A mathematical nature walk

Preface Acknowledgments Introduction At the beginning ... (General questions to challenge our powers of observation, estimation, and physical intuition): Q.1-Q.6: Rainbows Q.7: Shadows Q.8-9: Clouds and cloud droplets Q.10: Light Q.11: Sound Q.12-13: The rotation of the Earth Q.14: The horizon Q.15: The appearance of distant hills. In the "playground" (just to get our feet wet...): Q.16: Loch Ness-how long to empty it? Q.17: The Grand Canyon-how long to fill it with sand? Q.18: Just how large an area is a million acres? Q.19: Twenty-five billion hamburgers-how many have you eaten? Q.20: How many head of cattle would be required to satisfy the (1978) daily demand for meat in the United States? Q.21: Why could King Kong never exist? Q.22: Why do small bugs dislike taking showers? Q.23: How fast is that raindrop falling? Q.24: Why can haystacks explode if they're too big? In the garden: Q.25: Why can I see the "whole universe" in my garden globe? Q.26: How long is that bee going to collect nectar? Q.27: Why are those drops on the spider's web so evenly spaced? Q.28: What is the Fibonacci sequence? Q.29: So what is the "golden angle"? Q.30: Why are the angles between leaves "just so"? In the neighborhood: Q.31: Can you infer fencepost (or bridge) "shapes" just by walking past them? Q.32: Can you weigh a pumpkin just by carefully looking at it? Q.33: Can you determine the paths of low-flying ducks? In the shadows: Q.34: How high is that tree? (An estimate using elliptical light patches) Q.35: Does my shadow accelerate? Q.36: How long is the Earth's shadow? Q.37: And Jupiter's? And Neptune's? Q.38: How wide is the Moon's shadow?

In the sky: Q.39: How far away is the horizon (neglecting refraction)? Q.40: How far away is that cloud? Q.41: How well is starlight reflected from a calm body of water? Q.42: How heavy is that cloud? Q.43: Why can we see farther in rain than in fog? Q.44: How far away does that "road puddle" mirage appear to be? Q.45: Why is the sky blue? Q.46: So how much more is violet light scattered than red? Q.47: What causes variation in colors of butterfly wings, bird plumage, and oil slicks? Q.48: What causes the metallic colors in that cloud? Q.49: How do rainbows form? And what are those fringes underneath the primary bow? Q.50: What about the secondary rainbow? Q.51: Are there higher-order rainbows? Q.52: So what is that triple rainbow? Q.53: Is there a "zeroth" -order rainbow? Q.54: Can bubbles produce "rainbows"? Q.55: What would "diamondbows" look like? Q.56: What causes that ring around the Sun? Q.57: What is that shaft of light above the setting Sun? Q.58: What is that colored splotch of light beside the Sun? Q.59: What's that "smiley face" in the sky? Q.60: What are those colored rings around the shadow of my plane? Q.61: Why does geometrical optics imply infinite intensity at the rainbow angle?

In the nest: Q.62: How can you model the shape of birds' eggs? Q.63: What is the sphericity index? Q.64: Can the shape of an egg be modeled trigonometrically? Q.65: Can the shape of an egg be modeled algebraically? Q.66: Can the shape of an egg be modeled using calculus? Q.67: Can the shape of an egg be modeled geometrically? In (or on) the water Q.68: What causes a glitter path? Q.69: What is the path of wave intersections? Q.70: How fast do waves move on the surface of water? Q.71: How do moving ships produce that wave pattern? Q.72: How do rocks in a flowing stream produce different patterns? Q.73: Can waves be stopped by opposing streams? Q.74: How far away is the storm? Q.75: How fast is the calm region of that "puddle wave" expanding? Q.76: How much energy do ocean waves have? Q.77: Does a wave raise the average depth of the water? Q.78: How can ship wakes prove the Earth is "round"?

In the forest: Q.79: How high can trees grow?

Q.80: How much shade does a layer of leaves provide for the layer below?

Q.81: What is the "murmur of the forest"?

Q.82: How opaque is a wood or forest?

Q.83: Why do some trees have "tumors"? In the national park: Q.84: What shapes are river meanders?

Q.85: Why are mountain shadows triangular?

Q.86: Why does Zion Arch appear circular? In the night sky: Q.87: How are star magnitudes measured?

Q.88: How can I stargaze with a flashlight?

Q.89: How can you model a star?

Q.90: How long would it take the Sun to collapse?

Q.91: What are those small rings around the Moon?

Q.92: How can you model an eclipse of the Sun? At the end... Q.93: How can you model walking?

Q.94: How "long" is that tree?

Q.95: What are those "rays" I sometimes see at or after sunset?

Q.96: How can twilight help determine the height of the atmosphere?

Appendix 1: A very short glossary of mathematical terms and functions

Appendix 2: Answers to questions 1-15

Appendix 3: Newton's law of cooling

Appendix :4 More mathematical patterns in nature.

Introduction

At the beginning ...

In the "playground"

In the garden

In the neighborhood

In the shadows

In the sky

In the nest

In (or on) the water

In the forest

In the national park

In the night sky

At the end ...

Appendix 1: A very short glossary of mathematical terms and functions

Appendix 2: Answers to questions 1-15

Appendix 3: Newton's law of cooling

Appendix 4: More mathematical patterns in nature.