+
1
|
skin
|
login
|
edit
alpha
::
entropie
user:anonymous
_h1 information's entropy {show {@ src="data/plogp.jpg" height="350" width="700" title="For p=1/e ≠ 0.37, H = -K.p.log(p) gets its maximum value 1/e, with K = log(10) = 2.302585092994046"}} {pre °° {def H {lambda {:p} {- {* :p {log :p}}} }}°° {def H {lambda {:p} {- {* :p {log :p}}} }} °°{def D {lambda {:f :x} {{lambda {:f :x :dx} {/ {- {:f {+ :x :dx}} {:f {- :x :dx}} } {* 2 :dx} }} :f :x 0.0001} }} °° {def D {lambda {:f :x} {{lambda {:f :x :dx} {/ {- {:f {+ :x :dx}} {:f {- :x :dx}} } {* 2 :dx}} } :f :x 0.0001} }} °°{/ 1 {E}}°° {/ 1 {E}} °°{H {/ 1 {E}}}°° {H {/ 1 {E}}} °°{floor {{D H} {/ 1 {E}}}}°° {floor {{D H} {/ 1 {E}}}} } _p After Shannon and [[Brillouin|http://www.informationphilosopher.com/solutions/scientists/brillouin/]], the entropy of a an information given by an event whose probability is p is given by {b H(p) = -p.log(p)}. Its maximum - the zero of its derivee - is reached for {b p = 1/e} = {b {/ 1 {E}}}. _p The golden ratio is {b Φ1 = (1+sqrt(5))/2 = {/ {+ 1 {sqrt 5}} 2} or Φ2 = -(1-sqrt(5))/2 = {/ {- {sqrt 5} 1} 2}}. The number {b 1-Φ2 = {+ 1 {/ {- 1 {sqrt 5}} 2}}} is close to the previous value of p giving the maximum to the information's entropy. _p The golden ratio could be seen as a geometric approximation of a value linked to the information's entropy, to the {b quality of information}. Not a mysterious and magic Middle Age figure, but a simple (approached but useful) geometrical construction of a more general law related to some optimisation of information. Applying not only to painting and architecture, but also to music, poetry, love, esthetics, ... _p More to come ... _p Page under construction, meanwhile you may have a look to : _ul [[modulor|http://epsilonwiki.free.fr/alphawiki_2/?view=stock_others_modulor]] _ul [[entropie|http://epsilonwiki.free.fr/portail/?view=entropie]] (french) _ul [[golden ratio|http://epsilonwiki.free.fr/portail/?view=sectiondoree]] (french) _h4 fibonacci & golden rect _p A golden rect and its inner spiral can be built with Fibonacci numbers, for instance {b 1,2,3,5,8,13,21,34}. {div {@ style="display:none"} {def OX 700} {def OY 1500} {def gold {lambda {:fib :dx :dy :da} {span {@ style="position:absolute; left:0; top:0; width:{- {* :fib 10} 2}px; height:{- {* :fib 10} 2}px; border:2px solid red; background: cyan url('data/modulor/perfect_piggy.jpg') 50% 50%; transform: translate( {+ {OX} {* :dx 10}}px , {+ {OY} {* :dy 10}}px) rotate(:dadeg); -webkit-transform: translate( {+ {OX} {* :dx 10}}px , {+ {OY} {* :dy 10}}px) rotate(:dadeg); border-radius:0 {* :fib 10}px 0 0; "}} {span {@ style="position:absolute; left:0; top:0; width:{- {* :fib 10} 2}px; height:{- {* :fib 10} 2}px; border:2px solid red; transform: translate( {+ {OX} {* :dx 10}}px , {+ {OY} {* :dy 10}}px) rotate(:dadeg); -webkit-transform: translate( {+ {OX} {* :dx 10}}px , {+ {OY} {* :dy 10}}px) rotate(:dadeg); "}} }} } {gold 1 0 0 270} {gold 2 0 1 180} {gold 3 2 0 90} {gold 5 0 -5 0} {gold 8 -8 -5 270} {gold 13 -8 3 180} {gold 21 5 -5 90} {gold 34 -8 -39 0} {pre °° {def OX 700} {def OY 1300} {def gold {lambda {:fib :dx :dy :da} {span {@ style="position:absolute; left:0; top:0; width:{- {* :fib 10} 2}px; height:{- {* :fib 10} 2}px; border:2px solid red; background: cyan url('data/modulor/perfect_piggy.jpg') 50% 50%; transform: translate( {+ {OX} {* :dx 10}}px, {+ {OY} {* :dy 10}}px) rotate(:dadeg); "}} }} {gold 1 0 0 270} {gold 2 0 1 180} {gold 3 2 0 90} {gold 5 0 -5 0} {gold 8 -8 -5 270} {gold 13 -8 3 180} {gold 21 5 -5 90} {gold 34 -8 -39 0} °°}