Mathematics and Art Activity - Basic Plane Tessellation with GeoGebra

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1 1 Mathematics ad Art Activity - Basic Plae Tessellatio with GeoGebra Worksheet: Explorig Regular Edge-Edge Tessellatios of the Cartesia Plae ad the Mathematics behid it. Goal: To eable Maths educators to use GeoGebra to uderstad some of the mathematics that supports the costructio of regular plae tessellatios. Refereces to results that are aliged to the CAPS mathematics curriculum will also be discussed. Relevat Maths Keywords ad Cocepts: Tessellatio or Tilig, Euclidea Plae, Regular shape, Irregular shape, Polygo, Iterior agles, Exterior agles, Covex ad Cocave quadrilateral, Cogruet shapes, Regular Tessellatio, Irregular Tessellatio, Semi-regular Tessellatio. 1. Pre-Kowledge for explorig basic plae tessellatios: Polygos are 2-dimesioal shapes. They are made of straight lies (edges), ad the shape is "closed" (all the lies coect up i vertices). Regular Polygos are polygos that are equiagular (all agles are equal i measure) ad equilateral (all sides have the same legth).

2 2 You should also kow that: a whole tur aroud ay poit o a surface is 360 the sum of the iteral agles of ay triagle = 180 the sum of the iteral agles of ay quadrilateral = 360 the sum of the exteral agles of ay polygo=360 (oe whole tur) the sum of the iterior agles of a -sided regular polygo = ( - 2) 180 how to calculate or measure the iterior agles of regular polygos Iteral ad Exteral Agles of a Regular Polygo Example: The sum of the iteral agles of a regular Hexago (=6) is ( - 2) 180 =(6-2) 180 =720. Hece the iteral agles of a regular Hexago is = 120. Symmetry This is the property that a figure coicides with itself uder a isometry, where a isometry is a actio (movemet) the preserves size ad shape. There are three basic types of isometries that preset symmetry of a figure i a plae. Types of Symmetry: (a) Reflectioal symmetry. A object has reflectioal symmetry if you ca reflect it i a way such that the resultig image coicides with the origial. Hold a mirror up to it, its reflectio looks idetical. (b) Rotatioal symmetry. A object has rotatioal symmetry if it ca be rotated about a poit i such a way that its rotated image coicides with the origial figure before turig a full 360. (c) Traslatioal symmetry. A object has traslatioal symmetry if you move it alog a straight path without turig it to produce the same image.

3 3 Defiig a Tessellatio A tessellatio ca be defied as the coverig of a plae with a repeatig uit cosistig of oe or more shapes (regular or irregular) i such a way that: there are o ope spaces betwee ad o overlappig of the shapes that are used; the coverig process has the potetial to cotiue idefiitely (for a surface of ifiite dimesios Cartesia Plae). 2. Regular Tessellatios of the Plae Tessellatios i which oe regular polygo is used repeatedly are called regular tessellatios. Two key questios to cosider Which regular polygos will tessellate (or tile) the plae ad why? How may differet tessellatios are possible i each case? Some Termiology about Namig of Plae Tessellatios: Cosider the example of a edge-edge plae tessellatio i Figure 1. Although all the polygos are regular, there are more tha oe type of polygo which that are used to tessellate. This makes this a o-regular tessellatio (or tilig) of the plae. A vertex is a commo poit where sides (edges) of polygos meet. The cofiguratio of a vertex is the sequece of polygo orders that exist aroud it. Normally these orders are give i a sequece startig with the lowest order. The vertex cofiguratio of each vertex i the tilig show i Figure 1. is as each vertex is surrouded by two equilateral triagles, a square, aother equilateral triaglead fially a square. Figure Tessellatio Clearly the vertex cofiguratio of each vertex of a regular tessellatios of the plae will be idetical. There are oly three regular polygos that produce edge-edge regular tessellatios of the Cartesia Plae:

4 4 2.1 Equilateral triagles tessellate the plae: Use the tools to costruct a equilateral triagle. Select triagle ad use Copy ad Paste fuctios to create a few copies i the Graphics View area. Drag ad rotate aroud a commo vertex to show that a regular polyomial of order 3 (equilateral triagle) tessellates the plae. Note: The vertex cofiguratio of each poit of the tessellatio is idetical as 6 equilateral triagles surroud each vertex. Hece this edge-edge regular tessellatio could be described as a tilig of the plae. Some color patter variatios with regular tilig by equilateral tragles:

5 5 2.2 Squares tessellate the plae: Use the tools to costruct a square. Select the square ad use Copy ad Paste fuctios to create a few copies i the Graphics View area. Drag ad rotate aroud a commo vertex to show that a regular polyomial of order 4 (square) tessellates the plae. Note: The vertex cofiguratio of each poit of the tessellatio is idetical as 4 squares surroud each vertex. Hece this edge-edge regular tessellatio could be described as a tilig of the plae. Also ote that there are ifiite umber of ways that a square could tesslate the plae. This ca be doe by slidig each horizotal layer i relatio to the oe above or below it. Below we show a tilig of the plae which frequetly appears i bricklayig. Some color patter variatios with regular tilig by squares:

6 6 2.3 Hexagos tessellate the plae: Use the tools to costruct a regular hexago. Select the hexago ad use Copy ad Paste fuctios to create a few copies i the Graphics View area. Drag ad rotate aroud a commo vertex to show that a regular polyomial of order 6 (hexago) tessellates the plae. Note: The vertex cofiguratio of each poit of the tessellatio is idetical as 3 regular hexagos surroud each vertex. Hece this edge-edge regular tessellatio with hexagos could be described as a tilig of the plae. Some atural hexagoal packig structuresi a plae: Hoey bees ad their hives. Idetical circles i a hexagoal packig arragemet, the desest best packig possible.

7 7 3. A Petago or ay Regular Polygos of order >6 does ot tessellate the plae: Use the tools to costruct a regular petago. Select the petago ad use Copy ad Paste fuctios to create a few copies i the Graphics View area. Drag ad rotate aroud a commo vertex to show that a regular polyomial of order 5 (petago) does ot tessellates the plae. Petagos does ot fill the full cycle of 360 about each vertex. Why ot? The aswer lie i the patter of ier agles that are associated with regular polygos as reflected i Table 1. Order of Regular Polygo Commo Name Ier Agle Factor of 360? 3 Equilateral Triagle ( 2) 180 (3 2) Square ( 2) 180 (4 2) Petago ( 2) 180 (5 2) Hexago ( 2) 180 (6 2) Heptago ( 2) 180 (7 2) 180 Yes Yes No Yes No

8 8 8 Octago ( 2) 180 (8 2) Noago ( 2) 180 (9 2) 180 No No For a regular polygo to poduce a edg-edge tessellatio of the plae, the ier agle of the polygo must be a factor of 360. Hece oly equilateral triagles, squares ad hexagos will tesselate a plae. Exercise 1: Use the tools to costruct a regular petago with order >6. Select the polygo ad use Copy ad Paste fuctios to create a few copies i the Graphics View area. Drag ad rotate polygos aroud a commo vertex to show that a regular polyomial of order greater tha 6 does ot tessellates the plae. Exercise 2: Discuss ad explore the basic symmetrie of the Regular Tessellatios of a plae. Some atural examples of Irregular Tessellatios of surfaces: Scales o ski of reptiles Aimal Ski Girrafe Dried mud of water pool May others!!!!! Created by WA Olivier- May 2017