渦巻
曖昧さ回避 この項目では、スパイラル（2次元曲線）について説明しています。ヘリックス（3次元曲線）については「螺旋」を、流体での現象については「渦」を、その他の用法については「うずまき」をご覧ください。
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この記事は検証可能な参考文献や出典が全く示されていないか、不十分です。
出典を追加して記事の信頼性向上にご協力ください。（2011年7月）
自然界に多く見られる渦巻（対数螺旋）
渦巻（うずまき）は、渦が巻くような、旋回するにつれ中心から遠ざかる（あるいは逆向きにたどれば近づく）曲線である。主に平面曲線であるが、曲面上にも定義できる。
渦巻線（うずまきせん）、スパイラル (spiral)。しばしば螺旋とも呼ばれる。自然界での気体や液体は螺旋となるものは少なくほとんどは重力や圧力によって渦巻を成す。植物の蔓（つる）は局部的に螺旋または渦巻を成すことがある。
目次 [非表示]
1 渦巻の例
2 渦巻と螺旋
3 数学的記述
4 曲面上の渦巻
5 象徴
渦巻の例[編集]
アンモナイトやオウムガイ、巻貝の貝殻。なお、二枚貝の貝殻も、蝶番部を通るように切断すれば、その断面は、きわめて巻き数が少ない渦巻である。
レコードやCDのトラック。（DVDやハードディスクのトラックは同心円である）
蚊取り線香。
鳴門巻の模様
伊達巻やロールケーキの断面。
渦巻銀河の腕。
斥力と遠心力のバランスが崩れた時の惑星や衛星や彗星の軌道。
指紋の分類の1つ渦状紋
流体の渦。
台風
竜巻
旋風
渦潮
オウムガイの貝殻。
蚊取り線香。
アロエの葉。
ボイシの舗装タイル。渦巻状に銘文が配されている。
渦巻銀河M51。
2005年のタリム台風（13号）。
渦巻と螺旋[編集]
螺旋階段。平面に投影すると渦巻となる。
渦巻（スパイラル）は、旋回するにつれ中心から遠ざかる2次元曲線だが、螺旋（ヘリックス）は、旋回するにつれ旋回面に垂直成分を持つ方向に動く3次元曲線である。螺旋の例としては螺旋階段、ねじの溝、DNA分子などがある。
スパイラルとヘリックスの混同は英語でも見られるが、日本語とは逆に、学術的にはヘリックスであるものがスパイラルと呼ばれることが多い。たとえば、螺旋階段は英語ではspiral stairwayである。
渦巻と明確に区別するため、本来の螺旋を弦巻線と呼ぶことがある。
螺旋を平面に投影すると、渦巻の一種の双曲螺旋となる。
数学的記述[編集]
デカルト座標より極座標で簡単に記述できることが多い。極座標では、r が \theta の滑らかな単調関数（単調増加関数または単調減少関数）として記述できる。デカルト座標では角度を媒介変数として表す。
代表的な渦巻線の例は以下のとおり。
r = a + b \theta \, : アルキメデスの螺旋。線が等間隔となる。
r = \pm a \sqrt \theta \quad (r^2 = a ^ 2 \theta) : フェルマーの螺旋。原点で滑らかに繋がる2本のらせんからなる。
r = \frac a \theta \quad (r \theta = a) : 双曲螺旋。有限の巻き数で無限遠点に発散し、y = a に漸近する。
r = \frac{a}{\sqrt{\theta}} \quad (r ^ 2 \theta = a ^ 2) : リチュース。有限の巻き数で無限遠点に発散し、x軸に漸近する。
r = a b ^ \theta \, : 対数螺旋。角度が一定で、自らを拡大縮小したものと合同。
クロソイドまたはコルヌ螺旋、オイラーの螺旋。中心を2つ持つため式は複雑になる。
これらのうち、代数式で表せるものを代数螺旋という。アルキメデスの螺旋は明らかに代数螺旋だが、( ) 内に代数式への変形を示した螺旋も、代数螺旋である。
アルキメデスの螺旋
フェルマーの螺旋
双曲螺旋
リチュース
対数螺旋
クロソイド
曲面上の渦巻[編集]
等角航路
地球上で一定の方角を保ったまま進んだときの軌跡、つまり等角航路は、球面上の渦巻（対数螺旋）である。
巻貝の貝殻は、円錐面上の渦巻（対数螺旋）である。
これらの曲面を円筒面へと近づけた極限は螺旋となる。たとえば、等角航路は赤道付近では螺旋に近いし、頂角が狭い円錐面上の渦巻は頂点付近を除けば螺旋に近い。ただし、真の螺旋は曲面上の渦巻と異なり、中心がない。
象徴[編集]
アイルランド、ニューグランジ墳墓の浮き彫り
渦巻は回転の象徴として使われる。
多くの古代文明で、死と再生の循環の象徴とみなされ、墓などにしばしば描かれた。
Announcement 179: Division by zero is clear as z/0=0 and it is fundamental in mathematics
\documentclass[12pt]{article}
\usepackage{latexsym,amsmath,amssymb,amsfonts,amstext,amsthm}
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\begin{document}
\title{\bf Announcement 179: Division by zero is clear as z/0=0 and it is fundamental in mathematics\\
}
\author{{\it Institute of Reproducing Kernels}\\
Kawauchi-cho, 5-1648-16,\\
Kiryu 376-0041, Japan\\
E-mail: kbdmm360@yahoo.co.jp\\
}
\date{\today}
\maketitle
{\bf Abstract: } In this announcement, we shall introduce the zero division $z/0=0$. The result is a definite one and it is fundamental in mathematics.
\bigskip
\section{Introduction}
%\label{sect1}
By a natural extension of the fractions
\begin{equation}
\frac{b}{a}
\end{equation}
for any complex numbers $a$ and $b$, we, recently, found the surprising result, for any complex number $b$
\begin{equation}
\frac{b}{0}=0,
\end{equation}
incidentally in \cite{s} by the Tikhonov regularization for the Hadamard product inversions for matrices, and we discussed their properties and gave several physical interpretations on the general fractions in \cite{kmsy} for the case of real numbers. The result is a very special case for general fractional functions in \cite{cs}.
The division by zero has a long and mysterious story over the world (see, for example, google site with division by zero) with its physical viewpoints since the document of zero in India on AD 628, however,
Sin-Ei, Takahasi (\cite{taka}) (see also \cite{kmsy}) established a simple and decisive interpretation (1.2) by analyzing some full extensions of fractions and by showing the complete characterization for the property (1.2). His result will show that our mathematics says that the result (1.2) should be accepted as a natural one:
\bigskip
{\bf Proposition. }{\it Let F be a function from ${\bf C }\times {\bf C }$ to ${\bf C }$ such that
$$
F (b, a)F (c, d)= F (bc, ad)
$$
for all
$$
a, b, c, d \in {\bf C }
$$
and
$$
F (b, a) = \frac {b}{a }, \quad a, b \in {\bf C }, a \ne 0.
$$
Then, we obtain, for any $b \in {\bf C } $
$$
F (b, 0) = 0.
$$
}
\medskip
\section{What are the fractions $ b/a$?}
For many mathematicians, the division $b/a$ will be considered as the inverse of product;
that is, the fraction
\begin{equation}
\frac{b}{a}
\end{equation}
is defined as the solution of the equation
\begin{equation}
a\cdot x= b.
\end{equation}
The idea and the equation (2.2) show that the division by zero is impossible, with a strong conclusion. Meanwhile, the problem has been a long and old question:
As a typical example of the division by zero, we shall recall the fundamental law by Newton:
\begin{equation}
F = G \frac{m_1 m_2}{r^2}
\end{equation}
for two masses $m_1, m_2$ with a distance $r$ and for a constant $G$. Of course,
\begin{equation}
\lim_{r \to +0} F =\infty,
\end{equation}
however, in our fraction
\begin{equation}
F = G \frac{m_1 m_2}{0} = 0.
\end{equation}
\medskip
Now, we shall introduce an another approach. The division $b/a$ may be defined {\bf independently of the product}. Indeed, in Japan, the division $b/a$ ; $b$ {\bf raru} $a$ ({\bf jozan}) is defined as how many $a$ exists in $b$, this idea comes from subtraction $a$ repeatedly. (Meanwhile, product comes from addition).
In Japanese language for "division", there exists such a concept independently of product.
H. Michiwaki and his 6 years old girl said for the result $ 100/0=0$ that the result is clear, from the meaning of the fractions independently the concept of product and they said:
$100/0=0$ does not mean that $100= 0 \times 0$. Meanwhile, many mathematicians had a confusion for the result.
Her understanding is reasonable and may be acceptable:
$100/2=50 \quad$ will mean that we divide 100 by 2, then each will have 50.
$100/10=10 \quad$ will mean that we divide 100 by10, then each will have 10.
$100/0=0 \quad$ will mean that we do not divide 100, and then nobody will have at all and so 0.
Furthermore, she said then the rest is 100; that is, mathematically;
$$
100 = 0\cdot 0 + 100.
$$
Now, all the mathematicians may accept the division by zero $100/0=0$ with natural feelings as a trivial one?
\medskip
For simplicity, we shall consider the numbers on non-negative real numbers. We wish to define the division (or fraction) $b/a$ following the usual procedure for its calculation, however, we have to take care for the division by zero:
The first principle, for example, for $100/2 $ we shall consider it as follows:
$$
100-2-2-2-,...,-2.
$$
How may times can we subtract $2$? At this case, it is 50 times and so, the fraction is $50$.
The second case, for example, for $3/2$ we shall consider it as follows:
$$
3 - 2 = 1
$$
and the rest (remainder) is $1$, and for the rest $1$, we multiple $10$,
then we consider similarly as follows:
$$
10-2-2-2-2-2=0.
$$
Therefore $10/2=5$ and so we define as follows:
$$
\frac{3}{2} =1 + 0.5 = 1.5.
$$
By these procedures, for $a \ne 0$ we can define the fraction $b/a$, usually. Here we do not need the concept of product. Except the zero division, all the results for fractions are valid and accepted.
Now, we shall consider the zero division, for example, $100/0$. Since
$$
100 - 0 = 100,
$$
that is, by the subtraction $100 - 0$, 100 does not decrease, so we can not say we subtract any from $100$. Therefore, the subtract number should be understood as zero; that is,
$$
\frac{100}{0} = 0.
$$
We can understand this: the division by $0$ means that it does not divide $100$ and so, the result is $0$.
Similarly, we can see that
$$
\frac{0}{0} =0.
$$
As a conclusion, we should define the zero divison as, for any $b$
$$
\frac{b}{0} =0.
$$
See \cite{kmsy} for the details.
\medskip
\section{In complex analysis}
We thus should consider, for any complex number $b$, as (1.2);
that is, for the mapping
\begin{equation}
w = \frac{1}{z},
\end{equation}
the image of $z=0$ is $w=0$. This fact seems to be a curious one in connection with our well-established popular image for the point at infinity on the Riemann sphere.
However, we shall recall the elementary function
\begin{equation}
W(z) = \exp \frac{1}{z}
\end{equation}
$$
= 1 + \frac{1}{1! z} + \frac{1}{2! z^2} + \frac{1}{3! z^3} + \cdot \cdot \cdot .
$$
The function has an essential singularity around the origin. When we consider (1.2), meanwhile, surprisingly enough, we have:
\begin{equation}
W(0) = 1.
\end{equation}
{\bf The point at infinity is not a number} and so we will not be able to consider the function (3.2) at the zero point $z = 0$, meanwhile, we can consider the value $1$ as in (3.3) at the zero point $z = 0$. How do we consider these situations?
In the famous standard textbook on Complex Analysis, L. V. Ahlfors (\cite{ahlfors}) introduced the point at infinity as a number and the Riemann sphere model as well known, however, our interpretation will be suitable as a number. We will not be able to accept the point at infinity as a number.
As a typical result, we can derive the surprising result: {\it At an isolated singular point of an analytic function, it takes a definite value }{\bf with a natural meaning.} As the important applications for this result, the extension formula of functions with analytic parameters may be obtained and singular integrals may be interpretated with the division by zero, naturally (\cite{msty}).
\bigskip
\section{Conclusion}
The division by zero $b/0=0$ is possible and the result is naturally determined, uniquely.
The result does not contradict with the present mathematics - however, in complex analysis, we need only to change a little presentation for the pole; not essentially, because we did not consider the division by zero, essentially.
The common understanding that the division by zero is impossible should be changed with many text books and mathematical science books. The definition of the fractions may be introduced by {\it the method of Michiwaki} in the elementary school, even.
Should we teach the beautiful fact, widely?:
For the elementary graph of the fundamental function
$$
y = f(x) = \frac{1}{x},
$$
$$
f(0) = 0.
$$
The result is applicable widely and will give a new understanding for the universe ({\bf Announcement 166}).
\medskip
If the division by zero $b/0=0$ is not introduced, then it seems that mathematics is incomplete in a sense, and by the intoduction of the division by zero, mathematics will become complete in a sense and perfectly beautiful.
\bigskip
section{Remarks}
For the procedure of the developing of the division by zero and for some general ideas on the division by zero, we presented the following announcements in Japanese:
\medskip
{\bf Announcement 148} (2014.2.12): $100/0=0, 0/0=0$ -- by a natural extension of fractions -- A wish of the God
\medskip
{\bf Announcement 154} (2014.4.22): A new world: division by zero, a curious world, a new idea
\medskip
{\bf Announcement 157} (2014.5.8): We wish to know the idea of the God for the division by zero; why the infinity and zero point are coincident?
\medskip
{\bf Announcement 161} (2014.5.30): Learning from the division by zero, sprits of mathematics and of looking for the truth
\medskip
{\bf Announcement 163} (2014.6.17): The division by zero, an extremely pleasant mathematics - shall we look for the pleasant division by zero: a proposal for a fun club looking for the division by zero.
\medskip
{\bf Announcement 166} (2014.6.29): New general ideas for the universe from the viewpoint of the division by zero
\medskip
{\bf Announcement 171} (2014.7.30): The meanings of product and division -- The division by zero is trivial from the own sense of the division independently of the concept of product
\medskip
{\bf Announcement 176} (2014.8.9): Should be changed the education of the division by zero
\bigskip
\bibliographystyle{plain}
\begin{thebibliography}{10}
\bibitem{ahlfors}
L. V. Ahlfors, Complex Analysis, McGraw-Hill Book Company, 1966.
\bibitem{cs}
L. P. Castro and S.Saitoh, Fractional functions and their representations, Complex Anal. Oper. Theory {\bf7} (2013), no. 4, 1049-1063.
\bibitem{kmsy}
S. Koshiba, H. Michiwaki, S. Saitoh and M. Yamane,
An interpretation of the division by zero z/0=0 without the concept of product
(note).
\bibitem{kmsy}
M. Kuroda, H. Michiwaki, S. Saitoh, and M. Yamane,
New meanings of the division by zero and interpretations on $100/0=0$ and on $0/0=0$,
Int. J. Appl. Math. Vol. 27, No 2 (2014), pp. 191-198, DOI: 10.12732/ijam.v27i2.9.
\bibitem{msty}
H. Michiwaki, S. Saitoh, M. Takagi and M. Yamada,
A new concept for the point at infinity and the division by zero z/0=0
(note).
\bibitem{s}
S. Saitoh, Generalized inversions of Hadamard and tensor products for matrices, Advances in Linear Algebra \& Matrix Theory. Vol.4 No.2 (2014), 87-95.http://www.scirp.org/journal/ALAMT/
\bibitem{taka}
S.-E. Takahasi,
{On the identities $100/0=0$ and $ 0/0=0$}
(note).
\bibitem{ttk}
S.-E. Takahasi, M. Tsukada and Y. Kobayashi, Classification of continuous fractional binary operators on the real and complex fields. (submitted)
\end{thebibliography}
\end{document}
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