 Brandon Groeger  March 23, 2010  Chapter 18: Form and Growth  Chapter 19: Symmetry and Patterns  Chapter 20: Tilings

 Chapter 18: Form and Growth o Geometric Similarity and Scaling o Physical limits to Scaling  Chapter 19: Symmetry and Patterns o Numerical Symmetry and Fibonacci Numbers o The Golden Ratio o Rigid Motion o Patterns o Fractals  Chapter 20: Tilings o Types of tiling o Regular vs semiregular tiling o Tiling with irregular polygons

 Two objects are geometrically similar if they have the same shape.  A linear scaling factor of two geometrically similar objects is the ratio of any part of the second to the corresponding part in the first.

 Area and surface area change by the square of the scaling factor.  Volume changes by the cube of the scaling factor.  Example: Scaling by 3 o V =1 * 3 3 = 27 o SA = 6 * 3 2 = 54

 Pressure is the force per unit of area.  P = W/A  Question 1 o A) 3 3 * 500 = lbs o B) 3*3 = 9 sq. ft. o C)13500/9 =1500 lbs/ft vs. 500 lbs/ft  Steel has a crushing strength of about 7.5 million lbs/ft, meaning a 3 mi cube of steel would crush itself.

 Area-Volume Tension is a result of the fact that as an object is scaled up the volume increases faster than the areas of the cross-sections.  Gives theoretical limits for the height of trees, mountains and buildings, based on material strength.  Explains the differences in structure between animals of different sizes and the limits of these structures.

 Fibonacci Numbers o F 1 = 1, F 2 = 1, F n+1 = F n + F n-1 o 1, 1, 2, 3, 5, 8, 13, 21,34, 55, 89, 144, 233,…  Phyllotaxis is a spiral arrangement found on some plants. The ratio of spiral in one direction to spirals in the other are two Fibonacci numbers F n / F n+2

 Golden Ratio:  A golden rectangle has sides that are proportional to 1 and φ.  Can be found in ancient architecture

 The Greeks were interested in balance. They wanted to two segments so that the ratio of the sum of the segments to the larger segment was equal to the ratio of the larger segment to the smaller segment.  l = s + w, l/s = s/w = x = golden ratio.  Question 2 l ws

 The limit of F n+1 / F n as n approaches infinity is equal to the golden ratio.  1/1, 2/1, 3/2, 5/3, 8/5, 13/8…  1, 2, 1.5, 1.666, 1.6, 1.625,  golden ratio ≈

 The ratio of the diagonal of a pentagon with equal sides to the one of those sides is the golden ratio.  Many of the myths involving the golden ratio are false.

 A rigid motion is one that preserves the size and shape of figures.  In particular, any pair of points is the same distance apart after the motion as before.  Must be one of the following: o Reflection (across a line) o Rotation (around a point) o Translation (in a particular direction) o Glide Reflection (across a line)

 There are three main types of patterns across planes: o Rosette Patterns: no direction o Strip Patterns: only one direction o Wallpaper Patterns: more than one direction XXXXXXXXXXXXXXXXXX

 A fractal is a pattern that exhibits similarity at even finer scales.  Fractals can be used to mimic things in nature such as trees, leaves and snow flakes.

 A tiling or tessellation is a covering of an entire plane with non overlapping figures.

 Monohedral tiling uses only one size and shape tile.  Regular tiling is a type of monohedral tiling where the tile is a regular polygon.  Edge to edge tiling occurs when the edge of one tile completely coincides with the edge of the bordering tile.

 There are only three types of regular tilings, one with triangles, ones with squares, and ones with hexagons.  A semi regular tiling uses a mix of regular polygons with different number of sides but in which all vertex figures are alike and the same polygons are in the same order.

 Any triangle can tile a plane.  Any quadrilateral, even those that are not convex, can tile a plane.  Certain pentagons and hexagons can tile a plane.  A convex polygon with 7 or more sides cannot tile.

 Chapter 18 o Can you think of an applications of scaling? o Are there anyways to overcome the problems associated with scaling by large factors?  Chapter 19 o Are there other mathematical patterns in nature?  Chapter 20 o What applications does tiling have? o How would tiling work on a surface that is not a plane?

 (7 th edition)  Chapter 18: 4  Chapter 19: 25  Chapter 20: 8