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Tessellations Tessellations Jennifer Li and Maggie Smith Sonia Kovalevsky Day Mount Holyoke College Saturday, November 10, 2018 Tessellations everywhere Jennifer Li and Maggie Smith Tessellations April 18, 2018 2 / 39 What's the connection to Math? Mathematicians REALLY like patterns and symmetry! Jennifer Li and Maggie Smith Tessellations April 18, 2018 3 / 39 Tiles A tile is a geometric shape. Jennifer Li and Maggie Smith Tessellations April 18, 2018 4 / 39 Tiles A tile is a geometric shape. Jennifer Li and Maggie Smith Tessellations April 18, 2018 4 / 39 A tessellation covers the entire plane (infinite). No gaps and no overlaps! Tiles Tiles are the building blocks of a tessellation. Jennifer Li and Maggie Smith Tessellations April 18, 2018 5 / 39 A tessellation covers the entire plane (infinite). No gaps and no overlaps! Tiles Tiles are the building blocks of a tessellation. Jennifer Li and Maggie Smith Tessellations April 18, 2018 5 / 39 No gaps and no overlaps! Tiles Tiles are the building blocks of a tessellation. A tessellation covers the entire plane (infinite). Jennifer Li and Maggie Smith Tessellations April 18, 2018 5 / 39 Tiles Tiles are the building blocks of a tessellation. A tessellation covers the entire plane (infinite). No gaps and no overlaps! Jennifer Li and Maggie Smith Tessellations April 18, 2018 5 / 39 Polygons A polygon is a shape that is created by straight line segments. Jennifer Li and Maggie Smith Tessellations April 18, 2018 6 / 39 Polygons A polygon is a shape that is created by straight line segments. Jennifer Li and Maggie Smith Tessellations April 18, 2018 6 / 39 Regular Polygons In a regular polygon, all angles are equal and all side lengths are equal. Jennifer Li and Maggie Smith Tessellations April 18, 2018 7 / 39 Regular Polygons In a regular polygon, all angles are equal and all side lengths are equal. Jennifer Li and Maggie Smith Tessellations April 18, 2018 7 / 39 Types of Tessellations A regular tessellation is a symmetric tiling made up of regular polygons, all of the same shape. Jennifer Li and Maggie Smith Tessellations April 18, 2018 8 / 39 Regular Polygon Tessellations Equilateral Triangles Jennifer Li and Maggie Smith Tessellations April 18, 2018 9 / 39 Regular Polygon Tessellations Squares Jennifer Li and Maggie Smith Tessellations April 18, 2018 10 / 39 Activity: Regular Polygon Tessellations Activity Sheet: Tessellate the plane using the regular hexagon. Jennifer Li and Maggie Smith Tessellations April 18, 2018 11 / 39 Regular hexagons Vertex A vertex is a point where the corners of all polygons in a tessellation meet. Jennifer Li and Maggie Smith Tessellations April 18, 2018 12 / 39 Vertex A vertex is a point where the corners of all polygons in a tessellation meet. Regular hexagons Jennifer Li and Maggie Smith Tessellations April 18, 2018 12 / 39 Examples. n = 3 2 180 180 1 − = = 60 3 3 Each angle in an equilateral triangle is 60 degrees. Regular Polygons Each angle of an n-sided polygon equals 2 180 1 − n Jennifer Li and Maggie Smith Tessellations April 18, 2018 13 / 39 n = 3 2 180 180 1 − = = 60 3 3 Each angle in an equilateral triangle is 60 degrees. Regular Polygons Each angle of an n-sided polygon equals 2 180 1 − n Examples. Jennifer Li and Maggie Smith Tessellations April 18, 2018 13 / 39 Regular Polygons Each angle of an n-sided polygon equals 2 180 1 − n Examples. n = 3 2 180 180 1 − = = 60 3 3 Each angle in an equilateral triangle is 60 degrees. Jennifer Li and Maggie Smith Tessellations April 18, 2018 13 / 39 n = 4 2 180 180 1 − = = 90 4 2 Each angle in a square is 90 degrees. Regular Polygons Each angle of an n-sided polygon equals 2 180 1 − n Examples. Jennifer Li and Maggie Smith Tessellations April 18, 2018 14 / 39 Regular Polygons Each angle of an n-sided polygon equals 2 180 1 − n Examples. n = 4 2 180 180 1 − = = 90 4 2 Each angle in a square is 90 degrees. Jennifer Li and Maggie Smith Tessellations April 18, 2018 14 / 39 n = 9 2 180 180 1 − = 7 × = 140 9 9 Each angle in a nonagon is 140 degrees. Regular Polygons Each angle of an n-sided polygon equals 2 180 1 − n Examples. Jennifer Li and Maggie Smith Tessellations April 18, 2018 15 / 39 Regular Polygons Each angle of an n-sided polygon equals 2 180 1 − n Examples. n = 9 2 180 180 1 − = 7 × = 140 9 9 Each angle in a nonagon is 140 degrees. Jennifer Li and Maggie Smith Tessellations April 18, 2018 15 / 39 Can they fit without gaps and without overlapping? Which regular polygons tessellate the plane? How can we tessellate the plane with a regular n-sided polygon? Jennifer Li and Maggie Smith Tessellations April 18, 2018 16 / 39 Can they fit without gaps and without overlapping? Which regular polygons tessellate the plane? How can we tessellate the plane with a regular n-sided polygon? Jennifer Li and Maggie Smith Tessellations April 18, 2018 16 / 39 Which regular polygons tessellate the plane? How can we tessellate the plane with a regular n-sided polygon? Can they fit without gaps and without overlapping? Jennifer Li and Maggie Smith Tessellations April 18, 2018 16 / 39 Each polygon is n-sided: Which regular polygons tessellate the plane? At a vertex, there will be q regular polygons that meet: Jennifer Li and Maggie Smith Tessellations April 18, 2018 17 / 39 Each polygon is n-sided: Which regular polygons tessellate the plane? At a vertex, there will be q regular polygons that meet: Jennifer Li and Maggie Smith Tessellations April 18, 2018 17 / 39 Which regular polygons tessellate the plane? At a vertex, there will be q regular polygons that meet: Each polygon is n-sided: Jennifer Li and Maggie Smith Tessellations April 18, 2018 17 / 39 Which regular polygons tessellate the plane? At a vertex, there will be q regular polygons that meet: Each polygon is n-sided: Jennifer Li and Maggie Smith Tessellations April 18, 2018 17 / 39 Which regular polygons tessellate the plane? At each vertex, these angles must add to 360 degrees. Jennifer Li and Maggie Smith Tessellations April 18, 2018 18 / 39 Which regular polygons tessellate the plane? At each vertex, these angles must add to 360 degrees. Jennifer Li and Maggie Smith Tessellations April 18, 2018 18 / 39 2 q × 180 × 1 − = 360 n 2 q × 180 × 1 − n 360 = q × 180 q × 180 ¨ 2 q¡ × ¨180 × 1 − n 2 ¨ = q¡ × ¨180 q 2 2 1 − = n q 2 2 1 = + q n 1 1 1 + = q n 2 Which regular polygons tessellate the plane? 2 A total of q angles, each of 180 × 1 − degrees, sum to 360 degrees: n Jennifer Li and Maggie Smith Tessellations April 18, 2018 19 / 39 Which regular polygons tessellate the plane? 2 A total of q angles, each of 180 × 1 − degrees, sum to 360 degrees: n 2 q × 180 × 1 − = 360 n 2 q × 180 × 1 − n 360 = q × 180 q × 180 ¨ 2 q¡ × ¨180 × 1 − n 2 ¨ = q¡ × ¨180 q 2 2 1 − = n q 2 2 1 = + q n 1 1 1 + = q n 2 Jennifer Li and Maggie Smith Tessellations April 18, 2018 19 / 39 then a regular polygon tessellation is possible! Which regular polygons tessellate the plane? 1 1 1 If + = q n 2 Jennifer Li and Maggie Smith Tessellations April 18, 2018 20 / 39 then a regular polygon tessellation is possible! Which regular polygons tessellate the plane? 1 1 1 If + = q n 2 Jennifer Li and Maggie Smith Tessellations April 18, 2018 20 / 39 Which regular polygons tessellate the plane? 1 1 1 If + = q n 2 then a regular polygon tessellation is possible! Jennifer Li and Maggie Smith Tessellations April 18, 2018 20 / 39 An equilateral triangle has three sides, so n = 3. 2 Then q = 2 = 6 equilateral triangles meet at a vertex... 1 − 3 Is this correct? Yes! Which regular polygons tessellate the plane? 1 1 1 + = q n 2 Question. In an equilateral triangle tessellation, how many triangles must meet at a vertex? Jennifer Li and Maggie Smith Tessellations April 18, 2018 21 / 39 2 Then q = 2 = 6 equilateral triangles meet at a vertex... 1 − 3 Is this correct? Yes! Which regular polygons tessellate the plane? 1 1 1 + = q n 2 Question. In an equilateral triangle tessellation, how many triangles must meet at a vertex? An equilateral triangle has three sides, so n = 3. Jennifer Li and Maggie Smith Tessellations April 18, 2018 21 / 39 Is this correct? Yes! Which regular polygons tessellate the plane? 1 1 1 + = q n 2 Question. In an equilateral triangle tessellation, how many triangles must meet at a vertex? An equilateral triangle has three sides, so n = 3. 2 Then q = 2 = 6 equilateral triangles meet at a vertex... 1 − 3 Jennifer Li and Maggie Smith Tessellations April 18, 2018 21 / 39 Yes! Which regular polygons tessellate the plane? 1 1 1 + = q n 2 Question. In an equilateral triangle tessellation, how many triangles must meet at a vertex? An equilateral triangle has three sides, so n = 3. 2 Then q = 2 = 6 equilateral triangles meet at a vertex... 1 − 3 Is this correct? Jennifer Li and Maggie Smith Tessellations April 18, 2018 21 / 39 Yes! Which regular polygons tessellate the plane? 1 1 1 + = q n 2 Question. In an equilateral triangle tessellation, how many triangles must meet at a vertex? An equilateral triangle has three sides, so n = 3. 2 Then q = 2 = 6 equilateral triangles meet at a vertex..
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