Related papers: Hypercubic Self-Tilings
A triangulation of a surface is \emph{irreducible} if there is no edge whose contraction produces another triangulation of the surface. We prove that every irreducible triangulation of a surface with Euler genus $g\geq1$ has at most $13g-4$…
Proctor proved a formula for the number of lozenge tilings of a hexagon with side-lengths $a,b,c,a,b,c$ after removing a "maximal staircase." Ciucu then presented a weighted version of Proctor's result. Here we present weighted and…
A set of lines in $\mathbb{R}^n$ is called equiangular if the angle between each pair of lines is the same. We address the question of determining the maximum size of equiangular line sets in $\mathbb{R}^n$, using semidefinite programming…
F-boxes defined in [6] as hyper-boxes in N^{\infty} discrete space were applied here for the geometric description of the cobweb posetes Hasse diagrams tilings. The F-boxes edges sizes are taken to be values of terms of natural numbers'…
We establish asymptotic lower bounds for the number of elliptic curves over $\mathbb{Q}$ with prescribed entanglement of division fields, ordered by naive height. Such elliptic curves are obtained as $1$-parameter families arising from…
The square peg problem asks whether every continuous curve in the plane that starts and ends at the same point without self-intersecting contains four distinct corners of some square. Toeplitz conjectured in 1911 that this is indeed the…
A study of 7-fold tilings that use a set of three proto-rhombs in a substitution scheme to tile a large area. A set is discovered that is thought to be the most minimal or smallest one. The scheme uses 11 proto-rhombs to tile the next…
The concepts of tile number and space-efficiency for knot mosaics were first explored by Heap and Knowles (arXiv:1702.06462), where they determined the possible tile numbers and space-efficient layouts for every prime knot with mosaic…
We give bounds on the number of non-simple closed curves on a negatively curved surface, given upper bounds on both length and self-intersection number. In particular, it was previously known that the number of all closed curves of length…
A version of the Hardy-Littlewood circle method is developed for number fields K/Q and is used to show that non-singular projective cubic hypersurfaces over K always have a K-rational point when they have dimension at least 8.
The isoperimetric problem is one of the oldest in geometry and it consists of finding a surface of minimum area that encloses a given volume $V$. It is particularly important in physics because of its strong relation with stability, and…
John E. Littlewood posted the question {\em ``Is it possible in 3-space for seven infinite circular cylinders of unit radius each to touch all the others? Seven is the number suggested by counting constants.''} Boz\'oki, Lee, and R\'onyai…
A semi-regular tiling of the hyperbolic plane is a tessellation by regular geodesic polygons with the property that each vertex has the same vertex-type, which is a cyclic tuple of integers that determine the number of sides of the polygons…
Let $Q_d$ be the $d$-dimensional hypercube and $N=2^d$. We prove that the number of (proper) 4-colorings of $Q_d$ is asymptotically \[6e2^N,\] as was conjectured by Engbers and Galvin in 2012. The proof uses a combination of information…
We show that among any $n$ points in the unit cube one can find a triangle of area at most $n^{-2/3-c}$ for some absolute constant $c >0$. This gives the first non-trivial upper bound for the three-dimensional version of Heilbronn's…
In this paper, we prove that the general tiling problem of the hyperbolic plane is undecidable by proving a slightly stronger version using only a regular polygon as the basic shape of the tiles. The problem was raised by a paper of Raphael…
For a fixed integer h>=1, let G be a tripartite graph with N vertices in each vertex class, N divisible by 6h, such that every vertex is adjacent to at least 2N/3+h-1 vertices in each of the other classes. We show that if N is sufficiently…
An h-tiling on a finite simplicial complex is a partition of its geometric realization by maximal simplices deprived of several codimension one faces together with possibly their remaining face of highest codimension. In this last case, the…
Tile the unit square with $n$ small squares. We determine the minimum of the sum of the side lengths of the $n$ small squares, where the minimum is taken over all tilings of the unit square with $n$ squares.
In [B.Gruenbaum, G.C. Shephard, Spherical tilings with transitivity properties, in: The geometric vein, Springer, New York, 1981, pp. 65-98], they proved "for every spherical normal tiling by congruent tiles, if it is isohedral, then the…