Articles of tessellations

Very special geometric shape – parallelogram (No name yet?)

I suppose this geometric shape is something very ‘special’. I cannot clarify in short about being ‘special’, but I think this shape stands together with such special shapes like the square and the regular hexagon. Here it is: So it is a parallelogram based upon a square. Its height is equal to the side of […]

What is the analogon of the hexagonal grid in 3-dimensional space? Rhombic dodecahedral honeycomb?

Conjecture: The optimal way to divide 3-space into pieces of equal volume with the least total surface area is the rhombic dodecahedral honeycomb. Reasoning: “(The rhombic dodecahedral honeycomb) is the Voronoi diagram of the face-centered cubic sphere-packing, which is the densest possible packing of equal spheres in ordinary space.” (Wikipedia) This resembles very closely how […]

Not understanding this proof in Grünbaum-Shephard's Tilings and Patterns

I’m reading Grünbaum and Shephard’s Tilings and Patterns at the moment, and am kind of lost in the brevity of their statement and proof of Statement 10.1.1 (page 524 for anyone who has the book). Paraphrasing: If a monohedral tiling $\mathcal{T}$ is a $k$-composition of itself in a unique way then $\mathcal{T}$ must be non-periodic, […]

Maximum number of equilateral triangles in a circle

I am stuck with a question. Given a circle with radius $x$ cm, what is the maximum number of equilateral triangles of side length 1 cm that can fit in the circle without overlapping or overflowing the circle? $x$ is an integer. How do I approach this question?

What are the conditions for a polygon to be tessellated?

Upon one of my mathematical journey’s (clicking through wikipedia), I encountered one of the most beautiful geometrical concept that I have ever encountered in my 16 and a half years on this oblate spheroid that we call a planet. I’m most interested in the tessellation of regular polygons and their 3D counterparts. I’ve noticed that […]

Is a regular tessellation $\{p,q\}$ always possible on some closed surface $S$?

Suppose that we are given positive integers $p$, $q$, $V$, $E$, $F$, a closed surface $S$ and the Euler characteristic $\chi(S)$ of that surface. Suppose that we also know that the following relations hold: $$ qV=2E=pF\\ V-E+F=\chi(S).\tag{$\star$} $$ A regular tessellation $\{p,q\}$ is a tessellation by $p$-gons where $q$ tiles meet in every vertex. Or, […]

Dissecting an equilateral triangle into equilateral triangles of pairwise different sizes

It is know that a square can be dissected into other square such that no two of the squares have the same size. This is the simplest dissection of that kind: Is it also possible to dissect an equilateral triangle into equilateral triangles such that no two of them have the same size?

Floret Tessellation of a Sphere

I’m a programmer looking to create a 3D model of a Floret Tessellation of a sphere, like the one in this picture Class III 8,11 floret planar net (source) If anyone could point me in the right direction for an algorithm to calculate, and ideally group into petals and florets, the vertices, I would be […]

Can someone explain the math behind tessellation?

Tessellation is fascinating to me, and I’ve always been amazed by the drawings of M.C.Escher, particularly interesting to me, is how he would’ve gone about calculating tessellating shapes. In my spare time, I’m playing a lot with a series of patterns which use a hexagonal grid, and a tessellating design I saw on a chair […]

Nets of Geodesic spheres

I would realize the papercraft of a geodesic sphere like this: It is the dual of the one discussed in THIS OTHER QUESTION . Where can I find the printable nets, or the online resources to create them? In the other discussion I learned that there are 3 classes of possible tessellations. One of these […]