The Importance of Being Peripheral John D Barrow
The Importance of Being Peripheral John D. Barrow
Land Economy
Queen Dido’s Problem
Wiggliness
Isoperimetric Theorems Maximum area is enclosed by a circle (Perimeter)2 4 Area (2 r)2 = 4 r 2 Maximum volume is enclosed by a sphere (Area)3 36 (vol)2 (4 r 2)3 = 36 (4 r 3/3)2
When Small Boundaries Are Best Keeping warm Avoiding detection
Chilling Out Heat generation volume 2 Heat loss surface L Heating/Cooling L Is there a biggest possible computer? 3 L
Be big if you live at the North Pole
Huddles and Herds Keeping 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 Together Is the best policy Split A into A/2 + A/2 Perimeter of single A convoy is 2 A Perimeter of 2 A/2 convoys is 2 2 ½A And is Bigger by 2 = 1. 41. .
Fish-balling is bad for the group!
Division leads to more boundary cut
area A area 2 A area 4 A Likelihood of explosion Is increasing
Fire Storms Ignition of dust produces explosive spread of fire
Global Dimming? Sunlight scattering off atmospheric pollutants depends on surface area more pollutants more particles smaller droplets relatively more surface area more back-reflection of sunlight cooler Earth 2 -3% per decade in N lats 1 deg C rise in USA 3 days after 9/11
When Large Boundaries are Best Keeping cool Being seen Soaking up moisture Getting nutrients Dissolving fast
Cooking Times Heat diffusing through a cooking turkey Time area (size)2 (weight)2/3 because weight density (size)3 N 2 steps to random walk a straight line distance of N step-lengths T/ t=k 2 T so T/t T/d 2 and d 2 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 coastline D N(d) = M/d D = 1. 25 for the west coast of Britain D = 1. 13 for the Australian coast D = 1. 02 for the South African coast
Fractals A recipe for maximising surface Copy the same pattern over and over again on all scales Trees Flowers Human lungs Metabolic systems Jackson Pollock paintings
Romanesque Broccoli
Lungs small mass and volume but large surface interface
Fractals damp vibrations Lungs and coastlines What is its length? Fractal coastlines damp down waves and reduce erosion very efficiently
Universal metabolism Metabolic rate vs (mass)3/4 Kleiber’s Law
Puzzling ? ? ? Rate = Heat loss area L 2 3 Mass L So Metabolic rate (Mass)2/3 Not 3/4 (Mass)
Model as a fractal network in D dims that transports nutrients while minimising the energy lost by dissipation Rate (Mass)(D-1)/D 3/4 Rate (Mass) Fractal filling of 3 dims makes its information content like 4 dimensions
Black Holes R = 2 GM/c 2 2 2 Area = 4 R M Density 1/M 2
Black Holes Are Black Bodies They obey the Laws of Thermodynamics Thermal evaporation of energy with entropy given by the area and temperature by surface gravity (g)
The 2 nd Law of black-hole mechanics The total black hole area can never decrease The 2 nd Law of thermodynamics Entropy can never decrease 2 SBH Area M Information 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?
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