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.