Math in and out of the zoo Chris

Math in and out of the zoo Chris Budd

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

At Work? Hyperboloid of revolution

About town?

By the beach? Chevron folding caused by the geometry forced by the interaction of rock layers math can find the angle Singularities in rock folding described by the Swallow tail catastrophe:

Folded rocks have unlimited possible shapes!

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

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

Math guide to most Canadian animals

1. Fish: Artis Zoo Amsterdam, and hot fish

T: Temperature of the fish Sa: Outside air temperature I: Inside air temperature Heat gained = heat lost 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

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

Hitting the press! =

2. Penguins at….

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

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

• Blastoderm of lower density • Yolk is free to rotate

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

Modelling the conduction of the heat Q. Is turning needed to maintain an even temperature? Radius of egg R = 2 cm Temperature = T Thermal diffusivity Heat equation k=

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

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

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

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

People behave similarly in crowds but have attitude

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

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

Scramble crossing

Escape from the zoo!

4: But where are the bees?

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

First take your X-ray Source 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) Detector

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?

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

Good news … can now solve these equations rapidly using advanced numerical methods and compressed sensing techniques. And. . can then monitor the honey bees in high detail, and in real time HAEMOLYMPH VENTRICULUS 0. 05 mm

Or even in ancient times

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

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

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

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