Science > Outer Space and "Inner Space" Sciences

Ark - where are you headed?

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Richard:

--- Quote ---A complex number can be written in the form a + bi where a and b are real numbers (including 0) and i is the imaginary number i. Therefore a complex number contains two 'parts'; one that is real, and another part that is imaginary.
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--- Quote ---Complex numbers are often represented on a complex number plane (which looks very similar to a cartesian plane). On this  plane, the imaginary part of the complex number is measured by the vertical axis (on the cartesian plane, this is the y axis) and the real number part of  goes on the horizontal axis (where the 'x' values of coordinates normally go).
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I think I prefer copper plates.

kenlee:

--- Quote from: ark on April 11, 2012, 12:52:03 PM ---Groupoid in algebra clothing
When building a house, we start with the construction of a scaffolding.  Eiffel Tower is an example of an almost pure scaffolding. Even if today it houses a dozen of restaurants, nevertheless it is largely open to the winds does not attract an average human being to stay there for long.
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This makes a lot of sense to me (at least from the point of view of how I am interpreting it) since I would think that there would have to be a scaffolding before anything else. But I would think that the scaffolding and/or the groupoid would have to exist as a fact. Maybe the groupoid and/or the scaffolding is the first existing 'fact' and as such I would think that it would have to have both it's abstract and concrete representations.  Don't know if it's concrete representation (assuming that there is one) could be measured or detected with "pointer readings" but if this is the case then perhaps it can "kick back" and be detected? Just some more thoughts on this fwiw!

Bluelamp:
Already at 2x2 complex matrices, barely removed from a single bit.  One can go to Hilbert spaces, Lie Algebra, and Clifford Algebra from here; that's like quantum physics, classical physics, and binary math.

http://en.wikipedia.org/wiki/Pauli_matrices

--- Quote ---The Pauli matrices are a set of three 2 × 2 complex matrices which are Hermitian... In the language of quantum mechanics, hermitian matrices are observables, so the Pauli matrices span the space of observables of the 2-dimensional complex Hilbert space. In the context of Pauli's work,  is the observable corresponding to spin along the coordinate axis in R3.

The Pauli matrices (after multiplication by i to make them anti-hermitian), also generate transformations in the sense of Lie algebras: the matrices form... the Lie group of rotations of 3-dimensional space. Moreover, the algebra generated by the three matrices is isomorphic to the 3-dimensional Euclidean real Clifford Algebra.

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dant:
Interesting, I was reviewing The Wave: CH 24

--- Quote ---...
We know without a shadow of a doubt that the megalith builders utilized advanced mathematics, geometry and astronomy. Numbers appear to be the language by which we can translate right-brain perception into useful left-brain action. This is why the mystical traditions are written in mathematical codes.

The original splitting of the unity into two is described mathematically as the cosmos contracting infinitely, leaving a void, and everything else. This contraction or split made Divine apperception possible. Using this divine principle of creation, Gottfried Leibniz developed binary arithmetic in the 1700s, and this is the basis of all of our computer communications today. Two figures, 0 and 1 can express everything in the cosmos.
...

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Richard:
Still with the binary, I was lying in the bath letting my mind wander and I thought up a mechanical binary adding machine (haven't tried a subtracting machine).

Excuse my diagrams....they're in 2D due my lack of skills. What we have are hollow columns down which we drop a wheel with a platform attached on top. To make it easier let's call it a bean.

(image of wheel platform) (how does one get images up here?)

So what we have in the next two pictures is a chute down which we can drop as many beans as we wish. What happens is that the first bean is captured by the hole underneath column 1. The next bean (2) rolls over the first bean, hits a button or trip wire or shorts a connection, causing the first bean to fall and is captured by the 2nd hole. The third bean falls into the first hole. We have dropped three beans and our tally of closed holes is three. We can send down as many beans as we want and this process will count correctly.

(first and second grid pics)

If we want to represent the number of beans by dropping them down the upright columns, the last picture shows we already have the number 10 (8 + 2) to which we want to add the number 6 (4 + 2). So we drop beans down the 4 and 2 columns. Bean c falls straight into the hole while bean d falls onto bean b, rolls to the left and over beans d and a causing them all to fall and is captured by hole 16. (10 + 6 = 16)

(last pic)

What I found interesting about this is how there is a meaningful mechanism to the binary system and we can start looking at the process from a conceptual basis. We can talk about "states" and "transitions".

In the last example, the first "state" is the state of "10". By adding the beans we see a transition take place until the state of "16" is reached. Each hole is also in a closed or open state and their state at any particular moment gives meaning to the whole system

We could also look at the holes and define an "energy" to each of them depending on their position in relation to each other. And this gives me the idea that the positions of orbits in an atom or maybe in a solar system have a particular "energy" depending on their relation to each other. I wouldn't know if they would be in a binary relationship or perhaps in a relationship built upon prime numbers.

An interesting side thought. If beans are added only from the right hand sloping column, each of the beans would be intimately involved with the transition, falling through the holes. If, however, we added a "64", the 'smaller' beans wouldn't know they had just become members of a larger group. Sort of like a larger business buying a smaller one without making any personnel or operational changes vs a merger.

I don't know if this will spark any ideas but it was fun for me.