370 likes | 381 Views
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
E N D
The Importance of Being Peripheral John D. Barrow
Isoperimetric Theorems Maximum area is enclosed by a circle (Perimeter)2 4 Area (2r)2 = 4r2 Maximum volume is enclosed by a sphere (Area)3 36 (vol)2 (4r2)3 = 36 (4r3/3)2
When Small Boundaries Are Best Keeping warm Avoiding detection
Chilling Out Heat generation volume L3 Heat loss surface L2 Heating/Cooling L Is there a biggest possible computer?
Huddles and Herds Keeping warm
Trans-Atlantic Convoys Avoiding submarines Minimise perimeter or periscope image size
Sticking Together Is the best policy Split A into A/2 + A/2 Perimeter of single A convoy is 2A Perimeter of 2A/2 convoys is 22½A And is Bigger by 2 = 1.41..
area A area 2A area 4A 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 N2 steps to random walk a straight line distance of N step-lengths T/t=k2T so T/t T/d2andd2 t
How big can your boundary get ? Leads to as big a boundary as you wish for the same finite area
Numberof segments of length d needed to cover the coastline N(d) = M/dD 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
Lungs small mass andvolume but large surface interface
Fractalsdamp 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 L2 Mass L3 So Metabolic rate (Mass)2/3 Not (Mass)3/4
Model as a fractal network in D dims that transports nutrients while minimising the energy lost by dissipation Rate (Mass)(D-1)/D Rate (Mass)3/4 Fractal filling of 3 dims makes its information content like 4 dimensions
Black Holes R = 2GM/c2 Area = 4R2 M2 Density 1/M2
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 2nd Law of black-hole mechanics The total black hole area can never decrease The 2nd Law of thermodynamics Entropy can never decrease SBH Area M2 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