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The Mitre System


Origins of The Mitre System
by Adrian Fisher,EdPegg Jr, and MiroslavVicher -- Patent Pending

Tree of Life using the Mitre System copyright (c) 2000by Adrian Fisher.  Used with permission.
Back in 1998, I (EPJ) noticed the Fisher Pavers over at Adrian'ssite.  I considered these carefully designed heptagons and squares quite clever.  They could make some very nice designs, but I didn'tfind 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 (ChaosTiles) and 10-gon (Penrose System), so we started trying to find interesting ways to cut up n-gons.  All of the following can be extended into spectacular tessellations, but most cannot make squares or rectangles.

n-gon dissections copyright (c) 1998 by Adrian Fisherand 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 PeggJr.
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 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 original piece the Pyramid (hint: squint).  We called the new piece the Fin, and set about finding new squares.  We ran into difficulties while trying to find large nested squares.  Adrian solved the problem byusing some 30-60-90 triangles, but I didn't like it.  I suggested a piece that might work, but then later told Adrian that the piece wouldn'twork.  He wrote me back to tell me I was wrong -- the piece DID work! It worked wonderfully!

Nested Squares using the Mitre System copyright (c) 1998by 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 anysize -- 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.
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.  Dolook at Miroslav Vicher's site, for more.  For example, the following is one of over ten thousand dodecagon patterns he found.

Dodecagon using the Mitre System copyright (c) 1999 byMiroslav 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.

For Tiling and Paving Project Enquiries, and Tiling and Paving Manufacturers:
   Adrian Fisher, Victoria Lodge, 5 Victoria Grove, Portsmouth,PO5 1NE, England. Phone: +44 23 9235 5500 adrian@mazemaker.com

For Puzzles, Games, Recreational Mathematics, and other Creations:
Ed Pegg Jr, 529 S. Hancock, Colorado Springs CO 80903, United States. Phone: 719-Ed Pegg-9.  ed@mathpuzzle.com

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


All material on this site is copyright 2000 by Ed Pegg Jr.
  Can you find the Y2K error on this site?
  Copyrights of submitted materials stays with contributor and is used with permission. visitors since October 29, 1998.