345 research outputs found
Perfect Octagon Quadrangle Systems with an upper C4-system and a large spectrum
An octagon quadrangle is the graph consisting of an 8-cycle (x1, x2,..., x8) with two additional chords: the edges {x1, x4} and {x5, x8}. An octagon quadrangle system of order ν and index λ [OQS] is a pair (X,H), where X is a finite set of ν vertices and H is a collection of edge disjoint octagon quadrangles (called blocks) which partition the edge set of λKν defined on X. An octagon quadrangle system Σ=(X,H) of order ν and index λ is said to be upper C4-perfect if the collection of all of the upper 4-cycles contained in the octagon quadrangles form a μ-fold 4-cycle system of order ν; it is said to be upper strongly perfect, if the collection of all of the upper 4-cycles contained in the octagon quadrangles form a μ-fold 4-cycle system of order ν and also the collection of all of the outside 8-cycles contained in the octagon quadrangles form a ρ-fold 8-cycle system of order ν. In this paper, the authors determine the spectrum for these systems, in the case that it is the largest possible
ON THE SPECTRUM OF OCTAGON QUADRANGLE SYSTEMS OF ANY INDEX
An \emph{octagon quadrangle} is the graph consisting of a length cycle and two chords, and . An \emph{octagon quadrangle system} of order and index is a pair , where is a finite set of vertices and is a collection of octagon quadrangles (called blocks) which partition the edge set of , with as vertex set. In this paper we determine completely the spectrum of octagon quadrangle systems for any index , with the only possible exception of for
An Arithmetization of Logical Oppositions
An arithmetic theory of oppositions is devised by comparing expressions, Boolean bitstrings, and integers. This leads to a set of correspondences between three domains of investigation, namely: logic, geometry, and arithmetic. The structural properties of each area are investigated in turn, before justifying the procedure as a whole. Io finish, I show how this helps to improve the logical calculus of oppositions, through the consideration of corresponding operations between integers
strongly balanced 4 kite designs nested into oq systems
In this paper we determine the spectrum for octagon quadrangle systems [OQS] which can be partitioned into two strongly balanced 4-kitedesigns
The Spectrum of Balanced P^(3)(1, 5)-Designs
Given a 3-uniform hypergraph H(3), an H(3)-decomposition of the complete hypergraph K(3)_v is a collection of hypergraphs, all isomorphic to H(3), whose edge sets partition the edge set of K(3)_v. An H(3)-decomposition of K(3)_v is also called an H(3)-design and the hypergraphs of the partition are said to be the blocks. An H(3)-design is said to be balanced if the number of blocks containing any given vertex of K(3)_v is a constant. In this paper, we determine completely, without exceptions, the spectrum of balanced P(3)(1 5)-designs
Paperfolding morphisms, planefilling curves, and fractal tiles
An interesting class of automatic sequences emerges from iterated
paperfolding. The sequences generate curves in the plane with an almost
periodic structure. We generalize the results obtained by Davis and Knuth on
the self-avoiding and planefilling properties of these curves, giving simple
geometric criteria for a complete classification. Finally, we show how the
automatic structure of the sequences leads to self-similarity of the curves,
which turns the planefilling curves in a scaling limit into fractal tiles. For
some of these tiles we give a particularly simple formula for the Hausdorff
dimension of their boundary.Comment: 32 pages, 23 figure
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