Visualizing higher dimensions

The trick of visualizing higher dimension is: not to visualize it.

— Wikipedia

— Me@2011.08.19

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Besides trying to visualize, there are other methods to understand higher dimensions.

— Me@2018-10-28 04:28:01 PM

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What is the meaning of visualization?

— Me@2018-09-02 4:35 pm

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feel ~ receive all the data at once

(This definition is not totally correct, but is useful in the meantime.)

visual ~ feel at once through eyes

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you can visualize a 3D object ~ you can see all of a 3D object at once

you cannot visualize a 4D object ~ you cannot see all of a 4D object at once

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Actually, you can only visualize a 2D object, such as a square.

You cannot visualize a 3D object, such as a cube.

That’s why the screen of any computer monitor is 2 dimensional, not 3.

— Me@2018-10-28 04:32:41 PM

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2018.10.28 Sunday (c) All rights reserved by ACHK

Life, 3

生命 3

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We exist in time because time is change.

Growing is part of the definition of life. Growing is a kind of change.

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Also, without time/change, there would be no thinking and no thoughts.

— Me@2017-12-26 11:42 am

— Me@2018-05-23 10:05:03 PM

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time ~ change

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Time is logically necessary if change is necessary.

— Me@2018-02-04 09:07:48 PM

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2018.05.23 Wednesday (c) All rights reserved by ACHK

The language of Change 1.2

Energy conservation, 6.2 | Energy 5.2

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time ~ change

energy ~ the ability of _keeping_ changing

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constant velocity ~ the amount of an object’s change of position, measured with respect to its observer’s unit of change, is constant

s = \Delta x

v = \frac{s}{\Delta t} = \frac{\Delta x}{\Delta t}

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kinetic energy ~ the amount of the ability of keeping changing an object’s position

\frac{1}{2} m v^2 ~ the square of (the amount of change of position, relative to the observer’s unit of change)

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Energy difference is _not_ exactly a measurement of the amount of change, time interval is.

— Me@2018-02-20 09:39:30 AM

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2018.02.20 Tuesday (c) All rights reserved by ACHK

The language of Change

Energy conservation, 6 | Energy 5

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time ~ change

energy ~ the ability of causing change

Assuming

1. a system of one single particle

2. has only kinetic energy

3. and that kinetic energy is conserved.

conservation of energy ~ an object’s potential amount of change of position, measured with respect to its observer’s unit of change, is constant

s = \Delta x

v = \frac{s}{\Delta t} = \frac{\Delta x}{\Delta t}

— Me@2018-02-15 02:21:20 PM

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Note:

The above argument has a bug:

If the mass m is constant, the kinetic energy E_K should be proportional to velocity squared v^2, instead of velocity v.

E_K = \frac{1}{2} m v^2

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However, the above argument is still technically correct:

When E_K is constant, v^2 is constant. In turn, the magnitude of v also remains unchanged.

— Me@2018-02-19 09:37:24 PM

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2018.02.15 Thursday (c) All rights reserved by ACHK

Euler Formula

Exponential, 2
 

a^x

general exponential increase ~ the effects are cumulative
 
e^x

natural exponential increase ~ every step has immediate and cumulative effects

— Me@2014-10-29 04:44:51 PM
 

exponent growth

e^x = \lim_{n \to \infty} \left(1 + \frac{x}{n}\right)^n

~ compound interest effects with infinitesimal time intervals
 

multiply -1

~ rotate to the opposite direction

(rotate the position vector of a number on the real number line to the opposite direction)

~ rotate 180 degrees
 

multiply i

~ rotate to the perpendicular direction

~ rotate 90 degrees
 

For example, the complex number (3, 0) times i equals (0, 3):

3 \times i = 3 i
(3, 0) (0, 1) = (0, 3)
 

multiplying i

~ change the direction to the one perpendicular to the current moving direction

(current moving direction ~ the direction of a number’s position vector)
 

exponential growth with an imaginary amount

e^{i \theta} = \lim_{n \to \infty} \left( 1 + \frac{i \theta}{n} \right)^n

~ change the direction to the one perpendicular to the current moving direction continuously

~ rotate \theta radians

— Me@2016-06-05 04:04:13 PM
 
 
 
2016.06.08 Wednesday (c) All rights reserved by ACHK

注定外傳 2.6

Can it be Otherwise? 2.6 | The Beginning of Time, 7.3

還有,「宇宙」這個詞語,其實分析下去,是不合法的,因為「宇宙」的意思,就是「所有事物」。

而「所有」這個詞語的意思,是相對的,因為「所有」,即是「百分之一百」。

在沒有一個基數時,講「百分之一百」,其實不會知道,是指多少數量。同理,在沒有上文下理時,講「所有」,其實不太知道,是指什麼意思。例如,「所有人」即是有「多少人」呢?

沒有明確的上文下理,「所有人」自然沒有明確的意思。

詳情請參閱,我以往有關「所有」的文章,例如:

相反,如果有明確的上文下理,就自然有明確的意思。例如,『三十元中的百分之一百』,就很明顯是指,那三十元。

又例如,『這間屋的所有人』,都有明確的意思,因為有明確的範圍;有範圍,就可點人數:

凡是在這間屋內遇到的人,包括你自己,你都記下名字,直到在這間屋,再不找到新的人為止。那樣,你就可以得到,有齊『這間屋所有人』的名單。

『所有』,就是『場所之有』。

沒有明確的場所,就不知所「有」何物。

— Me@2016-05-18 11:40:31 AM

2016.05.18 Wednesday (c) All rights reserved by ACHK

Quantum Indeterminacy

注定外外傳 1

Quantum indeterminacy is the apparent necessary incompleteness in the description of a physical system, that has become one of the characteristics of the standard description of quantum physics.

Indeterminacy in measurement was not an innovation of quantum mechanics, since it had been established early on by experimentalists that errors in measurement may lead to indeterminate outcomes. However, by the later half of the eighteenth century, measurement errors were well understood and it was known that they could either be reduced by better equipment or accounted for by statistical error models. In quantum mechanics, however, indeterminacy is of a much more fundamental nature, having nothing to do with errors or disturbance.

— Wikipedia on Quantum indeterminacy

Quantum indeterminacy is the inability to predict the behaviour of the system with 100% accuracy, even in principle.

If everything is connected , quantum indeterminacy is due to the logical fact that, by definition, a “part” cannot contain (all the information of) the “whole”.

An observer (A) cannot separate itself from the system (B) that it wants to observe, because an observation is an interaction between the observer and the observed .  

In order to get a perfect prediction of a measurement result, observer (A) must have all the information of the present state of the whole system (A+B). However, there are two logical difficulties.

First, observer A cannot have all the information about (A+B).

Second, observer A cannot observe itself to get (all of) its present state information, since an observation is an interaction between two entities. Logically, it is impossible for something to interact with itself directly. Just as logically, it is impossible for your right hand to hold your right hand itself. 

So the information observer A can get (to the greatest extent) is all the information about B, which is only part of the system (A+B) it (A) needs to know in order to get a prefect prediction for the evolution of the system B.

— Me@2015-09-14 08:12:32 PM

2015.09.15 Tuesday (c) All rights reserved by ACHK

機遇再生論 1.3

『機遇再生論』的大概意思是,所有可能發生的事情,例如重生,在無限長的未來時間中,必會發生。

機遇再生論原始版本,有問題的字眼中,除了「所有」之外,還有「無限」。「無限」通常都是一個違法詞語。「無限」引起的問題,以前論述過,現不再詳談。請參閱「無限」系列的文章。

你可以嘗試移除「無限」這個詞語,只把「無限」的意思中,有意義的部分保留:

『機遇再生論』的大概意思是,所有可能發生的事情,例如重生,只要等足夠長的時間,總會發生。

但是,即使避開了「無限」,用了「足夠長」,仍然會有其他問題。「足夠長」這個詞語雖然不算違法,但是十分空泛,空泛到近乎沒有意義。

試想想,怎樣才為之「足夠長」呢?

— Me@2015.04.08

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2015.04.09 Thursday (c) All rights reserved by ACHK

Open Source MindOS, 2.1

朋友同事 5.1 | 已婚單身 2.1 | Singles 2.1

這段改編自 2010 年 4 月 10 日的對話。

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所以,我的其中一個擇偶條件是,她的心靈作業系統,是 open source 的。換句話說,即是「受教可改」。受教可改,就自然沒有「人格問題」。

「人格問題」不同於「缺點」。人不會「沒有缺點」,但可以「沒有人格問題」,即是「人格完整」。「人格問題」和「缺點」的分別,之前已經講過,所以,現在不再詳述。

簡單而言,個別不足之處,為之「缺點」,可以避開,因為其害處是局部的;思考或處事模式之誤,為之「人格問題」,必會碰釘,因為其害處無所不在。我之前講法是:

例如,我的其中一個缺點是『不懂烹飪』,但那不算是我的『人格缺失』。但是,如果我在既『烹飪欠佳』,又『不肯學習』的情況下,仍然要別人吃,我所煮的菜色的話,這性格就是我的『人格問題』。

又例如,『間中遲到五分鐘』不算是『人格缺失』。但是,如果『遲到五分鐘』後,竟然沒有任何歉意,亦沒有絲毫改正的意圖的話,這態度就是『人格缺失』。

又或者,你其實是有歉意的,但是『遲到五分鐘』已經成了你的習慣,再不是個別事件,這習慣就是『人格問題』。

比喻說,一個人的「一般缺點」,就好比一部電腦「個別應用程式」的問題。你可以把「個別應用程式」暫時關閉,甚至移除,然後安裝替代程式;而電腦中的其他程式,期間可以如常運作,不受影響。

一個人的「人格問題」,則有如「作業系統程式」的毛病,其害處你想避也避不到,直至你修改了,那「作業系統程式」的問題部分為止。

(安:「人格」其實即是「習慣」。「人格問題」,又即是「思考」或「處事」上的「壞習慣」,所以,其害處會如你所講,無所不在。)

差不多,不過要「同情地理解」才可以,因為「人格」和「習慣」,不算是百分百的同義詞。例如,我穿鞋時,每次都是先穿左邊的那一隻。那是我的「習慣」,但和我的「人格好壞」無關。

但是,如果我每次穿鞋都要花十分鐘,那就不只是我的「習慣」,而且是我的「人格問題」,因為那是有系統地浪費時間。

(安:無好壞處的「習慣」,就和「人格」無關。但是,有明顯壞處的的習慣,即是「壞習慣」,就是「人格缺失」。)

可以那樣說。

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人格

~ 性格

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~ 系統

~ 習慣

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人格

~ 性格系統

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— Me@2014.06.30

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2014.07.02 Wednesday (c) All rights reserved by ACHK