GoPiGo 小汽車︰格點圖像算術《投影幾何》【四‧平面國】《癸藏》

傳說當『平面國』之『點投派』想將『投影線』『抽象公設化』的時候,遭遇了一個很大的問題︰

自從喬治・康托爾 Georg Cantor 發展『集合論』至今,用集合 來談論『抽象系統』已是一種『傳統』。又因為『公理化』axiom 的盛行,今天的『數學』帶給人的印象常是一大套『抽象符號的總匯』,一本書裡到處填滿的是『空間』、『公理』、『假設』、『定義』、『引理』、『定理』、……。然而這卻是為使『定義』能『嚴謹』;『邏輯』能『明確』,希望『精簡』它『承載』的大量之『內容』,以致能達到『言之有物』與『理有所來。所以就算所知道的『術語』太少,常常會導致『閱讀』的困難,『面對』之且『克服』它終將會大有所穫。

金文大篆抽

甲骨文象

什麼是『抽象化』呢?『』字的本意是,在成長茂密的『莊稼』中『拔掉』一些過盛的『株苗』,使得『剩餘』的能夠更『結實飽滿』。而『』字就是『瞎子摸象』的象之象形

人事物的『事件』與『現象』通常都太過於『複雜』,很難那樣的『分析』和『認識 』它,假使將它用比較『簡約』的『概念』來『概括』,不但有助於『理解』,也能利於『論辨』之雙方,互相『發現』彼此之間思維可能之『誤謬』。

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金文大篆零

0

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民國39年發行的兩角鋁幣

一角蘭花硬幣

台灣硬幣5角正面

台灣硬幣5角反面

零的歷史』講述著人類試圖用著『已知』的等等來『推廣』,並 使它『一 般化 』 ,以致能夠解說『未明』之種種,想用著『簡易』的方式『精煉』所知之宇宙人生中的一切。終究總會有人問起『 3 - 3 』是多少?又為什麼不能『 2 - 5 』呢?然而『應用 0 』與『明白 0 』實有很大的『差別』,比方說『 x^0  應當是 1 』,因為『 \frac  {x^m} {x^n } = x^{m - n} 』當 m = n 時,就會有像 『 x^0 』這樣『形式 』 的數『發生』,同時很顯然『大小相同之數相除必然是一』,只不過要是 用在『 \frac{0}{0}』上 呢?難到是『 0 \ne 0』 嗎?假使我們思考如果『 x \times 0 = 0』而且『 \frac{0}{0} = 1 』;這樣由於『 (x  + 1) \times 0 = 0 』 又因『 \frac{0}{0} = 1 』,不可以得到『 x = x  + 1 』的嗎??也許『相容』於已知,又能『理則』之一致,一直 都是並不『易得』的啊!!因此『』之為『』的『特殊性』正在於它是數的『正負概念』之『分界』,已知所有的『』與『』也只有『 0 』這一個數是『 +  0 = -  0 = 0 』。

相較於『零』,『空集合』引發的爭論就大的多了?這又是為什麼呢?比方假使你『』要談論一些『什麼』?它總該是『』吧?不會是『空無一物』的吧?但是像『想寫尚未寫』的□書本,剛拿起『將裝還沒裝』的○袋子,到底□○是否能算是『』或是『沒有』 的呢?也許概念愈是『基本』,或許『辯論』就愈多!『 \phi  』的來歷好比是數系中的『 0 』,用於『指稱』著一個『特殊』的『集合』︰不管它是如同『獨角獸』一樣不『存在』半個元素的呢?還是它指稱『沒盛水』的『空杯子』的呢?舉例說吧,假使 一個貨幣收藏家構造了一個集合族 S_t =  \{ x | t  是時期,x 是當時面值小於一元的硬幣 \},這樣的集合構造『合法』的嗎?他說 S_{1980} = \phi  是『有效』的陳述嗎?其實集合雖是由它的元素所構成,但是集合與元素卻是兩個不同的概念。用『命了名』 的集合代表其內『元素』的『進出增減』之事實非常『普通』,以致有時人們會忘了其實『自己的名字』──  細胞共和國的國號 ── 就是這麼用的,它指稱著時光中變化的容顏,又不變的主體『我』!!

有人懷疑空集合可能是『無物』Nothing ?也許想一想『無水之杯』只是『無水』並非是『無杯』,就能明白的了。那麼果真有『無物』的嗎?有啊!比方說︰ S = \{ S \} 就是『誤謬』而『無物』,為什麼呢?假設 S \ne \phi,如果說 S 有某個元素 e,那 e 屬於 \{ S \} 嗎?顯然不屬於,這個 \{ S \} 集合只有一個元素就是 S,但是依據等式 S = \{ S \}e  它又不得不屬於 \{ S \},所以產生矛盾。假設 S  =  \phi,也就是說 \phi = \{ \phi\},只要知道 \{ \phi\} 有一個元素 \phi,當然不是空集合,就能知道它是個『虛假』陳述了。因此有了 \phi 能使『集合論』的『理論』之論述『精簡』與『定理』的表達『清晰』。

集合論用著 一些符號,來表達集合間的關係與運算,以及元素『屬於 \in 不屬於 \notin』某集合的基本關係。為了方便下面的論述,也許也避免『符號用意』的不同,所以先在此簡介一下。假使 S = \{ x_1, x_2,  x_3, x_4, x_5, \dots, x_k, \dots, x_n \}T = \{ y_1, y_2,  y_3, y_4, y_5, \dots, y_j, \dots, y_m \}

當然這些 x_k, k = 1 \dots n 都是 S 的元素,記作『 x_k \in S 』。如果談論 S 中的『每一個』、『任一個』或『所有的』元素,將以『\forall x\  x \in S 』表示;對等的『有一個』、『某一個』和『存在著』將用『\exists x\  x \in S 』表達。假使 TS 的『子集』記作『 T \subset  S 』,定義成『 \forall y\ y \in T \ \Rightarrow \ y \in S 』,這裡的 \Rightarrow 符號意謂邏輯可『推論』出的意思。建構一個『聯集』記作『 T \cup  S 』,是說『 \{ x | \ x \in T  或\vee  x \in S \} 』;同樣的構造一個『交集』記作『 T \cap  S 』,是講『 \{ x | \ x \in T  且\wedge  x \in S \} 』。所謂的笛卡爾的『乘積』──  座標系或 turple  有序元組 ──寫作『 S \times T 』,就是 \{ (x , y) |\  x \in S \wedge y \in T \}。至於集合的一般構造法 \{x | P x\}  可見之於《{x|x ∉ x} !!??》一文,為免於冗長起見其它的符號需要時再作引入。

人與自然關係

人際關係

自反

對稱

反對稱性波函數

邏輯樹

數學家是怎麼看待『關係』Relation 的呢?他說如果有一個集合叫 S,那定義在 S 上的二元關係『 R 』 就是『 S \times S 』的某個『子集』!!雖很『抽象』 ,分解的說︰
R 是一個有序二元組的集合。
R 是『論域』Domain S \times S 的『子集』。
如果 S 中之任意兩元素 s_1s_2 構成『 ( s_1, s_2 ) 』有序二元組,假使 ( s_1, s_2 ) \in R ,我們就說 ( s_1, s_2 )  擁有關係 R

如 果 Sn 個元素,那 S \times S  就有 n^2 個元素,且有 2^{n^2} 個子集,真是關係多於『牛毛』的啊!假使 R = \phi ,也就是說『在 S 集合裡,萬有元素皆無關』的勒!!這樣就可以追問關係 R 的集合能有什麼『性質』的嗎?比方說『自反性』︰
\forall x \ x \in S \ \Rightarrow \ (x, x) \in R,『反自反性』︰
\forall x \ x \in S \ \Rightarrow \ (x, x) \notin R。並及於『對稱性』︰
\forall x,y \ x ,y \in S \wedge (x,y) \in R \ \Rightarrow \ (y, x) \in R,『非對稱性』︰
\forall x,y \ x ,y \in S \wedge (x,y) \in R \ \Rightarrow \ (y, x) \notin R,『反對稱性』︰
\forall x,y \ x ,y \in S \wedge (x,y) \in R  \wedge (y,x) \in R\ \Rightarrow \ x = y,……種種『可定義』的性質。

一般 (x,y) \in R 可記作『 xRy 』,有時為了方便操作又寫成『 x \ \rightarrow \ y 』,這對『遞移性』關係的描述來講尤其是如此︰

\forall x,y,z \ x,y,z \in S \wedge \ x \rightarrow  y \ \wedge \ y \rightarrow z  \ \Rightarrow \ x \rightarrow z

x \rightarrow y \wedge y \rightarrow z  \ \Rightarrow \ x \rightarrow z 的表達不但可以導引想像,在論域脈絡清楚時── \forall x,y,z \ x,y,z \in S ──,不必寫那麼多的符號干擾思考。

如此當數學家說『函數f 的定義時︰

假使有兩個集合 ST,將之稱作『定義域』domain 與『對應域 』codomain,函數 fS \times T 的子集,並且滿足

\forall x \ x \in S \  \exists ! \ y \ y \in T \ \wedge \ (x,y) \in f

,記作 x \mapsto y = f (x),『 \exists \  !  』是指『恰有一個』,就一點都不奇怪了吧。同樣『二元運算』假使『簡記』成 X \times Y \mapsto_{\bigoplus} \  Z  ,X=Y=Z=S,是講︰

z = \bigoplus ( x, y) = x \bigoplus y,也是很清晰明白的呀!!

─── 摘改自 《Thue 之改寫系統《一》

 

既然吾國之傳統把 y=1 的『投影線』當作它和『視線』 y = \lambda \cdot x 之『交點』所形成之『所見點』為論述基礎,那麼觀者眾矣、觀點不同,要怎樣統括『點』、『線』的『概念』與『關係』耶??

那故事裡沒有後續之事,且不必捕風捉影,就直道『定義』為難吧 !!人們將如何融會所謂『對合』函數【函式?★】

Involution (mathematics)

In mathematics, an (anti-)involution, or an involutory function, is a function f that is its own inverse,

f(f(x)) = x

for all x in the domain of f.[1]

An involution is a function  f:X\to X that, when applied twice, brings one back to the starting point.

General properties

Any involution is a bijection.

The identity map is a trivial example of an involution. Common examples in mathematics of nontrivial involutions include multiplication by −1 in arithmetic, the taking of reciprocals, complementation in set theory and complex conjugation. Other examples include circle inversion, rotation by a half-turn, and reciprocal ciphers such as the ROT13 transformation and the Beaufort polyalphabetic cipher.

The number of involutions, including the identity involution, on a set with n = 0, 1, 2, … elements is given by a recurrence relation found by Heinrich August Rothe in 1800:

a0 = a1 = 1;
an = an − 1 + (n − 1)an − 2, for n > 1.

The first few terms of this sequence are 1, 1, 2, 4, 10, 26, 76, 232 (sequence A000085 in the OEIS); these numbers are called the telephone numbers, and they also count the number of Young tableaux with a given number of cells.[2] The composition gf of two involutions f and g is an involution if and only if they commute: gf = fg.[3]

Every involution on an odd number of elements has at least one fixed point. More generally, for an involution on a finite set of elements, the number of elements and the number of fixed points have the same parity.[4]

Euclidean geometry

A simple example of an involution of the three-dimensional Euclidean space is reflection against a plane. Performing a reflection twice brings a point back to its original coordinates.

Another is the so-called reflection through the origin; this is an abuse of language as it is not a reflection, though it is an involution.

These transformations are examples of affine involutions.

Projective geometry

An involution is a projectivity of period 2, that is, a projectivity that interchanges pairs of points. Coxeter relates three theorems on involutions:

  • Any projectivity that interchanges two points is an involution.
  • The three pairs of opposite sides of a complete quadrangle meet any line (not through a vertex) in three pairs of an involution (this is Desargues‘s Involution Theorem,[5] whose origins can be seen in Lemma IV of the lemmas to the Porisms of Euclid in Volume VII of the Collection of Pappus of Alexandria [6]).
  • If an involution has one fixed point, it has another, and consists of the correspondence between harmonic conjugates with respect to these two points. In this instance the involution is termed “hyperbolic”, while if there are no fixed points it is “elliptic”.

Another type of involution occurring in projective geometry is a polarity which is a correlation of period 2. [7]

Linear algebra

In linear algebra, an involution is a linear operator T such that  T^{2}=I. Except for in characteristic 2, such operators are diagonalizable with 1s and −1s on the diagonal. If the operator is orthogonal (an orthogonal involution), it is orthonormally diagonalizable.

For example, suppose that a basis for a vector space V is chosen, and that e1 and e2 are basis elements. There exists a linear transformation f which sends e1 to e2, and sends e2 to e1, and which is the identity on all other basis vectors. It can be checked that f(f(x)) = x for all x in V. That is, f is an involution of V.

This definition extends readily to modules. Given a module M over a ring R, an R endomorphism f of M is called an involution if f 2 is the identity homomorphism on M.

Involutions are related to idempotents; if 2 is invertible then they correspond in a one-to-one manner.

 

以及『透視性』講法的呢??!!

Perspectivity

In geometry and in its applications to drawing, a perspectivity is the formation of an image in a picture plane of a scene viewed from a fixed point.

Graphics

The science of graphical perspective uses perspectivities to make realistic images in proper proportion. According to Kirsti Andersen, the first author to describe perspectivity was Leon Alberti in his De Pictura (1435).[1] In English, Brook Taylor presented his Linear Perspective in 1715, where he explained “Perspective is the Art of drawing on a Plane the Appearances of any Figures, by the Rules of Geometry”.[2] In a second book, New Principles of Linear Perspective (1719), Taylor wrote

When Lines drawn according to a certain Law from the several Parts of any Figure, cut a Plane, and by that Cutting or Intersection describe a figure on that Plane, that Figure so described is called the Projection of the other Figure. The Lines producing that Projection, taken all together, are called the System of Rays. And when those Rays all pass thro’ one and same Point, they are called the Cone of Rays. And when that Point is consider’d as the Eye of a Spectator, that System of Rays is called the Optic Cone[3]

Projective geometry

In projective geometry the points of a line are called a projective range, and the set of lines in a plane on a point is called a pencil.

Given two lines  \ell and  m in a plane and a point P of that plane on neither line, the bijective mapping between the points of the range of  \ell and the range of  m determined by the lines of the pencil on P is called a perspectivity (or more precisely, a central perspectivity with center P).[4] A special symbol has been used to show that points X and Y are related by a perspectivity;  X \doublebarwedge Y . In this notation, to show that the center of perspectivity is P, write  X \ \overset {P}{\doublebarwedge} \ Y. Using the language of functions, a central perspectivity with center P is a function f_P \colon [\ell] \mapsto [m] (where the square brackets indicate the projective range of the line) defined by f_P (X) = Y \text{ whenever } P \in XY.[5] This map is an involution, that is, f_P (f_P (X)) = X \text{ for all }X \in [\ell].

The existence of a perspectivity means that corresponding points are in perspective. The dual concept, axial perspectivity, is the correspondence between the lines of two pencils determined by a projective range.

A perspectivity:  ABCD \doublebarwedge A'B'C'D',

Projectivity

The composition of two perspectivities is, in general, not a perspectivity. A perspectivity or a composition of two or more perspectivities is called a projectivity (projective transformation, projective collineation and homography are synonyms).

There are several results concerning projectivities and perspectivities which hold in any pappian projective plane:[6]

Theorem: Any projectivity between two distinct projective ranges can be written as the composition of no more than two perspectivities.

Theorem: Any projectivity from a projective range to itself can be written as the composition of three perspectivities.

Theorem: A projectivity between two distinct projective ranges which fixes a point is a perspectivity.

 

此或許 Elena Marchisotto 先生在其著作中

The Theorem of Pappus:A Bridge Between Algebra and Geometry

寫『從 Pappus 到 Desargues』之 路途遙遠乎!!??