Mathematics Masterclasses Stretching the

This is a valuable resource of non-syllabus material for mathematics in school education and science teachers at secondary school level, teenagers and parents. It contains written versions of Royal Institution masterclasses on a wide selection of topics in pure and applied mathematics, andvery little knowledge is assumed. Topics include chaos theory, meteorology, storage limitations of computers, population growth and decay, and the mechanics of dinosaurs. This book shows that mathematics can be fun!

1 A Little Bit of Chaos

1

(15)

Chris Bills

Liz Bills

1. Introduction

1

(1)

2. Complex numbers

2

(2)

3. The Argand diagram

4

(1)

4. Iterations

4

(2)

5. Prisoners and escapers

6

(3)

6. Julia sets

9

(1)

7. The Mandelbrot set

10

(1)

8. Conclusion

11

(5)

2 Nightmare on Large Number Street

16

(16)

Bill Dalton

1. Introduction

16

(1)

2. Hilbert's 10th problem

17

(1)

3. A Turing Machine

18

(4)

4. Conflicting computations

22

(2)

5. Binary numbers

24

(2)

6. Limitations on storage in computers

26

(6)

3 Games of Chance

32

(17)

David Harries

1. Starting activities

32

(1)

2. Probability

33

(4)

3. Expectation, arithmetic mean and standard deviation

37

(5)

4. Final thoughts

42

(1)

5. Discussion of experiments and questions

42

(7)

4 Weather

49

(11)

Brian Hoskins

1. What is weather?

49

(1)

2. The Earth and its atmosphere

49

(4)

3. The Coriolis force

53

(1)

4. The balance of forces

54

(1)

5. Motion around pressure centres

55

(1)

6. Some ideas on climate

56

(4)

5 Square Roots and Seventeengons

60

(29)

Frances Kirwan

1. Regular polygons

60

(2)

2. Construction of a regular pentagon

62

(7)

3. Compactible numbers and square roots

69

(8)

4. Polynomial equations

77

(4)

5. Format primes

81

(1)

6. Construction of the regular 17-gon using straight-edge and compasses

82

(7)

6 Discrete Mathematics and its Application to Ecology

89

(14)

Philip Maini

1. Introduction

89

(4)

2. The logistic map

93

(3)

3. Chaos

96

(2)

4. Summary

98

(5)

7 Channels, Pipes and Rivers

103

(21)

David Needham

1. Observations and examples

103

(1)

2. The flow of water

103

(2)

3. Mathematical model for flow in closed channels

105

(4)

4. Blockage in a closed channel

109

(3)

5. Flow in non-horizontal closed pipes

112

(3)

6. Applications

115

(3)

7. Open channel flow

118

(6)

8 Balloons and Bubbles

124

(25)

Michael Sewell

1. Experiments and questions

124

(1)

2. Equilibrium

124

(2)

3. Pressure on a patch

126

(3)

4. Equilibrium of a spherical patch

129

(2)

5. Tension in a soap film

131

(1)

6. Equilibrium of a patch of general shape

131

(1)

7. Equilibrium under zero excess pressure

132

(3)

8. Connected soap bubbles

135

(2)

9. Tension in rubber

137

(1)

10. Pressure peaks and pits

138

(2)

11. Comparison of tension and pressure values

140

(2)

12. Spherical and aspherical expansion

142

(1)

13. Cylindrical balloon

143

(6)

9 Dynamics of Dinosaurs

149

(27)

Michael Sewell

1. Introduction to some dinosaurs

149

(6)

2. Weighing dinosaurs

155

(7)

3. Dinosaur footprints

162

(4)

4. Speed of dinosaur

166

(4)

5. Athletic animals

170

(6)

10 Pythagorean Number Triples

176

(18)

David Stirling

1. Introduction

176

(3)

2. Diophantine equations

179

(6)

3. Pythagorean triples

185

(9)

11 Plato, Polyhedra and Weather Forecasting

194

(22)

Andy White

1. Setting the scene

194

(1)

2. Weather forecasting

195

(2)

3. Designing a new computer model

197

(1)

4. Polygons: irregular and regular

198

(3)

5. The Platonic solids

201

(3)

6. Almost regular arrangements of points on a sphere

204

(1)

7. Archimedean solids: the principle

205

(2)

8. Archimedean solids: the practice

207

(2)

9. Logic and visualization

209

(1)

10. Euler's formula

209

(7)

12 Water Waves

216

Winifred Wood

1. Introduction

216

(6)

2. Waves in deep water

222

(1)

3. Waves in shallow water

223

(2)

4. Groups of waves

225

(3)

5. Group speed

228

(2)

6. Other sea waves

230

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Mathematics Masterclasses Stretching the Imagination

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