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读书日不读书，看画。梁进老师的《数学与名画》终于出来了，为读书的同学带来一幅新式的眼镜，那就是用数学的眼光来看绘画，也从绘画里看数学的影子。梁老师“淌过”很多博物馆，用微分的眼睛解析过很多名画，于是有“世界名画中的数学”的系列博文（http://blog.sciencenet.cn/blog-39446-699737.html），然后有这本书，领着读者用新的眼光和心情来看绘画，乃至看艺术，看自然，看人生。因为我们和世间万象都是数学舞台上不断变换和演化着——当然，遵从一定的偏微分方程 (梁老师的专业)——的“点线面”，呈现为一幅幅不同风格和流派的图画。本书也许能改变“文艺青年”对数学的偏见，对科学的的偏见，对思维的偏见，让科学与艺术重新融合在头脑里。
时下流行“像艺术家一样思考”，似乎不是因为艺术家创造“美”——后现代派也创造“丑”，就让他们自己玩儿吧——而是因为他们“创造”了特殊的思维方式，令人眼红了。遗憾的是，人们闹着学艺术家的思维时，冷落了同样创造思维的数学家兄弟。这也难怪，不知从什么年月起，谁谁谁们就将艺术思维与科学思维分开了，将感性与理性对立了。这种“二分法”既夸张也荒唐，不但离间了科学和艺术，也让“理工男”与“文艺女”变成星河两岸的牛郎和织女，盈盈一水间，相望却无言。结果是我们的头脑越来越残缺。读这本书也许能惊奇一个平凡的事实：科学与艺术从来是相通的，“最抽象”的数学与“最具象”的绘画，有时简直像孪生兄弟。
5000多年前古埃及奥西里斯神庙的“生命之花”，就是“一群”圆圈的组合，像几何习题的插图。简单几何图形的重复、变形及其显现的对称和韵律，是几千年来不变的艺术“基元”。如果说绘画来源之一是自然物象，那么来源之二就是几何，绘画在不知不觉中成了数学的小兄弟。在文艺复兴时期，绘画更是自觉地融合科学精神。我们在达芬奇的手稿里可以看到时代的风尚：“先学科学，然后在那科学的指引下实践”（First study science, then follow the practice that born of thatscience）。他认为，绘画是科学活动，而且是最高级的活动。他的笔记手稿满是光线、阴影、透视、色彩，还有人体结构和“清流激湍”，几乎就是带插图的科学启蒙课本。
有趣的是，当科学风尚改变时，绘画风格似乎在跟着转变。二者虽无直接的因果，却从不同侧面代表了时代的文化生态。史家们常说时代思潮和时代文艺——如王国维说“凡一代有一代之文学”——我们同样可以说时代科学和时代数学。如果说古典时代科学与艺术的自然融合是因为那时的科学与文艺还不够“百花齐放”，那么近代的文艺风尚与科学风尚的呼应，则足以令人惊奇和惊喜了：从安格尔与德拉克罗瓦的线条-色彩之争，到莫奈的光影、塞尚的色块和毕加索的立体，都呼应着科学的风尚。如果说古典绘画抱着欧几里得几何的时空，那么立体画派更倾向非欧几何，更像拓扑学。（参见立体派画家Albert Gleiser和Jean Metzinger的《立体主义》。）当康定斯基让“色彩与形态”摆脱物象时，物理学也在从实验模型走向几何化（如相对论和规范场论），实现了哈代（华罗庚在剑桥的老师）所说的，数学家和画家一样，都创造“模式”。数学的概念和符号，犹如画家的线条和色彩，数学的结构犹如绘画的场景，而艺术“美”也就成为数学“真”的一个标准。这令人想起大诗人济慈因古希腊陶瓮而发的感叹：“美即是真，真即是美”（Beauty is truth, truth beauty）。大数学家外尔说，“我的工作总需要将真与美统一起来，当我不得不选择其一时，我通常选择美。”控制论创始人、自称“昔日神童”的维纳也在自传里表达过他的艺术式的数学体验：数学家最好的回报就是能爱上他发现的东西，犹如塞浦路斯国王爱上他自己塑造的雕像Pygmalion一样。
所以，数学晤对绘画，不是旁观者看画展，而是“会心人别有怀抱”，能发现画家自己都或许感觉了却并不明白的东西。我们看埃舍尔的画，会惊讶抽象的几何结构和空间变换以及难以言表的逻辑怪圈，竟能那么活泼泼地呈现出来。然而埃舍尔的数学并不好，从来就没及格。但他“莫名其妙就理解了数学”，似乎自己是数学家们“失散多年的兄弟”。（见恩斯特《魔镜，埃舍尔德不可能世界》）在不懂数学的埃舍尔的绘画里，我们还能看到相对论、黎曼几何和量子场论的“形象”——绘画在无意间生出数学，当然需要用数学的眼睛去看它；另一方面，数学法则本来就“存在于”自然世界和精神世界，绘画表现自然和自然激发的心情，当然也必然会隐藏数学。绘画过滤和抽象了世界，数学不过是再抽象一回罢了。越是抽象，越能虚怀地包容万象。
诗人画家王维画“雪中芭蕉”，曾引出科学家沈括的高见：“书画之妙，当以神会，难可以形器求也。”（《梦溪笔谈》卷十七）所谓“神会”，就是不能“心为形役”，而应以模式、韵律和精神去契合画的精神，而不是拿自然的物象去比较画面的形象。西方古典绘画是具象的，我们的写意山水也具象，即使缺一点“神会”也能看出几分模样；现代绘画没那么多“象”可以触摸了，必须换一种眼光去看。正如梁老师在解读康定斯基的抽象画时说的，我们需要从作品的“结构”看出它的“函数空间”。就是说，当我们用数学图像去看画，能自然把握它的元素、结构和韵律。例如我们看蒙德里安的“蓝色组合”，那些杂乱的大大小小深深浅浅重重叠叠的色块，并不代表什么具象的东西，却很好刻画了一种统计分布，不管什么东西的统计，我们觉悟了统计分布就好了，它可以存在于任何地方……
从某种意义说，数学就是脱离了物质世界的“抽象画”，它的形式和精神也就是绘画的形式和精神。正如康定斯基说的，“数是各类艺术最终的抽象表现。”我们用数学的眼睛来看绘画，只不过是与失散的兄弟重逢，尽管他们越来越陌生了。http://blog.sciencenet.cn/blog-279992-972240.html
\documentclass[12pt]{article}
\usepackage{latexsym,amsmath,amssymb,amsfonts,amstext,amsthm}
\numberwithin{equation}{section}
\begin{document}
\title{\bf Announcement 293: Parallel lines on the Euclidean plane from the viewpoint of division by zero 1/0=0}
\author{{\it Institute of Reproducing Kernels}\\
Kawauchi-cho, 5-1648-16,\\
Kiryu 376-0041, Japan\\
\date{\today}
\maketitle
{\bf Abstract: } In this announcement, for its importance we would like to declare that any parallel lines have the common point $(0,0) $ in the sense of the division by zero. From this fact we have to change our basic idea for the Euclidean plane and we will see a new world for not only mathematics, but also the universe.
\bigskip
\section{Introduction}
%\label{sect1}
By a {\bf natural extension} of the fractions
\begin{equation}
\frac{b}{a}
\end{equation}
for any complex numbers $a$ and $b$, we found the simple and beautiful 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 the 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 the extensions of fractions and by showing the complete characterization for the property (1.2):
\bigskip
{\bf Proposition 1. }{\it Let F be a function from ${\bf C }\times {\bf C }$ to ${\bf C }$ satisfying
$$
F (b, a)F (c, d)= F (bc, ad)
$$
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
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 $w=0$ ({\bf should be defined}). 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 (\cite{ahlfors}). Therefore, the division by zero will give great impacts to complex analysis and to our ideas for the space and universe.
However, the division by zero (1.2) is now clear, indeed, for the introduction of (1.2), we have several independent approaches as in:
\medskip
1) by the generalization of the fractions by the Tikhonov regularization or by the Moore-Penrose generalized inverse,
\medskip
2) by the intuitive meaning of the fractions (division) by H. Michiwaki,
\medskip
3) by the unique extension of the fractions by S. Takahasi, as in the above,
\medskip
4) by the extension of the fundamental function $W = 1/z$ from ${\bf C} \setminus \{0\}$ into ${\bf C}$ such that $W =1/z$ is a one to one and onto mapping from $ {\bf C} \setminus \{0\} $ onto ${\bf C} \setminus \{0\}$ and the division by zero $1/0=0$ is a one to one and onto mapping extension of the function $W =1/z $ from ${\bf C}$ onto ${\bf C}$,
\medskip
and
\medskip
5) by considering the values of functions with the mean values of functions.
\medskip
Furthermore, in (\cite{msy}) we gave the results in order to show the reality of the division by zero in our world:
\medskip
\medskip
A) a field structure containing the division by zero --- the Yamada field ${\bf Y}$,
\medskip
B) by the gradient of the $y$ axis on the $(x,y)$ plane --- $\tan \frac{\pi}{2} =0$,
\medskip
C) by the reflection $1/\overline{z}$ of $z$ with respect to the unit circle with center at the origin on the complex $z$ plane --- the reflection point of zero is zero,
\medskip
and
\medskip
D) by considering rotation of a right circular cone having some very interesting
phenomenon from some practical and physical problem --- EM radius.
\medskip
See also \cite{bht} for the relationship between fields and the division by zero, and the importance of the division by zero for computer science. It seems that the relationship of the division by zero and field structures are abstract in their paper.
Meanwhile, J. P. Barukcic and I. Barukcic (\cite{bb}) discussed recently the relation between the division $0/0$ and special relative theory of Einstein.
Furthermore, Reis and Anderson (\cite{ra,ra2}) extends the system of the real numbers by defining division by zero.
Meanwhile, we should refer to up-to-date information:
{\it Riemann Hypothesis Addendum - Breakthrough
Kurt Arbenz
https://www.researchgate.net/publication/272022137 Riemann Hypothesis Addendum - Breakthrough.}
\medskip
Here, we recall Albert Einstein's words on mathematics:
Blackholes are where God divided by zero.
I don’t believe in mathematics.
George Gamow (1904-1968) Russian-born American nuclear physicist and cosmologist remarked that "it is well known to students of high school algebra" that division by zero is not valid; and Einstein admitted it as {\bf the biggest blunder of his life} [1]：
1. Gamow, G., My World Line (Viking, New York). p 44, 1970.
For our results, see the survey style announcements 179,185,237,246, 247,250 and 252 of the Institute of Reproducing Kernels (\cite{ann179,ann185,ann237,ann246,ann247,ann250,ann252}).
At this moment, the following theorem may be looked as the fundamental theorem of the division by zero:
\bigskip
{\bf Theorem (\cite{mst}).} {\it Any analytic function takes a definite value at an isolated singular point }{\bf with a natural meaning.}
\bigskip
The following corollary shows how to determine the value of an analytic function at the singular point; that is, the value is determined from the regular part of the Laurent expansion:
\bigskip
{\bf Corollary.} {\it For an isolated singular point $a$ of an analytic function $f(z)$, we have the Cauchy integral formula
$$
f(a) = \frac{1}{2\pi i} \int_{\gamma} f(z) \frac{dz}{z - a},
$$
where the $\gamma$ is a rectifiable simple Jordan closed curve that surrounds one time the point $a$
on a regular region of the function $f(z)$.
}
\bigskip
The essential meaning of this theorem and corollary is given by that: the values of functions may be understood in the sense of the mean values of analytic functions.
\medskip
In this announcement, we will state the basic property of parallel lines by the division by zero on the Euclidean plane and we will be able to see that the division by zero introduces a new world and fundamental mathematics.
In particular, note that the concept of parallel lines is very important in the Euclidean plane and non-Euclidean geometry. The essential results may be stated as known since the discovery of the division by zero $z/0=0$. However, for importance, we would like to state clearly the details.
\section{The point at infinity}
We will be able to see the whole Euclidean plane by the stereographic projection into the Riemann sphere --- {\it We think that in the Euclidean plane, there does not exist the point at infinity}.
However, we can consider it as a limit like $\infty$. Recall the definition of $z \to \infty$ by $\epsilon$-$\delta$ logic; that is, $\lim_{z \to \infty} z = \infty$ if and only if for any large $M>0$, there exists a number $L>0$ such that for any z satisfying $L <|z|$, $M<|z|$. In this definition, the infinity $\infty$ does not appear.
{\it The infinity is not a number, but it is an ideal space point.}
The behavior of the space around the point at infinity may be considered by that around the origin by the linear transform $W = 1/z$(\cite{ahlfors}). We thus see that
\begin{equation}
\lim_{z \to \infty} z = \infty,
\end{equation}
however,
\begin{equation}
[z]_{z =\infty} =0,
\end{equation}
by the division by zero. The difference of (2.1) and (2.2) is very important as we see clearly by the function $1/z$ and the behavior at the origin. The limiting value to the origin and the value at the origin are different. For surprising results, we will state the property in the real space as follows:
\begin{equation}
\lim_{x\to +\infty} x =+\infty , \quad \lim_{x\to -\infty} x = -\infty,
\end{equation}
however,
\begin{equation}
[x]_{ +\infty } =0, \quad [x]_{ -\infty } =0.
\end{equation}
\section{Interpretation by analytic geometry}
We write lines by
\begin{equation}
L_k: a_k x + b_k y + c_k = 0, k=1,2.
\end{equation}
The common point is given by, if $a_1 b_2 - a_2 b_1 \ne 0$; that is, the lines are not parallel
\begin{equation}
\left(\frac{b_1 c_2 - b_2 c_1}{a_1 b_2 - a_2 b_1}, \frac{a_2 c_1 - a_1 c_2}{a_1 b_2 - a_2 b_1}\right).
\end{equation}
By the division by zero, we can understand that if $a_1 b_2 - a_2 b_1 = 0$, then the commom point is always given by
\begin{equation}
(0,0),
\end{equation}
even the two lines are the same. This fact shows that the image of the Euclidean space in Section 2 is right.
\section{Remarks}
For a function
\begin{equation}
S(x,y) = a(x^2+y^2) + 2gx + 2fy + c,
\end{equation}
the radius $R$ of the circle $S(x,y) = 0$ is given by
\begin{equation}
R = \sqrt{\frac{g^2 +f^2 -ac}{a^2}}.
\end{equation}
If $a = 0$, then the area $\pi R^2$ of the circle is zero, by the division by zero; that is, the circle is line
(degenerate).
Here, note that by the Theorem, $R^2$ is zero for $a = 0$, but for (4.2) itself
\begin{equation}
R = \frac{-c}{2} \frac{1}{\sqrt{g^2 + f^2}}
\end{equation}
for $a=0$. However, this result will be nonsense, and so, in this case, we should consider $R$
as zero as $ 0^2 =0$. When we apply the division by zero to functions, we can consider, in general, many ways.
For example,
for the function $z/(z-1)$, when we insert $z=1$ in numerator and denominator, we have
\begin{equation}
\left[\frac{z}{z-1}\right]_{z = 1} = \frac{1}{0} =0.
\end{equation}
However, in the sense of the Theorem,
from the identity
\begin{equation}
\frac{z}{z-1} = \frac{1}{z-1} + 1,
\end{equation}
we have
\begin{equation}
\left[\frac{z}{z-1}\right]_{z = 1} = 1.
\end{equation}
By the Theorem, for analytic functions we can give uniquely determined values at isolated singular points, however, the values by means of the Laurent expansion are not always reasonable. We will need to consider many interpretations for reasonable values.
In addition, the center of the circle (4.3) is given by
\begin{equation}
\left( - \frac{g}{a},- \frac{f}{a}\right).
\end{equation}
Therefore, the center of a general line
\begin{equation}
2gx + 2fy + c=0
\end{equation}
may be considered as the origin $(0,0)$, by the division by zero.
We can see similarly the 3 dimensional versions.
\medskip
We consider the functions
\begin{equation}
S_j(x,y) = a_j(x^2+y^2) + 2g_jx + 2f_jy + c_j.
\end{equation}
The distance $d$ of the centers of the circles $S_1(x,y) =0$ and $S_2(x,y) =0$ is given by
\begin{equation}
d^2= \frac{g_1^2 + f_1^2}{a_1^2} - 2 \frac{g_1 g_2 + f_1 f_2}{a_1 a_2} + \frac{g_2^2 + f_2^2}{a_2^2}.
\end{equation}
If $a_1 =0$, then by the division by zero
\begin{equation}
d^2= \frac{g_2^2 + f_2^2}{a_2^2}.
\end{equation}
Then, $S_1(x,y) =0$ is a line and its center is the origin $(0,0)$.
\bigskip
\bibliographystyle{plain}
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L. V. Ahlfors, Complex Analysis, McGraw-Hill Book Company, 1966.
\bibitem{bb}
J. P. Barukcic and I. Barukcic, Anti Aristotle - The Division Of Zero By Zero,
ViXra.org (Friday, June 5, 2015)
© Ilija Barukčić, Jever, Germany. All rights reserved. Friday, June 5, 2015 20:44:59.
\bibitem{bht}
J. A. Bergstra, Y. Hirshfeld and J. V. Tucker,
Meadows and the equational specification of division (arXiv:0901.0823v1[math.RA] 7 Jan 2009).
\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}
M. Kuroda, H. Michiwaki, S. Saitoh, and M. Yamane,
New meanings of the division by zero and interpretations on $100/0=0$ and on $0/0=0$,
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H. Michiwaki, S. Saitoh, and M.Yamada,
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\bibitem{mst}
H. Michiwaki, S. Saitoh and M. Takagi,
A new concept for the point at infinity and the division by zero z/0=0
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\bibitem{ra}
T. S. Reis and James A.D.W. Anderson,
Transdifferential and Transintegral Calculus,
Proceedings of the World Congress on Engineering and Computer Science 2014 Vol I
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\bibitem{ra2}
T. S. Reis and James A.D.W. Anderson,
Transreal Calculus,
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\bibitem{s}
S. Saitoh, Generalized inversions of Hadamard and tensor products for matrices, Advances in Linear Algebra \& Matrix Theory. {\bf 4} (2014), no. 2, 87--95. http://www.scirp.org/journal/ALAMT/
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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 operations on the real and complex fields, Tokyo Journal of Mathematics, {\bf 38}(2015), no. 2, 369-380.
\bibitem{ann179}
Announcement 179 (2014.8.30): Division by zero is clear as z/0=0 and it is fundamental in mathematics.
\bibitem{ann185}
Announcement 185 (2014.10.22): The importance of the division by zero $z/0=0$.
\bibitem{ann237}
Announcement 237 (2015.6.18): A reality of the division by zero $z/0=0$ by geometrical optics.
\bibitem{ann246}
Announcement 246 (2015.9.17): An interpretation of the division by zero $1/0=0$ by the gradients of lines.
\bibitem{ann247}
Announcement 247 (2015.9.22): The gradient of y-axis is zero and $\tan (\pi/2) =0$ by the division by zero $1/0=0$.
\bibitem{ann250}
Announcement 250 (2015.10.20): What are numbers? - the Yamada field containing the division by zero $z/0=0$.
\bibitem{ann252}
Announcement 252 (2015.11.1): Circles and
curvature - an interpretation by Mr.
Hiroshi Michiwaki of the division by
zero $r/0 = 0$.
\bibitem{ann281}
Announcement 281(2016.2.1): The importance of the division by zero $z/0=0$.
\bibitem{ann282}
Announcement 282(2016.2.2): The Division by Zero $z/0=0$ on the Second Birthday.
\end{thebibliography}
\end{document}
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