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# Boston University Symmetry in Geometry Mathematics Worksheet

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Topic 15 Classwork Assignment Problem 1: Classify each wallpaper pattern as either p411, p4gm, or p4mm. Label enough lines of reflection and centers of rotation in the patterns to support you answer. Use a black dotted line to designate the lines of reflection of reflections. Use a “×” to designate a 90° rotation. (a) (b) Pattern: (c) Pattern: Pattern: Problem 2: Classify each wallpaper pattern as either p311, p31m, or p3m1. Label enough lines of reflection and centers of rotation in the patterns to support you answer. Use a black dotted line to designate the lines of reflection of reflections. Use a “∆” to designate a 120° rotation. (a) (b) Pattern: Pattern: (c) Pattern: Problem 3: Classify each wallpaper pattern as either p611 or p6mm. Label enough lines of reflection and centers of rotation in the patterns to support you answer. Use a black dotted line to designate the lines of reflection of reflections. Use a “*” to designate a 60° rotation. (a) (b) Pattern: Pattern: Topic 15 Symmetry in Geometry: Wallpaper Patterns Part II Wallpaper Patterns Wallpaper patterns are formed by repetitions of a motif in such a way as to cover a plane. It can be shown that there are exactly seventeen wallpaper patterns. All wallpaper patterns are generated by a basic motif which is acted on by the four transformations: translations, rotations, reflections, and glide reflections. Since the wallpaper is considered to cover the entire plane, there must be translations in two different direction. 2 Classification of Wallpaper Patterns The ICN notation used to classify wallpaper patterns consists of four characters, each character determined based on the transformations in the pattern, as follows: ● The first character is a p or a c. There are only two wallpaper patterns of type c. ● The second character is n for the highest order of rotation: 1 for 0° rotation, 2 for 180° rotation, 3 for 120° rotation, 4 for 90°, or 6 for 60° rotation. ● The third and fourth characters are m if there are lines of reflection, g if there is no line of reflection but there is a glide reflection, and 1 otherwise. The angle at which these lines of symmetry meet also determines the pattern. (*) International Crystallographic Notation (**) In ICN notation, p stands for primitive cell, c stands for centered cell, m stands for 3 mirror , and g stands for glide reflection. Classification of Wallpaper Patterns Today We will consider wall paper patterns with 60°, 90°, and 120° rotations. 60° 90° 120° 4 Classification of Wallpaper Patterns ICN Notation p411 p4gm p4mm p311 p31m p3m1 p611 p6mm Short Form p4 p4g p4m p3 p31m p3m1 p6 p6m 5 Classification of Wallpaper Patterns The following is a flow chart for wallpaper patterns with a smallest angle of rotation equal to 90°. 45° Reflection 6 Classification of Wallpaper Patterns The following is a flow chart for wallpaper patterns with a smallest angle of rotation equal to 120°. 7 Classification of Wallpaper Patterns The following is a flow chart for wallpaper patterns with a smallest angle of rotation equal to 60°. 8 Wallpaper Patterns – p411 (p4) × Smallest Rotation Angle: 90° Reflection: No 9 Wallpaper Patterns – p4gm (p4g) × Smallest Rotation Angle: 90° Reflection: Yes 45° Reflection: No 45° Reflection 10 Wallpaper Patterns – p4mm (p4m) × Smallest Rotation Angle: 90° Reflection: Yes 45° Reflection: Yes 45° Reflection 11 Wallpaper Patterns – p311 (p3) ∆ Smallest Rotation Angle: 120° Reflection: No 12 Wallpaper Patterns – p31m (p31m) ∆ ∆ Smallest Rotation Angle: 120° Reflection: Yes All Rotation Centers on Reflection Lines: No 13 Wallpaper Patterns – p3m1 (p3m1) ∆ Smallest Rotation Angle: 120° Reflection: Yes All Rotation Centers on Reflection Lines: Yes 14 Wallpaper Patterns – p611 (p6) * Smallest Rotation Angle: 60° Reflection: No 15 Wallpaper Patterns – p6mm (p6m) * Smallest Rotation Angle: 60° Reflection: Yes 16 Topic 16 Classwork Assignment Problem 1: Label each polygon as either convex or non-convex. (a) (b) Convex or Non-Convex: Convex or Non-Convex: Problem 2: Complete the table for the following regular pentagons. Regular Polygon Number of Sides pentakaidecagon octakaidecagon icosagon tetrakaicosagon 15 18 20 24 Vertex Angle Problem 3: What is the vertex configuration for a regular tiling of hexagons? Vertex Configuration: Problem 4: Show kθ = 360° for a regular tiling of hexagons, where k is the number of regular polygons meeting at a vertex, and θ is the vertex angle. Problem 5: What is the vertex configuration for the following semiregular tiling? List the polygons in the semiregular tiling. Explain how the semiregular tiling conforms to all five rules for semiregular tilings. Vertex Configuration: Polygons: Rule 1: In a semiregular tiling of the plane, the sum of the vertex angles of the polygons meeting at each vertex must be exactly 360°. Rule 2: A semiregular tiling must have at least three and no more than five polygons meeting at each vertex. Rule 3: No semiregular tiling can have four or more different polygons meeting at a vertex. Thus, if a semiregular tiling has four or more polygons meeting at a vertex, there must be some duplicates. Rule 4: No semiregular tiling with exactly three polygons meeting at each vertex can have the vertex configuration k.n.m where k is odd unless n = m. Rule 5: No semiregular tiling with exactly four polygons meeting at each vertex can have the vertex configuration 3.k.n.m unless k = m. Problem 6: What is the vertex configuration for the following semiregular tiling? List the polygons in the semiregular tiling. Explain how the semiregular tiling conforms to all five rules for semiregular tilings. Vertex Configuration: Polygons: Rule 1: In a semiregular tiling of the plane, the sum of the vertex angles of the polygons meeting at each vertex must be exactly 360°. Rule 2: A semiregular tiling must have at least three and no more than five polygons meeting at each vertex. Rule 3: No semiregular tiling can have four or more different polygons meeting at a vertex. Thus, if a semiregular tiling has four or more polygons meeting at a vertex, there must be some duplicates. Rule 4: No semiregular tiling with exactly three polygons meeting at each vertex can have the vertex configuration k.n.m where k is odd unless n = m. Rule 5: No semiregular tiling with exactly four polygons meeting at each vertex can have the vertex configuration 3.k.n.m unless k = m. Topic 16 Symmetry in Geometry Tilings – Part I Polygons A polygon is a closed plane figure with straight edges. The edges are called sides, and the points where two edges meet are called vertices or corners. The interior of a polygon is sometimes called its body. Examples: An n-gon is a polygon with n sides; for example, a triangle is a 3-gon. 2 Polygons Polygons are commonly named by prefixes from Greek numbers. Number of Sides 3 4 5 6 7 8 9 10 11 12 Prefix tri quad penta hexa hepta octa nona deca hendeca dodeca Polygon triangle quadrilateral pentagon hexagon heptagon octagon nonagon decagon hendecagon dodecagon 3 Polygons A polygon is a convex if given any two points, A and B, in the polygon, the line segment AB lies in the polygon. A polygon that is not convex is called a non-convex polygon. Examples: Convex Polygon Non-Convex Polygon 4 Regular Polygons A polygon is regular if all of its sides and all of its angles are equal; i.e., if it is be both equilateral and equiangular. Examples: equilateral triangle square regular pentagon regular hexagon regular octagon regular nonagon regular decagon regular hendecagon A polygon that is not regular is irregular. regular heptagon regular dodecagon 5 Regular Polygons The vertex angle, θ, in degrees, of an n-sided regular polygon measures θ = Example: n−2 180° . n The vertex angle of an equilateral triangle is θ = 60°. 60° 60° 60° θ= n−2 180° n = 3 3−2 180° = n 3 = 1 180° 3 = 60° equilateral triangle n=3 6 Example: Regular Polygons Calculate the vertex angle, θ, of a square. Solution: 90° 90° 90° 90° square n=4 θ= n−2 180° n = 4 4−2 180° = n 4 The vertex angle of a square is θ = 90°. = 2 180° 4 = 90° 7 Vertex Angles of Some Regular Polygons Regular Polygon triangle square pentagon hexagon heptagon octagon nonagon decagon hendecagon dodecagon Number of Sides 3 4 5 6 7 8 9 10 11 12 Vertex Angle 60° 90° 108° 120° 128.57° 135° 140° 144° 147.27° 150° 8 Tilings A tesselation or tiling of the plane is a pattern of repeated copies of figures covering the plane so that the copies do not overlap and leave no gaps uncovered. The figures are called the tiles. 9 Regular Tilings A tiling is regular if it consists of repeated copies of a single regular polygon, meeting edge to edge so that at every vertex the same number of polygons meet. 10 Regular Tilings Squares, equilateral triangles and hexagons are the only three regular polygons that may be positioned to tile the plane in a regular pattern. Therefore, there are exactly three regular tilings. Regular Tiling of Squares Regular Tiling of Equilateral Triangles Regular Tiling of Hexagons 11 Regular Tilings: Vertex Configuration We denote a regular tiling by describing the number of sides of the polygons meeting at a vertex. Example: For a regular tiling of squares, we have: four sides four sides ● four sides four sides Vertex Configuration: 4.4.4.4 12 Example: Regular Tilings: Vertex Configuration What is the vertex configuration for a regular tiling of equilateral triangles? Solution: 3 3 3 ● 3 3 3 Vertex Configuration: 3.3.3.3.3.3 13 Regular Tilings: Vertex Configuration Regular Polygon triangle Regular Tiling Vertex Configuration 3.3.3.3.3.3 square 4.4.4.4 hexagon 6.6.6 14 Rule for Regular Tilings Rule: In a tiling of the plane, the sum of the vertex angles of the regular polygons meeting at each vertex must be exactly 360°. Therefore, for k regular polygons meeting at a vertex, each with a vertex angle θ, kθ = 360°. Example: For a regular tiling of squares, we have: k = 4 (Four squares meet at a vertex.) ● θ = 90° (the vertex angle of a squares is 90°.) kθ = 4 90° = 360° 15 Example: Regular Tilings Show that kθ = 360° for a regular tiling of equilateral triangles, where k is the number of regular polygons meeting at a vertex, and θ is the vertex angle. Solution: ● k = 6 (Six equilateral triangles meet at a vertex.) θ = 60° (The vertex angle of an equilateral triangle is 60°.) kθ = 6 60° = 360° 16 Semiregular Tilings A semiregular or Archimedean tiling is a tiling in which each tile is a regular polygon and each vertex is identical. There are exactly eight semiregular tilings. 17 Semiregular Tilings Semiregular Tiling of Hexagons and Triangles Semiregular Tiling of Triangles, Squares, and Hexagons Semiregular Tiling of Octagons and Squares Semiregular Tiling of Squares, Hexagons, and Dodecagons 18 Semiregular Tilings Semiregular Tiling of Triangles and Squares Semiregular Tiling of Triangles and Hexagons Semiregular Tiling of Triangles and Squares Semiregular Tiling of Triangles and Dodecagons 19 Semiregular Tilings: Vertex Configuration As with regular tilings, we denote semiregular tilings by describing the number of sides of the polygons meeting at a vertex. For semiregular tilings, we begin with the number of sides of the smallest polygon and list the number of sides of the remaining polygons in either clockwise or counterclockwise order. Example: For a semiregular tiling hexagons and triangles, we have: six sides three sides ● three sides six sides Vertex Configuration: 3.6.3.6 20 Example: Semiregular Tilings: Vertex Configuration What are the vertex configurations for the two semiregular tilings of triangles and squares? Solution: 4 4 ● 3 3 3 Vertex Configuration: 3.3.3.4.4 4 3 3 ● 3 4 Vertex Configuration: 3.3.4.3.4 21 Semiregular Regular Tilings: Vertex Configuration Regular Polygons Hexagons, Triangles Semiregular Vertex Tiling Configuration 3.6.3.6 Regular Polygons Semiregular Vertex Tiling Configuration Triangles, Squares 3.3.3.4.4 Octagons, Squares 4.8.8 Triangles, Squares 3.3.4.3.4 Triangles, Squares, Hexagons Squares, Hexagons, Dodecagons 3.4.6.4 Triangles, Hexagons 3.3.3.3.6 4.6.12 Triangles, Dodecagons 3.12.12 22 Rules for Semiregular Tilings Rule 1: In a semiregular tiling of the plane, the sum of the vertex angles of the polygons meeting at each vertex must be exactly 360°. Rule 2: A semiregular tiling must have at least three and no more than five polygons meeting at each vertex. Rule 3: No semiregular tiling can have four or more different polygons meeting at a vertex. Thus, if a semiregular tiling has four or more polygons at a vertex, there must be some duplicates. Rule 4: No semiregular tiling with exactly three polygons meeting at each vertex can have the vertex configuration k.n.m where k is odd unless n = m. Rule 5: No semiregular tiling with exactly four polygons meeting at each vertex can have the vertex configuration 3.k.n.m unless k = m. 23 Example: Semiregular Tilings Explain how the semiregular tiling conforms to all five 3 6 rules for semiregular tilings. ● Solution: 6 3 Vertex Configuration: 3.6.3.6 Polygons: triangle hexagon triangle hexagon Rule 1: In a semiregular tiling of the plane, the sum of the vertex angles of the polygons meeting at each vertex must be exactly 360°. triangle hexagon triangle hexagon 60° + 120° + 60° + 120° = 360° Rule 2: A semiregular tiling must have at least three and no more than five polygons meeting at each vertex. # of polygons meeting at each vertex = 4 Rule 3: No semiregular tiling can have four or more different polygons meeting at a vertex. Thus, if a semiregular tiling has four or more polygons meeting at a vertex, there must be some duplicates. 24 four polygons with two duplicates. triangle hexagon triangle hexagon Example: Semiregular Tilings Solution (continued): Vertex Configuration: 3.6.3.6 Polygons: triangle hexagon triangle hexagon 3 6 ● 6 3 Rule 4: No semiregular tiling with exactly three polygons meeting at each vertex can have the vertex configuration k.n.m where k is odd unless n = m. The semiregular tiling 3.6.3.6 has exactly four polygons meeting at each vertex. Therefore Rule 4 does not apply. Rule 5: No semiregular tiling with exactly four polygons meeting at each vertex can have the vertex configuration 3.k.n.m unless k = m. 3.6.3.6 3.k.n.m k=6 n=3 m=6 k=m 25 Example: Semiregular Tilings Which of the following semiregular tiling patterns violates Rule 4 for semiregular tiling configurations? 7 ● Solution: 3 4 42 ● 8 8 Vertex Configuration: 4.8.8 Vertex Configuration: 3.7.42 Rule 4: No semiregular tiling with exactly three polygons meeting at each vertex can have the vertex configuration k.n.m where k is odd unless n = m. k = 4 (even) 4.8.8 Since k is even, the semiregular tiling configuration 4.8.8 n=8 k.n.m does not violate Rule 4. m=8 k = 3 (odd) Since k is odd and n ≠ m, the semiregular tiling configuration 3.7.42 n=7 3.7.42 violates Rule 4. k.n.m 26 m = 42 Example: Semiregular Tilings Which of the following semiregular tiling patterns violates Rule 5 for semiregular tiling configurations? Solution: 4 3 4 3 ● ● 6 4 6 4 Vertex Configuration: 3.4.6.4 Vertex Configuration: 3.4.4.6 Rule 5: No semiregular tiling with exactly four polygons meeting at a vertex can have vertex configuration 3.k.n.m unless k = m. k = 4 Since k ≠ m, the semiregular tiling configuration 3.4.4.6 3.4.4.6 n = 4 violates Rule 5. 3.k.n.m m=6 3.4.6.4 3.k.n.m k=4 n=6 m=4 Since k = m, the semiregular tiling configuration 3.4.6.4 does not violate Rule 5. 27Purchase answer to see full attachment
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