thats a very simple to show you, Schemes and Mind Maps of Mathematical Methods

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Before we can understand how a glide reflection tessellation is created, it is important to review a few key mathematical concepts. The first is reflection : this is when an object is flipped across a line—called the line of reflection—without

changing its shape or size. By reflecting the object across this line, the original shape is transformed into a mirror image of itself. The reflected shape is also the same distance away from the line of reflection.

We see reflected shapes in our everyday lives. For example, reflections in smooth

water. See how the bodies of the birds in the photograph above look as if they have been “flipped” upside down in the surface of the water?

We can also reflect an object multiple times, over several lines. In the diagram to the right, notice how triangle A is flipped across line of reflection 1 to produce shape B.

Then, triangle B—that is, the center green triangle—is flipped across a second line of reflection, resulting in shape C.

Finally, we should briefly review the term “ tessellation .” A tessellation is a repeating pattern of shapes that can continue infinitely on a plane—a flat surface. This pattern must first have no gaps or holes between shapes , and second have no overlaps

between shapes.

Which of the following four patterns meet are tessellations?

Answer: B and C are tessellations, A and D are not—A because there are gaps

between the circles, and D because the shapes overlap.

Having reviewed the terms reflection, glide reflection, and tessellation, we can now understand how glide reflection tessellations are made! Firstly, a glide reflection tessellation is a pattern with no gaps or overlaps (e.g. a tessellation), made by

flipping (e.g. reflecting) and sliding (e.g. translating) a shape to repeat it. Similar to the glide reflection diagram we saw on a previous slide, here you can see how the arrow “A” is flipped across a line of reflection and moved parallel to it in order to produce the arrow “C.”

Sources:

[Slide 1] “Tropical Fish.” Arcadia’s Art Exhibit. http://cornflower.tripod.com/exhibit.html.

[Slide 2] “Reflections in Math: Definition and Overview.” Study.com. https://study.com/academy/lesson/reflections-in-math-definition-lesson-quiz.html

[Slide 3] “Glide Reflection.” Wikipedia.com. https://en.wikipedia.org/wiki/Glide_reflection

[Slide 3] “How Glide Reflections Work.” Dummies.com. https://www.dummies.com/education/math/geometry/glide-reflections-work/.

[Slide 4] “Tessellations.” MathEngaged.org. http://mathengaged.org/resources/activities/art- projects/tessellations/.

[Slide 5] “Arrow Tessellation.” in “M.C. Escher and Tessellations” (University of Waterloo: Centre for Education in Mathematics and Computing: 2015) https://www.cemc.uwaterloo.ca/events/mathcircles/2015-16/Fall/Junior78_Nov34- Solns.pdf.

[Slide 7] “Reflection Tessellation.” Melinda Kolk. 2020. Web.tech4learning.com. https://web.tech4learning.com/create-reflection-tessellations-in-wixie.

[Slide 9] “Regular Tiling No. 17.” M.C. Escher.

[Slide 10] “Horsemen.” M.C. Escher. 1946.