Date post: | 22-Dec-2015 |
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Where does an Oxford trained applied mathematician go to find some maths in action?
Maths masterclass students at a maze
Didcot Power Station?
€
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
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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!!!