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The Importance of Being Peripheral
John D. Barrow
Land Economy
Queen Didos Problem
Wiggliness
Maximum area is enclosed by a circle(Perimeter)2 4 Area(2r)2 = 4 r2Maximum volume is enclosed by a sphere(Area)3 36 (vol)2(4r2)3 = 36 (4r3/3)2Isoperimetric Theorems
When Small Boundaries Are BestKeeping warmAvoiding detection
Chilling OutHeat generation volume L3Heat loss surface L2Heating/Cooling LIs there a biggest possible computer?
Be big if you live at the North Pole
Huddles and HerdsKeeping warm
Avoid being on the edge of the herd
Trans-Atlantic Convoys Avoiding submarines
Minimise perimeter or periscope image size
One big one or many small ones?
Sticking TogetherIs the best policySplit A into A/2 + A/2Perimeter of single A convoy is 2APerimeter of 2A/2 convoys is22AAnd isBigger by 2 = 1.41..
Fish-balling is bad for the group!
Division leads to more boundarycut
area A area 2A area 4A Likelihood of explosionIs increasing
Fire StormsIgnition of dust produces explosive spread of fire
Global Dimming?Sunlight scattering off atmospheric pollutants depends on surface areamore pollutants more particles smaller droplets relatively more surface area more back-reflection of sunlight cooler Earth2-3% per decade in N lats1 deg C rise in USA 3 days after 9/11
When Large Boundaries are BestKeeping coolBeing seenSoaking up moistureGetting nutrientsDissolving fast
Cooking Times
Heat diffusing through a cooking turkeyTime area (size)2 (weight)2/3because weight density (size)3N2 steps to random walk a straight line distance of N step-lengthsT/t=k2T so T/t T/d2 and d2 t
How big can your boundary get ?
Leads to as big a boundary as you wish for the same finite area
Number of segments of length d needed to cover the coastlineN(d) = M/dDD = 1.25 for the west coast of BritainD = 1.13 for the Australian coastD = 1.02 for the South African coast
FractalsA recipe for maximising surfaceCopy the same pattern over and over again on all scalesTreesFlowersHuman lungsMetabolic systemsJackson Pollock paintings
Romanesque Broccoli
Lungs
small mass and volume but large surface interface
Fractals damp vibrationsLungs and coastlines
What is its length?Fractal coastlines damp down waves and reduce erosion very efficiently
Universal metabolism Metabolic rate vs (mass)3/4Kleibers Law
Puzzling ??? Rate = Heat loss area L2Mass L3So Metabolic rate (Mass)2/3Not (Mass)3/4
Model as a fractal network in D dims that transports nutrients while minimising the energy lost by dissipationRate (Mass)(D-1)/DRate (Mass)3/4Fractal filling of 3 dims makes its information content like 4 dimensions
Black Holes
R = 2GM/c2Area = 4R2 M2Density 1/M2
Black Holes Are Black BodiesThey obey the Laws of ThermodynamicsThermal evaporation of energy with entropy given by the areaand temperature by surface gravity (g)
The 2nd Law of black-hole mechanicsThe total black hole area can never decreaseThe 2nd Law of thermodynamicsEntropy can never decreaseSBH Area M2Information content SBH
Is there a universal holographic principle?The maximum information content of a region is determined by its surface area???S (Area)/4 = SBH
The edge of something to look into?
The Heat Death of the UniverseStotal grows
butSmax grows fasterStimeSmaxStotal
StimeSmaxStotalUniverse accelerates
The Importance of Being Peripheral*John D. Barrow
A fractal simulation
Cauliflower