Some Interesting Curves John D Barrow

1. Some Interesting Curves John D Barrow

2. Swiss Re Building 30 St. Mary Axe ‘The Gherkin’ Norman Foster & Partners

3. The Swiss Re Building 180 metres high 40 floors 2003 Design Factors Sky visible Low winds on ground Slow and smooth airflow Wedges bring in air and light Six on each floor, Offset creates spiral effect Helps bring air in All surfaces flat – cheaper!

4. Torre Agbar Barcelona (2005) 142m

5. Tatiana Tatiana’s House

6. The San Francisco Zoo Disaster

7. y Projectiles v x  u Launch velocity = (u, v) x = ut y = vt - ½ gt2 = vx/u – ½ gx2/u2 y vs x is a parabola dy/dt = 0 at maximum height: tmax = v/g ymax = ½ v2/g

8. Mt Etna

9. Crouching Tiger h V x V2 = g (h +(h2 + x2)) Over short distances on the flat a tiger can reach top speeds of more than 22 metres per sec (ie 50 miles per hr). From a 5 metre start it can easily reach a launch speed of 14 metres per sec. h = 3.8 metres x = 10 metres

10. Only V > 12 metres per sec launch speed needed for the tiger to clear the wall

11. The hanging chain, ‘catenaria’: Leibniz and Huygens 1691

12. y = Acosh(x/A) = ½ A{ex/A + e-x/A}

13. The portion AP is in equilibrium under the horizontal tension H at A, the tension F directed along the tangent at P, and the weight W of AP. If the weight of the string is w per unit length and s is the arc AP, W = ws; and from the force triangle, tan ψ = ws/H = s/c, where c = H/w is called the parameter of the catenary is determined by dy/dx = s/c With solutions    y = c cosh(x/c)s = c sinh(x/c) 

14. Half A Catenary The Rotunda was originally a tent put up in London as part of the festivities to celebrate the defeat of Napoleon. Designed by John Nash, It was moved to Woolwich in 1816 and converted in 1920 into a permanent structure. It is now the Royal Artillery Museum.

15. Inverted Catenary Arches The Gateway Arch, St Louis, MS 630 ft x 630 ft y = -127.7 ft cosh(x/127.7ft) + 757.7ft Robert Hooke 1671 Latin anagram (revealed in 1705): ‘As hangs the flexible chain, so inverted stand the touching pieces of an arch.’

16. Can You Ride A Bike With Square Wheels?

17. Stan Wagon Demonstrates For the rolling square: the shape of the road is the catenary truncated at x = + sinh-1(1)

18. For regular n-sided polygonal wheels the curve of the road is made from catenaries with y = -Acosh(x/A) A = Rcot(/n)

19. Clifton Suspension Bridge (1865)

20. Suspension Bridges are Parabolas Constant weight per unit length p

21. Gustave Eiffel’s Tower (1899) ‘moulded in a way by the action of the wind itself” 300m = 894 ft high f(x) x f(t) dt = f (x) x (x - t) f(t) dt  f(x) = Aebx

22. Watkin’s Great Wembley Folly Sir Edward Watkin, Chairman of Metropolitan Railway saw Eiffel’s 894 ft Tower He wanted a bigger one (1200 ft) on his land in Wembley Park Eiffel refused. Benjamin Baker completed stage 1 (155 ft) in Sept 1895. Opened 18th May 1896 but never went higher: marshy ground and shifting foundations. Tea shop for the new Underground Station, few visitors Declared unsafe in 1902. Demolished 1904-7. Iron sold for scrap. Wembley Stadium built on the site in the 1920s

23. Roller Coasters Millennium Force, Cedar Point

24. A Tale of Two Forces You feel Force of Gravity Weight = Mg  v You feel radially outward Centrifugal Force Mv2/r Mv2/r

27. Staying in your Seat at the Top Radius r Fall from height h under gravity from rest ½mVb2 = mgh At bottom: Vb =2gh Ascend to the top of the circular loop of radius r. Arriving there with speed Vt needs Energy = 2mgr + ½ mVt2 So: mgh = ½ mVb2 = 2mgr + ½mVt2 Net Upward Force on rider (mass m) at top = mVt2/r – mg > 0 So: we need Vt2 > gr or you fall out of the car ! h > 2.5r

28. Staying Conscious At the Bottom! If h > 2.5r you reach the bottom with speed Vb = (2gh) > (2g2.5r) = (5gr) The net downward force on you at the bottom will be Weight + Centrifugal force mg + mVb2 /r > mg + 5mg = 6mg A 6-g force will probably render you unconscious Oxygen would not get to the brain Circular roller coasters seem to fail their Risk Analysis

29. A Recipe for Success We want Vt2 /r big at the top to hold us in But Vb2 /r small at the bottom to reduce the g force on the riders Make ‘r’ small at the top and big at the bottom Ellipses first used in 1901 at Coney Island

30. Clothoid loop Shockwave roller coaster at Six Flags Over Texas, Arlington Werner Stengel’s first use of the Clothoid in 1975

31. Clothoid loop Loopen at Tusenfryd in Norway (Vekoma, Corkscrew, 1988)

32. Clothoid curvature varies linearly with arc length t

33. Motorway Junctions   An arc of a clothoid has variable curvature, proportional to the distance along the curve from the origin. It provides the smoothest link between a straight line and a circular curve. It is used in roads and railroads design: the centrifugal force actually varies in proportion to the time, at a constant rate, from zero value (along the straight line) to the maximum value (along the curve) and then back again to zero. A vehicle following the curve at constant speed will have a constant rate of angular acceleration.

34. At constant speed you can simply rotate the steering wheel at a constant rotation rate. If the bend was a different shape then you would need to keep adjusting the rate of movement of the steering wheel or the speed of the car

35. Möbius and His Bands August Möbius, notebook 1858

36. Möbius Belts, Tape-drives and Conveyor belts US Patent 3991631

37. The MöbiusUniversalRecycling Symbol Not a trademark! Gary Anderson, Student at USC, design competition winner, 1970

38. Taiwanese Recycling Symbol

39. Maurits Escher, woodcut Moebius Strip II (Red Ants), 1963

40. Robert Wilson, Fermilab, Batavia. Illinois