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  y 1 z
2
2
2
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:


st
s2
t 
(x, y)  s 
, 
,   s,t  
2 2
2
1 s
1 s 



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
A(Sa4  T 4 )  B  C (T  I)  D(T  Sg)  E V (sp(T)  sp(I))
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!

t

= 2
x
2
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 = 2cm
Temperature = T
Thermal diffusivity

Heat equation
7
2 1
1.4
10
m
s
k=
T
 k 2T
t
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
dr
v ,
dt

[Helbing]
dv
 f  f  fB
dt
f 

Local force: avoidance

1


0
v e  v
e 

p  r
p  r
Global force:
intent
Intended direction
Scramble crossing
Escape from the zoo!
4: But where are the bees?
X-ray CAT scan the beehive .. In real time
QuickTime™ and a
libx264 decompressor
are needed to see this picture.
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
R(  , )   f (  cos( )  s sin( ),  sin( )  s cos( )) ds
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
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
0.05mm
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!!!