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?

€

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

About town?

By the beach?

Singularities in rock folding described by the Swallow tail catastrophe:

€

(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 math can find the angle

Folded rocks have unlimited possible shapes!

Or maybe a trip to the Zoo?

3

2

5

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

Sg: Outside ground temperature

V: Fan velocity

€

A(Sa4 −T 4 ) + B =C (T − I) + D(T − Sg) + EV (sp(T) − sp(I))

Heat gained by solar radiation

Heat lost to air inside

Heat lost to ground

Cooling due to fan

Heat gained = heat lost

Fan velocity: V

Solve the formula to give the fish temperature T

Air temperature: Sa

T

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?

Can mathematics help us to optimise the turning strategy and save the penguins at ..

• Blastoderm of lower density

• Yolk is free to rotate

• 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 =

€

1.4 ×10−7 m2s−1

Q. Is turning needed to maintain an even temperature?

Heat equation

€

∂T∂t

= k∇ 2T

Too short!!! Consistent with results from incubator

20 minutes

2 minutes10 seconds

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

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.

People behave similarly in crowds but have attitude

• 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

€

drαdt

=vα ,dvαdt

= fα + fαβ + fαB

€

fα =1

τ αvα

0eα − vα[ ]

€

eα =pα − rαpα − rα

Intended direction

Global force: intent

Local force: avoidance

Mathematical formulae for these [Helbing]

Scramble crossing

Escape from the zoo!

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 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

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

Radon’s 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

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.

Or even in ancient times

At last .. A trip to the gift shop

Problem 5: What do you buy?

Robert Lang

Canadian Bull 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

€

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!!!