a perspective on mathematics, the pattern, and the abstract

Consider tiling the plane using only square tiles like this:

The 4-fold rotational symmetry of the square allows a tile to be placed in the 4 different orientations shown here:

Despite these constraints there are still a lot of different ways to tile the plane. Shown above are a few examples constructed with an algorithm using modular arithmetic. This essentially makes the tiles along different rows follow the same sequence but shifted over by some amount.

Each of the tilings shown are actually periodic and can tile the entire plane.

(e.g. 1, 2, 3, 4)

Tilings of this variety are called Truchet tilings.

Mathematica code:

T[r_, x_, y_] :=

Translate[

Rotate[

{EdgeForm[Thickness[0]], Polygon[{{1, 0}, {0, 0}, {0, 1}}]},

r, {.5, .5}],

{x, y}]

Manipulate[

Graphics[

Table[

T[Mod[a*x + b*y, m] Pi/2, x, y],

{x, 1, X, 1}, {y, 1, Y, 1}],

PlotRange -> {{1, X}, {1, Y - .1}}, ImageSize -> 500],

{{X, 29}, 1, 100, 1}, {{Y, 29}, 1, 100, 1},

{{m, 14}, 1, 100, 1},

{{a, 5}, 1, 100, 1},

{{b, 5}, 1, 100, 1}]

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