This is just humorous thought of mine…

If we have a “collection” where isn’t anything, is it a collection? If someone has got 100 books, the person has a collection of 100 books. But if a person hasn’t got books at all, does the person have a collection of books? No.

So, is the empty set as such a set? A collection where is nothing isn’t a collection.

But is it an empty collection? But is an empty collection a *collection* at all? đź™‚

I’ve been reading e-book â€ťIntroduction to Mathematical Philosophyâ€ť originally written by Bertrand Russell and published in the year 1901.

Among other interesting thoughts Russell gives thought to the definition of a number. This is something very interesting; I’ve been thinking myself strange things about number zero. Can zero be considered as a whole number? It doesn’t describe anything existing as whole. If the number of something is zero, this something doesn’t exist at all in somewhere, particularly not as whole.

As to definition of number, Russell discusses about classes. From an old Finnish book that discusses university level algebra, I recently learned the definition of zero as a class. In Russell’s book zero is defined as a class in slightly different way: Russell doesn’t say anything about the empty set, instead he mentions â€ťnull-classâ€ť. I think I will read this part of the book over and over again.

This is something fascinating…

Hopefully you got interested in this great book:

In my old article I have implementation for Cantor’s set without recursion. Now I implemented the Cantor’s set with recursion in JavaScript. This is better solution.

The definition of the Cantor’s set in the language of set theory is the following:

If this doesn’t mean anything to you, you might want to check the old post. đź™‚

Below is a picture from the output of the program of this post:

Below is the JavaScript listing from Notepad++ in full as png-file:

Back to the world of fractals…

One of the famous fractals is the Koch curve, described by Helge von Koch in the early 1900s.

The basic idea of this fractal is as follows:

- Take an equliteral triangle, build another triangle in the middle of each side of the shape, the new triangle having a base length of 1/3 of the length of the side.
- RepeatÂ
*ad infinitum.*

Turtle graphics are very handy in the implementation. The needed functions or methods areÂ *Forward*: Draw a line in given amount of units forwards to the current drawing angle. The second isÂ *Turn*: Turns the drawing angle in given amount of degrees.

With recursion we are now ready to go to build a snowflake based on the Koch curve.

Below is two videos of my implemantation:

..and the same with more iterations:

In my high school times I mostly listened to Amiga music and C64 music rather than â€ťrealâ€ť music. I will now take a look back in time to some of the best Amiga modules I listened to.

*Spell Amelioration *from Uncle Tom was one of my favorites. Personal and technically good ProTracker music module.

AlsoÂ *PoseidonÂ *from Uncle Tom was one of my favorites.

Of courseÂ *Klisje paa klisjeÂ *was a real good tune to listen to:

Jogeir Liljedahl’sÂ *Face Another DayÂ *had big enlightening impact on me:

Many chip tunes on Amiga really touched me. Below is a compilation of some of the most famous chip tunes including many of my favorites:

Many people have been touched by good Amiga music and remixed their favorite tunes with synthesizers and real instruments.

Below is my own so called chip tune made with OctaMED on Amiga:

Some of the Amiga musicians have become professonal musicians.

If you’ve had C64 and/or Amiga childhood and/or youth this was probably good nostalgic trip to you too.

This is perhaps funny sounding concept I came up… In sacred geometry everything is defined through geometry; what is post apocalyptic sacred geometry?

With the post apocalyptic sacred geometry I refer to imaginary new world, where geometry defines the â€ťrulesâ€ť of this world; what is changed now?

In post apocalyptic world the geometry on which everything is based, makes it impossible to feel pain, get hurt or harm *anyone *or *anything.* Some kind of imaginary sacred paradise through geometry that doesn’t allow anything bad or evil; these things don’t exist anymore and every possibility to do *anything *is good; and from good â€ťspiritâ€ť of every deed arises more good; one good thing is eventually more than just one thing, perhaps fractal-alike thing…

*Image courtesy of Danilo Rizzuti at FreeDigitalPhotos.net*

In this imaginary world immortality is something considered self-evident; nothing can harm anyone in anyway, because the nature of post apocalyptic sacred geometry.

Through sacred fractals one thing is endlessly interesting, there absolutely doesn’t exist possibility to get bored, which on the other hand is impossible through new post apocalyptic sacred geometry.

Not even smoking can harm anyone’s health in anyway in this imaginary world of post apocalyptic sacred geometry. đź™‚

In multiplying one (1) is neutral element: *a ** 1 = *a. *For example, 7 * 1 = 7. Number one keeps the identity of a number, which includes a number being even or uneven. But what about zero (0)?

0 * 1 = 0. Does one keep the identity of zero or does zero keep the identity of its own? The property of this identity is â€ťzeroingâ€ť property: *a * *0 = 0, were *a *whatever real number, including one and on the other hand zero.

In case -2 * 0, zero takes the whole identity of number -2: The number being negative and even; as a result we get â€ťjustâ€ť zero. Similar happens in 2 * 0 = 0.

“Unique Sphere Shows Standing Out”

*Image courtesy of Stuart Miles at FreeDigitalPhotos.net*

My two cents: Zero â€ťzeroesâ€ť any number except itself. It â€ťzeroesâ€ť the whole identity â€“ including a number being even or uneven â€“ of any number except from itself; in case 0 * 0 = 0 zero keeps the identity of its own, it doesn’t â€ťzeroâ€ť itself, which reflects the identity of zero itself, how it is neutral in a deep sense and meaning.