2. Maths in and out of the zoo Chris Budd

3. Where does an Oxford trained applied mathematician go to find some maths in action? Maths masterclass students at a maze

4. Didcot Power Station? Hyperboloid of revolution

5. By the beach? Singularities in rock folding described by the Swallow tail catastrophe: Chevron folding caused by the geometry forced by the interaction of rock layers

6. Or maybe a trip to the Zoo?

7. Some problems from the zoo: Fish, penguins, flocks, crowds, bees, and the gift shop …. Bristol Zoo

8. 1. Fish: Artis Zoo Amsterdam, and hot fish

10. Heat into tank Heat out of reservoir In detail!

11. Fan velocity Water temperature Solar temperature

12. Hitting the press! =

13. 2. Penguins at….

14. Preservation of rare bird species requires them to be bred in captivity One way is to incubate eggs artificially Need to control Temperature Humidity Turning of the egg Very sensitive to the turning strategy! Eggs are turned by mother every 20 minutes

15. Questions ….. Why do birds turn their eggs? Could we develop a mathematical model which will allow us to optimise the turning strategy and save the penguins at..

16. Blastoderm of lower density Yolk is free to rotate

17. Convection of heat X Penguins sit on the top of the egg! Conduction of heat … this is what the zoo believes! Dispersal of nutrients Removal of baby penguin poo Some possible reasons for turning eggs ….

18. Modelling the conduction of the heat Radius of egg R = 2cm Temperature = T Thermal diffusivity k = Q. Is turning needed to maintain an even temperature? Heat equation Thermal conduction timescale = 40 minutes Too short!!! Consistent with results from incubator

19. In fact … turning is actually needed to move the nutrients and remove the waste matter Monitor the turning using an artificial nylon egg … And then reproduce this in the incubator

20. 3. Birds of a feather flock together How do birds flock, fish shoal or people crowd? Each bird interacts with its nearest neighbours but the flock behaves like a single organism.

21. Flocking movie here and ABM simulation Equations for: alignment, vision, avoidance, intent

22. People behave similarly in crowds but have attitude

23. Global force : Intentions of the individual Local (social) force.. Avoidance strategy of people : or obstacles : Cohesion of families and groups Random effects Idea: Individual at location is acted on by several forces

24. Intended direction Global force Local force

25. Scramble crossing

26. Escape from the zoo!

27. Movie of an actual crowd here.

28. 4: But where are the bees?

29. X-ray CAT scan the beehive.. In real time

30. X-Ray Object ρ : Distance of the X-Ray from a fixed point θ : Angle of the X-Ray from a fixed line Measure attenuation of X-Ray R( ρ, θ ) Source Detector First take your X-ray

31. REMARKABLE FACT If we can measure R( ρ, θ ) accurately we can calculate the X-ray attenuation factor f(x,y) of the object at any point Knowing f tells us the structure of the object Mathematical theorem proved by Radon (1917) Took 60 years before computers and machines were developed by Cormack to use his results

32. Radon’s formula: basic equations of Tomography Radon’s formula leads to a large set of equations Problem … there are typically over 1 000 000 equations to solve Solving these (in a regularised form) tells us what f is!

33. VENTRICULUS HAEMOLYMPH 0.05mm And.. can then monitor the honey bees in high detail, and in real time Good news … can now solve these equations rapidly using the Conjugate Gradient Method!

34. At last.. A trip to the gift shop Problem 6: What do you buy?

35. Moose Stag Beetle Crease patterns are worked out using mathematics and obey strict mathematical rules. Eg. At any vertex the sum of all odd (even) angles is Maths can help you make the perfect gift

36. Can even use Origami to Trisect an Angle or double a cube!

37. I hope that you liked your trip to the zoo Good maths really is everywhere!!!