2Exploring Tessellations This Exploration of Tessellations will guide you through the following:Definition of TessellationRegular TessellationsSymmetry in TessellationsTessellations Around UsSemi-Regular TessellationsView artistic tessellations by M.C. EscherCreate your own Tessellation
3What is a Tessellation?A Tessellation is a collection of shapes that fit together to cover a surface without overlapping or leaving gaps.
4Tessellations in the World Around Us: Brick WallsFloor TilesCheckerboardsHoneycombsTextile PatternsArtCan you think of some more?
5Are you ready to learn more about Tessellations? CLICK on each topic to learn more…Regular TessellationsSemi-Regular TessellationsSymmetry in TessellationsOnce you’ve explored each of the topics above, CLICK HERE to move on.
6Regular 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:TriangleSquarePentagonHexagonOctagon
7Regular 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:TriangleSquarePentagonHexagonOctagonOnce you’ve discovered whether each of the regular polygons tessellate or not, CLICK HERE to move on.
8Regular Tessellations Does a Triangle Tessellate?The shapes fit together without overlapping or leaving gaps, so the answer is YES.
9Regular Tessellations Does a Square Tessellate?The shapes fit together without overlapping or leaving gaps, so the answer is YES.
10Regular Tessellations Does a Pentagon Tessellate?GapThe shapes DO NOT fit together because there is a gap. So the answer is NO.
11Regular Tessellations Does a Hexagon Tessellate?The shapes fit together without overlapping or leaving gaps, so the answer is YES.Hexagon Tessellation in Nature
12Regular Tessellations Does an Octagon Tessellate?GapsThe shapes DO NOT fit together because there are gaps. So the answer is NO.
13Regular Tessellations As it turns out, the only regular polygons that tessellate are:TRIANGLESSQUARESHEXAGONSSummary 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.
14Hexagon, Square & Triangle Semi-Regular TessellationsSemi-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 & TriangleSquare & TriangleHexagon, Square & TriangleOctagon & SquareOnce you’ve explored each of the semi-regular tessellations, CLICK HERE to move on.
15Semi-Regular Tessellations Hexagon & TriangleCan you think of other ways to arrange these hexagons and triangles?
16Semi-Regular Tessellations Octagon & SquareLook familiar?Many floor tiles have these tessellating patterns.
19Semi-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?
20Symmetry in Tessellations The four types of Symmetry in Tessellations are:RotationTranslationReflectionGlide ReflectionCLICK on the four types of symmetry above to learn more.Once you’ve explored each of them, CLICK HERE to move on.
21CLICK HERE to view some examples of rotational symmetry. Symmetry in TessellationsRotationTo 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 some examples of rotational symmetry.Back to Symmetry in Tessellations
25CLICK HERE to view some examples of translational symmetry. Symmetry in TessellationsTranslationTo 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 some examples of translational symmetry.Back to Symmetry in Tessellations
27CLICK HERE to view some examples of reflection symmetry. Symmetry in TessellationsReflectionTo 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 some examples of reflection symmetry.Back to Symmetry in Tessellations
30CLICK HERE to view some examples of glide reflection symmetry. Symmetry in TessellationsGlide ReflectionA 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 some examples of glide reflection symmetry.Back to Symmetry in Tessellations
32Glide Reflection Symmetry Back to Glide Reflections
33Symmetry in Tessellations Summary of Symmetry in Tessellations:The four types of Symmetry in Tessellations are:RotationTranslationReflectionGlide ReflectionEach of these types of symmetry can be found in various tessellations in the world around us, including the artistic tessellations by M.C. Escher.
34Exploring Tessellations We have explored tessellations by learning the definition of Tessellations, and discovering them in the world around us.
35Exploring Tessellations We have also learned about Regular Tessellations, Semi-Regular Tessellations, and the four types of Symmetry in Tessellations.
36Create 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 HandHow to create a Tessellation on the ComputerOnce 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.
37Exploring 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!