This Exploration of Tessellations will guide you through the following:

2. Exploring Tessellations This Exploration of Tessellations will guide you through the following: Definition ofTessellation RegularTessellations Symmetry inTessellations TessellationsAround Us Semi-RegularTessellations View artistictessellationsbyM.C. Escher Create yourownTessellation

3. What is a Tessellation? A Tessellation is a collection of shapes that fit together to cover a surface without overlapping or leaving gaps.

4. Tessellations in the World Around Us: Brick Walls Floor Tiles Checkerboards Honeycombs Textile Patterns Art Can you think of some more?

5. Are you ready to learn more about Tessellations? CLICK on each topic to learn more… Regular Tessellations Semi-RegularTessellations Symmetry inTessellations Once you’ve explored each of the topics above, CLICK HERE to move on.

6. Regular Tessellations Regular Tessellations consist of only one type of regular polygon. Do you remember what a regular polygon is? A regular polygon is a shape in which all of the sides and angles are equal. Some examples are shown here: Triangle Square Pentagon Hexagon Octagon

7. Regular Tessellations Which regular polygons will fit together without overlapping or leaving gaps to create a Regular Tessellation? Maybe you can guess which ones will tessellate just by looking at them. But, if you need some help, CLICK on each of the Regular Polygons below to determine which ones will tessellate and which ones won’t: Triangle Square Pentagon Hexagon Octagon Once you’ve discovered whether each of the regular polygons tessellate or not, CLICK HERE to move on.

8. Regular Tessellations Does a Triangle Tessellate? The shapes fit together without overlapping or leaving gaps, so the answer is YES.

9. Regular Tessellations Does a Square Tessellate? The shapes fit together without overlapping or leaving gaps, so the answer is YES.

10. Regular Tessellations Does a Pentagon Tessellate? Gap The shapes DO NOT fit together because there is a gap. So the answer is NO.

11. Regular Tessellations Does a Hexagon Tessellate? The shapes fit together without overlapping or leaving gaps, so the answer is YES. Hexagon Tessellationin Nature

12. Regular Tessellations Does an Octagon Tessellate? Gaps The shapes DO NOT fit together because there are gaps. So the answer is NO.

13. Regular Tessellations As it turns out, the only regular polygons that tessellate are: TRIANGLES SQUARES HEXAGONS Summary of Regular Tessellations: Regular Tessellations consist of only one type of regular polygon. The only three regular polygons that will tessellate are the triangle, square, and hexagon.

14. Semi-Regular Tessellations Semi-Regular Tessellations consist of more than one type of regular polygon. (Remember that a regular polygon is a shape in which all of the sides and angles are equal.) How will two or more regular polygons fit together without overlapping or leaving gaps to create a Semi-Regular Tessellation? CLICK on each of the combinations below to see examples of these semi-regular tessellations. Hexagon & Triangle Square & Triangle Hexagon, Square & Triangle Octagon & Square Once you’ve explored each of the semi-regular tessellations, CLICK HERE to move on.

15. Semi-Regular Tessellations Hexagon & Triangle Can you think of other ways to arrange these hexagons and triangles?

16. Semi-Regular Tessellations Octagon & Square Look familiar? Many floor tiles have these tessellating patterns.

17. Semi-Regular Tessellations Square & Triangle

18. Semi-Regular Tessellations Hexagon, Square, & Triangle

19. Semi-Regular Tessellations Summary of Semi-Regular Tessellations: Semi-Regular Tessellations consist of more than one type of regular polygon. You can arrange any combination of regular polygons to create a semi-regular tessellation, just as long as there are no overlaps and no gaps. What other semi-regular tessellations can you think of?

20. Symmetry in Tessellations The four types of Symmetry in Tessellations are: Rotation Translation Reflection Glide Reflection CLICK on the four types of symmetry above to learn more. Once you’ve explored each of them, CLICK HERE to move on.

21. Symmetry in Tessellations Rotation To rotate an object means to turn it around. Every rotation has a center and an angle. A tessellation possesses rotational symmetry if it can be rotated through some angle and remain unchanged. Examples of objects with rotational symmetry include automobile wheels, flowers, and kaleidoscope patterns. CLICK HERE to view someexamples of rotational symmetry. Back to Symmetry in Tessellations

22. Rotational Symmetry

23. Rotational Symmetry

24. Rotational Symmetry Back to Rotations

25. Symmetry in Tessellations Translation To translate an object means to move it without rotating or reflecting it. Every translation has a direction and a distance. A tessellation possesses translational symmetry if it can be translated (moved) by some distance and remain unchanged. A tessellation or pattern with translational symmetry is repeating, like a wallpaper or fabric pattern. CLICK HERE to view someexamples of translational symmetry. Back to Symmetry in Tessellations

26. Translational Symmetry Back to Translations

27. Symmetry in Tessellations Reflection To reflect an object means to produce its mirror image. Every reflection has a mirror line. A tessellation possesses reflection symmetry if it can be mirrored about a line and remain unchanged. A reflection of an “R” is a backwards “R”. CLICK HERE to view someexamples of reflection symmetry. Back to Symmetry in Tessellations

28. Reflection Symmetry

29. Reflection Symmetry Back to Reflections

30. Symmetry in Tessellations Glide Reflection A glide reflection combines a reflection with a translation along the direction of the mirror line. Glide reflections are the only type of symmetry that involve more than one step. A tessellation possesses glide reflection symmetry if it can be translated by some distance and mirrored about a line and remain unchanged. CLICK HERE to view someexamples of glide reflection symmetry. Back to Symmetry in Tessellations

31. Glide Reflection Symmetry

32. Glide Reflection Symmetry Back to Glide Reflections

33. Symmetry in Tessellations • Summary of Symmetry in Tessellations: • The four types of Symmetry in Tessellations are: • Rotation • Translation • Reflection • Glide Reflection • Each of these types of symmetry can be found in various tessellations in the world around us, including the artistic tessellations by M.C. Escher.

34. Exploring Tessellations We have explored tessellations by learning the definition of Tessellations, and discovering them in the world around us.

35. Exploring Tessellations We have also learned about Regular Tessellations, Semi-Regular Tessellations, and the four types of Symmetry in Tessellations.

36. Create Your Own Tessellation! • Now that you’ve learned all about Tessellations, it’s time to create your own. • You can create your own Tessellation by hand, or by using the computer. It’s your choice! • CLICK on one of the links below. You will be connected to a website that will give you step-by-step instructions on how to create your own Tessellation. • BOOKMARK the website so that you can come back to it later. How to create a Tessellation by Hand How to create a Tessellation on the Computer Once you’ve decided on whether your tessellation will be by hand or on the computer, and you have BOOKMARKED the website, CLICK HERE to move on.

37. Exploring Tessellations Before you start creating your own Tessellation, either by hand or on the computer, let’s take one final look at some of the artistic tessellations by M.C. Escher. The following pieces of artwork should help give you Inspiration for your final project. Good luck!

47. Resources • “Totally Tessellated” from ThinkQuest.org • Tessellations.com • MathAcademy.com • CoolMath.com • MathForum.org • ScienceU.com • MathArtFun.com • MCEscher.com Click to end