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 special because it is the only tiling that uses regular shapes and 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 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 ways to color a truncated square tiling. The colorings are named by the patterns around each point where the shapes meet. The patterns are called 122 and 123.

Circle Packing

The truncated square tiling can help us 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, making a neat and organized design. This shows 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 is called the Mediterranean pattern. It uses stone tiles with smaller squares placed diagonally. Other versions make the squares or octagons longer.

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

Related polyhedra and tilings

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

One related pattern is the tetrakis square tiling. It is the "dual" of the truncated square tiling. This means each point in the truncated square tiling matches a point in the tetrakis square tiling, but the shapes look different. The tetrakis square tiling is made by dividing each square into four triangles. This makes a pattern that looks like the UK flag, with triangles around each point.