Truncated square tiling

In geometry, the truncated square tiling is a special way to cover a flat surface with shapes. This pattern uses one square and two octagons meeting at each corner. It is unique because it is the only tiling made of regular shapes that includes an octagon. The pattern can be described using a special code called a Schläfli symbol, written as t{4,4}.

Conway calls this pattern a truncated quadrille because it comes from changing a simple pattern of squares, known as a square tiling. This change is called a truncation operation.

Other names for this beautiful pattern are Mediterranean tiling and octagonal tiling. In this version, the octagons have edges that change in length, and the squares are often smaller. There are 3 regular tilings and 8 semiregular tilings that can cover a flat surface in interesting ways.

Uniform colorings

There are two different ways to color a truncated square tiling. These colorings are named by looking at the patterns around each point where the shapes meet. The patterns are called 122 and 123.

Circle Packing

The truncated square tiling can be used to arrange circles in a special pattern called a circle packing. In this pattern, we place circles of the same size at the center of each point in the tiling. Each circle touches three other circles, creating a neat and organized design. This arrangement is an example of how shapes can fit together perfectly in geometry.

Main article: Circle packing

Further information: Kissing number

Variations

There are different ways to change the truncated square tiling. One common version, called the Mediterranean pattern, uses stone tiles with smaller squares placed diagonally. Other versions stretch out the squares or octagons.

The Pythagorean tiling is another pattern that looks similar. It uses large and small squares that are turned 45 degrees, and the octagons look more like squares with points in the middle of their sides. A weaving pattern also has the same layout, but the octagons look more like flat rectangles.

Related polyhedra and tilings

The truncated square tiling is connected to a group of patterns in geometry called uniform polyhedra and tilings. These patterns share a common shape at each point where their tiles meet, written as 4.2n.2n. When we look at a special 3D pattern called the bitruncated cubic honeycomb from a flat view, we see two copies of the truncated square tiling.

One closely related pattern is the tetrakis square tiling, which is the "dual" of the truncated square tiling. This means every point in the truncated square tiling has a matching point in the tetrakis square tiling, but the shapes are different. The tetrakis square tiling is made by dividing each square into four isosceles right triangles. This creates a pattern that looks like the UK flag, with triangles arranged around each point.