Maths in and out of the zoo

Chris Budd

Where does an Oxford trained applied mathematician go to find some maths in action?

Maths masterclass students at a maze

Didcot Power Station?

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x 2 + y 2 =1+ z2 Hyperboloid of revolution

By the beach?

Singularities in rock folding described by the Swallow tail catastrophe:

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(x,y) = s−st

1+ s2,s2

2+

t

1+ s2

⎛

⎝ ⎜

⎞

⎠ ⎟,−∞ < s, t < ∞

⎧ ⎨ ⎩

⎫ ⎬ ⎭

Chevron folding caused by the geometry forced by the interaction of rock layers

Or maybe a trip to the Zoo?

Some problems from the zoo:

Fish, penguins, flocks, crowds, bees, and the gift shop ….

Bristol Zoo

1. Fish: Artis Zoo Amsterdam, and hot fish

Heat into tank

Heat out of reservoir

In detail!

Fan velocity

Water temperature

Solar temperature

Hitting the press!

t∂∂

2

2

x∂∂=

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?

Could we develop a mathematical model which will allow us to optimise the turning strategy and save the penguins at ..

• Blastoderm of lower density

• Yolk is free to rotate

• 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 ….

Modelling the conduction of the heat

Radius of egg R = 2cm

Temperature = T

Thermal diffusivity k =

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1.4 ×10−7 m2s−1

Q. Is turning needed to maintain an even temperature?

Heat equation

Thermal conduction timescale

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∂T∂t

= k∇ 2T

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τ ≈R2

k= 40 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

How do birds flock, fish shoal or people crowd?

Each bird interacts with its nearest neighbours but the flock behaves like a single organism.

Flocking movie here and ABM simulation

Equations for: alignment, vision, avoidance, intent

People behave similarly in crowds but have attitude

• 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

€

fα

€

fαβ

€

rα

€

fαB

€

drαdt

=vα ,dvαdt

= fα + fαβ + fαB

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fα =1

τ αvα

0eα − vα[ ]

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eα =pα − rαpα − rα

Intended direction

Global force

Local force

Scramble crossing

Escape from the zoo!

Movie of an actual crowd here.

4: But where are the bees?

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

QuickTime™ and alibx264 decompressor

are needed to see this picture.

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

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

Radon’s formula: basic equations of Tomography

dsssfR ))cos()sin(),sin()cos((),( θθρθθρθρ +−=∫

Radon’s formula leads to a large set of equations

Problem … there are typically over 1 000 000 equations to solve

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R = A f

Solving these (in a regularised form) tells us what f is!

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(ATA + μ 2I) f = ATR

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!

At last .. A trip to the gift shop

Problem 6: What do you buy?

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

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π

Maths can help you make the Maths can help you make the perfect gift perfect gift

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!!!