Fivefold Tumbler: Growth Rings — Teal

A Penrose tiling, three generations deep.

A never-repeating pattern: This pattern never settles into a repeating rhythm. Part of Fivefold, our family of never-repeating patterns built on the number five.

This pattern takes a familiar fivefold never-repeating pattern and makes a new one. Cut every wide and narrow diamond along a few set lines, and the pieces, scaled down by the golden ratio, join up with their neighbors' pieces into smaller wide and narrow diamonds: a finer tiling of the same sort. Do it again and again and the pattern gets finer and finer.

Look closely at any large diamond and you will find the same tiling, one size smaller: chambers within chambers.

Never-repeating tilings are important puzzles in computation, too: no computer program can always tell whether a given set of tiles will cover an endless floor; a geometric version of the halting problem.

Even simple rules can result in extraordinary complexity: Some tiles can fit together to cover a floor forever, but only if the pattern never repeats. This pattern is one beautiful example.

Who it’s for

With connections to geometry, computation and the structure of materials, this pattern will resonate with mathematicians, computer scientists, physicists, materials scientists, architects and designers.

For long workdays, lazy weekend mornings and every walk in between.

The Tumbler

• 20 oz double-walled stainless steel, keeping drinks hot or cold for longer. About 18 cm (7 in) tall.
• Clear slide lid; fits most car cup holders.
• Printed all the way around.
• Hand wash only.

Every Zenzicube design is grown from a real piece of mathematics. Find more in the Mathematical Menagerie.