Related papers: Mathematics of Gozinta Boxes
We show that the volume of any Montesinos link can be bounded above and below in terms of the combinatorics of its diagram. This was known for Montesinos links with at most two tangles, and those with at least five tangles. We complete the…
Howards and Kobin give a sharp upper bound for crossing number of knots on rectangular mosaics. Here we extend the proof to create a new bound for hexagonal mosaics in all three natural settings and shorten the proof in the rectangular…
Suppose $k$ balls are dropped into $n$ boxes independently with uniform probability, where $n, k$ are large with ratio approximately equal to some positive real $\lambda$. The maximum box count has a counterintuitive behavior: first of all,…
We generalize the Cannonball Problem by introducing integer-valued and non-increasing arithmetic functions $w$. We associate these functions $w$ with certain polygons, which we call cannonball polygons. Through this correspondence, we show…
Given a non-empty bounded subset of hyperbolic space and a Kleinian group acting on that space, the orbital set is the orbit of the given set under the action of the group. We may view orbital sets as bounded (often fractal) subsets of…
It is possible to have a packing by translates of a cube that is maximal (i.e.\ no other cube can be added without overlapping) but does not form a tiling. In the long running analogy of packing and tiling to orthogonality and completeness…
We comment on, elaborate, and extend the work of Warren Ewens and Herbert Wilf, described in their http://www.pnas.org/content/104/27/11189.full.pdf about the maximum in balls-and-boxes problem. In particular we meta-apply their ingenious…
We treat the boundary of the union of blocks in the Jenga game as a surface with a polyhedral structure and consider its genus. We generalize the game and determine the maximum genus of the generalized game.
We develop a procedure for the complete computational enumeration of lattice $3$-polytopes of width larger than one, up to any given number of lattice points. We also implement an algorithm for doing this and enumerate those with at most…
In an additive group (G,+), a three-dimensional corner is the four points g, g+d(1,0,0), g+d(0,1,0), g+d(0,0,1), where g is in G^3, and d is a non-zero element of G. The Ramsey number of interest is R_3(G) the maximal cardinality of a…
Using computational algebraic geometry techniques and Hilbert bases of polyhedral cones we derive explicit formulas and generating functions for the number of magic squares and magic cubes.
In this paper, we recast a special case of Mahler'c conjecture by the maximum value of box splines. This is the case of polytopes with at most $2n+2$ facets. An asymptotic formula for univariate box splines is given. Based on the formula,…
The well--known Freiman--Ruzsa Theorem provides a structural description of a set $A$ of integers with $|2A|\le c|A|$ as a subset of a $d$--dimensional arithmetic progression $P$ with $|P|\le c'|A|$, where $d$ and $c'$ depend only on $c$.…
We review the current status of dimensions, as the result of a long and controversial history that includes input from philosophy and physics. Our conclusion is that they are subjective but essential concepts which provide a kind of…
In this paper, we show that the maximum number of points in $d\geq3$ dimensions determining exactly 2 distinct triangles is $2d$. We further show that this maximum is uniquely achieved by the vertices of the $d$-orthoplex. We build upon the…
We study the poset of Hamiltonian tori for polygon spaces. We determine some maximal elements and give examples where maximal Hamiltonian tori are not all of the same dimension.
The objective here is to find the maximum polygon, in area, which can be enclosed in a given triangle, for the polygons: parallelograms, rectangles and squares. It will initially be assumed that the choices are inscribed polygons, that is…
We consider tilings and packings of $\RR^d$ by integral translates of cubes $[0,2[^d$, which are $4\ZZ^d$-periodic. Such cube packings can be described by cliques of an associated graph, which allow us to classify them in dimension $d\leq…
In this paper we study circles tangent to conics. We show there are generically $184$ complex circles tangent to three conics in the plane and we characterize the real discriminant of the corresponding polynomial system. We give an explicit…
We present a novel proof that the maximum number of sets with 4 properties for 12 cards is 14 using the geometry of the finite field F_3^4, number theory, combinatorics, and graph theory. We also present several computer algorithms for…