3 Dimensional Logic?

@Reconstructo,

Reconstructo;155977 wrote:

These quaternions are something. If we forget for a moment their more practical use, and consider them apart from their potential relationship with real numbers....we find a crystalline system much like formal logic.

Multiplication is non-commutative. It matters which of the same two quaternions you use to "multiply" the other. But "multiplication" in this case is more like transformation.

The system is beautiful in itself and is in its way a 3 dimensional logic. I'm not saying it's useful, but it's a sculpture of pure thought.
Quaternion - Wikipedia, the free encyclopedia.

Imo it seems to overly complex things which should be simple.

@HexHammer,

HexHammer;159519 wrote:

Imo it seems to overly complex things which should be simple.

Well, it was inspired by the use of complex numbers, which made the number line 2 dimensional. So it was only natural that Hamilton would start to think of 3 and 4 dimensional options.

In the end, his system didn't prevail. But some software designers apparently use quaternions as a convenient way to simulate 3 dimensions.

The most interesting thing about quaternions is that, to be metaphorical, 2 times 3 does not equal 3 times 2. But this is because if you flip and object around in 3 dimensions, reversal is more complicated if you are only spinning around in 2. It should be noted that his system works. He struggled with it, but was struck with the solution while walking to church. There's a monument at the spot, and his notes are on an Irish postage stamp.

@Reconstructo,

Ah! Now it seeps it ^^

Well I do think it will have good use in the future, with more planetary sattelites ..etc.

@Reconstructo,

Reconstructo;155977 wrote:

Multiplication is non-commutative. It matters which of the same two quaternions you use to "multiply" the other. But "multiplication" in this case is more like transformation.

I have limited experience with quaternions in particular, but I understand it to be largely like vector mathematics, in which multiplication is multiplication in name only. You hit the nail on the head here with your description as a transformation, imo.

Do you have further experience with 3D or greater mathematics? Most of my later calculus study was all vectors, but this was a while ago. I have very little 4D+ math experience except code that I have written (long, long ago) in computer science courses.

@onetwopi,

onetwopi;162262 wrote:

I have limited experience with quaternions in particular, but I understand it to be largely like vector mathematics, in which multiplication is multiplication in name only. You hit the nail on the head here with your description as a transformation, imo.

Do you have further experience with 3D or greater mathematics? Most of my later calculus study was all vectors, but this was a while ago. I have very little 4D+ math experience except code that I have written (long, long ago) in computer science courses.

I have for fun, and not for profit, worked in 3 dimensions but not yet 4. I have tended to use arrays for this sort of thing, and I was especially immersed in programming in the days of yore, using pascal. Or if we go way back, Logo and Basic on the Apple IIe. Remember those? Lisp and APL seem quite fascinating at the moment, but I haven't learned them. APL seems amazing but a generally unpractical move. Have you seen it? Incredibly efficient, but made of a unique "hieroglyphic" syntax. I find games like Conway's Life to be quite fascinating. I wrote out an prog for it in 2 dimensions, and might do so in the 3, but with different rules, as the dimension factor radically increases the neighbor factor. I just got Wolfram's "a New Kind of Science." He also loves such things, and surely has more knowhow and resources than I ever will. As you may know, he is the creator of Mathematica.

As far as quaternions go, the "books" say that vectors were judged more handy than quaternions. I suppose quaternions did work enough just by opening the field of abstract algebra, or at least enlarging it. I'm no expert on the subject, though. Just a fan of it all. Especially the essence. I was driven into mathematics by Kant, Wittgenstein, Hegel, Leibniz, Spengler. Spengler writes a great chapter in Decline of the West on the cultural significance of the concept of number. Have you seen it?

page 41, if you are curious. The decline of the West - Google Books

@Reconstructo,

Reconstructo;162267 wrote:

I have for fun, and not for profit, worked in 3 dimensions but not yet 4. I have tended to use arrays for this sort of thing, and I was especially immersed in programming in the days of yore, using pascal. Or if we go way back, Logo and Basic on the Apple IIe. Remember those? Lisp and APL seem quite fascinating at the moment, but I haven't learned them. APL seems amazing but a generally unpractical move. Have you seen it? Incredibly efficient, but made of a unique "hieroglyphic" syntax. I find games like Conway's Life to be quite fascinating. I wrote out an prog for it in 2 dimensions, and might do so in the 3, but with different rules, as the dimension factor radically increases the neighbor factor. I just got Wolfram's "a New Kind of Science." He also loves such things, and surely has more knowhow and resources than I ever will. As you may know, he is the creator of Mathematica.

As far as quaternions go, the "books" say that vectors were judged more handy than quaternions. I suppose quaternions did work enough just by opening the field of abstract algebra, or at least enlarging it. I'm no expert on the subject, though. Just a fan of it all. Especially the essence. I was driven into mathematics by Kant, Wittgenstein, Hegel, Leibniz, Spengler. Spengler writes a great chapter in Decline of the West on the cultural significance of the concept of number. Have you seen it?

page 41, if you are curious. The decline of the West - Google Books

I did quite a bit of programming in BASIC, much more in C/C++ and Java in college. I did a bit of multidimensional work using arrays and dynamic links (forget the actual name of the data structure now). I had a prof in college that was a huge APL and could do massive array operations in single instructions. I worked with him some on some graphics type programs and music/computer interface programs.

I will need to grab Wolfram's new book and read the chapter you recommended. I will get back to you! Thanks for the awesome thread! (I'm such a nerd.)

@onetwopi,

onetwopi;162312 wrote:

I did quite a bit of programming in BASIC, much more in C/C++ and Java in college. I did a bit of multidimensional work using arrays and dynamic links (forget the actual name of the data structure now). I had a prof in college that was a huge APL and could do massive array operations in single instructions. I worked with him some on some graphics type programs and music/computer interface programs.

I will need to grab Wolfram's new book and read the chapter you recommended. I will get back to you! Thanks for the awesome thread! (I'm such a nerd.)

I love talking to my fellow nerds. I've looked at the syntax of C and I like it. It appeals to me with its minimalism. It was made by a man with an anarchist attitude. He didn't want to impose on programmers. I respect this. But then I also respect Wirth, the inventor of Pascal. Pascal is a beautiful little language. Of course I should really just say that structured programming is likable, for Pascal is just one example.

Of course I will always be fond of BASIC, because I can remember those first videos games conceived and created by yours truly. Not that they were especially sublime, but I built them from scratch. And that felt pretty good, especially at 11. My first was a sort of Choose Your Own Adventure game. Quite simple as far as programming goes. It's messing with those text graphics that made things exciting. A man can go far on if, goto, and arrays, yes? Of course goto is mentioned for its essence, not as an ideal way to loop...but rather as looping reduced to lowest terms.

I suspect you have more training than I do, as most of mine was high school. A little VB in tech school, but the class moved at a snail's pace. I was hired as a tutor, but it really wasn't much of an accomplishment. The Visual languages do so much work for you, of course, leaving only the logical bits. I guess this is nice, but it's sort of castrating. You are forced to play nice with Windows. If I had all the time in the world, I would learn machine language on some old chip. I would also like to have the skill to write a compiler and from there invent my own high level language. Ah, there's too much worth doing. I've got many crazy game ideas, as well, but one must choose choose choose.

I was reading Knuth about random algorithms. Apparently in some cases, randomness is the superior approach. Now that was entirely new to me. Erdos apparently is connected to this. Knuth would use randomness to grade papers, and also to explore the Bible, as he is a man of openminded faith. (You might be interested in his 3:16 project).

My latest programming fantasy was a 3-dimensional ecosystem in which simple animals move, feed, breed, and die. Maybe I would define them with an array of seven attributes, and with each reproduction there would be the small chance of mutation. One could make a little synthetic world and let them loose, the big and the small, the fast and slow. Run it for a million moves and see how the creatures have evolved. I feel that it's certainly doable. The time element would be tricky. You would probably need an array that included every single organism, and move them from 1 to n, one at a time. Seems to me that true "continuity" is impossible here. Of course I am reminded of the movie theater (I once worked as a projectionist) and of how convincing 24 frames per second is.

One last thing. I was reading CS Pierce and he mentioned the individual rods and cones of the eye, and how continuity is cooked up from this. It was a point I had never thought of. Even our human eyes are digital. Smile

@Reconstructo,

Reconstructo;162007 wrote:

Well, it was inspired by the use of complex numbers, which made the number line 2 dimensional. So it was only natural that Hamilton would start to think of 3 and 4 dimensional options.

In the end, his system didn't prevail. But some software designers apparently use quaternions as a convenient way to simulate 3 dimensions.

The most interesting thing about quaternions is that, to be metaphorical, 2 times 3 does not equal 3 times 2. But this is because if you flip and object around in 3 dimensions, reversal is more complicated if you are only spinning around in 2. It should be noted that his system works. He struggled with it, but was struck with the solution while walking to church. There's a monument at the spot, and his notes are on an Irish postage stamp.

Yes, quaternions are not commutative, AB is not equal to BA.

A. Cayley extended quaternions to octonions (8-D) and sedenions (16-D) numbers.

Octonions, are 8 dimentional hypercomplex numbers that are non-commutative and non-associative, A(BC) is not equal to (AB)C.

Sedenions, 16 dimentional hypercomplex numbers are non-commutative and non-associative and non-alternative and contains zero divisors, ie. (AA)B is not equal to A(AB) and AB=0 where neither A nor B is zero.

The further we extend Hamilton's quaternions the fewer 'field properties' remain.

Four dimentional 'vectorial' numbers are produced by the extension of complex numbers, ie. by constructing complex-complex numbers.

(a+bi+cj+dk) = (a+bi)+(c+di)j, where i^2=j^2=-1.
The multiplication table: ij=ji=k, ik=ki=-j, jk=kj=-i, k^2=+1, assures the standard 'field' postulates if we deny the multiplicative inverse of zero divisors.

These hypercomplex numbers, (complex-complex) numbers are: commutative, associative, alternative, distributive wrt addition, and contain zero divisors.
If we state that division is denied for zero and zero divisors, the these complex-complex numbers form a 'field', for any dimention (2^n) where n is >1.
Unlike the quaternion-like numbers these complex-complex numbers have elementary functions such as: sin(A), tan(A), log(A), e^A, etc..

@Owen phil,

i cannot see a point to this thread
.

@TuringEquivalent,

TuringEquivalent;165226 wrote:

i cannot see a point to this thread
.

An art exebition does not nessesarily have to have a reason to exibit items, I did not know of this concept, but now I do.

Imo it's very importaint to know of concepts, maybe once in ones life it may come in handy.

@HexHammer,

HexHammer;165242 wrote:

An art exebition does not nessesarily have to have a reason to exibit items, I did not know of this concept, but now I do.

Imo it's very importaint to know of concepts, maybe once in ones life it may come in handy.

What don ` t i know? This is a moronic topic. That poster engage in number mysticism, but actually know nothing of mathematics. O... look at the pretty numbers. Am i suppose to be ******* impress? I read gradate texts in mathematics, and that is surely more impressive than just being in stupid awe by looking at numbers.

@TuringEquivalent,

TuringEquivalent;165329 wrote:

What don ` t i know? This is a moronic topic. That poster engage in number mysticism, but actually know nothing of mathematics. O... look at the pretty numbers. Am i suppose to be ******* impress? I read gradate texts in mathematics, and that is surely more impressive than just being in stupid awe by looking at numbers.

Your attitude is not desireable, maybe in your haste of smudgeing other peoples creation, you forget to enjoy the beauty of simplicity.

@HexHammer,

HexHammer;165340 wrote:

Your attitude is not desireable, maybe in your haste of smudgeing other peoples creation, you forget to enjoy the beauty of simplicity.

Thanks Hex! So-and-so can read all the "graduate texts" (I doubt it!) he wants, and still miss the g.d. point.

When someone says they cannot see the point, they often pat themselves on the back for blindness. And call this wisdom.

The one thing I know we have in common is a suspicion of those who are always talking about their little idols rather than their big ideas.

@Reconstructo,

^ i guess i am a lier, and i worship idots?:whistling:

Now, you are being personal.:shifty:

@TuringEquivalent,

TuringEquivalent;165329 wrote:

What don ` t i know? This is a moronic topic. That poster engage in number mysticism, but actually know nothing of mathematics. O... look at the pretty numbers. Am i suppose to be ******* impress? I read gradate texts in mathematics, and that is surely more impressive than just being in stupid awe by looking at numbers.

Ah now, come on, friend.
1. I care nothing for number mysticism. I don't even believe in infinity.
2. But, yes, "look at all the pretty numbers!" I do love mathematics.
3. If you read graduate texts in mathematics, be grateful that you have such access. Why use it as a way to be hateful toward someone else who enjoys the subject?
4. I have been rude to you before. I'm sincerely sorry. I regret it entirely. I hope we can get along. And I think a friendly reply like this to an insulting post like the one above is proof of that.
5. Hey man, a little awe might be an appropriate response. Don't you find the clarity and precision of mathematics extraordinary?
6. I'm reading Frege's book on the Arithmetic. He too was interested in the roots.
7. I'm more interested in certain aspects of mathematics than in others. Do you find the number e impressive? I mean it was created as a limit. That's pretty special. And it was created largely in response to the problem of compound interest. e = lim n -> infinity (1 + 1/n)^n

That's beautiful! A number just an infinitesimal in magnitude above one multiplied by itself an infinite number of times. Of course this infinite is just a direction, a potential sort of infinite. We are never finished calculating e. Does this not at all fascinate you? I expect that you were already aware of most or all of this. But I wanted to present it my way, in order to show what I find inspiring about it.

I wish you well, and hope we can get along from now on. Sorry about our friction in the days of yore!:surrender:

---------- Post added 05-25-2010 at 12:15 AM ----------

HexHammer;165340 wrote:

Your attitude is not desireable, maybe in your haste of smudgeing other peoples creation, you forget to enjoy the beauty of simplicity.

Thank you, Hex. The "beauty of simplicity"! That's what I love about mathematics, or at least the simpler parts of it. Smile

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