Sometime in 2008 I heard music while sleeping. In fact it was me composing music in a dream. I always manage to do things in my dreams better than when I’m awake. The first time I composed music in my dream while sleeping, was when I was 15 years old.
The Amiga has only 4 hardware music channels. In my dream I had more advanced Amiga, that had 5 hardware channels and I composed with it music that I can’t really even imagine, when I’m awake — the music composed in my dream was beyond my ability to make music…
In 2008 I tried to put the notes together that I had heard in my dream. This is the result:
Sometimes when I’m programming in my dreams I feel that there is something, that I must program. I see the result of programming while sleeping, also the code and I experience myself thinking how to program something..
Sometimes when I’m over stressed in programming when I’m in fact sleeping, I suddenly realize that I’m not awake — I’m sleeping and I don’t have to do anything at all! …or do I still have to do all that? …even stress myself while sleeping and having a dream.
Those are moments when I experience somehow, that I have control over my dreams while sleeping…
Sometimes while sleeping and having a dream it takes a while to realize that I’m now not awake, this is only a dream. Though, I’ve started to think, that can I really say about my dreams, that they are only dreams…
I’ve had many nightmares where I’m haunted by demons.. Often in those nigthmares I’m also blind and can’t see my enemies — the demons — who attack me in number of ways… Once I fell off my bed to the floor and I woke up. In my dream I was dodging a demon impaling me…
It’s over midnight here.. Since the source code in one of my videos is not very readable, I decided to publish the Python source code of the video in this blog post.
import turtle def draw_curve(): window = turtle.Screen() turt=turtle.Turtle() turt.setpos(-300,-100) turt.clear() turt.speed('fastest') turt.hideturtle() vonKoch(turt,5,500) turt.left(120) vonKoch(turt,5,500) turt.left(120) vonKoch(turt,5,500) turt.left(120) vonKoch(turt,5,500) window.exitonclick() def vonKoch(turt,depth,length): if depth == 0: turt.forward(length) else: vonKoch(turt,depth-1,length/3) turt.right(60) vonKoch(turt,depth-1,length/3) turt.left(120) vonKoch(turt,depth-1,length/3) turt.right(60) vonKoch(turt,depth-1,length/3) draw_curve()
That’s it for now. Time to go to sleep.
There are finite number of letters. If we limit the length of a word to finite amount of letters, the number of all possible words is also finite.
If there were infinite number of letters in a word and this word consisted of infinite amount of sounds, the pronunciation of a word would never end.
As there are finite number of words, also the number of consepts formed from these words is finite.
On the other hand, if we would create consepts in mathematical way, we could have infinite number of consepts. Though, all of them couldn’t be explicitly ever written nor said in finite amount of time.
I’ve sometimes wonderd, that how many meaningful natural languages of humans’ from all of these finite number of words could be formed — also how many grammars. What we understand by “meaningful language” might be hard to define…
Let us examine numbers as distances from some number.
If we choose as starting point the zero (0), we have the usual case.
0 = |0 – 0|
1 = |0 – 1|
2 = |0 – 2|
n = |0 – n|
Each distance can be divided into parts, let’s think the number space as the familiar set R. For example the distance from zero to two can be divided in n parts, we get distances 2 / n between zero and two.
The point from which we started isn’t a distance but — like said — a point, that can’t be divided into smaller parts. Philosophically speaking this point is now in words of Democritus “atomos”, the smallest unit.
In the book “The way to geometry” it is said: “The magnitude of a point is zero.”
When we founded the numbers as distances in relation to zero, the zero point is indivisible; because in question is a point not a distance, dividing zero (a point) by any number, we get zero. Zero doesn’t get smaller by dividing.
Of course zero can’t be divided by itself. In distance interpretation philosophically speaking indivisible would be divided by indivisible.
Now zero is different from other numbers so that it is a point, the other numbers are distances from this point.
As distance interpretation we could start the numbers also in relation for example to number eight (8).
Now eight is a point, we get the other numbers as distances to number eight.
0 = ||8 – 8| – 0||
1 = ||8 – 8| – 1||
2 = ||8 – 8| – 2||
n = ||8 – 8| – n||
Here from number eight is “created new zero”. First we determine the distance of number eight from itself, then the distance of examined number from the determined distance earlier..
We could get the numbers (distances) and the zero point also by denoting:
0 = |8 – (8 + 0)|
1 = |8 – (8 + 1)|
2 = |8 – (8 + 2)|
n = |8 – (8 + n)|
When we see the numbers as distances from some number, we notice, that the distance can be divided into smaller parts, but the point from which we started, we can’t divide into smaller parts.
In our example when we started from eight, we examine first the distance of eight from itself, |8 – 8| = 0, which is and remains as zero.
The distance can be even or uneven, but the point from which the distance is determined can’t be even or uneven, that is the whole point of this post.
If we want also the negative numbers as distance interpretation, we simply put the minus sign before the determined distance, for example -2 = -|0 – 2| and this is how we can get the entire set R so that the distances from zero point can be divided into smaller parts, but zero point is and remains zero and is not a distance but a point. In the words of Euclid that which has no part. In my own words: “Indivisible”.
In this way the specialty of zero can be seen better.
The title is a bit absurd, we could examine the distance of any number from itself..
In the previous post I examined the property of distance being even or uneven. Let’s go back to distance of zero from itself. Formally the distance of zero from itself is |0 – 0| = 0. 0 mod 2 = 0, why isn’t this distance even?
The distance doesn’t exist, instead of distance we have a point. Euclid’s definition to a point is: “A point is that which has no part.” In Euclid words the “distance” of zero from itself, |0 – 0| = 0 has no part. Something like this isn’t even or uneven.
In an e-book called “The Way To Geometry” it says, that magnitude of a point is zero.
Philosophically speaking a point is what Democritus called “atomos” applied to geometry.
In short, there isn’t distance of zero from itself. The meaning of distance |0 – 0| = 0 is geometrically a point, something that has no part.
The main point is, that I see zero as a neutral number. What does that mean?
- Zero isn’t even or uneven
- Zero isn’t negative or positive
- Zero doesn’t have numerical opposite
The meaning of the video clip: “..achievements: Zero.”
Probably zero is my greatest achievement too… 🙂
Semantically zero is none, not nothing.
In multiplication zero overrides the neutral element property of one: 1 * a = a, but if a = 0, zero “zeroes” one. The number one doesn’t keep the identity of zero, but zero “zeroes” number 1’s neutral element property and keeps the identity of its own.
Let us examine the even and uneven property of zero from point of view of distances.
The distance of number 2 from zero is |0 – 2|= 2. 2 mod 2 = 0; 2 is even.
The distance of number 3 from zero is |0 – 3|= 3. 3 mod 2 = 1; 3 is uneven.
The distance of number 4 from zero is |0 – 4|= 4. 4 mod 2 = 0; 4 is even.
But what about our friend zero?
The distance of number zero from itself is |0 – 0| = 0. 0 mod 2 = 0, but now the meaning of zero is, that the distance doesn’t exist. There isn’t distance from zero to zero. The distance that doesn’t exist isn’t even or uneven. Therefore now zero can’t be considered even or uneven.
I came up last sleepless night with a fractal of my own. There are two versions of it. The first version has the following rules:
- Build a circle with radius of r
- Draw inside the previous circle a circle of radius of r / 2
- Draw circles below, above and sides of the previous circle, the radius of these smaller circles being r / 4
Choose for a new circle the circle that’s radius is r / 2. Repeat the previous steps ad infinitum.
This kind of fractal might be invented already, but I haven’t seen one ever before.
Version 2 of this fractal:
The previous has a bit more complicated rules.
I made two color version of the previous too:
The second one:
Perhaps this fractal of mine has at least some value of curiosity.
I try to find the time sometime in September to come back to this topic.