Exploring Tessellations With Regular And Irregular Polygons
Exploring Tessellations WithRegular and Irregular PolygonsFocus on After this lesson, youwill be able to.φ identify regularand irregularpolygons that canbe used to createtessellationsφ describe whycertain regularand irregularpolygons can beused to tessellatethe planeφ create simpletessellatingpatterns usingpolygonsMosaics are often made of repeatingpatterns of tiles. What patterns doyou see in the design?Many mosaic tile designs are madefrom shapes that cover the area, orthe plane, without overlapping orleaving gaps. These patterns arecalled tiling patterns or tessellations .Covering the plane in this way iscalled tiling the plane .Which shapes can you use to tile or tessellate the plane?tiling pattern a pattern that coversan area or planewithout overlappingor leaving gaps also called atessellation1.Copy the following table into your notebook.ShapeEquilateral triangleIsosceles triangletiling the plane using repeatedcongruent shapes tocover an area withoutleaving gaps oroverlapping also called tessellatingthe planeSquareRegular pentagonRegular hexagonRegular octagonIrregular quadrilateralIrregular pentagonIrregular hexagon446MHR Chapter 12Regular orIrregularPolygon?Measure ofEach InteriorAngleResult:Prediction:Will the shape Does the shapetile the plane? tile the plane?
Select an equilateral triangle block. Is this a regular or irregularpolygon? Record your answer in the table.b) Measure each interior angle and record your measurements inthe table.c) Predict whether the shape will tile the plane. Record yourprediction in the table.2. a)3.Trace the outline of the equilateral triangle. Move the triangle to anew position, so that the two triangles share a common side. Tracethe outline of the triangle again. Continue to see if the shape tilesthe plane. Record your conclusion in the table.4.Use the same method to find out if the isosceles triangle,square, regular pentagon, regular hexagon, and regularoctagon tile the plane. Record your results in the table.5.Cut out the shape of an irregular quadrilateral.a) Predict whether the shape will tile the plane.b) Try to tile the plane with the shape. Record your results inthe table.c) Repeat steps 5a) and 5b) using an irregular pentagon andan irregular hexagon of your own design.Reflect on Your FindingsWhat regular shapes tile the plane? Explain why some regularshapes tile the plane but others do not. Hint: Look at theinterior angle measures. Is there a pattern?b) Explain why some irregular shapes tile the plane but othersdo not.6. a) set of pattern blocks, or cardboard cutoutsof pattern blockshapesprotractorcardboard cutouts ofan isosceles triangle,a regular pentagon,and a regular octagoncardboardscissorsrulerpenta means 5hexa means 6octa means 8LiteracyLinkThe term planemeans a twodimensional flatsurface that extendsin all directions.12.1 Exploring Tessellations With Regular and Irregular Polygons MHR447
Example: Identify Shapes That Tessellate the PlaneDo these polygons tessellate the plane? Explain why or why not.a)b)96º90º90º90º90ºShape A116º116º106º106ºShape BSolutiona) Arrange the squares along a common side. The rotated squaresdo not overlap or leave gaps when you try to form them into atessellation. Shape A can be used to tessellate the plane.90º90º 90º90ºCheck:Each of the interior angles where the vertices of the polygons meetis 90 . The sum of the four angles is 90 90 90 90 360 .This is equal to a full turn. The shape can be used to tessellatethe plane.b)Arrange the pentagons along a common side. The irregularpentagons overlap or leave gaps when you try to form them intoa tessellation. Shape B cannot be used to tessellate the plane.96º96º 96º96ºWhat other possiblearrangements of thepentagons can you find?Do they overlap orleave gaps?Check:Each of the interior angles where the vertices of the polygons meetis 96 . The sum of the four angles is 96 96 96 96º 384º.This is more than a full turn. The shape cannot be used to tessellatethe plane.448MHR Chapter 12
Which of the following shapes can be used to tessellate the plane?Explain your ºc)50º120º120º70º60º A tiling pattern or tessellation is a pattern that covers a planewithout overlapping or leaving gaps. Only three types of regular polygons tessellate the plane. Some types of irregular polygons tessellate the plane. Regular and irregular polygons tessellate the plane when theinterior angle measures total exactly 360 at the point wherethe vertices of the polygons meet.90º 90º90º 90º90º 90º 90º 90º 360º105º 75º105º 75º105º 75º 75º 105º 360º1.Draw three types of regular polygons that tessellate the plane.Justify your choices.2.What are two types of irregular polygons that can be used totessellate the plane? Explain your choices to a friend.3.Megan is tiling her kitchen floor. Should she choose ceramictiles in the shape of a regular octagon? Explain how you know.12.1 Exploring Tessellations With Regular and Irregular Polygons MHR449
9.For help with #4 to #7, refer to the Example onpage 448.4.Patios are often made from interlockingrectangular bricks. The pattern shownbelow is called herringbone.Do these regular polygons tessellatethe plane? Explain why or why not.a)b)On grid paper, create two different patiodesigns from congruent rectangular bricks.5.Use this shape to tessellate the plane.Show and colour the result on grid paper.6.Tessellate the plane with an isoscelestriangle. Use colours or shading to createan interesting design on grid paper.7.Describe three tessellating patterns thatyou see at home or at school. Whatshapes make up the tessellation?8.Jared is painting a mosaic on one wallof her bedroom that is made up oftessellating equilateral triangles. Describetwo different tessellation patterns thatJared could use. Use triangular dot paperto help you describe the tessellations.450MHR Chapter 1210.Some pentagons can be used to tessellatethe plane.a) Describe a pentagon that will tessellatethe plane. Explain how it tessellatesthe plane.b) Compare your pentagon with those ofyour classmates. How many differenttessellating pentagons did you and yourclassmates find?11.A pentomino is a shape made up of fivesquares. Choose two of the followingpentominoes and try to make a tessellationwith each one. Do each of yourpentominoes make a tessellation? Explainwhy or why not.
12.Sarah is designing a pattern for the hoodand cuffs of her new parka. She wants touse a regular polygon in the design andthree different colours. Use grid paper tocreate two different designs that Sarahmight use. Colour your designs.Describe the dual of the originalsquare tessellation.b) Draw a tessellation of regularhexagons. Draw and describe its dual.c) Draw a tessellation of equilateraltriangles. Draw and describe its dual.a)14.13.The diagram shows a tessellation ofsquares. A dot has been added to thecentre of each square. The dots are joinedby dashed segments perpendicular tocommon sides. The result is anothertessellation, which is called the dual ofthe original tessellation.Identify two different regular polygonsthat can be used together to create atessellating pattern. Draw a tessellationon grid paper using the two polygons.Many Islamic artists make very intricate geometricdecorations and are experts at tessellation art.MATH LINKThis tiling pattern is from Alhambra, a Moorishpalace built in Granada, Spain. Four different tileshapes are used to create this pattern.a) Describe the four shapes. Are they regularor irregular polygons?b) Use templates to trace the shapes ontocardboard or construction paper.c)Cut out ten of each shape and use some orall of them to create at least two differenttile mosaics. Use each of the four shapes inyour mosaics.Web LinkTo generate tessellations on the computer, goto www.mathlinks8.ca and follow the links.12.1 Exploring Tessellations With Regular and Irregular Polygons MHR451
Constructing Tessellations UsingTranslations and ReflectionsFocus on After this lesson, youwill be able to.φ identify howtranslations andreflections can beused to create atessellationφ create tessellatingpatterns using twoor more polygonstransformation a change in a figurethat results in adifferent position ororientation set of pattern blocks, or cardboard cutoutsof pattern blockshapesrulerscissorsglue sticktapecardboard orconstruction paper452MHR Chapter 12In section 12.1 you created simple tessellating patterns using regular andirregular polygons. Tessellations can also be made by combining regularor irregular polygons and then transforming them. Do you recognize thepolygons used in this tessellation? What transformations were used tocreate the pattern?How can you create a tessellation using transformations?1.Draw a regular hexagon on a piece ofpaper using a pattern block or cardboardcutout. Cut out the hexagon and glue it toa sheet of cardboard or construction paper.2.Draw two equilateral triangles on a piece of paperusing a pattern block or cardboard cutout. Makesure that the side lengths of the triangles are thesame as the side lengths of the hexagon. Cut outthe triangles and glue them to a sheet of cardboardor construction paper so that they are attached tothe sides of the hexagon as shown.3.Cut out the combined shape. Trace the shape on a new sheet of paper.
4.Translate the shape so that the hexagon fits into the space formed bythe two triangles. Trace around the translated shape and repeat twomore times. What other ways can you translate the shape?5.Translate the combined piece vertically and horizontally so thatthe base of the hexagon is now at the top of one of the triangles.Reflect on Your FindingsDescribe how to use translations to create tessellations.b) What other transformations could you use to get the samepattern as in #5? Explain the difference.6. a)Example: Identify the TransformationWhat polygons and whattransformations are usedto create this tessellation?b) Does the area of thetessellating tile changeduring the tessellation?a)12.2 Constructing Tessellations Using Translations and Reflections MHR453
Solutiona) The tessellation is made froma tessellating tile consisting ofa hexagon with two squaresand two equilateral triangles.The tessellating tile is thentranslated vertically andhorizontally. This tessellationis created using translations.b)The area of the tessellating tile remains the same throughoutthe tessellation. There are no gaps or overlapping pieces.What transformation was used tocreate this tessellation? Explainyour reasoning. Tessellations can be made with two or more polygons as long as theinterior angles where the vertices of the polygons meet total exactly 360 . Two types of transformations commonly used to create tessellations aretranslationsreflections The area of the tessellating tile remains the same after it has beentransformed to create a tessellation.1.454Brian missed today’s class. How would you explain to him why some tessellatingpatterns made using translations could also be made using reflections?MHR Chapter 12
2.Ashley and Vijay are trying to figure out how thistessellation was made. Whose answer is correct?Explain.Ashley says:Vijay says:The tessellation is basedon reflecting the bluetriangles across the reddodecagon.The tessellation is basedon translating the reddodecagon with 2 bluetriangles.b)5.The diagram shows a garden pathmade from irregular 12-sided bricks.a)b)c)c)A dodecagon is a12-sided polygon.What type of transformation couldbe used to create each tessellationin #3?Identify the two regular polygons usedto create each tessellation.a)Link4.For help with #3 and #4, refer to the Example onpages 453–454.3.Literacyd)e)Explain why the 12-sided bricktessellates the plane.Use grid paper to design an irregularten-sided brick that could be used tomake a path.Explain why your ten-sided bricktessellates the plane.Use grid paper to design an irregularsix-sided brick that could be used totessellate the plane.Explain why your six-sided bricktessellates the plane.12.2 Constructing Tessellations Using Translations and Reflections MHR455
6.7.8.Simon is designing a wallpaper pattern thattessellates. He chooses to use the letter “T”as the basis of his pattern. Create twotessellations using the three coloured lettersshown.9.Priya is designing a kitchen tile that usestwo different regular polygons. She thenuses two different translations to create atessellation. Use grid paper to design a tilethat Priya could use. Show how it tilesthe plane.Barbara wants to make a quilt using thetwo polygons shown. Will she be able tocreate a tessellating pattern using theseshapes? Explain.MATH LINKMany quilt designs are made using tessellating shapes.This quilt uses fabric cut into triangles that are sewntogether to form squares. The squares are then translatedvertically and horizontally.Design your own quilt square using one or more regulartessellating polygons. Create an interesting design basedon patterns or colours.456MHR Chapter 12An equilateral triangle is called a reptile(an abbreviation for “repeating tile”)because four equilateral triangles can bearranged to form a larger equilateraltriangle.“reptile”Which of these figures are reptiles? Usegrid paper to draw the larger figure foreach reptile.a)b)c)d)
Constructing TessellationsUsing RotationsFocus on After this lesson, youwill be able to.φ identify howrotations can beused to create atessellationφ create tessellatingpatterns usingtwo or morepolygonsProfessor RonaldResch of theUniversity of Utahbuilt the world’slargest pysanka from3500 pieces ofaluminum. It islocated in Vegreville,Alberta; weighs2300 kg; is 9.4 mhigh, 7 m long, and5.5 m wide; andturns in the wind likea weather vane!Pysanky is the ancient EasternEuropean art of egg decorating.The Ukrainian version of pysankyis the most well known. The namecomes from the verb to write,because artists use a stylus to writewith wax on the eggshell. Canyou see how rotations are used tomake the patterns on these eggs?How can you create tessellations using rotations?1.Draw an equilateral triangle with sidelengths of 4 to 5 cm on a piece of paper.Cut out the triangle and glue it to a sheetof cardboard or construction paper tocreate a tile.2.Trace around your tile on a piece of paper. tracing paperscissorsglue sticktapecardboard orconstruction paper coloured pencils12.3 Constructing Tessellations Using Rotations MHR457
3.Rotate the tile 60 about one vertex until the edge of the tile fallsalong the edge of the previous tracing as shown. Trace around thetile again.4.Repeat #3 until a full turn has been made.a) What shape did you create?b) How many times did you have to rotate the tile to createthis shape?5.Add colour and designs to the tessellation to make a piece of art.6.How could you continue to use rotations to make a largertessellation?60ºReflect on Your FindingsDescribe how to use rotating polygons to create tessellations.b) What types of polygons can be used to make tessellations basedon rotations? Explain.7. a)Example: Identify the TransformationWhat polygons and whattransformation could be usedto create this tessellation?SolutionThe tessellating tile is made upof a regular hexagon that has beenrotated three times to make a completeturn. The three hexagons forming thistile can be translated horizontallyand diagonally to enlarge the tessellation.What polygons and transformations could beused to create this tessellation? Explain howyou know.458MHR Chapter 12What othertransformation(s) couldcreate this tessellation?360º
Tessellations can be made with two or more polygons as long asthe interior angles where the polygons meet total exactly 360 . Rotations can be used to create tessellations.1.When creating a tessellation using rotations, why is it important forthe sum of the angle measures at the point of rotation to equal 360 ?Explain.2.Describe to a partner how to use rotating polygons to create tessellations.For help with #3 and #4, refer to the Example onpage 458.3.5.Examine the piece of stained glass.Identify the polygons used to create eachtessellating tile.a)b)Describe the transformation(s) usedto make this pattern.b) If you were using this pattern to tilethe plane, what modifications wouldyou have to make?a)c)4.What transformations could be usedto create each tessellation in #3?6.Design your own stained-glass window ongrid paper. Describe the steps you followedto create the pattern.7.Create a tessellation using two differentregular polygons and rotations.12.3 Constructing Tessellations Using Rotations MHR459
8.Which of the following shapes tessellate?Explain how you know a shape will orwill not tessellate.ABECFGDHTessellationsInvolving ThreeRegular PolygonsShape1Triangle (60 )1Square (90 )0Pentagon(108 )0Hexagon(120 )0Octagon(135 )0Dodecagon(150 )2Number ofSides(3, 12, 12)60 2(150) 360 Sum ofAngles9.The diagram shows one arrangement ofthree or more polygons that can be usedto create tessellations using rotations.One triangle and two dodecagons canbe used because the angles at eachvertex total 360 where they join. Thisis represented as (3, 12, 12). The tableshows the features of this tessellation,for Shape 1.a)Shape Shape Shape234Copy the table into your notebook.Complete the table for Shape 2 forthe diagram shown.488Shape 2312Shape 1Complete the table for Shapes 3 and 4,using different combinations of three ormore regular polygons that total 360 .c) Create construction paper or cardboardcutouts of the regular polygons frompart b). Try to tessellate the plane using thecombinations that you believe will work.b)12MATH LINKCreate your own pysanka design based on tessellating one or morepolygons. Use at least one rotation in your design. Trace your designon grid paper, and colour it. Make sure it is the correct size to fit on anegg. If you have time, decorate an egg with your pysanka design.460MHR Chapter 12Web LinkTo see examples ofpysankas, go towww.mathlinks8.caand follow the links.
Creating Escher-StyleTessellationsFocus on After this lesson, youwill be able to.φ create tessellationsfrom combinationsof regular andirregular polygonsφ describe thetessellations interms of thetransformationused to createthemIn the previous sections, you created tessellating patterns using regularand irregular polygons. When Escher created his tessellations, he did soin a variety of ways. Look at the two Escher works. What is differentabout the tessellations?How do you make Escher-style tessellations?1.Draw an equilateral trianglewith 6-cm sides on a blank pieceof paper. Cut out the triangleand glue it to a sheet ofcardboard or construction paper.Cut out the triangle again.2.Inside the triangle, draw a curvethat connects two adjacent vertices.Cut along the curve to remove apiece from one side of the triangle.3.Rotate the piece you removed 60 counterclockwise about the vertex at thetop end of the curve. This rotation movesthe piece to another side of the triangle.Tape the piece in place to completeyour tile. rulerscissorsglue stickcardboard orconstruction paper tape coloured pencils12.4 Creating Escher-Style Tessellations MHR461
4.To tessellate the plane, draw around the tile ona piece of paper. Then, rotate and draw aroundthe tile over and over until you have a designyou like.5.Add colour and designs to the tessellation tomake a piece of art.6.Repeat steps 1 through 5 using a parallelogram and translationsto create another Escher-style drawing.Reflect on Your Findings7.You can use transformations to create Escher-style tessellations justas you did with regular and irregular polygons.a) Describe how to use rotations to create Escher-style tessellations.b) What do you notice about the sum of the angle measures at thevertices where the tessellating tiles meet?c) How does the area of the modified tile compare with the areaof the original polygon? Explain.Example: Identify the Transformation Used in a TessellationWhat transformation was used to create each of the following tessellations?Tessellation A462MHR Chapter 12Tessellation BThe leadinggeometer of thetwentieth centurywas a professor atthe University ofToronto namedDonald Coxeter(1907–2003). He metM.C. Escher in 1954and gave Eschersome ideas for his art.
SolutionTessellation A is made up of triangles that havebeen rotated to form a hexagon. This tessellationis made using rotations.Tessellation B is made up of figuresthat alternate gold to black andthen repeat horizontally across thedrawing. This tessellation is madeusing translations.What transformation was usedto create this tessellation? Explainyour answer. You can create Escher-style tessellations using the same methods youused to create tessellations from regular or irregular polygons:Start with a regular or irregular polygon.The area of the tessellating tile must remain unchanged—any portionof the tile that is cut out must be reattached to the tile so that it fitswith the next tile of the same shape.Make sure there are no overlaps or gaps in the pattern.Make sure interior angles at vertices total exactly 360 .Use transformations to tessellate the plane.12.4 Creating Escher-Style Tessellations MHR463
1.When creating a tile for an Escher-style tessellation, the originalpolygon is cut up. How do you know the area of the original polygonis maintained?2.Rico believes that he can use this tile to createan Escher-style tessellation. Is he correct?Explain.3.Tessellations must have no gaps or overlaps. What other twoproperties must be maintained when creating Escher-styletessellations?b)For help with #4 to #7, refer to the Example onpages 462–463.4.Identify the transformations used to createeach tessellation.a)5.464MHR Chapter 12Identify the original shape from whicheach tile was made for each tessellationin #4.
6.Identify the transformations used to createeach tessellation.a)8.Create an Escher-style tessellation usinga scalene triangle with translations.9.Create an Escher-style tessellation usingan equilateral triangle with rotations.10.Create an Escher-style tessellation usingsquares with rotations and translations.11.Escher also used impossible figures in hisart, as shown.b)7.Identify the original shape from whicheach tile was made for each tessellationin #6.What impossible figures were used inthe drawing?b) Research other examples of Escher’sart that include impossible figures.a)MATH LINKUse an Escher-style tessellation to create a design fora binder cover, wrapping paper, a border for writingpaper, or a placemat.Web LinkTo see examples of Escher’sart, go to www.mathlinks8.caand follow the links.12.4 Creating Escher-Style Tessellations MHR465
The diagram shows a tessellation of squares. A dot has been added to the centre of each square. The dots are joined by dashed segments perpendicular to common sides. The result is another tessellation, which is called the dual of the original tessellation. a) Describe the dual of the original square tessellation
Exploring Tessellations: art origami by Tracy Cheng Sunday 9:30 – 11:30 (B) Learn how to design your own tessellation patterns and/or origami tessellations! Geology of Snickers Candy Bars by Jaz Harris Sunday 4:30 – 5:30 (F) Let's discover the composition of
twist and tess l 11 content 14 introduction 17 why a book about tessellations? 18 how to use the book 19 what is a tessellation 20 tessellations and origami (a little bit of history) 20 a bit of mathematics 22 basic techniques 25 materials 25 papers 27 necessary tools 29 how to fold a tessellation 29 cp interpretation 30 preparing the grid 30 add the pleats outside the grid (pre-crease)
2. Angle Measurement 3. Interior Angles 4. Polygons Make Great Tessellations 5. Round and Round the Vertex: Naming a Pattern 6. Patterns of Polygons 7. Color Makes A Difference 8. Alphabet Reflective Symmetry 9. Example of Symmetry 10. ReplicatorTiles Strike! 11. Tessellating With Irregular Polygons: Add color to two squares 12.
Discovering New Tessellations Using Dynamic Geometry Software Ma. Louise Antonette N. De Las Peñas mdelaspenas@ateneo.edu . uniform tilings have been found by computer and published on the web by Galebach [2]. The k-uniform tilings for k 6 have not bee
at origami tessellations, it’s likely that there will be many ways to change the directions of the folds and end up with quite di erent looking tessellations, doing this for the brick wall pattern just gives an idea of what might be possible. 1.1. counting folds at a vertex. In order to count how many crease patterns there are, rst consider .
Worksheet 29: Friday November 20 Tessellations: Tiling The Plane A tiling of the plane or tesselation is a pattern that covers the plane with non-overlapping figures A periodic tiling is one in which there exists at least two translations in non-parallel directions in which the tiling is
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We view a near-regular texture (NRT) as a statistical distortion of a regular, wallpaper-like congruent tiling, possibly with individ-ual variations in tile shape, size, color and lighting. Near-regular textures can depart from regular tiling along different axes of ap-pearance, and thus could have (1) a regular structural layout but
