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Part VI: On size and Growth Part VI: On size and Growth

Part VI: On size and Growth - PowerPoint Presentation

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Part VI: On size and Growth - PPT Presentation

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

tiling ratio golden chapter ratio tiling chapter golden scaling patterns tile regular area symmetry fibonacci plane sides types numbers

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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