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.

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!

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.

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.

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.

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.

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.

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.

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).

When one take the time derivative of the oscillator, . 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 . 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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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

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.

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

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.

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.

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.

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.

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.

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.