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Math in and out of the zoo Chris Budd

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Where does a mathematician go to find some maths when they are not in their office? At play?

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At Work? Hyperboloid of revolution

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About town?

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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 math can find the angle

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Folded rocks have unlimited possible shapes!

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Or maybe a trip to the Zoo? 3 2 5 1 1

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

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Math guide to most Canadian animals

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1. Fish: Artis Zoo Amsterdam, and hot fish

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T: Temperature of the fish Sa: Outside air temperature I: Inside air temperature Sg: Outside ground temperature V: Fan velocity Heat gained by solar radiation Heat lost to air inside Heat lost to ground Cooling due to fan Heat gained = heat lost

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Fan velocity: V Solve the formula to give the fish temperature T Air temperature: Sa T

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Hitting the press! =

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2. Penguins at….

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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

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Questions ….. Why do birds turn their eggs? Can mathematics help us to optimise the turning strategy and save the penguins at..

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Blastoderm of lower density Yolk is free to rotate

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Conduction of heat … this is what the zoo believes! Dispersal of nutrients Removal of baby penguin poo Some possible reasons for turning eggs ….

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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

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Too short!!! Consistent with results from incubator 20 minutes 2 minutes10 seconds

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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

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3. Birds of a feather flock together Birds flock, fish shoal and people crowd

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Math describes this through equations for: alignment, vision, avoidance, intent Each bird interacts with its nearest neighbours but the flock behaves like a single organism.

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People behave similarly in crowds but have attitude

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Global force: Intentions of the individual Repulsive force.. Avoidance strategy of people or obstacles : Cohesive force of families and groups Idea: Individual in crowd is acted on by several forces Put these forces together to work out the crowd behaviour

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Intended direction Global force: intent Local force: avoidance Mathematical formulae for these [Helbing]

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Scramble crossing

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Escape from the zoo!

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4: But where are the bees?

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

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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

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REMARKABLE FACT If we can measure R( ρ, θ ) accurately for enough X-rays we can calculate the density f(x,y) of the object Mathematical theorem proved by Radon (1917) Knappe Kop?

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Radons formula: basic equations of Tomography Radons formula leads to a set of equations for f Problems … there are over 1 000 000 equations to solve, and the information must be incomplete for short radiation times

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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 advanced numerical methods and compressed sensing techniques.

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Or even in ancient times

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At last.. A trip to the gift shop Problem 5: What do you buy? Robert Lang

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Canadian Bull MooseStag 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

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

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

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