contain a triangle [22, Proposition 3.2], the following characterization follows from an easy induction on th... |

9 |
Computational geometry software
- Amenta
- 1997
(Show Context)
Citation Context ...s and to go beyond. Many features of the polymake system are realized by interfacing to these programs. For an overview over existing software systems in the area of Computational Geometry see Amenta =-=[2]-=-. Here we describe polymake Version 1.3, released Feb 12, 1999. 1 2 Exploring polymake 2.1 First Steps Suppose you have asnite set of points in some vector space R d . Their convex hull is a polytope ... |

5 | Reconstructing a non–simple polytope from its graph - Joswig |

2 |
cddplus, Version 0.76a, http://www.ifor.math.ethz.ch/staff/fukuda/ cdd home/cdd.html
- Fukuda
- 1999
(Show Context)
Citation Context ...to the third, fourth, and eighth coordinate. There are very many tools available which allow to deal with various aspects of polytope theory on a computer. Among these are Avis' LRS [4], Fukuda's CDD =-=[17]-=-, Christof and Loebel's PORTA [11], The Geometry Center's GEOMVIEW [28], and many more. The purpose of polymake is to combine the features of these (and other) programs and to go beyond. Many features... |

1 |
Version 3.2, ftp://mutt.cs.mcgill.ca/pub/C/lrs.html
- lrs
- 1998
(Show Context)
Citation Context ...jection of P 00 onto the third, fourth, and eighth coordinate. There are very many tools available which allow to deal with various aspects of polytope theory on a computer. Among these are Avis' LRS =-=[4]-=-, Fukuda's CDD [17], Christof and Loebel's PORTA [11], The Geometry Center's GEOMVIEW [28], and many more. The purpose of polymake is to combine the features of these (and other) programs and to go be... |

1 |
4-polytopes with few vertices, Geom. Dedicata 66
- Cubical
- 1997
(Show Context)
Citation Context ...ymake random.poly SIMPLICIAL SIMPLICIAL(true) Recently, there seems to be a growing interest in cubical polytopes, i.e. polytopes whose proper faces are (combinatorial) cubes, cf. Blind and Blind [8] =-=[9]-=-, see also the references in [23]. Given that the graph of a cubical polytope does not contain a triangle [22, Proposition 3.2], the following characterization follows from an easy induction on the di... |

1 |
Polytope Base, Version 0.2, http://www.math.washington.edu/ ~bremner/PolytopeBase
- Bremner
- 1999
(Show Context)
Citation Context ...aces to the programs listed below. See Ziegler [34, Section 1.2] for an introduction to convex hull computation by FourierMotzkin elimination. See also Avis, Bremner and Seidel [5] as well as Bremner =-=[10]-=- for a thorough discussion of various convex hull computation methods and their complexity. 4.1.1 CDD This package has been developed by Fukuda. It implements the double description (or Fourier-Motzki... |

1 |
et al., POV-Ray: Persistence of Vision, Version 3.1
- Young
- 1999
(Show Context)
Citation Context ...,5 2 7 0,1 Figure 11: Ane Gale diagram of a 4-dimensional 0=1-polytope with 8 vertices. drawn between any two points of the same color. 4.4.4 POVRAY We also oer an interface to the ray-tracer POVRAY [=-=15-=-]. This allows the user to produce high-quality pictures of his/her favorite 3-polytopes. Schlegel diagrams can also be visualized in this way. This interface greatly diers from the interface to GEOMV... |

1 |
TOPCOM, Version 0.2.0
- Rambau
- 1999
(Show Context)
Citation Context ...the technical level. Let us sketch some of the things we have in mind. In the near future polymake will learn a little bit about oriented matroids and triangulations by interfacing to Rambau's TOPCOM =-=[30]-=-. This allows for exact volume computation and plenty more. One of the next versions of polymake will provide the user with a shell-type environment, similar to most Computer Algebra Systems. It will ... |