Roger Penrose — the English mathematical physicist who shared the 2020 Nobel Prize in Physics for the discovery that black hole formation is a robust prediction of the general theory of relativity — is also famous for something more terrestrial - Penrose Tiling. Here’s a look at the other side of the Nobel laureate:Tile High: Back in the 1970s, Roger Penrose created a set of tiles that could be used to cover an infinite plane in a pattern that never repeats. His work changed our basic understanding of design, showing how infinite variations could be created within a highly ordered environment. Patterns of Penrose
This post explains a new Grasshopper tool I developed. It is a Penrose Tiling Generator. I utilized subdivision and triangular mapping.
A Penrose tiling is an example of non-periodic tiling. They are named after mathematician and physicist Roger Penrose, who investigated these sets in the 1970s.
Using squares or triangles to tile a surface is easy. Place them side by side until you're done, but what about pentagons? They don't line up easily, and they leave many strange shapes in between. Scientists and Mathematicians have tried though the ages to experiment with pentagon tessellations. Sir Roger Penrose wondered if the pentagon could be used to create a tile pattern using inscribed arcs to continue a non repeating pattern. Penrose took the pentagon and divided it up and arrived at two shapes: the kite and the dart. These shapes can be used to create all sorts of intriguing combinations to stretch the mind and to have lots of fun as you play with them. The reverse side of all the tiles is blank to add more choices for your design. The object is to keep the archs connected as you add another tile so they flow from tile to tile, but you can have fun by combining them any way you would like. There are 55 kites and 34 darts in each set. They are laser cut using toy grade baltic birch plywood 1/8" thick. The connecting sides are 1" long. Have fun as you experiment with the Penrose tiles. Not for children under the age of 3.
girih and Penrose
I made this Christmas Tree Skirt as a wedding present for my niece and nephew-in-law: It's based on a Penrose tiling, which is an aperiodic tiling derived from tiling pentagons. (I'll let wikipedia explain it). I used a combination of machine piecing and English paper piecing to construct it using two types of diamonds: and I lay them out in this base tiling: Then I do two tricky things that exploit some of the niftier properties of Penrose tilings: matching rules and inflation/deflation. These need to be explained in more detail... Constructing Penrose Tiling using Matching Rules These rhombi (plural of rhombus, a fancy word for diamond) can be tiled in a lot of ways that are not actually Penrose tilings. They can be tiled in ways that are periodic, or completely non-periodic: and you can certainly make some lovely quilts doing this: This was made by Domesticat This was made by Dorothea However, these are not strictly Penrose tilings. To make a quilt that has that quirky stars-with-ripples effect that the aperiodic tiling creates, you need to follow the matching rules: My first plan was to design a quilt that copied the curved design as shown in the above diagram (but just the pink lines): NOTE: Penrose tiling is beyond the powers of Electric Quilt, so I designed these using Geometer's Sketchpad instead. I made this pot holder as a test to see if it was feasible to make the entire full sized quilt this way: Although this was strictly feasible, I found that all the curved lines and sharp points made this too tricky for my skill level. So I decided to tweak the matching rules into straight lines: This way I can machine piece these straight lines first and paper piece the rhombi. Constructing Penrose Tiling using Inflation/Deflation Properties One of the mind-bending properties of a Penrose tiling is that if all the rhombi in a given tiling are broken up like so: The resulting tiling is also a Penrose tiling of smaller tiles. In this example I deflate this simple tiling one level: Now I can deflate this tiling down another level (I shaded a couple of rhombi from the previous level to illustrate this): You can continue this indefinitely. Here's 3 and 4 level deflation: The flip side of this is inflation; any tiling can be combined into larger tiles by reversing this process. So any tiling, no matter how large, is only a subset of a single rhombus (mind = blown). This inspired me to design a quilt that exploits this property. I first thought it would be interesting to have a quilt that uses different sized rhombi: Then I thought it would be cool if I designed a quilt where all the rhombi were the same size, but by using different colors and/or values (light and dark), an inflated level of Penrose tiling would emerge: The combination of this design with the matching rule design above became the final design for the Christmas tree skirt: Next Post - Construction Phase >
A Penrose tiling (Wikipedia), named for British mathematical physicist Sir Roger Penrose, who investigated them in the 1970s. A Penrose tiling is "aperiodic," or, simply put, produces a pattern that does not repeat itself no matter how far you extend it across the plain. All Penrose tilings are aperiodic, but not all aperiodic tilings are Penrose tilings. Lots of bright creative folks have installed custom Penrose tile floors. Here's a selection of a few of my faves from around the web. I couldn't find anybody online who's selling pre-cut Penrose prototiles, so it looks like anybody who wants to do it themselves has to cut their own. Or, if somebody is feeling entrepreneurial...
Discover the profound connections between Penrose tilings, quantum error correction, and fractal patterns found throughout nature, art, and sacred geometry.
I just finished another Penrose tile quilt: First - Credit where it's due... If you do a Google Image search on 'Penrose tile quilt' the top three results will be this awesome quilt: This lovely quilt was made by Serena Mylchreest nearly 20 years ago. This obviously was an inspiration for my design. The element of the Mylchreest design that I borrowed is the way the rhombi are divided in half (fat rhombi divided lengthwise, and the thin rhombi split on the narrow diagonal) so it is tiled with triangles instead of rhombi. Then the critical design element is to select colors of different values for the 'light half' and 'dark half' of the rhombi to create the nifty 3-D effect. I was pleased with the way my last Penrose tile quilt turned out when I first came up with the notion to machine piece the patches before I English paper pieced the rhombi. So I played around with some other machine piecable patterns and this was one of the results. When I designed the layout, I wanted to make a Penrose tiling that was not radially symmetrical through the center like my last quilt (and most Penrose tile quilts). I find their quasi-periodic nature one of the more mind bending elements of Penrose tilings, and I feel this is not obvious when it is radially symmetrical. I began with this simple layout: The yellow dots indicate the eventual location of the yellow stars. The black square roughly indicates the final border of the quilt. Then I deflate the rhombi 3 levels thusly: I explain inflation/deflation of Penrose tiles more thoroughly in this post. After removing the superfluous rhombi, we're left with this base layout: Then we add these machine piece rhombi: One last design decision I made was to inflate just five of the rhombi to make the one large star. Two other designs I came up with while playing with machine-pieced rhombi:
Discover the profound connections between Penrose tilings, quantum error correction, and fractal patterns found throughout nature, art, and sacred geometry.
I just finished another Penrose tile quilt: First - Credit where it's due... If you do a Google Image search on 'Penrose tile quilt' the top three results will be this awesome quilt: This lovely quilt was made by Serena Mylchreest nearly 20 years ago. This obviously was an inspiration for my design. The element of the Mylchreest design that I borrowed is the way the rhombi are divided in half (fat rhombi divided lengthwise, and the thin rhombi split on the narrow diagonal) so it is tiled with triangles instead of rhombi. Then the critical design element is to select colors of different values for the 'light half' and 'dark half' of the rhombi to create the nifty 3-D effect. I was pleased with the way my last Penrose tile quilt turned out when I first came up with the notion to machine piece the patches before I English paper pieced the rhombi. So I played around with some other machine piecable patterns and this was one of the results. When I designed the layout, I wanted to make a Penrose tiling that was not radially symmetrical through the center like my last quilt (and most Penrose tile quilts). I find their quasi-periodic nature one of the more mind bending elements of Penrose tilings, and I feel this is not obvious when it is radially symmetrical. I began with this simple layout: The yellow dots indicate the eventual location of the yellow stars. The black square roughly indicates the final border of the quilt. Then I deflate the rhombi 3 levels thusly: I explain inflation/deflation of Penrose tiles more thoroughly in this post. After removing the superfluous rhombi, we're left with this base layout: Then we add these machine piece rhombi: One last design decision I made was to inflate just five of the rhombi to make the one large star. Two other designs I came up with while playing with machine-pieced rhombi:
Discover the profound connections between Penrose tilings, quantum error correction, and fractal patterns found throughout nature, art, and sacred geometry.
Marble floors exude richness, quality and sophistication. There is an ancient history of marble flooring, and much of this architectural work still exists today from antiquity. Some of it is still in excellent condition after thousands of years. Marble floors are found in historic structures from the ancient Greek Parthenon to Grand Central Station in New York City (below). Patterns can be created by using different colored stone, and inlaying them for contrast and juxtaposition. The surface of the marble floor is then polished smooth, for an even flooring surface. Because marble flooring is so sleek, shiny and reflective, it can often be difficult to keep spotless. Streaks, smudges, scuffs and dust will show up clearly and immediately. Regular cleaning is a must. A soft dust mop works well for daily maintenance, with at least a weekly sweeping and a damp mop. Marble floors need to be professionally cleaned and polished at least once a year to repair any cracks or dried-out areas and help it keep its sheen. Since marble is so durable, heavy, dramatic and used in places like palaces, folks should get the hint that it's also pretty expensive. Installation of the floor will also be costly, and should not be undertaken by anyone who does not know what he's doing. The maintenance alone takes its toll, with constant cleaning and regular resealing which should also be done by professionals. In addition to the high cost, marble floors are very cold on bare feet. Slippers or throw rugs are highly recommended during winter months Below are the marble floors of the Mamluk 14th century madrasa and mausoleum of Sultan Hasan in Cairo, Egypt. Here is a panel from the Duomo’s marble floor in Siena, Italy. Around 1920, Italian masons realized a less expensive method for producing marble floor, in a technique now known as “Terrazzo.” First, a concrete substrate is built from a slab at least 3 inches thick. Then a wet mixture of sand-based mortar is applied to the substrate. While this mortar is still wet, marble chips and pigment are placed on top of the wet mortar. This mixture is then tamped or consolidated in place, and subsequently polished. The beauty of terrazzo is that different colored patterns can readily be incorporated, with high precision and accuracy. Terrazzo does not have the same high sheen as marble slabs, so usually a filler or additive is applied to the terrazzo to give it a more lustrous finish. Here is a Terrazo floor from the Texas State Capitol building: In American architecture today, a marble floor sometimes seems pretentious, in the same vein as a “McMansion” of the nouveau riche. However, judicious use of marble tastefully applied, properly installed and adequately cared for can add real beauty to any building.
An array of wooden tiles assembled into an intriguing pattern flecked with stars forms a striking contrast to the regular arrangement of bricks making up the wall on which it hangs on the third floor of Avery Hall, home of the mathematics department at the University of Nebraska-Lincoln. Constructed by Nebraska mathematician Earl S. Kramer from diamond-shaped cherry and maple tiles and installed in March 2005, the wall piece represents a patch of one of the infinite number of ways in which to arrange fat and skinny diamonds into an aperiodic pattern characteristic of a Penrose tiling. The two types of tiles for assembling such a Penrose tiling are rhombs (each rhomb has four sides of equal length) with acute angles of 36 and 72 degrees. Matching rules specify the ways in which these rhombs must be assembled edge to edge to create an aperiodic tiling (one in which the tiling cannot be lifted and placed back onto itself with all points displaced but still looking the same). The particular tiling pattern depicted in the wall piece is one of two Penrose rhomb arrangements that have the dihedral automorphism group d5, featuring rotations of order five and reflections across a line, readily apparent in the design. For another artistic representation of a Penrose tiling, see "Tessellation Tango." Reference: Peterson, I. 2001. Fragments of Infinity: A Kaleidoscope of Math and Art. Wiley. Photos by I. Peterson
In the world of mosaic art, patterns play a crucial role in creating visually captivating and intellectually stimulating pieces. One such pattern that has garnered considerable attention and admiration is Penrose tiling. Named after the British mathematician and physicist Sir Roger Penrose, this intricate design offers an endless array of possibilities for mosaic artists. In this blog post, we will explore the fascinating world of Penrose tiling, its mathematical foundations, and how it can be beautifully integrated into mosaic art. What is Penrose Tiling? Penrose tiling refers to a non-periodic tiling pattern composed of two distinct shapes—often referred to as "kites" and "darts." Unlike regular tiling patterns that repeat periodically, Penrose tiling covers a plane without repeating itself. This unique attribute makes it particularly intriguing for both mathematicians and artists, offering a blend of order and unpredictability that captivates the eye and the mind. The Artistic Appeal Penrose tiling captivates with its unique aesthetic and intricate patterns. Here's why it's so visually interesting: Endless Variety: Unlike traditional patterns, Penrose tiling never repeats, offering endless visual interest and surprise. Golden Beauty: The design often incorporates the golden ratio, historically linked to aesthetic beauty and harmony. Unique Symmetry: It features fivefold rotational symmetry, adding a unique and uncommon appeal. These features make Penrose tiling a favorite among artists looking to create visually stunning and unconventional designs. Penrose Tiling in Mosaic Art Incorporating Penrose tiling into mosaic art offers a unique opportunity to create pieces that are both visually stunning and intellectually stimulating. Here are some ways mosaic artists can utilize Penrose tiling in their work: Creating Dynamic Visual Effects One of the most striking features of Penrose tiling is its ability to create dynamic visual effects. The non-repeating nature of the pattern ensures that each section of the mosaic offers a new and intriguing visual experience. This can be particularly effective in large installations, where the viewer can spend considerable time exploring the intricate details. Versatility in Design Penrose tiling can be adapted to suit a variety of artistic styles and themes. Whether you're aiming for a modern, abstract piece or a more classical design, the versatility of Penrose tiling allows for endless creativity. Artists can experiment with different color schemes, materials, and textures to create unique and personalized works of mosaic art. Symbolic Significance The mathematical properties of Penrose tiling imbue it with a sense of mystery and depth. For those who appreciate the intersection of art and science, incorporating Penrose tiling into a mosaic can be a way to symbolize the beauty of mathematical order and the infinite possibilities of human creativity. Examples of Penrose Tiling in Mosaic Art Numerous artists and architects have successfully integrated Penrose tiling into their works, creating pieces that are both aesthetically pleasing and intellectually engaging. Here are a few notable examples: Public Installations Public spaces such as parks, plazas, and building facades offer excellent opportunities for large-scale Penrose tiling mosaics. These installations can transform ordinary spaces into extraordinary visual experiences, inviting viewers to explore and interact with the artwork. Interior Design Penrose tiling can also be used in interior design to create stunning floors, walls, and ceilings. The unique pattern can add a sense of sophistication and intrigue to any space, making it a popular choice for high-end residential and commercial projects. Artistic Pieces For individual artists, Penrose tiling offers a rich source of inspiration for smaller, more intimate works. Whether it's a framed mosaic, a decorative tabletop, or a piece of wearable art, the versatility of Penrose tiling ensures that each creation is a one-of-a-kind masterpiece. Penrose tiling represents a fascinating intersection of mathematics and art, offering endless possibilities for creativity and innovation. As a mosaic art company, we are continually inspired by the timeless beauty and intellectual depth of this unique pattern. Whether you're an artist seeking new ideas or a lover of fine art looking to explore something different, Penrose tiling offers a world of wonder waiting to be discovered. If you're interested in learning more about our work or commissioning a piece that incorporates Penrose tiling, browse our collection on our website!
An exhibit of mathematical art reveals the aesthetic side of math.
The Penrose collection of patterned ceramic tiles offer an authentic and beautiful decorative tile perfect for kitchens, bathrooms, hallways and entrances alike. In a classic square format, the Penrose White patterned ceramic offers rustic tones of off white and cream. Rectified: No (see installation information for further details) R Rating: R9 Variation: Low variation Pattern Repeat: 10 tiles Tiles Per Box: 5 tiles PEI Rating: 4 Composition: Ceramic
