Animations

Number Theory

Real Numbers Are Dull

summary: 

Real numbers are dead dull in an animation, nailed to the origin, the only differences involving when they blink.

description: 

The real number -1 in blue is added to the real number 3 in yellow to generate the real number 2 in green.

The global economy is fueled by real numbers. This is somewhat frightening when you realize how excedingly dull real numbers are in an animation. They are paralyzed, able to exists in only one place, the origin 0, 0, 0. Their variation comes from blinking at different times from now. Minus real numbers are in the past, positives in the future, and only one point may exist at the origin at time zero or now. This set of blinking lights is totally ordered: a real number at the origin will either be before, after or at the same time as another point.

command: 
q_graph -dir vp -out real -loop 0 -box 4 -command 'echo -1 0 0 0' -color blue-command 'echo 3 0 0 0' -color yellow -command 'q_add -1 0 0 0 3 0 0 0' -color green
youtube embed: 
Outside video: 
math
equation: 

(q + q^*)/2 = (t, 0, 0, 0)

tags
Math Tag: 
real numbers
Programming Tag: 
command line quaternions
q_add

Complex Numbers Move Straight

summary: 

Complex numbers are constrained to move with their basis vectors, unable to explore all of spacetime.

description: 

Complex numbers have some freedom to move in space. The 3 straight lines indicate a choice of Cartesian basis vectors, but other basis vectors could have been chosen. Complex numbers are used extensively in quantum mechanics. Yet the obvious limitations in the animations suggest we should rebuild the foundations of complex-valued quantum mechanics. Sounds like a lot of work!

command: 
q_graph -out complex -dir vp -loop 0 -box 5 -command 'q_add_n -3 4 0 0 .002 -0.008 0 0 1000' -color yellow -command 'q_add_n 4 0 4 0 -.006 0 -0.008 0 1000' -color blue -command 'q_add_n -6 0 0 -6 .012 0 0 .012 1000' -color green
youtube embed: 
math
equation: 

(2 q + q^* + (i q i)^*) / 2 = (t, x, 0, 0)
(2 q + q^* + (j q j)^*) / 2 = (t, 0, y, 0)
(2 q + q^* + (k q k)^*) / 2 = (t, 0, 0, z)

tags
Math Tag: 
real numbers
complex numbers
Programming Tag: 
command line quaternions
q_add_n

Quaternion Addition = An Inertial Observer

summary: 

Quaternion addition results in constant linear motion, the only motion covered in special relativity.

description: 

Quaternion addition is the simplest of all math actions one can do. The animation is also easy to understand: a dot moves at a steady pace.

command: 
q_graph -dir vp -out addition -loop 0 -box 2 -command 'q_add_n -2 2 -2 .5 .004 -.003 .0035 -.001 1000'
youtube embed: 
Outside video: 
math
equation: 

q_a+n*q_b=q_{abn}

tags
Math Tag: 
addition
addition
addition
addition
addition
addition
addition
addition
addition
addition
addition
addition
addition
addition
addition
addition
addition
Programming Tag: 
command line quaternions
q_add_n

The Norm Requires a Clock

summary: 

The norm of spacetime is only about time.

description: 

The input is in yellow. The conjugate of the input is in blue, a 3D spatial reflection. The norm that results from multiplying yellow and blue is in green. Nailed to the origin, no ruler is needed, only a watch to record how long into the future is any spacetime norm.

command: 
q_graph -box 6 -loop 0 -dir vp -out norm -command 'q_add_n -2 -3 1 2 .004 .003 -.0005 -.003 1000' -color yellow -command 'q_add_n -2 -3 1 2 .004 .003 -.0005 -.003 1000 | q_conj' -color blue -command 'q_add_n -2 -3 1 2 .004 .003 -.0005 -.003 1000 | q_norm' -color green
youtube embed: 
Outside video: 
math
equation: 

norm(q) = q^* q

tags
Math Tag: 
norm
norm
norm
Programming Tag: 
command line quaternions
q_norm
q_add_n

Polynomials

2nd Order Polynomials

summary: 

A second order polynomial may have as many as two events appear in an animation at one moment in time.

description: 

The input in yellow goes directly through the origin. The polynomial in red never makes it to positive time. Near the origin, a second red dot appears to run at the other for an annihilation, which should become a familiar sight. Because the coefficients are reals, and the input goes through the origin, the polynomial is in exactly the same line as the input. There is nothing to "knock it" into another place in space.

command: 
q_graph -out poly_2_origin -dir poly -box 6 -command 'q_add_n -3 -3 -3 -3 .005 .005 .005 .005 1000 | q_poly -n 2 -c 1 -n 1 -c 3 -n 0 -c -2' -color red -command 'q_add_n -3 -3 -3 -3 .005 .005 .005 .005 1000' -color yellow
math
equation: 

As q goes from (-3, -3, -,3 -3) to (2, 2, 2, 2):
q^3 + 3 q - 2

tags
Math Tag: 
polynomials
Programming Tag: 
command line quaternions
q_add
q_poly

3rd Order Polynomials

summary: 

A third order polynomial can have three events appearing in an animation at once (this is not required, but represents a maximum).

description: 

The input which travels through the origin is shown in yellow. A cubic polynomial is in red. The polynomial continues its march down and to the left after a brief moment of indecision. The coefficients are all real, so the red and yellow are all in the same line.

command: 
q_graph -out poly_3_origin -dir poly -box 5 -command 'q_add_n -3 -3 -3 -3 .005 .005 .005 .005 1000 | q_poly -n 3 -c 1 -n 1 -c 3 -n 0 -c -2' -color red -command 'q_add_n -3 -3 -3 -3 .005 .005 .005 .005 1000' -color yellow
math
equation: 

Input: q ranges from (-3, -3, -,3 -,3) to (2, 2, 2, 2)
Output: q^3 + 3 q -2

tags
Math Tag: 
polynomials
Programming Tag: 
command line quaternions
q_add
q_poly

5th Order Polynomials

summary: 

A fifth order polynomial can change its directions as this one does three times.

description: 

The input is in yellow and goes through the origin. The polynomial in red has real coefficients. Perhaps a different polynomial could switch directions 5 times, although I have yet to construct one.

command: 
q_graph -out poly_5_origin -dir poly -box 5 -command 'q_add_n -3 -3 -3 -3 .005 .005 .005 .005 1000 | q_poly -n 5 -c 1 -n 1 -c 3 -n 0 -c -2' -color red -command 'q_add_n -3 -3 -3 -3 .005 .005 .005 .005 1000' -color yellow
math
equation: 

Input: (-3, -3, -3, -3) to (2, 2, 2, 2)
Output: q^5 + 3 q - 2

tags
Math Tag: 
polynomial
Programming Tag: 
command line quaternions
q_add
q_poly

2nd Order Polynomials With Quaternion Coefficients

summary: 

A polynomial can be shifted the input line with quaternion coefficients. This is an example of a subtle shift, but it would be trivial to make much more radical changes to the polynomial.

description: 

The yellow line is the input, the red line a 2nd order polynomial with real coefficients, and the green line is the same polynomial with quaternion coefficients. The coefficients are only subtly different as shown in the animation. The green line is never quite in line with the yellow/red line.

command: 
q_graph -out poly_2_q_coeffients -dir poly -box 6 -command 'q_add_n -3 -3 -3 -3 .005 .005 .005 .005 1000 | q_poly -n 2 -c "1 0.1 0 0" -n 1 -c "3 0 .2 0" -n 0 -c "-2 .01 .01 .01"' -color green -command 'q_add_n -3 -3 -3 -3 .005 .005 .005 .005 1000 | q_poly -n 2 -c 1 -n 1 -c 3 -n 0 -c -2' -color red -command 'q_add_n -3 -3 -3 -3 .005 .005 .005 .005 1000' -color yellow
math
equation: 

Input: (-3, -3, -3, -3) to (2, 2, 2, 2)
Ouput: (1, 0.1, 0, 0) q^2 + (3, 0, .2, 0) q - (-2, .01, .01, .01)

tags
Math Tag: 
polynomials
Programming Tag: 
command line quaternions
q_add
q_poly

3rd Order Polynomials with Quaternion Coefficients

summary: 

The green line takes a slightly bigger step away from its real coefficient counterpart. Polynomials can be curved in space due to their quaternion coefficients.

description: 

The input in yellow passes through the origin. The red input has real coefficients, and stays exactly in line with the yellow. The green line with its complex coefficients curves though space, but has the same large scale features as the red line - a reverse s shape.

command: 
q_graph -out poly_3_q_coeffients -dir poly -box 5 -command 'q_add_n -3 -3 -3 -3 .005 .005 .005 .005 1000 | q_poly -n 3 -c "1 0.1 0 0" -n 1 -c "3 0 .2 0" -n 0 -c "-2 .01 .01 .01"' -color green -command 'q_add_n -3 -3 -3 -3 .005 .005 .005 .005 1000 | q_poly -n 3 -c 1 -n 1 -c 3 -n 0 -c -2' -color red -command 'q_add_n -3 -3 -3 -3 .005 .005 .005 .005 1000' -color yellow
math
equation: 

Input: (-3, -3, -3, -3) to (2, 2, 2, 2)
Output in red: q^3 + 3 q - 2
Output in green: (1, .1, 0, 0) q^3 + (3, 0, .2, 0) q - (2, .1, .1, .1)

tags
Math Tag: 
polynomials
Programming Tag: 
command line quaternions
q_add
q_poly

3rd Order Polynomial Not Through the Origin

summary: 

When the input does not go straight through the origin, the behavior of the polynomial becomes hard to guess in both the animations and complex planes.

description: 

The yellow line is the input that goes through the origin for red, a cubic polynomial with real coefficients. The blue line is an input that does not go through the origin, making the curved orange line. That curve appears to line in a plane defined by the yellow and blue lines. The shape of the orange line is distinct from that of the red line. What goes in can dramatically change what comes out. The sharp bend in the t-x graph for the orange line was unexpected.

command: 
q_graph -out poly_3_off_diagonal -dir poly -box 5 -command 'q_add_n -3 -3 -3 -3 .006 .005 .004 .003 1000 | q_poly -n 3 -c 1 -n 1 -c 3 -n 0 -c -2' -color orange -command 'q_add_n -3 -3 -3 -3 .005 .005 .005 .005 1000 | q_poly -n 3 -c 1 -n 1 -c 3 -n 0 -c -2' -color red -command 'q_add_n -3 -3 -3 -3 .005 .005 .005 .005 1000' -color yellow
math
equation: 

Input for red: (-3, -3, -3, -3) to (2, 2, 2, 2)
Input for orange: (-3, -3, -3, -3) to (3, 2, 1, 0)
Output for red and orange: q^3 + 3 q - 2

tags
Math Tag: 
polynomials
Programming Tag: 
command line quaternions

Graphs

Graph theory details how things are connected. There are nodes that are connected by edges. The edges can either be directional or undirectional. Here is the graph for real number multiplication:

real_graph.png

One times one makes a loop bringing you back to one. This may be the most important loop in all of mathematical physics since it provides a mechanism for particles to keep on being themselves, just getting a little older. The same type of number (reals) can loop with itself.

complex_graph.png

Contrast this with the graph for complex numbers. The only edges out to the imaginary node and back are directional. 1 times i gets to i, but i must be multiplied by -i to get back to one, a different road. Graph theory provides a reason why real numbers are different from imaginary numbers (I wish this had been pointed out at the start). There is a loop, but involves the imaginary vertex working with the real edge.

q_graph_ham.gif

The graph for Hamilton's quaternions continue the theme of imaginary numbers. The undirectional edges are all loops using real numbers. At least on this site, the real number is time so this graph was animated to make the suggestion more concrete. The three imaginary nodes are what is needed for the three dimensions of space. Things out in space can be reflected in mirrors, which flip handedness, a signal for directional edges. Every node has the four types of edges, but only the real number has a loop which matches the color of the node.

q_graph_even.gif

If all the edges are undirectional, the graph applies to hypercomplex numbers, or what I call the California representation of quaternion multiplication.

When you do graph theory with clay and pipe cleaners, the simplicity of the California representation is clear: only one pipe cleaner goes between each node instead of two. Nature must use this sort of math tool often, which I think it does for inertia.

Random Graphs in Time and 3D Space

summary: 

Some 6000 nodes were generated, and if close enough in both time and space, were linked together. There is no pattern here, and none was expected.

description: 

A random graph in spacetime looks both random and transient. It might be interesting to know how many of these short paths have branches. For this initial sketch, I did not provide a means of tracking the more complicated graphs.

command: 
q_graph -dir networks -out random_3d_6000 -box 1.2 -command "generate_random_network -nodes 6000"
math
equation: 

Link nodes if dt < 0.05 and |dx|+|dy|+|dz| < 0.15

tags
Math Tag: 
graphs
Programming Tag: 
command line quaternions
generate_random_network

Random Graphs in Time and 2D Space

summary: 

Random graphs in a plane can be relevant for where to place cell phone towers, so this topic is studied commercially.

description: 

There are 6000 nodes used in this animation. How many survived by being close enough to another node is not known.

command: 
q_graph -dir networks -out random_3d_6000_y -box 1.2 -command generate_random_network -nodes 6000 -fix_y -1
math
equation: 

Link nodes if dt < 0.05 and |dx|+|dz| < 0.15, y = -1

tags
Math Tag: 
graphs
Programming Tag: 
command line quaternions
generate_random_network

Spacetime Reversal

Space Reversal Uses Mirrors

summary: 

Reversal in space is only about mirrors.

description: 

There are three directions in space, and three complex basis vectors. Flipping a space variable, or equivalently taking the conjugate of a complex basis vector, means using a mirror around the point (t, 0, 0, 0). Whatever time it is, a space reflection makes another point appear on the "other side". What was left handed now looks right handed. If you see pairs of points swing together around the origin, suspect the work of space reflection or conjugation.

command: 
q_graph -dir vp -out space_reversal -loop 0 -box 5 -command 'q_add_n -5 -5 -5 -5 0.005 0.005 0.005 0.005 1000' -color yellow -command 'q_add_n -5 5 5 5 0.005 -0.005 -0.005 -0.005 1000' -color blue
youtube embed: 
Outside video: 
math
equation: 

q\to q'=q^*

tags
Physics Tag: 
space reversal
Math Tag: 
conjugates
conjugates
Programming Tag: 
command line quaternions
q_add_n
q_conj

Time Reversal Need Memory

summary: 

Time reversal requires memories of where something was, so it can be undone.

description: 

The look of time reversal is familiar. It requires a linear sequence of ordered events to be played back in exactly the reverse order. Time represents the real numbers, a totally ordered set. This set property grants the ability to run things backwards.

command: 
q_graph -dir vp -out time_reversal -loop 0 -box 5 -command 'q_add_n -5 -5 -5 -5 0.005 0.005 0.005 0.005 1000' -color yellow -command 'q_add_n -5 -5 -5 -5 0.005 0.005 0.005 0.005 1000 | q_conj | q_x_scalar -1' -color green
youtube embed: 
Outside video: 
math
equation: 

q\to q'=-q^*

tags
Physics Tag: 
time reversal
Math Tag: 
conjugates
conjugates
Programming Tag: 
command line quaternions
q_add_n
q_conj
q-x_scalar

Space and Time Reversal Are Different

summary: 

Time and space reversal do not look the same which makes the difference between real and imaginary numbers concrete.

description: 

Time requires memory, space needs mirrors. These do not look the same animated. If animation starts off like it ends up, then time reversal is in play. While time reversal can involve a minimum of one event on a screen, space reflection requires matching pairs of events. In math, imaginary basis vectors are represented as a 90 degree rotation in the complex plane. There is no difference except in label between real and complex numbers. The complex plane misleads. Real numbers have a graph that is undirectional - one times one is one - so real numbers can live without the imaginaries. Imaginary graphs have a directional graph - 1 times i is i, while i times -i is 1 - and there is no way to do multiplication with only an imaginary basis.

command: 
q_graph -dir vp -out space_and_time_reversal -loop 0 -box 5 -command 'q_add_n -5 -5 -5 -5 0.005 0.005 0.005 0.005 1000' -color yellow -command 'q_add_n -5 -5 -5 -5 0.005 0.005 0.005 0.005 1000 | q_conj | q_x_scalar -1' -color green -command 'q_add_n -5 -5 -5 -5 0.005 0.005 0.005 0.005 1000 | q_conj' -color blue1
youtube embed: 
Outside video: 
math
equation: 

\begin{align*} q\to q' &= q^* \quad \textup{space reversal}\\ q\to q' &= -q^* \quad \textup{time reversal} \end{align*}

tags
Physics Tag: 
space reversal
time reversal
Math Tag: 
conjugates
conjugates
Programming Tag: 
command line quaternions
q_conj
q_x_scalar
q_add_n

Spacetime Reversal Goes Forward

summary: 

While space reflections require a mirror, and time reflection need memory, a reflection in both space and time can look the same as no reflection at all!

description: 

The input is in yellow, going from (-5 -5 -5 -5) to zero. The space reflection in blue, (-5, 5, 5, 5) to zero, is a mirror operation around the origin. The time reflection, (5, -5, -5 -5) to zero, in green requires you recall how the yellow input collection of events came onto the stage, so the green back it out. The reflection of both time and space, (5, 5, 5, 5) to zero, in red looks like a continuation of the yellow. Deep in quantum field theory they tell the odd tale of antiparticles going backward in time looking like particles going forward in time. Such an animated story now looks more reasonable.

command: 
q_graph -dir vp -out spacetime_reversal -loop 0 -box 5 -command 'q_add_n -5 -5 -5 -5 0.005 0.005 0.005 0.005 1000' -color yellow -command 'q_add_n -5 -5 -5 -5 0.005 0.005 0.005 0.005 1000 | q_conj' -color blue -command 'q_add_n -5 -5 -5 -5 0.005 0.005 0.005 0.005 1000 | q_conj | q_x_scalar -1' -color green -command 'q_add_n -5 -5 -5 -5 0.005 0.005 0.005 0.005 1000 | q_x_scalar -1' -color red1
youtube embed: 
Outside video: 
math
equation: 

\begin{align*} q\to q' &= q* \quad\textup{space reversal}\\ q\to q' &= -q* \quad\textup{time reversal}\\ q\to q' &= -q \quad \textup{spacetime reversal} \end{align*}

tags
Physics Tag: 
space reversal
time reversal
spacetime reversal
Math Tag: 
conjugates
conjugates
Programming Tag: 
command line quaternions
q_conj
q_x_scalar

Simple Harmonic Oscillators

Everything is a simple harmonic oscillator, so if we pay attention to the details, we may learn something from these animations.

A Classical Simple Harmonic Oscillator

summary: 

A spring with no losses due to friction will oscillate forever.

description: 

The motion happens to occur along one axis only. The animation has the look we expect of springs, stretching out when going fastest, compressing at the endpoints. Just what everyone expects to see. Try to predict what the velocity animation looks like (my guess was wrong).

command: 
q_graph -out sho-x -dir sho -box 10 -command 't_function -mqs 2 -mqs 1 -t_function polynomial -t_function one -x_function cos -x_function polynomial -y_function zero -y_function zero -z_function zero -z_function zero 1 2 1 2 1 3 0 0 0 1 0 0 -shift "-11 5 0 0" -pi 8 -n_steps 1600' -color yellow
math
equation: 

(t, x, y, z) = (t, cos (2 t + 3), 0, 0)

tags
Physics Tag: 
simple harmonic oscillator
Math Tag: 
cosine
Programming Tag: 
command line quaternions
t_function

The Velocity of a Classic SHO

summary: 

The scalar term is equal to 1 for every velocity because the problem is classical.

description: 

When one take the time derivative of the oscillator, \frac{d t}{d t} = 1. I puzzled over that obvious result for days, since its meaning was not clear. Then I recalled in relativistic physics, one takes the derivative with respect to the interval tau, so \frac{d t}{d \tau} = \gamma. For tiny values of velocity, gamma will equal 1. What people often graph is the velocity parameterized by time. They don't wish to treat the 4-velocity like a 4-velocity. If you keep the books consistent, you can get fun surprises.

command: 
q_graph -out sho-x_and_v -dir sho -box 10 -command 't_function -mqs 2 -mqs 1 -t_function polynomial -t_function one -x_function cos -x_function polynomial -y_function zero -y_function zero -z_function zero -z_function zero 1 2 1 2 1 3 0 0 0 1 0 0 -shift "-11 5 0 0" -pi 8 -n_steps 1600' -color yellow -command 't_function -mqs 2 -mqs 1 -t_function one -t_function one -x_function sin -x_function polynomial -y_function zero -y_function zero -z_function zero -z_function zero -shift "0 5 0 0" -pi 8 -n_steps 1600 -- 1 2 1 2 1 3 0 0 0 -2 0 0' -color blue
math
equation: 

(t, x, y, z) = (t, cos( 2 t + 3), 0, 0)
(\gamma, \gamma \beta_x, \gamma \beta_y, \gamma \beta_z) = (1, -2 sin( 2 t + 3), 0, 0)

tags
Physics Tag: 
simple harmonic oscillator
Math Tag: 
sine
cosine
Programming Tag: 
command line quaternions
t_function

Velocity and Acceleration of a Classical SHO

summary: 

The first term of the 4-acceleration is frozen at 0, an observations whose implications I do not understand.

description: 

The oscillator is in yellow, its first time derivative in blue, and second time derivative in red. I understand why the velocity has a fixed scalar value equal to 1, that is what gamma is for low velocities. Acceleration in special relativity must be handled with care. The math is easy: tack the derivative of a constant, and zero will result. The implications of that math are not clear to me.

command: 
q_graph -out sho-x_and_v_and_a -dir sho -box 10 -command 't_function -mqs 2 -mqs 1 -t_function polynomial -t_function one -x_function cos -x_function polynomial -y_function zero -y_function zero -z_function zero -z_function zero 1 2 1 2 1 3 0 0 0 1 0 0 -shift "-11 5 0 0" -pi 8 -n_steps 1600' -color yellow -command 't_function -mqs 2 -mqs 1 -t_function one -t_function one -x_function sin -x_function polynomial -y_function zero -y_function zero -z_function zero -z_function zero -shift "0 5 0 0" -pi 8 -n_steps 1600 -- 1 2 1 2 1 3 0 0 0 -2 0 0' -color blue -command 't_function -mqs 2 -mqs 1 -t_function zero -t_function zero -x_function cos -x_function polynomial -y_function zero -y_function zero -z_function zero -z_function zero -shift "0 5 0 0" -pi 8 -n_steps 1600 -- 1 2 1 2 1 3 0 0 0 -4 0 0' -color red
math
equation: 

(t, x, y, z) = (t, cos(2 t + 3), 0, 0)
(\gamma, \gamma \beta_x, \gamma \beta_y, \gamma \beta_z) = (1, -2 sin(2 t + 3, 0, 0)
(a_t, a_x, a_y, a_z) = (0, -4 cos(2 t + 3, 0, 0)

tags
Physics Tag: 
simple harmonic oscillator
Math Tag: 
sine
cosine
Programming Tag: 
command line quaternions
t_function

3 Frequency classical SHO

summary: 

The 3 complex planes in a quaternion can be tuned to different frequencies and amplitudes.

description: 

The simple harmonic oscillator is in yellow, its velocity in blue, and acceleration in red. The scalar value for the a low velocity system is equal to one while the scalar acceleration is zero. The differences between both frequencies and amplitudes change the relative lengths of the blue and red lines, velocity and acceleration respectively. In this example, the frequency decreases while the amplitude increase going from x to y to z.

command: 
q_graph -out sho-xyz_and_v_and_a -dir sho -box 10 -command 't_function -mqs 2 -mqs 1 -t_function polynomial -t_function one -x_function cos -x_function polynomial -y_function cos -y_function polynomial -z_function cos -z_function polynomial 1 2 1 2 1 3 2 1 0 1 6 8 -shift "-11 5 0 0" -pi 8 -n_steps 1600' -color yellow -command 't_function -mqs 2 -mqs 1 -t_function one -t_function one -x_function sin -x_function polynomial -y_function sin -y_function polynomial -z_function sin -z_function polynomial -shift "0 5 0 0" -pi 8 -n_steps 1600 -- 1 2 1 2 1 3 2 1 0 -2 -6 -4' -color blue -command 't_function -mqs 2 -mqs 1 -t_function zero -t_function zero -x_function cos -x_function polynomial -y_function cos -y_function polynomial -z_function cos -z_function polynomial -shift "0 5 0 0" -pi 8 -n_steps 1600 -- 1 2 1 2 1 3 2 1 0 -4 -6 -2' -color red
math
equation: 

(t, x, y, z) = (t, cos(2 t + 3), 6 cos(t + 2), 8 cos(t/2 + 1))
(\gamma, \gamma \beta_x, \gamma \beta_y, \gamma \beta_z) = (1, -2 sin(2 t + 3), -6 sin(t + 2), -4 sin(t/2 + 1))
(a_t, a_x, a_y, a_z) = (0, -4 cos(2 t + 3), -6 cos(t + 2), -2 cos(t/2 + 1))

tags
Physics Tag: 
simple harmonic oscillator
Math Tag: 
sine
cosine
Programming Tag: 
command line quaternions
t_function

Trig Functions

Sine and Cosine for a Fixed Point in Space is a Circle

summary: 

When a position in space is fixed, the sine and cosine functions circle that point, the angle depending on the exact values of x, y, and z.

description: 

Sines and cosines have to do with circles. By fixing x, y, and z, the circle stays fixed. What direction the line in space points to is arbitrary.

The line in yellow is the input for the The length of the line in space is the amplitude.

command: 
q_graph -loop 0 -box 25 -dir vp -out sin-cos_xyz_constant -command 'q_add_n -50 1 2 1 .05 0 0 0 2000' -color yellow -command 'q_add_n -50 1 2 1 .05 0 0 0 2000 | q_sin' -color red -command 'q_add_n -50 2 1 2 .01 0 0 0 10000 | q_cos | q_x_scalar 2' -color blue
math
equation: 

\begin{align*} \sin(t,\vec{R}) &= (\sin(t) \cosh(|R|), \cos(t) \sinh(|R|) \frac{\vec{R}}{|R|})\\ \cos(t,\vec{R}) &= (\cos(t) \cosh(|R|), \sin(t) \sinh(|R|) \frac{\vec{R}}{|R|}) \end{align*}

tags
Physics Tag: 
simple harmonic oscillator
Math Tag: 
trig functions
sine
cosine
Programming Tag: 
command line quaternions
q_add_n
q_sin
q_cos

Sine and Cosine from -5 to 5

summary: 

Quaternions ranging from (-5, -5, -5, -5) to (5, 5, 5, 5) were fed into a sine and cosine function, with these odd looking results.

description: 

The linear input is in yellow, the odd sine function is in red, the even cosine in blue. The cosine has only one apparent line diving into the origin because that line is a doublet. I thought I knew what cosines and sines looked like, but it feels like there is a frightening large amount of unknown diversity possible for these formerly familiar friends. The cause of the oddness maybe due to varying t, x, y, and z at the same time.

command: 
q_graph -loop 0 -box 10 -dir vp -out sin-cos -command 'q_add_n -10 -10 -10 -10 .01 .01 .01 .01 2000' -color yellow -command 'q_add_n -10 -10 -10 -10 .01 .01 .01 .01 2000 | q_sin' -color red -command 'q_add_n -10 -10 -10 -10 .01 .01 .01 .01 2000 | q_cos' -color blue
math
equation: 

\begin{align*} \sin(t,\vec{R}) &= (\sin(t) \cosh(|R|), \cos(t) \sinh(|R|) \frac{\vec{R}}{|R|})\\ \cos(t,\vec{R}) &= (\cos(t) \cosh(|R|), \sin(t) \sinh(|R|) \frac{\vec{R}}{|R|}) \end{align*}

tags
Physics Tag: 
simple harmonic oscillator
Math Tag: 
trig functions
sine
cosine
Programming Tag: 
command line quaternions
q_add_n
q_sin
q_cos

Sines and Cosines Over Long Periods of Time in Spacetime

summary: 

Over long periods of time with smaller changes in space, sine and cosine functions make spirals.

description: 

In classical physics, the amount of change in time measured in the same units as space vastly exceeds changes in space. In other words, relativistic velocities are low. In these animations, time changes by 100 while changes in space are only 20.

command: 
q_graph -loop 0 -box 10 -dir vp -out sin-cos-50t -command 'q_add_n -50 -10 -10 -10 .05 .01 .01 .01 2000' -color yellow -command 'q_add_n -50 -10 -10 -10 .005 .001 .001 .001 20000 | q_sin' -color red -command 'q_add_n -50 -10 -10 -10 .005 .001 .001 .001 20000 | q_cos' -color blue
math
equation: 

\begin{align*} \sin(t,\vec{R}) &= (\sin(t) \cosh(|R|), \cos(t) \sinh(|R|) \frac{\vec{R}}{|R|})\\ \cos(t,\vec{R}) &= (\cos(t) \cosh(|R|), \sin(t) \sinh(|R|) \frac{\vec{R}}{|R|}) \end{align*}

tags
Physics Tag: 
simple harmonic oscillator
Math Tag: 
trig functions
sine
cosine
Programming Tag: 
command line quaternions
q_add_n
q_sin
q_cos

Sines in Many Directions

summary: 

When the input never moves in space, the output of oscillating points is easier to understand.

description: 

Each of these sets of events starts out pointing in a different direction. Yet the x, y, and z values of the input is never altered. This is why you can spot the spatial origin, the point in the center of all the moving points. It would be simple enough to shift these arbitrary oscillators around precisely the origin to arbitrary oscillators around arbitrary points - just add in an arbitrary value as a last step.

Colored gum ball physics
Let's make animal balloons!
command: 
q_graph -loop 0 -box 12 -dir vp -out sines_xyz_constant -command 'q_add_n -50 2 2 1 .05 0 0 0 2000 | q_sin' -color red -command 'q_add_n -50 -2.5 0 .5 .05 0 0 0 2000 | q_sin' -color blue -command 'q_add_n -50 0 1.9 -2.3 .05 0 0 0 2000 | q_sin' -color green -command 'q_add_n -50 2 .8 2.2 .05 0 0 0 2000 | q_sin' -color orange -command 'q_add_n -50 1.5 -1 -1.8 .05 0 0 0 2000 | q_sin' -color black -command 'q_add_n -50 -1.5 -2.4 0 .05 0 0 0 2000 | q_sin' -color aqua -command 'q_add_n -50 -1.8 1.5 1.6 .05 0 0 0 2000 | q_sin' -color purple
math
equation: 

\sin(t,\vec{R}) = (\sin(t) \cosh(|R|), \cos(t) \sinh(|R|) \frac{\vec{R}}{|R|})

tags
Physics Tag: 
simple harmonic oscillator
Math Tag: 
trig functions
sine
Programming Tag: 
command line quaternions
q_add_n
q_sin

The Standard Model Symmetries

The Group U(1), Electromagnetism's Gauge Symmetry

summary: 

The group for the unit circle in a complex plane, U(1), can be at an arbitrary angle in 3D space. The transverse waves of EM have this symmetry.

description: 

If one picks a quaternion at random, normalize it, then take n powers of that number, one ends up with this animation. It looks like tilted circles in the complex planes. The motion is fasted at the creation and annihilation. The velocity of the dots is slowest when the two have their largest separation. This group is symmetric in both time and space reflection. Recall that time reflection requires recall, memories of the path taken, while space reflection involves mirrors.

<p>On pleasing aspect of this animation is that it starts to make sense of a transverse wave. Mapping that wave to electric or magnetic fields will require considerably more work.

command: 
q_graph -dir vp -out group_u1 -loop 0 -box 1.1 -command "g_u1 1 2 3 4 1000"
youtube embed: 
Outside video: 
math
equation: 

\frac{A}{|A|} \in U(1)

tags
Physics Tag: 
standard model
electromagnetism
gauge symmetry
U(1)
Math Tag: 
groups
U(1)
Programming Tag: 
command line quaternions
g_u1

The Group SU(2), Weak Force Symmetry

summary: 

The group of unitary quaternions is the symmetry underlying the weak force of radioactive decay. Who would have thought the symmetry looks like this?

description: 

If one takes the vector part of a quaternion and takes the exponential, the norm is always equal to 1. The animation starts out at 8 points, those for exp(0, +/-1, +/-1, +/-1). These points grow into each other until they form a sphere. That sphere then shrinks to the point (1, 0, 0, 0), the furthest into the future the exponential can reach.

command: 
q_graph -dir vp -out group_SU2 -loop 0 -box 1.1 -command 'q_random_n_11 50000 | q_vector | q_exp'
youtube embed: 
Outside video: 
math
equation: 

exp(A - A^*)\in SU(2)

tags
Physics Tag: 
standard model
gauge symmetry
weak force
Math Tag: 
groups
SU(2)
Programming Tag: 
command line quaternions
g_su2

Group U(1)xSU(2) for Electroweak Symmetry

summary: 

The group U(1)xSU(2) can be represented using all four parts of a quaternion, three in the exponential, the fourth as a normalized quaternion. The group covers the entire unit sphere, but has a bias for the past.

description: 

Quaternions do not commute in general, but they will commute if two quaternions point in the same direction. The common way to represent the group U(1) is with a normalized complex number. The same thing can be done with a quaternion. This will commute with a unitary quaternion if they both use the same quaternion pointing in the same direction. Electroweak symmetry uses all the degrees freedom available in a quaternion.

command: 
q_graph -dir vp -out group_u1xsu2 -loop 0 -box 1.1 -command 'q_group -group U1xSU2 -n 20000'
youtube embed: 
math
equation: 

\frac{A}{|A|} exp(A - A^*) \in U(1) \times SU(2)

tags
Physics Tag: 
standard model
electroweak force
gauge symmetry
electromagnetism
weak force
Math Tag: 
groups
U(1)
SU(2)
U(1)xSU(2)
Programming Tag: 
command line quaternions
q_group
q_random_n_11

The Group SU(3), Symmetry of the Strong Force

summary: 

The group SU(3) is created by taking the Euclidean product of two electroweak symmetries. Nature may need less tools than the standard model suggests.

description: 

This groups is the completely uniform unit quaternion sphere, starting from t=-1, expanding to its maximal size at t=0, then contracting to t=+1. For an observer is now at the center of their private Universe - (0, 0, 0, 0) - when they see an event, no matter what the cause, the event can be scaled to fit on this sphere. The norm of any event in the unit sphere is exactly 1, even if with rulers and atomic clocks a big or small sized measurement could be made.

<p>Because the symmetries U(1), SU(2) and U(1)xSU(2) are formally subgroups of SU(3), there is no need for a larger group to unify these groups. A rather large effort is still required to connect to all we know of the standard model.

command: 
q_graph -dir vp -out group_su3 -loop 0 -box 1.1 -command q_group -group U1xSU2xSU3 -n 50000
youtube embed: 
Outside video: 
math
equation: 

(\frac{A}{|A|} exp(A - A^*))^* \frac{B}{|B|} exp(B - B^*) \in SU(3)

tags
Physics Tag: 
standard model
gauge symmetry
Math Tag: 
groups
Programming Tag: 
command line quaternions
q_group
g_su3

The Lorentz Group: Circles and hyperbolas

summary: 

The Lorentz group has one subgroup of 3D rotations, and another for inertial reference frame boosts where time effectively rotates into a spatial dimension.

description: 

The spacial rotations are in yellow, while the boosts are in red. The hyperbolas can cover all of spactime while the rotations are limited to a circle about the origin. Out at infinity, there will be two points that approach the yellow circle. They split so that one set of red points can be there are the creation and annihilation of the yellow points, and another set can greet the yellow points when they are furthest apart, about to change directions.

command: 
q_graph -box 3 -dir trig -out circle -command 'g_u1_n 4 1 2 3 1000' -color yello -command 'g_u1h_n 4 1 2 3 4000' -color red [note: the function for generating the hyperbola has not been released, and its name will probably change]
math
equation: 

for the circle: \frac{q}{\sqrt{q q*}} = \frac{(t, \vec{R})}{t^2 + R^2}
for the hyperbola: \frac{q}{\sqrt{\pm scalar(q q)}} = \frac{(t, \vec{R})}{t^2 - R^2}

tags
Physics Tag: 
Lorentz transformations
special relativity
Math Tag: 
Lorentz group
Programming Tag: 
command line quaternions
g_u1

Right Triangles

Future Time-like Right Triangle

summary: 

Starting from an observer at the origin, the observer in watches the departure of another who drifts away until an ending bling (my jargon for many points appearing all at the same time).

description: 

There are right triangles in the tx and ty planes because there is motion along x and y, not z. The animation looks like the straightest of all possible lines, part of the fun of analytical animations. The green circle has a radius equal to the blue line. Because trigonometry is all about the relationship between circles and triangles, this may prove helpful. Notice how the origin is as far away as possible in space from the green lines. When the blue events touch the green, the blue events cease.

command: 
./q_graph -box 1.3 -dir triangles -out timelike_future -command 'g_u1_n 1 -.7 -.2 0 2000 | q_x_scalar 1.23693' -color green -command 'constant_linear_motion 100 0 0 0 0 1 0 0 0' -color red -command 'constant_linear_motion 100 0 0 0 0 1 -.7 -.2 0' -color blue -command 'constant_linear_motion 100 1 0 0 0 1 -.7 -.2 0' -color yellow
math
equation: 

Triangle:
(0, 0, 0, 0), (1, 0, 0, 0), and (1, -.7, -2, 0)
Circle:
Center: (0, 0, 0, 0)
Intersects: (1, -.7, -2, 0)

tags
Physics Tag: 
inertial observers
Math Tag: 
triangles
Programming Tag: 
command line quaternions

Past Time-like Right Triangles

summary: 

Bling is the first thing, the separation between the outside event and the observer, which collide and annihilate each other.

description: 

This is precisely the future time-like triangle run in reverse.

command: 
./q_graph -box 1.3 -dir triangles -out timelike_past -command 'g_u1_n -1 .7 .2 0 2000 | q_x_scalar 1.23693' -color green -command 'constant_linear_motion 100 0 0 0 0 -1 0 0 0' -color red -command 'constant_linear_motion 100 0 0 0 0 -1 .7 .2 0' -color blue -command 'constant_linear_motion 100 -1 0 0 0 -1 .7 .2 0' -color yellow
math
equation: 

Triangle formed between events: (0, 0, 0, 0), (-1, 0, 0, 0), and (-1, 0.7, 0, 0)

tags
Physics Tag: 
inertial observers
Math Tag: 
triangles
Programming Tag: 
command line quaternions

Space-like Right Triangle

summary: 

Beginning with a bling indicating the separation between the origin and the outside source, a non-physical line is drawn to the event.

description: 

Events separated by a spacelike interval cannot signal between each other. The triangle can be drawn, can be animated, but cannot be realized in the physical world. It is the observer at the origin who appears only for an instant.

command: 
./q_graph -box 1.3 -dir triangles -out spacelike_right -command 'g_u1_n 0.53 0.7 .2 0 2000 | q_x_scalar 1.23693' -color green -command 'constant_linear_motion 100 0 0.961522 0.274720 0 0.728009 0.961522 0.274720 0' -color purple -command 'constant_linear_motion 100 0 0 0 0 0.728009 0.961522 0.274720 0' -color blue -command 'constant_linear_motion 100 0 0 0 0 0 0.961522 0.274720 0' -color yellow
math
equation: 

Triangle between events: (0, 0, 0, 0), (0, 1, 0, 0), and (0.7, 0, 0, 0)

tags
Physics Tag: 
inertial observers
Math Tag: 
triangles
Programming Tag: 
command line quaternions

Static 3D Triangle

summary: 

A static 3D triangle is all bling - appears on the screen for just a moment. The spacial triangle does not look like one in any of the complex graphs because time is constant.

description: 

What looks like a triangle in 3D space no longer looks like a triangle in a complex plane. The triangle is fleeting.

command: 
./q_graph -box 1.2 -dir triangles -out 3d_triangle -command 'constant_linear_motion 100 0 0 0 0 0 1 .5 .3' -color red -command 'constant_linear_motion 100 0 0 0 0 0 -.7 .1 -.3' -color blue -command 'constant_linear_motion 100 0 1 .5 .3 0 -.7 .1 -.3' -color yellow -loop 0
math
equation: 

Triangle formed between:
(0, 0, 0, 0), ( 0, 1, .5, .3), and (0, -0.7, 0.1, -0.3)

tags
Physics Tag: 
inertial observers
Math Tag: 
triangles

4 Right Triangles

summary: 

4 right triangles in one animation, two time-like, two space-like.

description: 

It looks like a pinwheel, kind of. It is hard for me to process all this dynamic information. Four right triangles should be easy! Some simple issues: these are all along the same line in space.

With practice, it is possible to follow one triangle through the origin to the other side, where time is positive. Look for the points where the blue ball merges with the green circle (only blue reaches the circle). Notice that the red events never take a step away from the origin.

Spend some time looking at the shadows. At any giving time, there are only 2 triangles being animated.

command: 
./q_graph -box 1.3 -dir triangles -out t4 -command 'g_u1_n -1 .7 .2 0 2000 | q_x_scalar 1.23693' -color green -command 'constant_linear_motion 100 0 0 0 0 -1 0 0 0' -color red -command 'constant_linear_motion 100 0 0 0 0 -1 .7 .2 0' -color blue -command 'constant_linear_motion 100 -1 0 0 0 -1 .7 .2 0' -color yellow -command 'constant_linear_motion 100 0 0 0 0 1 0 0 0' -color red -command 'constant_linear_motion 100 0 0 0 0 1 -.7 -.2 0' -color blue -command 'constant_linear_motion 100 1 0 0 0 1 -.7 -.2 0' -color yellow -command 'constant_linear_motion 100 0 0.961522 0.274720 0 0.728009 0.961522 0.274720 0' -color purple -command 'constant_linear_motion 100 0 0 0 0 0.728009 0.961522 0.274720 0' -color blue -command 'constant_linear_motion 100 0 0 0 0 0 0.961522 0.274720 0' -color yellow -command 'constant_linear_motion 100 0 -0.961522 -0.274720 0 -0.728009 -0.961522 -0.274720 0' -color purple -command 'constant_linear_motion 100 0 0 0 0 -0.728009 -0.961522 -0.274720 0' -color blue -command 'constant_linear_motion 100 0 0 0 0 0 -0.961522 -0.274720 0' -color yellow
math
equation: 

4 triangles:
(0, 0, 0, 0), ( 1, 0, 0, 0), ( 1,-0.7,-0.2, 0)
(0, 0, 0, 0), (-1, 0, 0, 0), (-1, 0.7, 0.2, 0)
(0, 0, 0, 0), (0, 0.961, 0.274, 0), ( 0.728, 0.961, 0.274, 0)
(0, 0, 0, 0), (0,-0.961,-0.274, 0), (-0.728,-0.961,-0.274, 0)

tags
Physics Tag: 
inertial observers
Math Tag: 
triangles
Programming Tag: 
command line quaternions

Cubes and Lattices

4D Wire Cube

summary: 

A 4D wire cube spends most of its time going from -1 to +1 in time, saving the 3D box for start and end.

description: 

A 4D wire cube has 4^2 = 16 vertices. We are familiar with the 3 spacial vertices. The two for time are in the past (-1) and the future (+1) from time now (0). Most of the animation is spent going from the past set of 8 spatical vertices to the future set of 8 vertices. When going from one wire vertex to another, only one of the four values may change. A 4D solid vertex would appear at time t = -1, then disappear at +1, never to be seen again. Solid objects in spacetime are transient.

command: 
q_graph -out 4D_wire_cube -dir vp -loop 0 -box 1.1 -meta_command generate_cube
youtube embed: 
Outside video: 
math
equation: 

permutations of (\pm 1, \pm 1, \pm 1, \pm 1)

tags
Programming Tag: 
command line quaternions
generate_cube

Quantum Mechanics

The 2 images on the right hand side of these animations may be a way to visually represent quantum mechanics. The wave function is the superposition of all possible states a system governed by a complex-valued wave function can be in. The story is complete, even if uncertain due to the complex numbers which are not a totally ordered set. The upper right image is literally a superposition of every frame that appears in the animation that is front and center. Below all that is possible is a random sampling of those what is possible. All that can be is contrasted with what happens to be, a core issue in the foundations of quantum mechanics.

But why must this be so? We can write complex valued equations which are accurate representations of measurements we make of Nature, yet those equations blissfully ignore if one collection of events could cause another set of event to occur. The equation in fact describes clusters of events which are independent of each other due to a spacelike separation. Such an equation is valid math. Yet to make it apply to Nature, we need to take the norm of the expression. When we take this step, the resulting expression is not a description of one event that appears after another as happens in classical physics. Instead the norm is the probability that an event will be seen at a location in spacetime.

The Wave Function of a Wave Equation

summary: 

The wave function of a wave looks simple, but simple can be deceptive.

description: 

The wave function of a wave equation in spacetime does not move in a bumpy wave through space. It moves along military straight lines (unless you choose different coordinates, which is perfectly valid). The movement starts with pair creation, an agreed apon parting of ways. Movement is slowest at the maximal separation. There is a rush to collide and destroy. Because the animation going in looks like the one going out, there is time reflection. Because there are always two points dancing toward or away from each other, there is a reflection in space. Much of the mystery in interference experiments of quantum mechanics centers around this symmetric spacetime function.

command: 
q_graph -out amp -dir int10 -box 1.6 -command 't_function -t_function cos -x_function sin -y_function zero -z_function zero -n_steps 199 -pi 4 -n_t_cycles 300 -n_t_step 0 1 1 0 0' -color yellow
math
equation: 

\phi = (cos(\omega t), sin(\omega t), 0, 0)

tags
Physics Tag: 
simple harmonic oscillator
Math Tag: 
sine
cosine
Programming Tag: 
command line quaternions
t_function

The Wave Function of a Wave Equation Shifted

summary: 

A phase shift is added to a wave function, but you cannot tell due to the periodic boundary condition.

description: 

A change in the phase was included. Each event above another event has a different starting time due to the phase shift, but given enough time, all the same locations in spacetime are experienced, so the pattern looks the same.

command: 
q_graph -out amp_shifted -dir int14 -box 1.6 -command 't_function -t_func cos -x_func sin -y_func zero -z_func zero -n_steps 1000 -pi 10 -n_t_cycles 1000 -n_t_step 0.0314 1 1 0 0 | q_add 0 0 -1.5 0 | q_add_n_m 0 0 0.003 0 1000 1000' -color yellow
math
equation: 

\phi = (cos(\omega t + \delta), sin(\omega t + \delta), k \delta, 0) \, \textrm{with} \, \delta: 0 \to 10 \pi

tags
Physics Tag: 
quantum mechanics
wave function
wave equation
Programming Tag: 
command line quaternions
t_function
q_add
q_add_n_m

The Wave Function of a Wave Equation Shifted and Marked

summary: 

Each and every photon is identical to every other, but by coloring in those where t = 0, we can cheat and see the phase.

description: 

Part of the great mystery of quantum mechanics is that all particles are identical. There is no adding a tag or painting one red while the rest are yellow. As a programmer, we can cheat, do something not allowed in Nature, and mark all those where t = 0 in red. The shift is the same as before, but now we can spot its trail.

What is so tricky in quantum mechanics is not the vectors - those we can always point at with our fingers. Instead it is the scalars that provide the challenge, the unpointables. Each scalar is connected to three vectors to make 3 complex numbers, but the scalar can be shared by other events. The scalar become the thread within a pattern of events, and between separate patterns of events. It is wonderfully ironic that the simplest core component can be so confusing by playing many roles.

command: 
q_graph -out amp_shifted_marked -dir int14 -box 1.6 -command 't_function -t_func cos -x_func sin -y_func zero -z_func zero -n_steps 0 -pi 10 -n_t_cycles 1000 -n_t_step 0.0314 1 1 0 0 | q_add 0 0 -1.5 -.1 | q_add_n_m 0 0 0.003 0 1 1000' -color red -command 't_function -t_func cos -x_func sin -y_func zero -z_func zero -n_steps 1000 -pi 10 -n_t_cycles 1000 -n_t_step 0.0314 1 1 0 0 | q_add 0 0 -1.5 0 | q_add_n_m 0 0 0.003 0 1000 1000' -color yellow
math
equation: 

\phi = (cos(\omega t + \delta), sin(\omega t + \delta), k ~ \delta, 0) \, \textrm{with} \, \delta: 0 \to 10 \pi
\textrm{red} = (cos(\delta), sin(\delta), k ~ \delta, 0) \, \textrm{with} \, \delta: 0 \to 10 \pi

tags
Physics Tag: 
quantum mechanics
wave function
wave equation
phase
Math Tag: 
sine
cosine
Programming Tag: 
command line quaternions
t_function
q_add
q_add_n_m

The Wave Function Squared of a Wave Equation with Phase Shifts

summary: 

The product of one wave equation with another whose phase has been shifted results in an interference pattern.

description: 

The product of one wave function with another that has been shifted is animated. The order of this product is done both ways, resulting in the red and blue animations. In the 3 complex planes, we see 3 patterns. The circle in the t-y plane indicates that a wave equation with harmonic boundary conditions is on display. The straight line in the t-z plane shows nothing is happening except the march of time. Really the same happens in the t-x plane, but the change in phase has been included.

command: 
q_graph -out interference_ty_red_blue -dir int14 -box 1.6 -command '221 "t_function -t_function cos -x_function zero -y_function sin -z_function zero -n_step 1799 -pi 124 -n_t_cycles 1000 -n_t_step 0 1 0 1 0 | q_conj" "t_function -t_function cos -x_function zero -y_function sin -z_function zero -n_step 1799 -pi 124 -n_t_cycles 1000 -n_t_step 0.031 1 0 1 0" | q_x | q_add 0 -1.5 0 0 | q_add_n_m 0 0.003 0 0 1800 1000' -color red -command '221 "t_function -t_function cos -x_function zero -y_function sin -z_function zero -n_step 1799 -pi 124 -n_t_cycles 1000 -n_t_step 0.031 1 0 1 0 | q_conj" "t_function -t_function cos -x_function zero -y_function sin -z_function zero -n_step 1799 -pi 124 -n_t_cycles 1000 -n_t_step 0 1 0 1 0" | q_x | q_add 0 -1.5 0 0 | q_add_n_m 0 0.003 0 0 1800 1000' -color blue
math
equation: 

\phi'^* \phi = (cos(\omega t + \delta), 0, -sin(\omega t + \delta), 0)(cos(\omega t), 0, sin(\omega t), 0) + (0, k \delta, 0, 0) \, \textrm{with} \, \delta: 0 \to 10 \pi in red
\phi^* \phi'= (cos(\omega t), 0, -sin(\omega t), 0)(cos(\omega t + \delta), 0,  sin(\omega t + \delta), 0) + (0, k \delta, 0, 0) \, \textrm{with} \, \delta: 0 \to 10 \pi in blue

tags
Physics Tag: 
wave function
wave equation
interference
Math Tag: 
sine
cosine
Programming Tag: 
command line quaternions
t_function
q_add
q_add_n_m

Gamma Matrices

summary: 

The gamma matrices are a tool to systematically look through all possible paths through spacetime given 4 numbers.

description: 

The path in yellow is multiplied on both sides by all 16 combinations of the basis vectors on the left and the right. An Italian physicist named DeLeo figured out how to map the gamma matrices (also referred to as the Dirac matrices) to the triple product ("Quaternions and Special Relativity", J. Math. Phys., 37:6, 2955-2968, 1996). A team in Mexico, José López-Bonilla, L. Rosales-Roldán, and A. Zúñiga-Segundo detailed the process - and made me aware of the connection via email. The gamma matrix machinery can be hard to appreciate, there being all kinds of combinations of matrices and spinors that play roles. With quaternions the story is much more straightforward: multiply on the left and the right by (1, i, j, k).

Let's look at 1 triple product, i (t, x, y, z) k = (z, -y, -x, t). The algebra is simple, but the results are odd. This function swaps the value of time into the z position, and visa versa. The values of x and y trade places and signs. While you and I might like to consider values for time and space to be solid, in relativistic quantum field theory, Nature has a need to take what ever numbers are in the house and systematically shuffle them so that the sum of all possible paths can be calculated. This animation shows four points coming in form four directions, all paths possible with these 4 numbers. The 16 paths can be seen. Now the work done by the 16 Dirac matrices does not seem so utterly abstract.

command: 
q_graph -out gamma -dir gamma -box 2.5 -loop 0 -command 'q_add_n 2.9 3.1 3.2 2.8 -0.006 -0.0059 -0.0061 -0.0062 1000' -color yellow -command 'q_add_n 2.9 3.1 3.2 2.8 -0.006 -0.0059 -0.0061 -0.0062 1000| gamma -almost' -color blue
youtube embed: 
Outside video: 
math
equation: 

( 1 |  i |  j | k) (t,x,y,z) ( 1 |  i |  j | k)

tags
Physics Tag: 
quantum field theory
gamma matrices
Programming Tag: 
command line quaternions
q_add_n
gamma

Simple Lorentz Boosts

summary: 

Two inertial observers, looking at the same collection of events, will see significantly different animations depending on their velocies so long as the difference in speed is a significant fraction of the speed of light.

description: 

The events in yellow move at a nice steady rate. The line in blue represents a boost along the x axis only. In the tx complex plane, the blue line is compressed in time but overlaps the yellow line. The line in red has been boosted in both the x and y axis. There is no complex plane where the red line is colinear with the yellow because the change is distributed in two planes. The reason neither red nor blue is colinear in either ty or tz planes is that only time is changed by the boost. The red line has a highest boost, so appears on the screen for a smaller time.

command: 
q_graph -out boost -dir vp -loop 0 -box 4 -command 'q_add_n -5 -5 -5 -5 0.010 0.010 0.010 0.010 1000' -color yellow -command 'q_add_n -5 -5 -5 -5 0.010 0.010 0.010 0.010 1000 |q_boost -vx .5' -color blue -command 'q_add_n -5 -5 -5 -5 0.010 0.010 0.010 0.010 1000 | q_boost -vx .3 -vy .4' -color red
math
equation: 

(t, \vec{R}) \rightarrow (t', R') = (\frac{t}{\sqrt{1 - \beta^2}} - \frac{\vec{\beta} \cdot \vec{R}}{\sqrt{1 - \beta^2}},\vec{R} \times \frac{\vec{V}}{|V|} + \frac{1}{\sqrt{1 - \beta^2}}(\vec{R} - \vec{R} \times \frac{\vec{V}}{|V|} - \vec{\beta} t))

tags
Physics Tag: 
Lorentz boost
Math Tag: 
Lorentz group
Programming Tag: 
command line quaternions
q_add_n
q_boost