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Connectivity in the regular polytope representation

Rodney James Thompson, Peter van Oosterom
2009 Geoinformatica  
This paper describes a basis for the robust geometrical construction of spatial objects in computer applications using a complex called the "Regular Polytope".  ...  In order to be able to draw inferences about real world phenomena from a representation expressed in a digital computer, it is essential that the representation should have a rigorously correct algebraic  ...  Open Access This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided  ... 
doi:10.1007/s10707-009-0094-3 fatcat:m5rdogvhurb33j6dw64vgllgtq

Mathematically provable correct implementation of integrated 2D and 3D representations [chapter]

Rodney Thompson, Peter van Oosterom
2008 Lecture Notes in Geoinformation and Cartography  
The concept of the 'Regular Polytope' has been designed to facilitate the search for a rigorous closed algebra for the query and manipulation of the representations of spatial objects within the finite  ...  This paper explores the practicalities of implementing this approach both in terms of the database schema and in terms of the algorithmic implementation of the connectivity and topological predicates and  ...  Fig. 15. 3 3 Regular polytope O defined as the union of convex polytopes C 1 and C 2 Fig. 15. 4 4 Modes of Connectivity in 3D Fig. 15. 5 5 The Regular Polytope model.  ... 
doi:10.1007/978-3-540-72135-2_15 fatcat:q5uidmznq5aole3lksocy37et4

Integrated Representation of (Potentially Unbounded) 2D and 3D Spatial Objects for Rigorously Correct Query and Manipulation [chapter]

Rodney James Thompson, Peter van Oosterom
2010 Lecture Notes in Geoinformation and Cartography  
This paper explores a restricted, but useful representation, which supports a rigorous unsorted logic within the finite precision arithmetic of computer hardware: the regular polytope.  ...  mathematical model, whereby the representation is assumed to be defined by real-numbered coordinates in 2D or 3D space.  ...  Section 3 provides definitions of the basic primitives of our approach, resulting in the Regular Polytope. The practical question of connectivity and its definition is addressed in Sect. 4.  ... 
doi:10.1007/978-3-642-12670-3_11 fatcat:fxnni7fq3reqpmdnc6x56xe7ia

Restrictedly-Regular Map Representation of n-Dimensional Abstract Polytopes

Antonio Breda D'Azevedo
2018 Zenodo  
Regularity has often been present in the form of regular polyhedra or tessellations; classical examples are the nine regular polyhedra consisting of the five Platonic solids (regular convex polyhedra)  ...  However we show that it is possible to make a "surfacification" of the n-dimensional polytope, that is, it is possible to construct a restrictedly-marked map representation of the abstract polytope on  ...  The rooted Γ n -slice (Fig. 1a) for Q is the above n-polygon with a distinguished flag (in black): The monodromy group of the (n − 1)-hypermap (which corresponds to the connection group of the n-polytope  ... 
doi:10.5281/zenodo.2021629 fatcat:6tlkeluk4zazbpiv7tslhhqdgm

Polytopes of high rank for the alternating groups

Maria Elisa Fernandes, Dimitri Leemans, Mark Mixer
2012 Journal of combinatorial theory. Series A  
In this paper, for each d 3, a string C-group with d generators, isomorphic to an alternating group of degree n is constructed (for some n 9), or equivalently an abstract regular d-polytope, is produced  ...  There is a well-known correspondence between abstract regular polytopes and string C-groups.  ...  for Research and Development in Mathematics and Applications (University of Aveiro) and the Portuguese Foundation for Science and Technology ("FCT-Fundação para a Ciência e a Tecnologia"), within project  ... 
doi:10.1016/j.jcta.2011.07.006 fatcat:pacvrnvcmrddtk3b5ba2fsqreu

The Classification of Regular Solids

T. M. Madden, S. A. Robertson
1995 Bulletin of the London Mathematical Society  
In the present terminology, this associates either one or two regular solids (that are not polytopes) to some (but not all) regular polytopes.  ...  A convex body that is regular in this new sense is called a regular solid. Thus the set ^ of all regular polytopes is a subset of the set 5^ of all regular solids.  ...  Then there exists a connected compact Lie group G with symmetric space representation ft: G -> SO(K) such that G-and G-orbits in Vcoincide.  ... 
doi:10.1112/blms/27.4.363 fatcat:zwlqtwtrznbhto2junyos23f3q

Mapification of n-dimensional abstract polytopes and hypertopes

Antonio Breda d'Azevedo
2020 Ars Mathematica Contemporanea  
The restrictedly-marked map representation of an abstract polytope is a cellular embedding of the flag graph of a polytope.  ...  We illustrate this construction with the 4-cube, a regular 4-polytope with automorphism group of size 384. This paper pays a tribute to Lynne James' last work on map representations.  ...  For a further reading on polytopes we address the reader to the classical book by McMullen and Schulte [13] . 2 Restrictedly-marked maps Lynne James in [9] introduced maps representations and associate  ... 
doi:10.26493/1855-3974.1869.9f6 fatcat:bty6lxff7bh7jjxgtq5lihweyi

CPR graphs and regular polytopes

Daniel Pellicer
2008 European journal of combinatorics (Print)  
This paper studies C-group permutation representation graphs, or for short, CPR graphs. C-groups are the groups of abstract regular polytopes.  ...  Moreover, we illustrate their use by constructing regular polyhedra with alternating groups A n as the automorphism groups.  ...  The cycle representation of φ consists only of cycles of length l in each connected component C of G 0,1 , and no vertex fixed under φ.  ... 
doi:10.1016/j.ejc.2007.01.001 fatcat:atffciul3nbvneuj2h37fwngby

Representing the Sporadic Archimedean Polyhedra as Abstract Polytopes [article]

Michael Hartley, Gordon Williams
2009 arXiv   pre-print
We present the results of an investigation into the representations of Archimedean polyhedra (those polyhedra containing only one type of vertex figure) as quotients of regular abstract polytopes.  ...  Representations of the 14 sporadic Archimedean polyhedra (including the pseudorhombicuboctahedron) as quotients of regular abstract polyhedra are obtained, and summarised in a table.  ...  Also, in the examples studied to date, nite polytopes have all yielded representations as the quotients of nite regular polytopes.  ... 
arXiv:0910.2445v1 fatcat:jw64xwd5bnccjln57r2kogwfce

Representing the sporadic Archimedean polyhedra as abstract polytopes

Michael I. Hartley, Gordon I. Williams
2010 Discrete Mathematics  
Also, in the examples studied to date, finite polytopes have all yielded representations as the quotients of finite regular polytopes.  ...  Outline of topics Section 2 reviews the structure of abstract polytopes and their representation as quotients of regular polytopes and discusses two new results on the structure of the quotient representations  ... 
doi:10.1016/j.disc.2010.01.012 fatcat:f7dn25oejbbmvpz5dxgmxif6fm

Fix Your Features: Stationary and Maximally Discriminative Embeddings using Regular Polytope (Fixed Classifier) Networks [article]

Federico Pernici, Matteo Bruni, Claudio Baecchi, Alberto Del Bimbo
2019 arXiv   pre-print
vertices of three regular polytopes available in R^d, namely: the d-Simplex, the d-Cube and the d-Orthoplex.  ...  These regular polytopes have the maximal amount of symmetry that can be exploited to generate stationary features angularly centered around their corresponding fixed weights.  ...  The different connectivity of the three regular polytope classifiers has a direct influence on the evaluation of the loss.  ... 
arXiv:1902.10441v2 fatcat:vcvmv2b3yvcclnawf6hfrnfwhq

The string C-group representations of the Suzuki, Rudvalis and O'Nan sporadic groups

Dimitri Leemans, Jessica Mulpas
2021 The Art of Discrete and Applied Mathematics  
The new rank three algorithms also permit us to get all string C-group representations of rank three for the Conway group Co 2 and the Fischer group Fi 22 .  ...  We use these algorithms to classify all string C-group representations of the Suzuki, Rudvalis and O'Nan sporadic groups.  ...  (P3) P is strongly connected, that is, each section of P (including P itself) is connected (in the sense given below).  ... 
doi:10.26493/2590-9770.1405.4ce fatcat:7vep26jmu5edtgcmd73xfu3euq

Hexagonal Extensions of Toroidal Maps and Hypermaps [chapter]

Maria Elisa Fernandes, Dimitri Leemans, Asia Ivić Weiss
2018 Discrete Geometry and Symmetry  
Many new examples are produced that are regular or chiral, even when the extensions are polytopal. We also construct a new infinite family of finite nonlinear hexagonal extensions of the tetrahedron.  ...  The rank 3 concept of a hypermap has recently been generalized to a higher rank structure in which hypermaps can be seen as "hyperfaces" but very few examples can be found in literature.  ...  ), through CIDMA -Center for Research and Development in Mathematics and Applications, within project UID/MAT/04106/2013.  ... 
doi:10.1007/978-3-319-78434-2_8 fatcat:o56kgvfhrfervi75riaebifsd4

Edge-Ramsey theory

K. Cantwell
1996 Discrete & Computational Geometry  
It is known that the n-dimensional measure polytopes (the higher-dimensional analogs of a cube), the n-dimensional simplex, and the regular polyhedra in R 2 and R 3 are Ramsey.  ...  In this article it is shown that the n-dimensional measure polytope and the n-dimensional cross polytope are edge-Ramsey.  ...  Lemma 4 . 2 . 42 If K is a regular polygon with n sides, the line segments connecting adjacent vertices can be translated so they lie on the line segments connecting the origin to the vertices of a regular  ... 
doi:10.1007/bf02711499 fatcat:rmdw4wlgczebje5nt4ar2duksy

Page 10 of Mathematical Reviews Vol. 2, Issue 1 [page]

1941 Mathematical Reviews  
An interesting connection is established between these repre- sentations and Frame’s representation by collineations in the finite geometry PG(3, 4).  ...  Four tables at the end give explicitly various generators of [3**1]’, [3%*"] and [3**1]. J. S. Frame (Providence, R. I.). Coxeter,H.S.M. Regular and semi-regular polytopes. I. Math.  ... 
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