These are examples of geometric tilings ... in walls and floors everywhere. In math, tilings are often appreciated for their regular patterns. But mathematicians also find beauty in irregularity. It’s ...

A new 13-sided shape is the first example ... in mathematics), you could find a position where the floor looks exactly the same as before, proving that it's a repeating pattern.

It is also one of the hardest problems in mathematics ... without ever creating a repeating design. In these special cases, called aperiodic tilings, there’s no pattern that you can copy ...

Researchers have discovered a new 14-sided shape called the Spectre that can be used to tile a surface without ever creating a repeating pattern ... in the world of mathematics.

Infinitely many copies of a 13-sided shape can be arranged with no overlaps or gaps in a pattern that ... investigations—for example, shapes that form non-repeating tilings could help design ...

but never make a repeating pattern. Periodic tilings have translational symmetry: a honeycomb pattern, for example, can be repeated forever and looks identical after being shifted in any of six ...