Command Line Quaternions

Analytic Animation

Quaternions are numbers with 4 parts: one for time, three for space. This project hopes to create command line functions that generate thousands of points of quaternions, to be fed into animation software. Think: analytical animation! Here are a few examples...

Addition

In the graph below, the straight line represents the addition of the same quaternion to a starting quaternion.
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In physics, this motion would be called an inertial reference frame because the observer moves at a constant velocity in spacetime.

The three "flat" graphs on the left are complex planes, ploting time against x, y, and z. The quaternion animation is top center. On the right is a superposition of all possible states. Below the superpoisition of all states is a random sampling of those states. The graphs on the right are inspired by ideas that arise in quantum mechanics, namely the wavefunction as a representation of all possible states, and the act of measuring which leads to a collapse of the wave function.

Space reversal

Yellow is input, from txyz=(-5, -5, -5, -5) to (0, 0, 0, 0).
Blue is a spatial reversal, from txyz=(-5, 5, 5, 5) to (0, 0, 0, 0)
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Think mirror reflection.

Space and time reversal

Yellow is input, from txyz=(-5, -5, -5, -5) to (0, 0, 0, 0).
Blue is a spatial reversal, from txyz=(-5, 5, 5, 5) to (0, 0, 0, 0)
Green is a time reversal, from txyz=(5, -5, -5, -5) to (0, 0, 0, 0)
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Think mirror for space reflection.
Think memory for time reflection.

Sine and Cosine

Yellow is input, from txyz=(-5, -5, -5, -5) to (5, 5, 5, 5).
Red is sine.
Blue is cosine.
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Sine and cosine all point in exactly the same direction as the input stream. Cosine comes in to the input at a right angle, while sine parallels the input.

Multiplying two streams of quaternions

One input stream is in yellow.
One input stream is in blue.
The product is in green.
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Notice how curved the images appear on the complex summary graphs, but the superposition looks straight. In the animation, two green spheres appear at once, how a "curve on a complex plane" looks!

Quaternion Exponentials

The input stream is in red +,
the output as a red ~circle.
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The input can be positive or negative. The input does not move much in space at all. The output exponential is only for postive time. The duration of the animation is 20 seconds, so for the first 10 seconds, there is no red circle. Notice how the jumps in space get much bigger for later times. That is the exponential in action. If the 3-vector input is changed, that will change the direction out of the origin that the exponential will travel upon.

The Norm

Take a stream of quaternions in yellow
Take the conjugate, which is mirror reflection in blue.
The result is the norm which always sits on the origin in green.
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Spatial Rotations

Take a stream of quaternions in yellow
Multiply on the right by q = 0 1 3 1, and on the left by q's inverse.
The result is a rotation in blue.
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The line points in a new direction, but the points are traveling at a different speed. Since the rotation quaternion used had more in the y direction, the left and right values changed the most.

Lorentz Boost

Take a stream of quaternions in yellow
Boost along the x axis by 9/10th the speed of light in blue.
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In the y and z summary graphs in the left column, the blue line gets more vertical. For the boost along x (bottom center), the points become more concentrated, but remain on the line.

Group Theory

Groups are fun to look at! This one is known as S1, your basic circle.
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Lots of empty spacetime. What would fill up spacetime? A group known as U(1)xSU(2)xSU(3) appears to fill up spacetime smoothly, which is amazing, since it is the same group that underlies the standard model of particle physics.
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Right now, there is no underlying reason for the symmetries in the standard model. Filling up quaternion spacetime may be the cause.

4D cube

In 3D, there are 8 vertices, and one draws lines between them. In 4D, there are 16 vertices, and you do the same thing.
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The image looks the same as the 3D image for 2 brief moments, when time does not change. When time is the dimension that is changing, the spatial variables do not, hences the steady is space 8 points. Those points are NOT at a vertex, it is one of the inbetween connector lines.

Mini HOWTO

Download & install first.
Read, then do.

Strengths of the command line interface:

Strengths of quaternion animations:

Development style:

What works:

What does not work:

Links

The SourceForge.net page.

Dynamic graphs, an introduction to this area of study.