Pì The Radian Estimates of  Gabriel’s Horn

2. Pì The Radian Estimates of  Gabriel’s Horn

3. In nature  is found in meandering rivers. • Measure the actual length of a river from the source to the mouth. Then measure the direct length in a straight line from source to mouth. On average the actual length equals  multiplied by the direct length. • The actual length is about 3·14 times the direct length. Why is this true?

4. Rivers tend to bend, because of more erosion on the outer edge. If they bend too much, the degree of bending is restricted because the bends join up, straightening the river again. • Einstein explained that balance between these two opposing factors leads to an average ratio of π (3·14) between actual and direct length.

6. 1 7 22 7 __ ___ π= 3·14 or 3 or • π is unfortunately irrational. • This means that no matter how many decimal places are taken no pattern can be found. It is usually estimated as: • Your calculator estimates π = 3·141592654

7. r r 1 radian The Radian

8. The Radian • There are 2π radians in a full rotation – once around the circle. • There are 360° in a full rotation. • There are about 6·28 radians in one full circle. • 1 radian is approximately equal to 57·3º.

9. Estimates for  • A little known verse of the Bible reads And he made a molten sea, ten cubits from the one brim to the other: it was round all about, and his height was five cubits: and a line of thirty cubits did compass it about. (I Kings 7, 23)  = 3

10. Estimates for  • Archimedes over 2000 years ago used regular polygons inscribed in a circle (with radius one) to estimate . • By ever increasing the number of sides of the polygon, he argued that the perimeter of the polygon gets closer to value of the circumference of the circle. • His method of exhaustion is the same concept as the modern notion of infinite limits. The use of linear approximations of curves is exactly the same idea that is the basis of differential calculus.

11. 223 7 22 7 ____ ___ < π < Estimates for  Using a 96-sided polygon

12. æ ö 2 2 4 4 6 6 8 8 10 ... p = ´ ´ ´ ´ ´ ´ ´ ´ 2 ç ÷ 1 3 3 5 5 7 7 9 9 è ø Estimates for  • The English mathematician John Wallis in 1855 showed that  could be calculated from the infinite series

13. 1 x 1 x Draw the graph of the function f (x) = . Gabriel’s Horn

14. y x Gabriel’s Horn This graph can be turned into a 3D shape by rotating it 360°, a full circle, on the x-axis. This produces a shape like a horn.

15. Gabriel’s Horn • If the horn has a radius of 1 m at the wide end, then the total volume of the horn is π m3.

16. Gabriel’s Horn • If the horn has a radius of 1 m at the wide end, then the total volume of the horn is π m3. • Remember this horn goes on forever to the right, just getting thinner and thinner. • This causes a problem. The total volume of the horn is π m3, but the surface area is infinite. • The proof of this requires integration (part of calculus).