Brandon Groeger March 23 2010 Chapter 18 Form and Growth Chapter 19 Symmetry and Patterns Chapter 20 Tilings Outline Chapter 18 Form and Growth Geometric Similarity and Scaling Physical limits to Scaling ID: 423200
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Slide1
Part VI: On size and Growth
Brandon GroegerMarch 23, 2010
Chapter 18: Form
and Growth
Chapter 19: Symmetry and Patterns
Chapter 20:
TilingsSlide2
Outline
Chapter 18: Form and GrowthGeometric Similarity and ScalingPhysical limits to ScalingChapter 19: Symmetry and Patterns
Numerical Symmetry and Fibonacci Numbers
The Golden Ratio
Rigid Motion
Patterns
Fractals
Chapter 20:
Tilings
Types of tiling
Regular
vs
semiregular
tiling
Tiling with irregular polygonsSlide3
Geometric Similarity and Scaling
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.Slide4
Scaling Area and Volume
Area and surface area change by the square of the scaling factor.Volume changes by the cube of the scaling factor.Example: Scaling by 3V =1 * 33 = 27
SA = 6 * 3
2
= 54Slide5
Scaling a Steel Cube
Pressure is the force per unit of area.P = W/AQuestion 1A) 33
* 500 = 13500 lbs
B) 3*3 = 9 sq. ft.
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.Slide6
More Scaling
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. Slide7
Numerical Symmetry and Fibonacci Numbers
Fibonacci NumbersF1 = 1, F2
= 1, F
n+1
= F
n
+ F
n-1 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 Fn / Fn+2Slide8
The Golden Ratio
Golden Ratio: A golden rectangle has sides that are proportional to 1 and φ.
Can be found in ancient architectureSlide9
Why the Golden Ratio?
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
w
sSlide10
The Golden Ratio and Fibonacci Numbers
The limit of Fn+1 / Fn 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, 1.615
golden ratio
≈ 1.618034Slide11
Golden Ratio Trivia
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.Slide12
Symmetry and Rigid Motion
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:Reflection (across a line)
Rotation (around a point)
Translation (in a particular direction)
Glide Reflection (across a line)Slide13
Patterns
There are three main types of patterns across planes:Rosette Patterns: no directionStrip Patterns: only one directionWallpaper Patterns: more than one direction
XXXXXXXXXXXXXXXXXXSlide14
Fractals
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.Slide15
Tilings (Tessellation)
A tiling or tessellation is a covering of an entire plane with non overlapping figures.Slide16
Types of Tiling
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.Slide17
Regular vs. Semiregular Tiling
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.Slide18
Tiling with Irregular Polygons
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. Slide19
Discussion
Chapter 18Can you think of an applications of scaling?Are there anyways to overcome the problems associated with scaling by large factors?Chapter 19Are there other mathematical patterns in nature?
Chapter 20
What applications does tiling have?
How would tiling work on a surface that is not a plane?Slide20
Homework
(7th edition)Chapter 18: 4Chapter 19: 25Chapter 20: 8