Mitre2

The Mitre System set and Frames. Patented by Adrian Fisher and Ed Pegg Jr.

Origins

Core Shapes and Supporting Shapes of The Mitre System. Patent Pending by Adrian Fisher and Ed Pegg Jr. For more on the shapes, click here.

by Adrian Fisher, Ed Pegg Jr, and Miroslav Vicher

Back in 1998, I (EPJ) noticed the Fisher Pavers over at Adrian's site. I considered these carefully designed heptagons and squares quite clever. They could make some very nice designs, but I didn't find them chaotic enough. So, I started playing with Autocad LT, trying to find something better.

These images of the Fisher Paver System are copyright (c) 2000 by Adrian Fisher. Used with permission. My first finding was a pentagon that could be put together to make an 18-gon. This turned out to be the Hirschhorn Medalion. When I tried putting two medalions together, I only needed one extra piece to tile the plane chaotically.

This image of Ed Pegg's Chaos Tile system is copyright (c) 1998 by Ed Pegg Jr. I shared this with Adrian a bit tauntingly. "Your Fisher Pavers are nice, but they can't make chaotic patterns." This began a long correspondence. My Chaos Tiles were nice, but doing the edging would be dreadful. He suggested that this was one of the reasons that the Penrose Tiles weren't widely used. If you look at the edges of the rectangle above, you'll see a tile-worker's nightmare. After a lot of e-mails and phone calls, the following Rules of Decorative Bricks were decided upon:
1. No fault lines across the tessellation in any direction.
2. Ability to make squares and rectangles.
3. No sharp angles -- they break too easily
4. Under a modular layout, the ability to lay out a design with minimal cutting.
5. Spirals and other curves should be possible by using differently colored bricks.
6. The tiling should be able to interface with the modules of other tiling systems.
7. The tiling should use few pieces.
8. The ability to make both periodic and nonperiodic designs is desirable.
9. The pieces must be of similar size. (Dodecagons and triangles won't work for weight-bearing floors)

Semiregular Dodecagon-Triangle design by Archimedes. Can you find all 8 semiregular tessellations? Both of us were intrigued by the results of dividing the 18-gon (Chaos Tiles) and 10-gon (Penrose System), so we started trying to find interesting ways to cut up n-gons. All of the following designs can be extended into spectacular tessellations, but most cannot make squares or rectangles. It is well known that any n-gon can be divided into rhombi, but we were interested in using a minimal number of pieces. Greg Frederickson talks about this a bit in his dissections book. What is the largest n-gon that can be divided nicely, like these? To avoid an obvious answer, the minimal angle on any piece must be greater than 30 degrees.

n-gon dissections copyright (c) 1998 by Adrian Fisher and Ed Pegg Jr. The 20-gon isn't regular, the others are. We had a lot of fun tossing designs back and forth at each other. In the meantime, I contracted Kadon Enterprises to lasercut some of the pieces of Chaos Tiles for me. Soon, I had a bag of botched pieces (my fault - I sent the wrong angles). Turns out that this piece could make 24-gons, when squares were added. More importantly, half of this piece had a very nice property -- It could make squares! Thus started the Mitre System. The date was 18 December 1998 by this point. Earlier, I had discovered an almost square with a single piece. Adrian was finding lots of fascinating patterns, but didn't see the square. So I sent it to him.

Design C1853 copyright (c) 1998 by Adrian Fisher. Used with permission. 20-piece Square copyright (c) 1998 by Ed Pegg Jr. He responded almost immediately with the Terrazzo Tiling, which used the same pieces to make a larger square. I didn't expect this.

Terrazzo Tiling design copyright (c) 1998 by Adrian Fisher. Used with permission. We called this original piece the Pyramid (hint: squint). We eventually decided that the best way to fill in the small square was to extend the existing piece to make a new piece we called the Fin. While trying to find large nested squares, we ran into difficulties. Adrian solved the problem by using some 30-60-90 triangles, but I didn't like it. I suggested the shape of a piece that might work, but then later sent Adrian the sad news that it wouldn't work. He wrote me back to tell me I was wrong -- the piece DID work! It worked wonderfully! Indeed, the three pieces allowed us to make all manner of nested shapes.

Nested Squares using the Mitre System copyright (c) 1998 by Ed Pegg Jr. We called the new piece the Mitre. Using all three pieces, a wide variety of squares and rectangles of can be made, all with elaborate, beautiful designs. We also found a variety of supporting pieces that worked very well with the core shapes. Later, we filed a patent on the system. Investigating patents at the US Patent & Trademark Office was interesting. Patent 4133152 is the Penrose Tiling system.

A variety of squares using the Mitre System copyright (c) 1999 by Ed Pegg Jr, Adrian Fisher, and Miroslav Vicher. Used with permission. In addition to discovering a variety of squares, we also found hexagons and dodecagons. Due to a quirk in the construction, the same set of pieces can be used to almost perfectly fill large rectangles of any size -- if the rectangle is large enough. This feature allows tile to be laid with much less cutting. Hexagonal layouts are possible with many patterns.

A variety of hexagons using the Mitre System copyright (c) 1999 by Ed Pegg Jr and Adrian Fisher. Used with permission. For hexagonal tilings, the Bell shape proved very useful. Depending on the needs of a particular tiling design, a number of supporting shapes were formalized. Follow this link to a page detailing more shapes.

I'd gotten to know Miroslav Vicher when he started solving a variety of difficult tiling problems, such as Eternity. After Adrian and I filed the patent, we hired Mira to look for solutions with this system using his program, and he found thousands. Do look at Miroslav Vicher's site, for more. For example, the following is one of over ten thousand dodecagon patterns he found. He found many squares and rectangles that eluded us, and a wide variety of beautiful tessellations.

Dodecagon using the Mitre System copyright (c) 1999 by Miroslav Vicher. Used with permission. Much interest has been shown in the Mitre System for decorative design. This page is a mere taste of what we've found. After our contact information, you'll see an example of a modularized Mitre System layout. There are more designs by Adrian Fisher available.
For actual paving and house design, this page describes the different scale systems.

For all inquiries, please contact:
Adrian Fisher, Victoria Lodge, 5 Victoria Grove, Portsmouth, PO5 1NE, England.
Phone: +44 23 9235 5500 Email: adrian@mazemaker.com

Fountain copyright (c) 2000 by Adrian Fisher. Used with permission.

New Tessellation Systems You can see Semiregular tessellations here (by M. Vicher). Pentagons that tile the plane are here. Chaim Goodman-Strauss wrote a page on tessellations. There's a program called Tessellmania. M C Escher liked them. Andrew Crompton likes them. Robert Fathauer sells tessellations at www.tessellations.com. Kadon Enterprises also sells them. Patent 4,133,152 was awarded to Sir Roger Penrose for a tessellation. Adrian Fisher has patent 5,945,181. There are 187 patented tessellations listed at the US Patent Office. Both Chaos Tiles and the Mitre System have been awarded a patent.

But none of the above links gives either of the following tessellations, both done with the same shape. This quadrilateral can tile the plane with either five-fold symmetry or six-fold symmetry.

There are many other interesting single tile tessellations. For example, Michael Dowle has done an extensive study on Iamond tessellations. Here is a new page by him on Enneiamond tessellations. You can also look at Michael's other tessellation pages on Small Iamonds, Hexiamonds,Heptiamonds, Octiamonds, and Radial/Random Tessellations. In a similar vein, Michael Reid has a page on rectifiable polyominoes. The Poly Pages by Andrew Clarke has a section on tessellations.

When I made the Chaos Tiles, I found a large number of interesting tessellations based on the nonagon. Here's one of them.

By putting paths on these pieces, complex looking celtic knots can be made. How many different pieces are required for the following figure? For the really adventuresome puzzle solvers, I'd like a flawless celtic knot for the 30-gon, with a minimal number of pieces.

One of the bonuses of The Mitre System is the seemingly infinite number of beautiful patterns. Miroslav Vicher found many. Here is one I found, with fans, pyramids, corks, and a few triangles.

I liked that pattern, but I felt it was possible to use fewer shapes. Here is a Mitre System tiling that just uses two shapes, and an occasional third.

To me, it seems like there is still much to know about tessellations. Much beauty remains to be discovered. Time passes... and now it's the next day. This new shape, in purple, seemed promising. It was. The below pattern uses only pyramids and the new shape, dubbed the Millstone.