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# Your Ultimate Guide to Penrose Tiling

Penrose tiling is inspired by mathematics and features a distinctive design that draws your eye. Read our guide to learn the history of Penrose tiling and how you can use it in your home.

Written by Updated 07/23/2024

Penrose tiling, or aperiodic tiling, is one of the hottest home decor trends. It’s chic and modern and consists of non-overlapping polygons or other shapes. Penrose tiling has several variations. The most popular options are two different rhombi or quadrilaterals dubbed kites and darts.

If you plan to retile a bathroom or are looking for fresh and attractive new tiling, Penrose tiling is an excellent option. Our guide covers the history and mathematical basis of Penrose tiling and its use in architecture, design, and home renovation.

## Understanding Penrose Tiling

Penrose tiling is unique in that it has mathematical and architectural appeal. It’s self-similar, meaning it converts to equivalent Penrose tilings with tiles of different sizes in an infinite pattern. This repeating pattern creates visual interest in any space and gives the tiling endless applications in architecture over non-periodic tilings. Penrose tiling is popular in bathrooms. Part of what makes Penrose tiling so beautiful is that it doesn’t repeat and features an infinite plane. While it can have rotational symmetry and reflection symmetry, it doesn’t have translational symmetry.

The Penrose tiling pattern is unique from a mathematical perspective. Several properties and common features of Penrose tiling involve what’s known in mathematics as the golden ratio.

## The History of Penrose Tiling

Mathematician and physicist Roger Penrose introduced the theorem behind Penrose tiling in 1974. His initial sketch featured an aperiodic set of six prototiles. Penrose drew inspiration from mathemetician and astronomer Johannes Kepler, who demonstrated that attempts to tile the plane with regular pentagons would lead to gapping. Pentagrams could neatly fill any gaps.

Penrose was able to find matching rules for the shapes in his sketches. He reduced the number of prototiles to two and discovered kite and dart tiling. Penrose and his colleagues, including mathematician John H. Conway, investigated and expanded the properties of Penrose tilings in the years that followed. Their findings were publicized by Martin Gardner in January 1977.

Dutch mathematician N. G. de Bruijn offered up two separate methods to construct Penrose tilings in the early 1980s:

• The “cut and project method” is a two-dimensional projection from a five-dimensional cubic structure.
• The “multigrid method” represents the Penrose tilings as dual graphs of arrangements of five families of parallel lines.

## The Mathematical Basis of Penrose Tiling

Penrose tiling is a perfect example of aperiodic tiling. It incorporates what’s known in mathematics as the golden ratio, which is the ratio of chord lengths to side lengths in a regular pentagon. Penrose tiling features local pentagonal symmetry, with points in the tiling surrounded by a symmetric configuration of tiles, and has fivefold rotational symmetry about the quasicrystals’ center point. This pattern also features five mirror lines of reflection symmetry passing through the point, with at most one center point of global fivefold symmetry.

Many of the common features of Penrose tilings follow a hierarchical pentagonal structure called inflation and deflation or composition and decomposition. Penrose tiling has a scaling self-similarity that was initially discovered by Penrose.

The mathematics behind the pattern might seem complex. The most important thing to remember is that Penrose tiling is aperiodic, or consisting of irregular occurrences and that a set of tiles in this pattern doesn’t contain large periodic regions or patches.

## The Kite and Dart Pattern

There are three types of Penrose tiling, with the kite and dart pattern the most popular for a tile or accent wall area. This pattern features two Robinson triangles, after the mathematician R. M. Robinson. Here are a few points regarding the kite and dart pattern and its use in interior design:

• Form a basic kite and dart pattern by constructing a regular pentagon and then creating its diagonals.
• The kite can be bisected along its axis of symmetry to form a pair of acute Robinson triangles.
• The kite and dart pattern often forces the placement of certain tiles, depending on how the tiles meet.

In addition to kite and dart tiling, there are the original pentagonal Penrose tiling and rhombus tiling. The kite and dart pattern is most often seen in home interiors, possibly due to the greater degree of visual interest.

## Penrose Tiling in Architecture and Design

While periodic and aperiodic patterns and the algorithm behind them can sound complicated, they result in beautiful architecture and interior design. Penrose tiling is prominent in Islamic architecture and modern high-rise buildings, and more homeowners are embracing these types of tiles for their home’s interior.

Penrose tiling is based on mathematical theory. Still, you don’t have to be a math genius to make it work for your home. You can use premade patterns and rotations for your tiling project, and online apps make finding a pattern easier.

Many homeowners choose contrasting hexagon tile colors to create more visual interest, but opting for one color or shades of one color highlights the pattern itself. Black and white is often used in upscale buildings and homes. Think about the aesthetic you want to create in the room you’re working on and go from there. If you want something vibrant and eye-catching that excites the senses, consider blue and orange or red and black. Shades of blue may be a good choice if you want a soothing atmosphere.

You can visit your local home supply store or tile shop to determine how Penrose tiling would look in your space or if you need additional visualization before committing to a ceramic or porcelain tile. Most stores will allow you to bring home some samples. You can put the samples where the tile would go to visualize the colors and patterns in your space.

## Our Conclusion

Penrose tiling is stylish and unique, with mathematical and architectural appeal. The signature Penrose pattern comprises an entire plane of non-overlapping polygons or other shapes. There are variations of Penrose tiling, including two different rhombi or two different quadrilaterals known as kites and darts.

Consider Penrose tiling If you’re looking for a chic, elegant way to add visual interest to your home or if you’re considering installing tile floors. You’ll be embracing a mathematical principle that has been around for decades and opting for a design style that has become a favorite with a wide range of homeowners.

### What is the Penrose tiling method?

The Penrose tiling method is an example of aperiodic tiling, non-overlapping polygons, or other shapes with endless applications in architecture. Roger Penrose introduced Penrose tiling in 1974, and the tiling pattern continues to be studied extensively.

### Is Penrose tiling infinite?

Penrose tiling, like other aperiodic patterns, doesn’t repeat itself. Penrose tiling isn’t infinite and can have rotational symmetry and reflection symmetry, even though it doesn’t have translational symmetry.

### What is the pattern that never repeats?

The pattern that never repeats is known as an “einstein tile.” The 13-sided figure is the first to fill an entire infinite surface with an original pattern.

### What is the golden ratio of Penrose tiles?

In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. In terms of Penrose tiling, the Golden Ratio is the ratio of chord lengths to side lengths in a regular pentagon.

### Can I use Penrose tiling in my home?

Penrose tiling is an excellent fit for most homes, especially for bathrooms, accent walls, or places that call for striking visuals. Consider Penrose tiling as a backsplash if you prefer not to incorporate bold patterns.