69 research outputs found
Lower bounds for the simplexity of the n-cube
In this paper we prove a new asymptotic lower bound for the minimal number of
simplices in simplicial dissections of -dimensional cubes. In particular we
show that the number of simplices in dissections of -cubes without
additional vertices is at least .Comment: 10 page
There are only two nonobtuse binary triangulations of the unit -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 this so-called simplexity of the unit cube is now
known to be , respectively. In this paper, we study
triangulations of with simplices that only have nonobtuse dihedral
angles. A trivial example is the standard triangulation into simplices. In
this paper we show that, surprisingly, for each there is essentially
only one other nonobtuse triangulation of , and give its explicit
construction. The number of nonobtuse simplices in this triangulation is equal
to the smallest integer larger than .Comment: 17 pages, 7 figure
Asymptotically efficient triangulations of the d-cube
Let and be polytopes, the first of "low" dimension and the second of
"high" dimension. We show how to triangulate the product
efficiently (i.e., with few simplices) starting with a given triangulation of
. Our method has a computational part, where we need to compute an efficient
triangulation of , for a (small) natural number of our
choice. denotes the -simplex.
Our procedure can be applied to obtain (asymptotically) efficient
triangulations of the cube : We decompose , for
a small . 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 and , we can triangulate
with simplices, instead of the 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
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 Lower Bound for the Simplexity of then-Cube via Hyperbolic Volumes
AbstractLet T(n) denote the number of n -simplices in a minimum cardinality decomposition of the n -cube into n -simplices. For n≥ 1, we show that T(n) ≥H(n), where H(n) is the ratio of the hyperbolic volume of the ideal cube to the ideal regular simplex. H(n) ≥12·6n/2(n+ 1)−n+12n!. Also limn→∞n [H(n)]1/n≈ 0.9281. Explicit bounds for T(n) are tabulated for n≤ 10, and we mention some other results on hyperbolic volumes
Simplexity of the n-cube
Major: Mathematics and Classics
Faculty Mentor: Dr. Su-Jeong Kang, Mathematics
The process of dividing shapes into triangles is called triangulation, and it is possible to abstract the idea of a triangle to higher dimensions, where it will be called a simplex in n-dimensions, or an n-simplex. I studied this process of generalized triangulation, or decomposition, in order to find an optimal decomposition of a 5-cube to help improve the bounds on the general case of an n-cube
Lattice Delone simplices with super-exponential volume
In this short note we give a construction of an infinite series of Delone
simplices whose relative volume grows super-exponentially with their dimension.
This dramatically improves the previous best lower bound, which was linear.Comment: 7 pages; v2: revised version improves our exponential lower bound to
a super-exponential on
Products of Foldable Triangulations
Regular triangulations of products of lattice polytopes are constructed with
the additional property that the dual graphs of the triangulations are
bipartite. The (weighted) size difference of this bipartition is a lower bound
for the number of real roots of certain sparse polynomial systems by recent
results of Soprunova and Sottile [Adv. Math. 204(1):116-151, 2006]. Special
attention is paid to the cube case.Comment: new title; several paragraphs reformulated; 23 page
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