| Symmetries,_Patterns_and_Tessellations Constructed with Geometer's Sketchpad by Allan Bergmann Jensen. |
| Symmetry_and_Pattern__The_Art_of_Oriental_Carpets Resources compiled by the Math Forum. |
| Symmetry_and_the_Shape_of_Space With interactive demos using Geometer's Sketchpad (requires Macintosh). |
| Symmetry_and_the_Shape_of_Space Chaim Goodman-Strauss, Arkansas. With interactive demos using Geometer's Sketchpad (requires Macintosh). |
| TESS Tessellation software. 11 rosette, all 7 frieze and all 17 wallpaper groups included. |
| Tessellated,_Interlocking_Concrete_Pavers_and_Molds Pictures of some concrete tilings. |
| Tessellating_Wallpaper_Patterns Original tessellating desktop wallpaper tiles. |
| Tessellation_and_Photography A study for the endless possibilities of tessellating the plane based on the work of Dutch graphic artist M.C. Escher. |
| Tessellation_Links Compiled by Suzanne Alejandre for the Math Forum. |
| Tessellation_Tutorials_by_Suzanne_Alejandre Tutorials and templates for making tessellations using ClarisWorks, the Geometer's Sketchpad, HyperCard, HyperStudio, and straightedge and compass, including step-by-step instructions for classroom |
| Tessellations A company selling foam puzzles, many based on planar tessellations (tilings) and fractals. |
| Tessellations__Technology_and_Culture Examples and Java applets. |
| Tessellations_Using_Geometer\'s_Sketchpad Description of a class project. |
| Tessellations_with_Java Examples and source code. |
| Tiling_Plain_and_Fancy Tiling from the mathematical and historical viewpoint by Steve Edwards. Provides a number of examples of various types of tiles that tile the plane and includes periodic tiling plus Penrose's aperiod |
| Tilings_and_Geometric_Ornament Applying principles of computer graphics to the creation of geometric ornament, as a continuation of the tradition of ornamental design using modern tools and algorithms. |
| Totally_Tessellated An introduction to tessellations explaining the basic underlying mathematics, and with many examples from real life. |
| 17_Wall_Paper_Symmetry_Groups_to_Create_a_Regular_Division_of_the_Plane Graphics. |
| Wallpaper_Groups Images and descriptions of the 17 plane symmetry groups. |
| William\'s_Geometric_Construction_Page Activities to create patterns using simple resources, such as making equilateral triangle from a circle. Suitable for upper primary and lower secondary ages. |
| Aanderaa_Instruments Instruments and Sensors for Land, Sea and Air |
| Adcon_Telemetry_GmbH Provide meteorological sensors, but specialty is wireless telemetry. |
| Advanced_Designs_Corp_ Doppler Weather Radar and Weather Data Displays, Doppler Radar, Weather Radar, Doppler, Radar |
| Advanced_Sensing_Products ASP is a manufacturer standard PRTs, precision RTDs, and different types of thermocouples. |
| Alltec_Corp_ A manufacturer of lightning protection and grounding systems. |
| AllWeatherInc A source of weather sensors, environmental monitoring systems, data collection products, and display and recording devices. |
| Applied_Technologies,_Inc_ Manufacturer and supplier of scientific instrumentation, software, and systems for meteorological and atmospheric research. |
| Atmospheric_Radar_Systems_(ATRAD) Atmospheric Radar systems for scientific and commercial applications. |
| Axinum_Innovations_&_Technologies Manufacture of Siphoning Rain Gauges and Data Loggers |
| B__K__Consimpex_Meteorological_Systems Satellite remote sensing & metereological system integrators in India. |
| Belfort_Instruments Offer a variety of weather instruments for the measurement of Wind, Precipitation, Temperature, Humidity, Pressure, and Visibility. |
| BOLTEK A leader in affordable thunderstorm tracking technology and Lightning Detection Systems. |
| Campbell_Scientific_Ltd_ Professional weather monitoring products, including a full line of automated, research-grade weather stations. |
| Casella_CEL Manufacturer and supplier of meteorological equipment, weather stations, rain gauges, solar equipment and wind speed and direction. |
| Climatronics_Corp_ Meteorological systems for ambient surface and upper air weather stations and networks. Offering sensors for wind, temperature, humidity, rainfall, pressure, solar, ice, clouds and visibility. |
| Coastal_Environmental_Systems Coastal Environmental designs and manufactures weather stations, meteorological, environmental monitoring instrumentation, and data acquisition systems. |
| Columbia_Weather_Systems Makers of the Capricorn weather stations. Systems include wind speed and direction, temperature, barometric pressure, humidity, rainfall, leaf wetness, solar radiation and soil moisture. |
| Comptus_Inc_ Manufacturer of meterological instruments for measuring wind, rain, snow and other weather elements. |
| Corr-Tek_Idrometria Environmental Monitoring: Products for Sampling and Monitoring Water |
| Davis_Instruments_Corp_ Weather instrumentation for amateur and commercial boaters, farmers, meteorologists, hobbyists. Introductory Flash option. |
PDF VersionThe Penrose Tiling at Miami Universityby Dr. David Kullman, Mathematics and Statistics Department In September, 1976, Miami University held its Fourth Annual Mathematics and Statistics Conference. The theme that year was Recreational Mathematics and among the featured speakers were John Conway, Elwyn Berlekamp, David Klarner, Doris Schattschneider, Leon Bankoff, and Wolfgang Haken. Conway had just published his book, On Numbers and Games, and Haken had announced a computer-assisted proof of the Four Color Theorem only a month earlier. On the spur of the moment, to fill in for a speaker who had cancelled at the last minute, Conway offered to give a talk about "Penrose's Puzzle Pieces." (He just happened to have some transparencies with him.) This was the first time that most of us in the audience had heard about this intriguing set of two tiles, kites and darts, that tile the plane only nonperiodically. That is, no region of the tiling will tile by means of translations alone. The following January, Martin Gardner, in his "Mathematical Games" column in Scientific American presented "for the first time" a description of the Penrose tiles, including many of Conway's results concerning them. Meanwhile Bachelor Hall, which would become the new home of Miami's Department of Mathematics and Statistics, was still on the drawing board. Although we would share the facility with the departments of English and Communication, some of the mathematics faculty hoped that the design would include something uniquely mathematical. Milton Cox (who would become Chairman of the MAA Ohio Section ten years later) proposed to the University administration that a Penrose tiling would be a fitting decoration for the Bachelor Hall courtyard. After seeing the article in Scientific American, the vice president who oversaw the building projects and the architect agreed. Milt was appointed to come up with a specific design. (Perhaps it is more than coincidence that Milt's middle name is Dart and that he had an aunt whose last name was Kite.) The way Milt tells the story, he took the magazine article, some outlines of Penrose tilings, and his daughter's colored pencils with him on a trip to Boston, where he examined the tilings in the Massachusetts Statehouse. He decided that Penrose's cartwheel design was the most interesting, and he chose the colors of wedgewood blue, cream, red, and white to complement the red brick exterior of the building. The architect selected terrazzo, which is made by embedding small chips of marble or granite in mortar and then polishing the surface after it has dried, as the tiling medium. A brass framework was fabricated in the outline of the tiling, and the colored tiles were poured individually. Work on the terrazzo began in the summer of 1979, while both Milt and ourdepartment chair, Elwood Bohn, were on vacation. When they returned, itwas discovered that the brass frame had been rotated 90 degrees so that thetiling's axis of symmetry did not coincide with the axis of symmetry ofthe building. Work was immediately stopped while the powers that be scratched theirheads. One idea that was rejected was to rotate the building 90 degrees.Fortunately, only about 25% of the tiles had actually been poured, so itwas decided to chisel them out and start over, with the frame properlyoriented. Only a couple of days' work were lost in the process. Another minor flaw developed when a drain had to be installed in the very center of the tiling, to help carry rain water out of the courtyard. Here is an example of the beauty of pure mathematics having to be tempered by constraints of the real world. Bachelor Hall was dedicated in the fall of 1979, and the Penrose tilinghas been a focal point ever since. Its overall diameter is approximately22 feet, with individual tiles measuring 10.5 by 17 inches. Every semesterstudents can be seen measuring the tiles or sketching the design as partof some class assignment in mathematics, art, or architecture. We believethat this was the first Penrose tiling to be incorporated in the design ofa public building and it may still be the only one executed in terrazzo.(In 1993 another cartwheel Penrose tiling, 15 feet in diameter, was constructed of ceramic tiles at Carleton college in Northfield, MN.)Presented at the MAA Ohio Section Meeting Shawnee State University, October 24, 1997Return to Top PageReturn to MiamiLINK