*等角螺旋 equiangular spiral [#n892493a]
The equiangular spiral has a lot longer history than the science of mathematics. The spiral has been produced for thousands of years in the shape of the nautulis shell, the arrangement of sunflower seeds in the sunflower, among various other natural phenomena.
The spiral has been known by a variety of names corresponding to one or another of its fetures. By Descartes, who discussed it in 1638, it was designated the equiangular spiral, because the angle at which a radius increasess in geometrical progression as its polar angle increasess in arithmetical progression, it has been called the geometrical spiral. Descartes started from the property s = a.r. Torricelli, who died in 1647, worked on it independently and used for a definition the fact that the radii are in geometric progression if th angles increase uniformly. From this he discovered the relation s = a.r; that is to say, he found the rectification of the curve. Halley noting that the lenghts of the segments cut off from a fixed radius by successive turns of the curve were continued proportion, named it the proportional spiral. Jacob Bernoulli (1654-1705), who was so fascinated by the mathematical beauty of the curve that he asked that it might be engraved on his tombstone, called it the logarithmic spiral. Bernoulli (1654-1705) who requested that the curve be engraved upon his tomb with the phrase "Eadem mutata resurgo" ("I shall arise the same, though changed.")
--星雲 M51
*対数螺旋:logarithmic spiral [#i638f5bf]
それは中心からの距離が増加するにつれてらせんの間隔が幾何級数(または等比数列: Geometricpr pgression )的に増加します。等角らせん(equiangular spiral)、成長らせん(growth spiral)、及び{バーニューリー(ベルヌーイ;Bernoulli)}らせんあるいは"驚異のらせん(spira mirabilis,The marvellous spiral)"としても知られているこの曲線の豊富な特徴は、17世紀の哲学者{デイカールト(デカルト;Descartes)}によって発見されて以来、数学者たちを魅了しています。興味深いことにこの抽象的な形は自然界には上の印象的な視覚比較で示されるよりももっと豊富にあります。たとえば、対数らせんが表現するのは、ひまわりの種の配列、オウム貝の形そして、もちろん、{コーラフラウア(カリフラワー;cauliflower)}があります。
*歴史的記述 デカルトとヤコブ ベルヌーイ [#h8030d73]
After the discovery of analytical geometry by Rene Descartes(1596-1650) in 1637, the custom of representing various curves with the help of equations came into vogue.
Logarithmic spiral was one of those curves which at that time'drew the attention of mathematicians. During that time the polar equation of logarithmic spiral was written as In r = θ where 'In' stands for natural logarithm, i.e. logarithm with base e (then called as hyperbolic logarithm). At present that equation is written in the form r = e^aθ,where θ is measured in radians (one
radian is approximately equal to 57 degrees). 
The constant 'a' represents the rate of increase of the spiral. When a > 0, then r increases in the anti-clockwise sense and left-handed spiral is generated .


θ=log r/log a
aは定数だから、log aも定数で1/log a をkとすると
θ=k log r
-17 世紀の数学者ヤコブ・ベルヌーイは等角螺旋のもつ様々な美しい性質に感動し, 自分の墓石には等角螺旋を刻んで欲しいという遺言を残しました. しかし石屋の手違いまたは手抜きのために彼の墓石にはアルキメデス螺旋(r = aθ) が刻まれてしまった。
-対数螺旋の動径と接線のなす角度がθに関わりなく一定である。等角螺旋THE EQUIANGULAR SPIRALとも呼ばれる
*いろいろな螺旋と呼び名 [#f50d3841]
Equiangular spiral (also known as logarithmic spiral, Bernoulli spiral, and logistique) describe a family of spirals. It is defined as a curve that cuts all radii vectors at a constant angle. 
A special case of equiangular spiral is the circle, where the constant angle is 90 Degrees.

*定義そして長さ、曲率、正接角 [#f4390a66]
The equation of Equiangular (or logarithmic spiral in Polar Coordinates is given by


where r is the distance from the Origin, is the angle from the x-Axis, and a and b are arbitrary constants. It can be expressed parametrically using
The rate of change of Radius is
and the Angle between the tangent and radial line at the point (r,θ)


So, as b->0 then φ->π/2 and the spiral approaches a Circle. 

If P is any point on the spiral, then the length of the spiral from P to the origin is finite. In fact, from the point P which is at distance r from the origin measured along a Radius vector, the distance from P to the Pole along the spiral is just the Arc Length. 

The Arc Length, Curvature, and Tangential Angle of the logarithmic spiral are

基本性質:曲率半径Rが弧長(路長)s に比例する。
*性質 Properties of Logarithmic Spiral [#i145e29e]
--1. The most important property of a logarithmic spiral is that r (i.e. the distance from the origin) increases proportionately with increase of θ.
The reason for this is e^aφ acts as common ratio in the relation
--2. If from any point on a logarithmic spiral one starts spiralling inwards along the curve then an infinite number of complete rotations are required to reach the origin, but the distance traversed will be finite . This property was discovered by Evangelista Toricelli(160?-1647), pupil of Galileo Galilei (1564-1642).This discovery of Toricelli was the first instance of rectification, i.e. the calculation of the length of an arc of a non-algebraic curve.
--3. Any line segment drawn through the origin always intersects a logarithmic spiral at equal angles.If we put a = 0 in the equation of an equiangular spiral, then we get r = 1 which is the equation of a unit circle. So, circle is a special type of equiangular spiral whose rate of growth is zero.

*螺旋の例題 [#vcfde456]
--[証明]軌跡はOを極とした極座標(r,θ)を考える。AはBに向かって走るのですから常にABが接線になります.すると∠OAB=π/4となる。r =f (θ)とし,接線と動径のなす角をωとすれば, r/r’=tanωなので r’=r 。これは「微分方程式」で「変数分離型」である。dr/dθ=r ∫1/rdr=∫dθとなるので logr=θ+定数となる。この解は r=a・e^θ となり、等角螺旋である。
正方形の1辺の長さをsとすれば、A点では r=s/√2 θ=π/4なので a定数が決まる。
s/√2=a e^π/4 より a=s/√2e^(-π/4)
曲線の長さLは,媒介変数表示x=f ( t ),y=g ( t )の場合は,t =αからt =βまでの曲線の長さは
 L=∫α->β√(dx/dt)^2+(dy/dt)^2 dt
x=r cosθ,y=r sinθですから,積の微分を使って, dx/dθ= dr/dθ・cosθ-r sinθ, = dr/dθ・sinθ+r cosθ であるから
 (dx/dt)^2+(dy/dt)^2=(dr/dθ・cosθ-r sinθ)^2+(dr/dθ・sinθ+r cosθ)^2=(dr/dθ)^2+r^2
長さは dθ=dtとして
 L=∫α->β √((dr/dθ)^2+r^2) dθ
 L=∫α->β √2・rdθ=∫α->β √2・a・e^θdθ=√2・a(e^β-e^α)
求める区間はα=-∞ からβ=π/4 であるので、
さきに求めたa=s/√2e^(-π/4)を代入すれば L=sとなる。

*黄金比、黄金分割と螺旋 [#d6a13cf8]
The book Dynamic Symmetry of Jay Hambidge(published in 1926) has also influenced several artists for a long time. Hambidge has written the book keeping'golden section' or 'golden ratio'  in his mind. Rectangles whose lengths and breadths are in the ratio φ:1(where φ is the golden number) are known as 'golden rectangles'. According to many artists, dimensions
(length and width) of golden rectangles are most beautiful from the aesthetic point of view. For this reason,golden section has played a major role in architecture.
It is interesting to note that if the breadth of a golden rectangle is taken as the length of another rectangle then that rectangle will also be a golden rectangle.

--A golden rectangle is one whose side lengths are in the golden ratio, 1: φ (one-to-phi), that is, 1:(1+√5)/2 or approximately 1:1.618.


*螺旋の長さを求める [#fb8802ff]
 r=e^(aθ) 0≦θ≦2π
 長さは s=∫[0→2π]√{e^(2aθ)+a^2e^(2aθ)}dθ
 0≦θ≦2π の区間では
*渦巻き銀河(子持ち銀河 M51)の運動方程式 [#ya46b78c]
-負のポテンシャルエネルギーが螺旋星雲を形づくる。時空の歪み:Time-space torsion

*ベルヌーイ試行と大数の法則 [#kd51c84a]
ヤコブ・ベルヌーイ(Jacques Bernoulli 1654-1705)は確率論の基礎になる大数の法則を見つけた

ベルヌーイ試行(Bernoulli Trials)とは白か黒か・表か裏か・成功か失敗か等、どちらかの事象が独立して起る確率を公式化したものである。


*参考 [#e25a73d9]
-[[オイラー・スパイラル 数学的歴史>http://www.eecs.berkeley.edu/Pubs/TechRpts/2008/EECS-2008-111.html]]

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