Tessellations and Symmetries
Date: 06/07/97 at 21:51:15 From: Joey Fang Subject: Is it possible to make a tessellation with glide reflection, rotation, and translation all in one object? Dr. Math, My homework is to make a tessellation with a rotation, reflection, and translation all in one shape. Is this possible? I've used the TesselMania program from MECC, but it doesn't design 3-way tessellations, only 2-way ones. My teacher said it was possible, but I'm stuck trying to make one. Please help me out. Thanks a lot, Joey
Date: 06/10/97 at 00:05:37 From: Doctor Sarah Subject: Re: Is it possible to make a tessellation with glide reflection, rotation, and translation all in one object? Hi Joey - With the help of Suzanne Alejandre, whose Web tutorials on tessellations are hosted by the Math Forum, we passed your question along to an expert, Prof. Susan Addington of the Math Department at California State University, San Bernardino - see her home page: http://www.math.csusb.edu/faculty/susan/home.html Prof. Addington is the Director of the California Math Show, a portable, interactive math exhibit based on the idea of symmetry. She and technology teacher Alejandre have collaborated on web pages about symmetry in tessellations: http://mathforum.org/sum95/suzanne/wheremath.html Here's how Prof. Addington answered your question: Yes, you can have tessellations that include three types of symmetries, or even all four types (reflection, rotation, translation, glide reflection). There are two main ideas in the reason why: 1. In a symmetric pattern, if you have two symmetries, then you have the combination (also called composition or product) of the symmetries. For example, if your pattern has symmetries of rotation by 45 degrees and rotation by 90 degrees, then it also has rotation by 45 + 90 = 135 degrees. This pattern would also have rotation by 45, 90, 135, 180, 225, 270, 315, and 0 degrees. (Why?) 2. All 4 types of symmetry can be made by combining reflections. - A rotation is the composition of two reflections in intersecting lines. - A translation is the composition of two reflections in parallel lines. - A glide reflection is the composition of a translation and a reflection, so it is the composition of 3 reflections. For some exercises in understanding these facts, see: http://mathforum.org/sum95/suzanne/rex.html So if you build a tessellation that has, say reflections in two parallel lines and a perpendicular line, then the combinations will give you translations, glide reflections, and a 180-degree rotation. TesselMania doesn't include any tessellations with reflection symmetry. You might want to try the free program Kali from the Geometry Center. You can run it over the web at http://www.geom.uiuc.edu/java/Kali or download a Macintosh version from: http://www.geom.uiuc.edu/software/download/kali.html A page with full information about all possible combinations of symmetries (there are 17 combination types) is: http://aleph0.clarku.edu/~djoyce/wallpaper/ (Some parts are fairly technical (college level), but you can get an overview of the subject.) -Doctor Sarah, The Math Forum Check out our web site! http://mathforum.org/dr.math/
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