Maths in and out of the zoo Chris Budd

1. Maths in and out of the zoo Chris Budd

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

3. Didcot Power Station? Hyperboloid of revolution

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

5. Or maybe a trip to the Zoo?

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

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

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

10. Water temperature Fan velocity Solar temperature

11. Hitting the press! =

12. 2. Penguins at….

13. 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

14. 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 ..

15. Blastoderm of lower density • Yolk is free to rotate

16. Some possible reasons for turning eggs …. • Convection of heatX 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

17. 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

18. 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

19. 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.

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

21. People behave similarly in crowds but have attitude

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

23. Global force Intended direction Local force

24. Scramble crossing

25. Escape from the zoo!

26. Movie of an actual crowd here.

27. 4: But where are the bees?

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

29. First take your X-ray Source X-Ray Detector 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)

30. 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

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

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

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

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

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

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