There are only two nonobtuse binary triangulations of the unit n -cube

Brandts, Jan; Dijkhuis, Sander; Haan, Vincent de; Kříek, Michal

doi:10.1016/j.comgeo.2012.09.005

Cited by 3 publications

(11 citation statements)

References 19 publications

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“…Transversing the tree level by level corresponds to an enumeration of all the rationals Q ∩ (0, 1). 3 The circled integers displayed in Figure 4 below each vertex equal the sum of numerator and denominator of the fraction belonging to that vertex. At level k these integers correspond to the absolute values of the determinants of each of the 2 k matrices H λ of size (k + 4) × (k + 4).…”

Section: Main Results Obtained From Analyzing the Generated Datamentioning

confidence: 99%

Enumeration and investigation of acute 0/1-simplices modulo the action of the hyperoctahedral group

Brandts

Cihangir

2017

Special Matrices

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Abstract:The convex hull of n + a nely independent vertices of the unit n-cube I n is called a / -simplex.It is nonobtuse if none its dihedral angles is obtuse, and acute if additionally none of them is right. In terms of linear algebra, acute / -simplices in I n can be described by nonsingular / -matrices P of size n × n whose Gramians G = P P have an inverse that is strictly diagonally dominant, with negative o -diagonal entries [6,7]. The rst part of this paper deals with giving a detailed description of how to e ciently compute, by means of a computer program, a representative from each orbit of an acute / -simplex under the action of the hyperoctahedral group Bn [17] of symmetries of I n . A side product of the investigations is a simple code that computes the cycle index of Bn, which can in explicit form only be found in the literature [11] for n ≤ . Using the computed cycle indices for B , . . . , B in combination with Pólya's theory of enumeration shows that acute / -simplices are extremely rare among all / -simplices. In the second part of the paper, we study the / -matrices that represent the acute / -simplices that were generated by our code from a mathematical perspective. One of the patterns observed in the data involves unreduced upper Hessenberg / -matrices of size n × n, block-partitioned according to certain integer compositions of n. These patterns will be fully explained using a so-called One Neighbor Theorem [4]. Additionally, we are able to prove that the volumes of the corresponding acute simplices are in one-to-one correspondence with the part of Kepler's Tree of Fractions [1,24] that enumerates Q ∩ ( , ). Another key ingredient in the proofs is the fact that the Gramians of the unreduced upper Hessenberg matrices involved are strictly ultrametric [14,26] matrices.

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Section: Main Results Obtained From Analyzing the Generated Datamentioning

confidence: 99%

Enumeration and investigation of acute 0/1-simplices modulo the action of the hyperoctahedral group

Brandts

Cihangir

2017

Special Matrices

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“…From this follows the so-called one neighbor theorem, which states that all (n−1)-facets F of S are interior to the cube, and that each is shared by at most one other acute 0/1-simplex in I n . See [6] for an alternative proof of that fact. If S is merely a nonobtuse 0/1-simplex, the support of P only contains a doubly stochastic pattern, and moreover, P can be partly decomposable.…”

Section: Outlinementioning

confidence: 99%

“…, F n are one-dimensional, the corresponding 0/1simplex is a so-called orthogonal simplex, as it has a spanning tree of mutually orthogonal edges. Orthogonal 0/1-simplices played an important role in the nonobtuse cube triangulation problem, solved in [6]. We briefly recall them in Section 5 and put them into the novel context of Section 4.…”

Section: Outlinementioning

confidence: 99%

“…The orthogonal 0/1-simplices can be defined recursively as follows [6]. The cube edge I 1 is an orthogonal simplex.…”

Section: Orthogonal Simplices and Their Matrix Representationsmentioning

confidence: 99%

“…Note that this is the maximum number of right dihedral angles a simplex can have, as Fiedler proved in [13] that any simplex has at least n acute dihedral angles. Orthogonal simplices are useful in many applications, see [6,9] and the references therein. We will restrict our attention to orthogonal 0/1-simplices.…”

Section: Orthogonal Simplices and Their Matrix Representationsmentioning

confidence: 99%

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On the combinatorial structure of $0/1$-matrices representing nonobtuse simplices

Brandts¹,

Cihangir²

2018

Appl.Math.

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A 0/1-simplex is the convex hull of n + 1 affinely independent vertices of the unit n-cube I n . It is nonobtuse if none its dihedral angles is obtuse, and acute if additionally none of them is right. Acute 0/1-simplices in I n can be represented by 0/1-matrices P of size n × n whose Gramians G = P ⊤ P have an inverse that is strictly diagonally dominant, with negative off-diagonal entries.In this paper, we will prove that the positive part D of the transposed inverse P −⊤ of P is doubly stochastic and has the same support as P . In fact, P has a fully indecomposable doubly stochastic pattern. The negative part C of P −⊤ is strictly row-substochastic and its support is complementary to that of D, showing that P −⊤ = D − C has no zero entries and has positive row sums. As a consequence, for each facet F of an acute 0/1-facet S there exists at most one other acute 0/1-simplexŜ in I n having F as a facet. We callŜ the acute neighbor of S at F .If P represents a 0/1-simplex that is merely nonobtuse, the inverse of G = P ⊤ P is only weakly diagonally dominant and has nonpositive off-diagonal entries. Consequently, P −⊤ can have entries equal to zero. We show that its positive part D is still doubly stochastic, but its support may be strictly contained in the support of P . This allows P to have no doubly stochastic pattern and to be partly decomposable. In theory, this might cause a nonobtuse 0/1-simplex S to have several nonobtuse neighborsŜ at each of its facets.In the remainder of the paper, we study nonobtuse 0/1-simplices S having a partly decomposable matrix representation P . We prove that if S has such a matrix representation, it also has a block diagonal matrix representation with at least two diagonal blocks. Moreover, all matrix representations of S will then be partly decomposable. This proves that the combinatorial property of having a fully indecomposable matrix representation with doubly stochastic pattern is a geometrical property of a subclass of nonobtuse 0/1-simplices, invariant under all n-cube symmetries. We will show that a nonobtuse simplex with partly decomposable matrix representation can be split in mutually orthogonal simplicial facets whose dimensions add up to n, and in which each facet has a fully indecomposable matrix representation. Using this insight, we are able to extend the one neighbor theorem for acute simplices to a larger class of nonobtuse simplices.

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On 0/1-polytopes with nonobtuse triangulations

Cihangir

2015

Applied Mathematics and Computation

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There are only two nonobtuse binary triangulations of the unit n -cube

Cited by 3 publications

References 19 publications

Enumeration and investigation of acute 0/1-simplices modulo the action of the hyperoctahedral group

Enumeration and investigation of acute 0/1-simplices modulo the action of the hyperoctahedral group

On the combinatorial structure of $0/1$-matrices representing nonobtuse simplices

On 0/1-polytopes with nonobtuse triangulations

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