Ordinary

Posted on December 14, 2021 by rpflee

The power of a man’s virtue should not be measured by his special efforts, but by his ordinary doing.

— Blaise Pascal

2021.12.14 Tuesday ACHK

改變因果鏈起點, 1.2

Posted on December 12, 2021 by rpflee

這段改編自 2010 年 4 月 24 日的對話。

但是，只有人類這個物種，有大規模的知識傳授。而大規模的知識傳授，則自然需要大量時間。

幸而，雖然人生苦短，但仍剛剛足夠長，去行一個「學習、發掘和傳授」循環。在進化接力賽中，人類每一代剛好有足夠時間，去完成交收接力棒的工序。在知識上，每一代人也可以，以上一代人的終點，作為自己的起點。

簡單而言，即是「接棒接得切」。

（安：何謂「接棒接得切」呢？）

意思是，人類整體之知識，可以累積。例如，你現在 30 歲。雖然有 30 年的人生經驗，但是那毋須用，三十年去傳授。你的後輩，可能只需花一年，就可以學到，那 30 年的所有重點。

你可以想像，如果人類的生命太短，例如平均得 3 年的話，那就既沒有足夠時間，去累積人生經驗，更沒有足夠時間，去傳授下一代。那樣，人類整體之知識，就不可累積。

第二種假想情況是，你現時 30 年的人生經歷，要花足足 20 年，才傳授得到。那樣，人類的整體智力，就會累積得超慢。

（安：第二種情況，我不太明白。）

我改一改那些數字。如果你 80 年的人生經歷，要花足足 79 年，才傳授得到的話，你是否會有，那額外附加的 79 年壽命呢？

你相當於要有，起碼 159 年的壽命，才能完成整個，知識交棒的工序。

幸而，現實不是那樣。現實是，人有足夠的生命，去累積有用的人生經驗；而且，傳授所花的年期，遠少於親身經歷所花。

簡單來說，發掘知識時間極長；複製知識時間極短。

— Me@2021-12-11 07:18:17 PM

重頭愛你

Posted on December 5, 2021 by rpflee

Love me once again

—

2021.12.05 Sunday ACHK

Ex 1.21 The dumbbell, 3.4

Posted on November 26, 2021 by rpflee

Structure and Interpretation of Classical Mechanics

e. Make a Lagrangian ( $\displaystyle{= T - V}$ ) for the system described with the irredundant generalized coordinates $\displaystyle{x_{cm}}$ , $\displaystyle{y_{cm}}$ , $\displaystyle{\theta}$ and compute the Lagrange equations from this Lagrangian. They should be the same equations as you derived for the same coordinates in part d.

~~~

Eq. 1.95:

$\displaystyle{ L'(t;q,c,F; \dot q, \dot c, \dot F)}$

$\displaystyle{ = \sum_\alpha \frac{1}{2} m_\alpha \left( \partial_0 f_\alpha (t, q, c) + \partial_1 f_\alpha (t, q,c) \dot q + \partial_2 f_\alpha (t, q, c) \dot c \right)^2 }$

$\displaystyle{ - V(t, f(t,q,c)) - \sum_{\{ \alpha, \beta | \alpha < \beta, \alpha \leftrightarrow \beta \}} \frac{F_{\alpha \beta}}{2 l_{\alpha \beta}} \left[ c_{\alpha \beta}^2 - l_{\alpha \beta}^2 \right] }$

Eq. 1.97:

$\displaystyle{ L''(t,q,\dot q)}$

$\displaystyle{ = \sum_\alpha \frac{1}{2} m_\alpha \left( \partial_0 f_\alpha (t, q) + \partial_1 f_\alpha (t, q) \dot q \right)^2 - V(t, f(t,q,l)) }$

71

Consider a function $\displaystyle{g}$ of, say, three arguments, and let $\displaystyle{g_0}$ be a function of two arguments satisfying $\displaystyle{g_0(x, y) = g(x, y, 0)}$ .

Then $\displaystyle{(\partial_0 g_0)(x, y) = (\partial_0 g)(x, y, 0)}$ .

The substitution of a value in an argument commutes with the taking of the partial derivative with respect to a different argument. In deriving the Lagrange equations for $\displaystyle{q}$ we can set $\displaystyle{c = l}$ and $\displaystyle{ \dot c = 0}$ in the Lagrangian, but we cannot do this in deriving the Lagrange equations associated with $\displaystyle{c}$ , because we have to take derivatives with respect to those arguments.

[guess]

$\displaystyle{ M = m_1 + m_2 }$

$\displaystyle{ \frac{1}{\mu} = \frac{1}{m_1} + \frac{1}{m_2} }$

$\displaystyle{ V = 0 }$

$\displaystyle{ T }$

$\displaystyle{ = \frac{1}{2} M v^2 + \frac{1}{2} I \omega^2 }$

$\displaystyle{ = \frac{1}{2} (m_0 + m_1) (\dot x_{CM}^2 + \dot y_{CM}^2 ) + \frac{1}{2} \left( m_1 r_1^2 + m_2 r_2^2 \right) \dot \theta^2 }$

$\displaystyle{ = \frac{1}{2} (m_0 + m_1) (\dot x_{CM}^2 + \dot y_{CM}^2 ) + \frac{1}{2} \left[ m_1 \left( \frac{m_2 l}{m_1 + m_2} \right)^2 + m_2 \left( \frac{m_1 l}{m_1 + m_2} \right)^2 \right] \dot \theta^2 }$

$\displaystyle{ = \frac{1}{2} (m_0 + m_1) (\dot x_{CM}^2 + \dot y_{CM}^2 ) + \frac{1}{2} m_1 m_2 \left( \frac{1}{m_1 + m_2} \right)^2 ( m_2 + m_1 ) l^2 \dot \theta^2 }$

$\displaystyle{ = \frac{1}{2} (m_0 + m_1) (\dot x_{CM}^2 + \dot y_{CM}^2 ) + \frac{1}{2} \frac{m_1 m_2}{m_1 + m_2} l^2 \dot \theta^2 }$

$\displaystyle{ = \frac{1}{2} (m_0 + m_1) (\dot x_{CM}^2 + \dot y_{CM}^2 ) + \frac{1}{2} \mu l^2 \dot \theta^2 }$

[guess]

— Me@2021-11-24 09:58:31 PM

Impossible to break

Posted on November 21, 2021 by rpflee

The difference between a machine possible to break and a machine impossible to break is that when a machine impossible to break breaks, it is impossible to fix.

— based on Douglas Adams

— Me@2021-11-21 08:03:16 PM

Copyright

Posted on November 16, 2021 by rpflee

On copyright, no one, neither the creators nor the users, should have absolute rights.

Respect must be mutual.

— Me@2011.08.28

相聚零刻 2.2

Posted on November 10, 2021 by rpflee

尋覓 2.2.3.6.2

男士其實要到 28 歲時，思想才帶點成熟，通常。

然後，那個哲學家又說… … 千萬不要說出去，以免得罪人。

然後，那個哲學家又說，至於女士，則於 18 歲時，就已經成熟。

聽不聽得出，他在投訴什麼呢？

他的意思是，女士在思想上，於 18 歲時，就已經成熟；自此以後，就不再成熟。（當然，那只是大概而然；每個人也不同。）

他舉例：

一位女士可以照顧小孩，與其玩耍，長達數天也不厭倦，而樂在其中。但是，一般男士也不會。

女士，就是介乎於，男士與小孩之間的物體。

那個哲學家，覺得那是缺點。但我覺得不一定；那也可以是優點，因為，我們可以視照顧小孩，為女性的天職；然後，再視照顧女性，為男性的天職。

實情是，有很多職業是，需要照顧小孩子的，例如幼稚園教師。如果一位男士，做幼稚園教師的話，你總會覺得很奇怪。

那樣，我想講的重點是什麼呢？

假設你現在已經有拖拍，對象和你同年，即是都是大概 20 歲。那樣，他就很可能仍是，傻仔一個。

我不是針對他。我是泛指，一般的年輕男子。我年輕時，也是傻仔人士一名。我當年已經企圖不傻。曾幾何時，我每星期看一本書。但是，現在回顧當年的我，仍然九分低能。

所以，你現在毋須太緊張。

現在的你不用，急於找到對象。或者，到你 25 歲時，才遇到 28 歲的他，會好一點。那時，那個他已經是，思想成熟和人格完整的男士。

那不是指，你可以控制。那只是指，理想的劇情怎麼樣。

（CPK: 廿五歲的我，才認識 28 歲的他？會不會太遲？）

其實會。理想劇情，應該這樣說：

你倆可能自小，已經認識對方。到你 25 歲時，才與他發展成戀人。

— Me@2021-11-09 10:15:46 PM

Do You Remember Love?

Posted on October 30, 2021 by rpflee

可有記起愛？

— Me@2021-10-30 07:24:36 AM

2021.10.30 Saturday ACHK

機會率驗算 2

Posted on October 27, 2021 by rpflee

Proofreading probability

There are 5 women and 6 men in a bus. But there are only 4 seats.

Assume that all the seats will be occupied and all possible seating arrangements have the same probability to appear.

What is the probability that 2 or more men will have seats?

~~~

Probability method gets the final probability by multiplying several fractions of probability together.

P method:

This method considers each seat one by one: what is the probability that this seat will get, for example, an M?

Let $\displaystyle{x}$ be the number of men that will have seats. Also, instead of writing $\displaystyle{C^n_r}$ , we use the standard notation $\displaystyle{{n \choose r}}$ .

$\displaystyle{P(x \ge 2)}$
$\displaystyle{= 1 - P(x = 0) - P(x = 1)}$
$\displaystyle{= 1 - P(FFFF) - P(MFFF) - P(FMFF) - P(FFMF) - P(FFFM)}$

$\displaystyle{= 1 - P(FFFF) - {4 \choose 1} P(FFFM)}$

$\displaystyle{P(FFFF) = \left( \frac{5}{11} \right) \left( \frac{4}{10} \right) \left( \frac{3}{9} \right) \left( \frac{2}{8} \right)}$

$\displaystyle{P(FFFM) = \left( \frac{5}{11} \right) \left( \frac{4}{10} \right) \left( \frac{3}{9} \right) \left( \frac{6}{8} \right)}$

$\displaystyle{P(x \ge 2)}$

$\displaystyle{= \frac{6360}{7920}}$

$\displaystyle{= \frac{53}{66}}$

Statistics method gets the final probability by dividing the number of ways of getting the desired result by the number of all possible results.

S method 1:

This method considers all the seats at once: what is the number of ways that they will get, for example, 3F and 1M?

In other words, instead of the seats, this method focuses on the people: what is the number of ways that 3F and 1M will be chosen?

In other words, instead of the choosing process, this method focuses on counting the results of the choosing.

Let $\displaystyle{N(...)}$ = the number of ways of …

$\displaystyle{P(x \ge 2)}$

$\displaystyle{= \frac{N(\text{seats choose 2 or more men})}{\text{total number of ways}}}$

$\displaystyle{= \frac{\text{total number of ways} - N(\text{4 F among 5}) - N(\text{choose 3 F among 5 and then 1 M among 6})}{\text{total number of ways}}}$

$\displaystyle{= \frac{N(\text{seats choose any 4 people}) - N(\text{4 F among 5}) - N(\text{choose 3 F among 5}) \times N(\text{1 M among 6})}{N(\text{seats choose any 4 people})}}$

$\displaystyle{= \left[{11 \choose 4} - {5 \choose 4} - {5 \choose 3} {6 \choose 1} \right] / \left[ {11 \choose 4} \right]}$

$\displaystyle{= \frac{ 330 - 5 - 10 \times 6 }{330}}$

$\displaystyle{= \frac{ 53 }{ 66 }}$

S method 2:

This method considers each seat one by one: what is the number of ways that this seat will get, for example, an M?

In other words, instead of the results, this method focuses on choosing process itself.

Note that this method uses permutation instead of combination.

$\displaystyle{P(x \ge 2)}$

$\displaystyle{= \frac{\text{total number of ways} - N(FFFF) - N(MFFF) - N(FMFF) - N(FFMF) - N(FFFM)}{\text{total number of ways}}}$

$\displaystyle{= \frac{(11)(10)(9)(8) - (5)(4)(3)(2) - (6)(5)(4)(3) - (5)(6)(4)(3) - (5)(4)(6)(3) - (5)(4)(3)(6)}{(11)(10)(9)(8)}}$

$\displaystyle{= \frac{(11)(10)(9)(8) - (5)(4)(3)(2) - 4(6)(5)(4)(3)}{(11)(10)(9)(8)}}$

$\displaystyle{= \frac{6360}{7920}}$

$\displaystyle{= \frac{ 53 }{ 66 }}$

— Me@2021-10-02 05:18:02 PM

Logical arrow of time, 10

Posted on October 3, 2021 by rpflee

Two distinguishable macrostates can both evolve into one indistinguishable macrostate.

— Me@2013-08-11 11:08 AM

Ultimate Freedom, 2

Posted on September 26, 2021 by rpflee

Imagination is not an empirical or superadded power of consciousness, it is the whole of consciousness as it realizes its freedom.

— Jean-Paul Sartre

Your mental world/software world lies your ultimate freedom.

It is where anything not forbidden can happen.

It is where anything logically consistent can happen.

— Me@2016-11-20 09:14:06 AM

— Me@2021-09-26 09:22:54 PM

改變因果鏈起點

Posted on September 23, 2021 by rpflee

這段改編自 2010 年 4 月 24 日的對話。

所以，我們先企圖進攻那些大題目，無論它們長遠是否，對你直接有用；因為，過程之中，自然會引發很多技術細節。而那些技術細節，很多是你直接可用的。

原初辛苦上山的目標是，傳說中，山頂上的寶藏。但是，上到山頂後，才發現最珍貴的寶藏，反而是沿途找到的那些。

（安：我覺得你對我最大的價值，反而是：

因為你鼓勵我，先企圖進攻的那些大題目中，很多也是你自己，曾經進攻過的，所以，你從它們之中篩選出來，提議我去研究的，必定是你覺得對我來說，最有價值的那些。

如果不是那樣的話，我可能要麼就是，開始研究某個大題目時，就覺得徬徨，而無心機繼續；要麼就是，真的花了一年半載，研究某個大題目後，發覺原來事倍功半，或者價值不大。）

那其實正正就是，「教育」的精髓，「學習」的意思。

為何不事事親身去經歷呢？　

因為人生苦短。

透過吸收他人的經歷，可以改變因果鏈的起點。

人在某個意思之下，其實已經有了永生。個別的人會死亡，但人類整體而然，可存在很久。只要不絶種，就是得永生。對於任何物種，都可以這樣說。

但是，只有人類這個物種，有大規模的知識傳授。而大規模的知識傳授，則自然需要大量時間。

— Me@2021-09-22 10:24:46 PM

1990s, 5.2

Posted on September 19, 2021 by rpflee

— Photomyne colorization

— Me@2021-09-19 06:13:39 PM

Ex 1.21 The dumbbell, 3.3

Posted on September 18, 2021 by rpflee

Structure and Interpretation of Classical Mechanics

c. Make a change of coordinates to a coordinate system with center of mass coordinates $\displaystyle{x_{cm}}$ , $\displaystyle{y_{cm}}$ , angle $\displaystyle{\theta}$ , distance between the particles $\displaystyle{c}$ , and tension force $\displaystyle{F}$ . Write the Lagrangian in these coordinates, and write the Lagrange equations.

~~~

[guess]


(define (KE-particle m v)
  (* 1/2 m (square v)))

(define ((L-free-constrained m0 m1 l) local)
  (let* ((extract (extract-particle 2))
     (p0 (extract local 0))
     (q0 (coordinate p0))
     (qdot0 (velocity p0))
  
     (p1 (extract local 1))
     (q1 (coordinate p1))
     (qdot1 (velocity p1))
  
     (F (ref (coordinate local) 4)))
 
    (- (+ (KE-particle m0 qdot0)
          (KE-particle m1 qdot1))
       (U-constraint q0 q1 F l))))

(define ((extract-particle pieces) local i)
  (let* ((indices (apply up (iota pieces (* i pieces))))
         (extract (lambda (tuple)
                    (vector-map (lambda (i)
                                  (ref tuple i))
                                indices))))
    (up (time local)
        (extract (coordinate local))
        (extract (velocity local)))))

(define (U-constraint q0 q1 F l)
  (* (/ F (* 2 l))
     (- (square (- q1 q0))
        (square l))))

(let ((L (L-free-constrained 'm_0 'm_1 'l))
      (q-rect (up (literal-function 'x_0)
                  (literal-function 'y_0)
                  (literal-function 'x_1)
                  (literal-function 'y_1)
                  (literal-function 'F))))
  (show-expression
   ((compose L (Gamma q-rect)) 't)))

$\displaystyle{ \frac{1}{2} m_0 \left( \dot x_0^2 + \dot y_0^2 \right) + \frac{1}{2} m_1 \left( \dot x_1^2 + \dot y_1^2 \right) + \frac{F}{2 l} \left( l^2 - y_1^2 + 2 y_0 y_1 - x_1^2 + 2 x_0 x_1 - y_0^2 - x_0^2 \right) }$

$\displaystyle{ = \frac{1}{2} m_0 \left( \dot x_0^2 + \dot y_0^2 \right) + \frac{1}{2} m_1 \left( \dot x_1^2 + \dot y_1^2 \right) - \frac{F}{2 l} \left[ (y_1 - y_0)^2 + (x_1 - x_0)^2 - l^2 \right] }$


(define ((q->r m0 m1) local)
  (let ((q (coordinate local)))
    (let ((x_cm (ref q 0))
          (y_cm (ref q 1))
          (theta (ref q 2)) 
          (c (ref q 3))
	      (F (ref q 4))
	      (M (+ m0 m1)))
      (let ((x0 (- x_cm (* (/ m1 M) c (cos theta))))
            (y0 (- y_cm (* (/ m1 M) c (sin theta))))
            (x1 (+ x_cm (* (/ m0 M) c (cos theta))))
            (y1 (+ y_cm (* (/ m0 M) c (sin theta)))))
        (up x0 y0 x1 y1 F)))))

(let ((q (up (literal-function 'x_cm)
	         (literal-function 'y_cm)
	         (literal-function 'theta)
	         (literal-function 'c)
	         (literal-function 'F))))
  (show-expression (q 't)))

 
(show-expression
 (up 't
     (up 'x_cm 'y_cm 'theta 'c 'F)
     (up 'xdot_cm 'ydot_cm 'thetadot 'cdot 'Fdot)))


(show-expression
  ((q->r 'm_0 'm_1) 
     (up 't
         (up 'x_cm 'y_cm 'theta 'c 'F)
         (up 'xdot_cm 'ydot_cm 'thetadot 'cdot 'Fdot))))


(let ((q (up (literal-function 'x_cm)
             (literal-function 'y_cm)
             (literal-function 'theta)
             (literal-function 'c)
             (literal-function 'F))))
  (show-expression ((q->r 'm_0 'm_1) ((Gamma q) 't))))


(show-expression
  ((F->C (q->r 'm_0 'm_1)) 
     (up 't
         (up 'x_cm 'y_cm 'theta 'c 'F)
         (up 'xdot_cm 'ydot_cm 'thetadot 'cdot 'Fdot))))

(define (L-cm m0 m1 l)
  (compose
   (L-free-constrained m0 m1 l) (F->C (q->r m0 m1))))

(show-expression
 ((L-cm 'm_0 'm_1 'l)
     (up 't
         (up 'x_cm 'y_cm 'theta 'c 'F)
         (up 'xdot_cm 'ydot_cm 'thetadot 'cdot 'Fdot))))

$\displaystyle{ \frac{1}{\mu} = \frac{1}{m_0} + \frac{1}{m_1} }$

$\displaystyle{ L_{cm} }$

$\displaystyle{ = \frac{ ( c^2 \dot \theta^2 + \dot c^2 ) l m_0 m_1 + (l m_0^2 + 2 l m_0 m_1 + l m_1^2) (\dot x_{cm}^2 + \dot y_{cm}^2) + F ( l^2 - c^2 )(m_0 + m_1) }{2 l (m_0 + m_1)} }$

$\displaystyle{ = \frac{ ( c^2 \dot \theta^2 + \dot c^2 ) l m_0 m_1 + l (m_0 + m_1)^2 (\dot x_{cm}^2 + \dot y_{cm}^2) + F ( l^2 - c^2 )(m_0 + m_1) }{2 l (m_0 + m_1)} }$

$\displaystyle{ = \frac{1}{2} ( c^2 \dot \theta^2 + \dot c^2 ) \mu + \frac{1}{2} (m_0 + m_1) (\dot x_{cm}^2 + \dot y_{cm}^2) + \frac{1}{2l} F ( l^2 - c^2 ) }$


(show-expression
 (((Lagrange-equations
    (L-cm 'm_0 'm_1 'l))
   (up (literal-function 'x_cm)
       (literal-function 'y_cm)
       (literal-function 'theta)
       (literal-function 'c)
       (literal-function 'F)))
  't))


(show-expression
 (((Lagrange-equations
    (L-cm 'm_0 'm_1 'l))
   (up (literal-function 'x_cm)
       (literal-function 'y_cm)
       (literal-function 'theta)
       (lambda (t) 'l)
       (literal-function 'F)))
  't))

[guess]

— Me@2021-09-17 06:35:51 AM

Meta numbers

Posted on September 13, 2021 by rpflee

Numbers are meta objects.

Infinities are meta numbers.

— Me@2017-02-03 05:20:51 PM

Year 2000 Reloaded

Posted on September 12, 2021 by rpflee

These are what I kept reminding myself during my Age of University.

1. I think, therefore I am

A lot of times, something seems to be a deadend.

However, once I have faith that a solution may be possible and start to think, most of the time, that “deadline” is actually resolvable.

2. Nothing Less

A lot of times, an action seems to be insignificant.

However, actually, every action may have infinite consequences, especially when you cultivate it.

3. The Aladdin Factor

A key to get a meaningful life is to dare to ask other people or yourself for answers, advice, or favours, when necessary.

Don’t be afraid to ask, as long as a request is ethical, reasonable, and polite.

— Me@2011.08.17

— Me@2021-09-11

相聚零刻 2.1

Posted on September 10, 2021 by rpflee

尋覓 2.2.3.6.1

男士 20 歲時，還找不到女朋友，可能是好事，因為，男士其實要到 28 歲時，思想才帶點成熟，通常。

那是某一個哲學家講的。我當年在 25 歲時，就已經聽到這一點。到 28 歲時，真的明白了一些，以前不明白的東西。（當然，任何年齡也可以這樣說，如果一個人不斷進步的話。）

例如，說話時，為什麼會「食螺絲」（舌頭打結）呢？

是因為，企圖在同一刻講，超過一個字。只要稍為放慢說話速度，就可以化解。

又例如，生活中，為什麼會諸事不順呢？

是因為，企圖在同一格時間中，放多於一件事。

然後，那個哲學家又說… … 千萬不要說出去，以免得罪人。

然後，那個哲學家又說，至於女士，則於 18 歲時，就已經成熟；自此以後，就不再成熟了。

聽不聽得出，他在投訴什麼呢？

他的意思是，女士在思想上，於 18 歲時，就已經成熟；自此以後，就不再成熟。（當然，那只是大概而然；每個人也不同。）

女士，就是介乎於，男士與小孩之間的物體。

那個哲學家，覺得那是缺點。但我覺得不一定；那也可以是優點。

例如，善良之女士，可以永久可愛。男士則不可以。男士青春期時，聲線會變。變聲後，不可能再可愛。

女士要可愛。男士要可靠。

無論在外表或性格上，女士的職責是，保持可愛，尋找可靠；男士的職責則是，保持可靠，保護可愛。

— Me@2021-07-26 05:12:55 PM

— Me@2021-09-07 05:15:56 PM

Pier, 2

Posted on September 1, 2021 by rpflee

— Me@2021-09-01 04:41:10 PM

2.10 A spacetime orbifold in two dimensions, 3

Posted on August 31, 2021 by rpflee

A First Course in String Theory

(b) Draw a spacetime diagram, indicate the $\displaystyle{x^+}$ and $\displaystyle{x^-}$ axes, and sketch the family of curves

$\displaystyle{x^+ x^- = a^2}$ ,

where $\displaystyle{a > 0}$ is a real constant that labels the various curves.

…

~~~

— Me@2021-08-31 08:42:40 PM

Visualizing higher dimensions, 2

Posted on August 23, 2021 by rpflee

Geometry is global.

Space is what we can see at once.

Dynamics is local.

Time is what we cannot see at once.

— Me@2017-02-07 10:11:34 PM

If we could see, for example, several minutes at once, that several minutes would become a spatial dimension.

In other words, that dimension is visualized for us.

— Me@2017-02-03 07:31:25 AM

Physics Town

continues

Ordinary

改變因果鏈起點, 1.2

重頭愛你

Ex 1.21 The dumbbell, 3.4

Impossible to break

Copyright

相聚零刻 2.2

Do You Remember Love?

機會率驗算 2

Logical arrow of time, 10

Ultimate Freedom, 2

改變因果鏈起點

1990s, 5.2

Ex 1.21 The dumbbell, 3.3

Meta numbers

Year 2000 Reloaded

相聚零刻 2.1

Pier, 2

2.10 A spacetime orbifold in two dimensions, 3

Visualizing higher dimensions, 2