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Connectivity in the regular polytope representation
2009
Geoinformatica
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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]
2008
Lecture Notes in Geoinformation and Cartography
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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]
2010
Lecture Notes in Geoinformation and Cartography
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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
2018
Zenodo
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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
2012
Journal of combinatorial theory. Series A
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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
1995
Bulletin of the London Mathematical Society
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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
2020
Ars Mathematica Contemporanea
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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
2008
European journal of combinatorics (Print)
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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]
2009
arXiv
pre-print
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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
2010
Discrete Mathematics
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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]
2019
arXiv
pre-print
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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
2021
The Art of Discrete and Applied Mathematics
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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]
2018
Discrete Geometry and Symmetry
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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
1996
Discrete & Computational Geometry
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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
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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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