Articles of tiling

Graph Relatives for Tessellation of the Hyperbolic Plane

I’m trying to get into the theory about the Modular group. Among the “Paracompact hyperbolic uniform tilings in [∞,3] family” in the section “Tessellation of the hyperbolic plane” I found the Order-3 apeirogonal tiling: The figure reminded my on Frucht’s graph, which seems to have the similar structural features. Are there also graphs for the […]

Checkerboard with one gap can be covered by triominos?

A checkerboard with $2^{n}\times 2^{n}$ squares from which one square has been removed can be covered exactly by “triominos”. Form is one square up with $2$ below them.

What is the first $w$ such that a rectangle, $R_{w\times w-1}$ is minimally-square-partitioned by less than $w$ squares.

Motivated by: Tiling an orthogonal polygon with squares, How to prove that the minimum square partition of a 3X2 rectangle has 3 squares, Minimum square partitions for 4×3 and 5×4 rectangles, What is the minimum square partition of an almost-square rectangle?. Let: An almost-square-rectangle be a rectangle that has a width $w$ and height $h=w-1$. […]

Decomposing a circle into similar pieces

Is it possible to decompose a circle into finitely many similar disjoint pieces, one of which contains the circle’s center in its interior?

An Olympiad Problem (tiling a rectangle with the L-tetromino)

An L block that is 3 unit blocks high and 2 unit blocks wide . It is true that if an n by m rectangle can be covered by such L blocks with out overlap that we would require an even amount of L blocks, why?

A tiling puzzle/question

My teacher gave us a riddle that goes like this: You have a $7\times 7$ square and $16$ $3\times 1$ tiles. Of the $16$ tiles, $15$ are straight and $1$ is crocked (“L” shaped). When you tile the square with these tiles you should get that one unit is left un-tiled (because $7 \times 7=49$ […]

Tiling Posters on a Wall

I’m a noob, and I’m not a mathematician (Although I will be a Math major next semester). My question is: I have 68 maps I would like to use as posters on my wall at home. They are all rectangles, and all different dimensions. Is there an algorithm (or better yet, a website or other […]

Penrose tilings as a cross section of a $5$-dimensional regular tiling

Could somebody explain to me how a penrose tiling , which is not periodic, can be a cross section of a regular tiling in $5$ dimensions, which is periodic? It does not make sense to me how a periodic tiling can produce an aperiodic cross-section. Also, are there any examples of periodic $3$-dimensional tilings that […]

Cutting a $m \times n$ rectangle into $a \times b$ smaller rectangular pieces

How many $a \times b$ rectangular pieces of cardboard can be cut from $m \times n$ rectangular piece of cardboard so that the amount of waste(“left over” cardboard) is a minimum? This question was given to me by my Mathematics Teacher as a “brain teaser”. At first I divided $m \times n$ by $a \times […]

Decidability of tiling of $\mathbb{R}^n$

Given a polytope of dimension $n$, is there some general way to determine if it can tile $\mathbb{R}^n$?