# Questions tagged [tessellations]

For question on Tessellations, the process of creating a two-dimensional plane using the repetition of a geometric shape with no overlaps and no gaps.

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### consistent vertex configurations for Archimedean tiling

I am trying to find if there is a simple rule for vertex configurations $n_1.n_2\ldots n_k$ that define Archimedean tilings (equivalently called uniform/semi-regular/vertex-transitive tilings). In ...
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### Non-monotileable amenable groups

We say a subset $A$ of a group $G$ is a monotile for $G$ if $G$ is a disjoint union of right translates of $A$. In his article Monotileable Amenable Groups, B. Weiss gives lots of examples of amenable ...
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### Do Schläfli symbols unambiguously represent gemetric shapes?

According to Wikipedia, the tesseract is the four-dimensional analogue of the cube and has the Schläfli symbol {4,3,3}, and Wikipedia features the following visualization: However, when looking at it,...
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### Explanation of using wallpaper groups with vertex figure for k-uniform tilings

In my project I need to implement various uniform tiling of a 2D-plane, so some time ago I started to dig a little bit into sources related to subject. From what I understand, any k-uniform ...
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### Tessellation of n-gons into parallelograms

Which regular n-gons can be tiled by (or tessellated into) a finite number of non-overlapping parallelograms? I have enumerated a few cases below where I know the answer: Case $n = 3$, there is ...
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### How many tiling?

Recently, I study tiling of the plane with regular polygon which is edge-to edge. If we restrict to the two triangles and two hexagons, then we can slide rows of tiles so that in adjacent rows, we ...
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### How to obtain translations vector in periodic tiling

A tiling of the plane, $\mathcal{T}$, is a family of sets- called tiles- that cover the plane without gaps or overlaps. Assume that tiles are regular polygons and tiling is edge-to-edge. A tiling is ...
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### Filling space with polycube snakes

One of Martin Gardner's "Mathematical Games" columns (Scientific American June 1981, pp24–29; reprinted in The Last Recreations (1997), pp274–283, and The Colossal Book of Mathematics (2001) ...
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### Is "Escherian metamorphosis" always possible?

This question is motivated by Escher's series of Metamorphosis woodcuts (see e.g. here), where one tesselating tile is gradually transformed into another. Basically, this is a precise way of asking ...
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### How many fixed polyominoes does it take to force an aperiodic tiling of the plane?

A longstanding open problem asks whether there is a single connected tile that only tiles the plane aperiodically. As far as I know, this is still open even in the case of polyominoes: we do not know ...
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### If a (possibly nonconvex) pentagon tiles the plane, can it do so periodically?

From the classification of monohedral tilings with convex pentagons, we know that all convex pentagons which tile the plane can do so periodically; I'd like to know whether the same result is known to ...
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