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

Triangulations of the cube into a minimal number of simplices without additional vertices have been studied by several authors over the past decades. For $3\leq n\leq 7$ this so-called simplexity of the unit cube $I^n$ is now known to be $5,16,67,308,1493$, respectively. In this paper, we study triangulations of $I^n$ with simplices that only have nonobtuse dihedral angles. A trivial example is the standard triangulation into $n!$ simplices. In this paper we show that, surprisingly, for each $n\geq 3$ there is essentially only one other nonobtuse triangulation of $I^n$, and give its explicit construction. The number of nonobtuse simplices in this triangulation is equal to the smallest integer larger than $n!({\rm e}-2)$.Comment: 17 pages, 7 figure

Asymptotically efficient triangulations of the d-cube

Let $P$ and $Q$ be polytopes, the first of "low" dimension and the second of "high" dimension. We show how to triangulate the product $P \times Q$ efficiently (i.e., with few simplices) starting with a given triangulation of $Q$. Our method has a computational part, where we need to compute an efficient triangulation of $P \times \Delta^m$, for a (small) natural number $m$ of our choice. $\Delta^m$ denotes the $m$-simplex. Our procedure can be applied to obtain (asymptotically) efficient triangulations of the cube $I^n$: We decompose $I^n = I^k \times I^{n-k}$, for a small $k$. Then we recursively assume we have obtained an efficient triangulation of the second factor and use our method to triangulate the product. The outcome is that using $k=3$ and $m=2$, we can triangulate $I^n$ with $O(0.816^{n} n!)$ simplices, instead of the $O(0.840^{n} n!)$ achievable before.Comment: 19 pages, 6 figures. Only minor changes from previous versions, some suggested by anonymous referees. Paper accepted in "Discrete and Computational Geometry

COMPLEXITY * SIMPLICITY * SIMPLEXITY

“In the midst of order, there is chaos; but in the midst of chaos, there is order”, John Gribbin wrote in his book Deep Simplicity (p.76). In this dialectical spirit, we discuss the generative tension between complexity and simplicity in the theory and practice of management and organization. Complexity theory suggests that the relationship between complex environments and complex organizations advanced by the well-known Ashby’s law, may be reconsidered: only simple organization provides enough space for individual agency to match environmental turbulence in the form of complex organizational responses. We suggest that complex organizing may be paradoxically facilitated by a simple infrastructure, and that the theory of organizations may be viewed as resulting from the interplay between simplicity and complexity. JEL codes:

Extremal properties for dissections of convex 3-polytopes

A dissection of a convex d-polytope is a partition of the polytope into d-simplices whose vertices are among the vertices of the polytope. Triangulations are dissections that have the additional property that the set of all its simplices forms a simplicial complex. The size of a dissection is the number of d-simplices it contains. This paper compares triangulations of maximal size with dissections of maximal size. We also exhibit lower and upper bounds for the size of dissections of a 3-polytope and analyze extremal size triangulations for specific non-simplicial polytopes: prisms, antiprisms, Archimedean solids, and combinatorial d-cubes.Comment: 19 page

A list of websites and reading materials on strategy & complexity

The list has been developed based on a broad interpretation of the subject of ‘strategy & complexity’. Resources will therefore more, or less directly relate to ‘being strategic in the face of complexity’. Many of the articles and reports referred to in the attached bibliography can be accessed and downloaded from the internet. Most books can be found at amazon.com where you will often find a number of book reviews and summaries as well. Sometimes, reading the reviews will suffice and will give you the essence of the contents of the book after which you do not need to buy it. If the book looks interesting enough, buying options are easy

Reframing the systemic approach to complex organizations as intangible portfolios

The aim of this paper is to pave the way towards the inclusion of mainstream sociological approaches (based on Luhmann’s approach) for the studies of firms-organizations. In social sciences we can observe that the theoretic consequences of a paradigm shift is signiicantly represented by the evolution of systemic thinking from Parsons to Luhmann. This shift implies the change from the vision of systemic organizations as “structures” to that of systemic organizations as “communication flows”. The milestone of systemic approach in management maybe found in the research and applied works of Anthony Staford Beer with his Viable System Model (VSM) that today faced a relevant reconiguration by Golinelli and the Italian school on Viable Systemic Approach (VSA). The paradigm shift in this ield has been smoother than in sociology, and didn’t imply the discard of the concept of organization as a structure. This because, in management sciences, the perspective and, consequently, the subject of study is the organization and its structure. We think this paradigm shift is possible also in management sciences, if we consider the whole organization as a structured information low creating a dematerialized structure. Our research question is: “Is it possible to apply in business sciences the fundamental concepts that caused the paradigm shift in sociology?” To answer to this question we discuss about ontology of the firm and of the concept of value in order to understand to what extent intangible communication lows are called upon to be involved in a new deinition of structure