math in and out of the zoo chris budd where does a mathematician go to find some maths when they are...
TRANSCRIPT
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!!!