13 research outputs found
Concrete polytopes may not tile the space
Brandolini et al. conjectured that all concrete lattice polytopes can
multitile the space. We disprove this conjecture in a strong form, by
constructing an infinite family of counterexamples in .Comment: 6 page
CONCRETE POLYTOPES MAY NOT TILE THE SPACE
Brandolini et al. conjectured in (Preprint, 2019) that all concrete lattice polytopes can multitile the space. We disprove this conjecture in a strong form, by constructing an infinite family of counterexamples in â„ť3
Spectral sets and weak tiling
A set is said to be spectral if the space
admits an orthogonal basis of exponential functions. Fuglede
(1974) conjectured that is spectral if and only if it can tile the
space by translations. While this conjecture was disproved for general sets, it
was recently proved that the Fuglede conjecture does hold for the class of
convex bodies in . The proof was based on a new geometric
necessary condition for spectrality, called "weak tiling". In this paper we
study further properties of the weak tiling notion, and present applications to
convex bodies, non-convex polytopes, product domains and Cantor sets of
positive measure
On the nonexistence of k reptile simplices in â„ť^3 and â„ť^4
A d-dimensional simplex S is called a k-reptile (or a k-reptile simplex) if it can be tiled by k simplices with disjoint interiors that are all mutually congruent and similar to S. For d = 2, triangular k-reptiles exist for all k of the form a^2, 3a^2 or a^2+b^2 and they have been completely characterized by Snover, Waiveris, and Williams. On the other hand, the only k-reptile simplices that are known for d ≥ 3, have k = m^d, where m is a positive integer. We substantially simplify the proof by Matoušek and the second author that for d = 3, k-reptile tetrahedra can exist only for k = m^3. We then prove a weaker analogue of this result for d = 4 by showing that four-dimensional k-reptile simplices can exist only for k = m^2
Elections with Three Candidates Four Candidates and Beyond: Counting Ties in the Borda Count with Permutahedra and Ehrhart Quasi-Polynomials
In voting theory, the Borda count’s tendency to produce a tie in an election varies as a function of n, the number of voters, and m, the number of candidates. To better understand this tendency, we embed all possible rankings of candidates in a hyperplane sitting in m-dimensional space, to form an (m - 1)-dimensional polytope: the m-permutahedron. The number of possible ties may then be determined computationally using a special class of polynomials with modular coefficients. However, due to the growing complexity of the system, this method has not yet been extended past the case of m = 3. We examine the properties of certain voting situations for m ≥ 4 to better understand an election’s tendency to produce a Borda tie between all candidates