| 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. |
| Diplex_Instruments Manufacture, import and distribute meteorological and scientific instruments. Specialist in thermometers. |
| DWC_(Danish_Wind_Controller_A/S) Rugged wind sensors for critical environmental conditions. |
| Dynalab Manufacturer of Meteorological sensors and Automatic Weather stations |
| Dynamax,_Inc_ Provides instruments and sensors for plant, water, environment, and hydrology. |
| E+E_Elektronik Humidity and temperature transmitters, sensors, and data loggers that can be used for meteorology. |
We find such repeated patterns on wallpaper everywhere in our daily life. Have you considered how many kinds of repetitions there are for patterns on a plane? Do you think there are five or six kinds? No, there are seventeen! We can show this using the mathematics of group theory. The following tables show examples in traditional Japanese patterns: (A) Without rotations. p1 
(Bingata) pm 
(Yakata-Mon) pg 
(Hana-Usagi-Kinran) cm 
(Seikaiha) (B) With rotations of 180 degrees. Without rotations of 90, 60 degrees. p2 
(Ariso-Donsu) pmm 
(Yoshiwara-Tsunagi) pgg 
(Fundo-Tsunagi) cmm 
(Takeda-Bishi) pmg 
(Shikan-Jima) (C) With rotations of 90 degrees. p4 
(Rokuyata-Koshi) p4m 
(Sippo-Tsunagi) p4g 
(Sayagata) (D) With rotations of 120 degrees. Without rotations of 60 degrees. p3 
(Onaga-Mitsudomoe)p31m
(Bisyamon-Kikko) p3m1
(Kasane-Kikko-ni-Wa) (E) With rotations of 60 degrees. p6 
(Kagome) p6m 
(Asanoha) Above we have shown the mathematical symbols to distinguish the seventeen kinds of patterns and representative traditional Japanese patterns. For long, I could not check an example of type p3 among traditional Japanesepatterns. However, in the end of 2002, Professor Amiko Tone and ProfessorSezo Kondo at Kyoto Furitsu University informed me that examples of typep3 among traditional Japanese patterns actually exist. I express thanksfor both professors.In this table we describe the Japanese nomenclature for the names of thepatterns. p1Bingatagata = shape, Bingata = a generic name of traditional dyed patterns of the Okinawan IslandspmYakata-MonYakata = big house, Mon = pattern, Yakata-Mon = a symbol of the Ise Shrine pgHana-Usagi-Kinran Hana = flower, Usagi = rabbit, Kinran = the name of method of weaving with golden thread in the texturecmSeikaihaSei = blue, kai = ocean, ha = wavep2Ariso-Donsu Ariso = abbreviation of ara-iso, ara = wild, iso = rocky beach, Donsu = the name of method of weaving in texturepmmYoshiwara-TsunagiYoshiwara = the name of a downtown area in Edo, Tsunagi = connected objects, Edo = the former name of TokyopggFundo-TsunagiFundo = weight for a balance, Tsunagi = connected objectscmmTakeda-BishiTakeda = the name of a daimio (feudal lord) in the 16th century, Bishi = rhombuspmgShikan-JimaShikan = the name of a kabuki actor, Jima = stripesp4Rokuyata-KoshiRokuyata = the name of a character in kabuki play, Koshi = latticep4mSippo-Tsunagi Sippo = seven kinds of treasure, Tsunagi = connected objectsp4gSayagataSaya = silk textile with beautiful patterns, gata = shape or patternp3Onaga-MitsudomoeO = tail, naga = long, Mitsu = three, Mitsudomoe = the name of a Japanese traditional crest repeated in the pattern p31m Bisyamon-KikkoBisyamon = the name of a Buddhist god, Kikko = carapace of a tortoisep3m1Kasane-Kikko-ni-Wa Kasane = repeated, Kikko = carapace of a tortoise, ni = and, Wa = ringp6KagomeKago = basket, me = eye or holep6mAsanohaAsa = hemp, no = of, ha = leafThe examples shown here of pmg and p4 are related to kabuki plays.The patterns were used not only on the costumes but were dyed onsouvenir hand towels and distributed to fans.In particular, the name of Shikan-Jima demonstrating pmg has a double meaning as a joke. Shikan is the name of an actor. (Later he changed his name to Nakamura Utaemon. In Japan sometimes the same name is handed down in a family.Shikan is the second Nakamura Utaemon.)At the same time "shi" means "four" and "kan" means "ring" or "handle"; and "Jima", stirpes: as in the pattern offour stripes and the handle of a chest.Kasane-Kikko-ni-Wa, the p3m1 example, was often used on noh costumes.Let us give a more detailed mathematical explanation.
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