Parametric functions

Just out of interest, here are the standard functions we are used to, but converting the second differential of the function to curvature instead. Or equivalently, converting gradient to path angle:

x

x^2

x^3

x^4

tan x

1/x

exp x

log x

x^x

normal dist

8.5 * normal dist

But one of the more interesting is probably the sine function, which looks quite different just by scaling up:

sin x

pi/2 sin x

2.11 sin x

2.405 sin x

2.66 sin x

pi sin x

4.7 sin x

5.24 sin x

The coefficient in the figure eight sine wave is more precisely the first zero of the Bessel function J_0. Credit to Greg Egan for pointing that out.

You can also render the factorial function as a continuous curve. However to prevent lots of overlap it helps to divide the angle by four:

In addition to standard asymptotes, this depiction of functions can also in somes cases visualise functions with a domain larger than the Reals. 

For instance, if we take a standard cubic function:

and make it periodic every omega on the Surreal domain by setting:

where r_0 is the cofficient of omega^0 in the Conway normal form of x:

then we can visualise it as:

The green lines are values outside of the Reals.

This example is rather artificial as it explicitly creates an omega-periodic function. It is not clear whether any "standard" function on the Reals that is non-periodic can become periodic on the Surreals. It definitely isn't the case for polynomial functions. 

3DOF Wheels

In this post I described a type of void-sponge based on the 6 regular polychora (5-cell, 8-cell, 16-cell, 24-cell, 120-cell and 600-cell). A nice property of these is their symmetry to 4D rotations projected stereographically into 3D, in other words transformations that rotate around a circular ring. 

If we apply this rotation with translation you get a nice 'swimming' motion:

You could imagine that if there was such a flexible object then it could traverse through water because the outer region pushing downwards is larger than the inner region pushing upwards. 

And due to its symmetry it can move in any direction in 3D. The principle axes require the least expansion and contraction, and are shown here with it moving in each axis direction in turn:

You can think of it as a 3 degree-of-freedom version of a wheel. It can move a payload in 3DOFs where a ball-robot can move it in 2D, a wheel moves a payload in 1D and something like a table leg moves a payload in 0D i.e. nowhere, it just supports the payload.

But there are 6 DOFs of Mobius transformations, so the structure can do more than just translations. If we offset the circular rotation then it should turn as it pushes through the water, since the outer edge is pushing down faster on the right side than the left in this animation:

Even though there are 6 DOFs in Mobius transformations, the three rotation degrees of freedom are rigid so uncontrollable by the structure itself. The remaining degrees of freedom allow it to get around, underwater in the above case, but also on land.

For example, something like this transformation could allow the structure to move forwards, by pushing more of its mass forward:

Notice that this is different from a rigid roll, notice the change in size of the large circular edge.

It could also transform into more of a 1DOF wheel shape:

and there is still freedom to distort the wheel while keeping its rim circular, in order to push in a particular direction by offsetting the centre of mass:

We can do all of these things with the other regular polychora-based void-sponges. Here is the swimming motion for the 24-cell sponge:

The largest holes are the best locations for new directions as they minimise the expansion and contraction required. The 24-cell has more symmetry than the 8-cell, so change its movement with a choice of six different directions naturally, compared to four for the 8-cell (translate left, right, fwd, bwd).

Incidentally, the 'rotation around a circular ring axis' transformation is called an elliptic Mobius transformation. But since this is somewhat misleading terminology I prefer the term poloidal rotation, which couples nicely with its counterpart toroidal rotation (along the ring): https://en.wikipedia.org/wiki/Toroidal_and_poloidal_coordinates.

You could also call it vortex rotation or vortex circulation, but that is more suggestive of a vector field.

There is another form of locomotion that seems better suited to burrowing underground as the poloidal motion above would require displacing a lot of earth. The motion is a hyperbolic Mobius transformation:

Unlike the elliptic transformation, this contains unbounded contraction (at the top) and expansion at the bottom. It would be harder to generate biophysically, but not impossible, it would need the physical cells to redistribute as it moves. Or in other words, the cells would need to coordinate the holes to move locally upwards without moving the cells upwards.

As with the swimming motion, this method can burrow along any of the three axes, and turn its heading. It is also possible with the other void-sponges too, such as the 8-cell version.

There is one Mobius transformation not mentioned yet, the parabolic transformation. This is equivalent to addition on the Riemann sphere, it leaves one point fixed on the structure's surface. This could be useful for turning on the spot when burrowing.  

basic inversive tree-shell

The tree-shell is probably the hardest structure to make with inversive geometry because it requires the structure to 'conspire' to meet along a line or curve. Other structures such as cluster-trees and tree-sponges only require single points to meet.

Ideally I'd like to find one that is nowhere differentiable, in particular that every surface patch contains approximations of the whole structure under conformal transformations. 

However due to the difficulty in this post I'll make one that is mainly smooth spherical surfaces. Nevertheless, it makes for a nice looking shape.

It is based on the 2D tree-solid from a previous post, but extended to 3D, and looks like this:

The 8 legs is arbitrary, as long as it is even. You can go higher like 10 here, but start to get a little self-intersection:

The method is a replacement fractal under Mobius transformations. There are three types of structure being used. Each type is constructed from a combination of the  types as shown:

 

Type 0:

Type 1:

Type 2:

It is a tree-shell because it would normally be a tree of recursive hemispheres, but the hemispheres meet along edges as seen between the two types on the right side of the above pictures. This forms a water-tight basin, at multiple scales, which makes it a shell.

Apart from the making a nowhere differentiable tree-shell (which would have a very different construction) I think there is probably a way to improve the above structure...

Notice the type 2 shape in the bottom image, the rightmost limb is significantly bigger than the limb next to it. That's because we use only three types, one meets the parent sphere at an angle of zero (the big one) and the others meet at an angle of 45 degrees.

I think it may be possible to interpolate between the small limb (left) and the big limb (right) so the meeting angle gradually drops to zero. 

Shader: https://www.shadertoy.com/view/tcycWD

Extra anti-twisters

The anti-twister is an interesting mechanism that gives a physical interpretation for a physical state that returns to the same state after the inner part turns 720 degrees.

This has connections with the quaternion sandwich product as the qvq^-1 is similar to the RTR^-1 transformation of the anti-twister.

One thing missing in the analogy is that the quaternion sandwich product can be applied in two ways. The quaternion is an isoclinic rotation, so it rotates by equal magnitude angles on two orthogonal planes in 4D. To create the double-cover of the 3D rotation one of these rotations is cancelled out in the sandwich product. The alternative sandwich product (using an alternative multiplicatino table, or possible using the complement of q) has the cancelled out rotation having the opposite sign.

The anti-twister's twist matrix T rotates around the y axis by an angle y in radius that depends on the radius x. The usual function is a smooth step from y=pi at x=0 down to y=0 at say x=2.

We can create two different anti-twisters if we make the object being rotated 720 degrees not the centre, but a unit sphere. Both the centre and the distance space is unrotated. The two types depend on which direction we rotate the inner space relative to the outer space. 

If we rotate in the same direction we get this anti-twister:

which corresponds to this y-axis rotation profile with respect to radius: 

If we rotate in the opposite direction we get this anti-twister:

which corresponds to this rotation profile:

Quantum physics folk wisdom

There is so much 'maths magic' attributed to quantum physics that I thought I'd give my own take on myth vs reality. The takeaway for me is that the biggest hurdle is the Many Worlds Interpretation, so let's start with that, then work through the others. I might add more over time, there's so many.

Myth: The universe splitting into billions every nanosecond is ridiculous

Reality: This is the wrong picture. In the Many Worlds Interpretation there is a huge universal wave function that represents the entire multiverse. This wave function acts perfectly deterministically according to the known laws of Quantum Field Theory, such as the Dirac equation.

"worlds" (meaning independent child universes) are the apparent effect when one part of the wave function becomes decoherent from the rest. Imagine for example an ocean wave (which can carry information in its undulations) hitting and defraction around a rock to cause two orthogonal waves diagonally inwards on the far side of the rock. These two 'child universes' are now independent and act as superpositions, each carrying and evolving that same information differently over time.

In this scenario the ocean does not get bigger at any point, we just have a splitting of information from one to two now-independent wave fronts. 

Myth: Quantum physics is stochastic

Reality: each scientist in each child "world" sees their observed particle at a different definite location. They are each seeing a small section of a wave function and a section of a wave function looks like a wave packet: a particle. This gives the appearance of stochasticity because all the other options have split off and are inaccessible to each version of the scientist. 

So in the Many Worlds Interpretation stochasticity is phenomenological, not fundamental. 

Myth: An entire multiverse to explain just the wave function collapse is wildly excessive

Reality: we have no constraint on the size of unobservable regions of reality, nature could be much bigger or much smaller, but we do have an unexplained mechanism in wave function collapse, and an unexplained gap between quantum and classical scales. Many Worlds reduces the number of unexplained things, and it is the number of unexplained things that we need to be conservative with, not the size of nature. 

Myth: Quantum physics is about discrete things

Reality: it is often about bounded fields and bounded fields produce standing waves. If you pluck a guitar string it only has a few discrete vibration patterns called harmonics, you can create the first couple by softly placing your finger on the string halfway or a third of the way down the string before plucking. 

This is the only source of discreteness in quantum physics. All the equations are continuous equations, as are the equations of guitar strings or waves in a pond.

Myth: Space and time must be discrete due to the Planck scale

Reality: Nothing in the Standard Model of particle physics claims or requires this. We do expect things to be different at the Planck scale, but it doesn't prevent waves from being smaller wavelength, it is only a problem to *sense* those details due to the energy required causing your sensor to turn into a black hole!

Anyone who claims quantum physics has a minimum scale needs to remember the fundamental role of Lorentz invariance in both General Relativity and Quantum Field Theory: any length is an arbitrarily smaller length to some other observer going at a different speed, due to length contraction. 

Myth: Quantum physics is inherently weird because it uses imaginary numbers

Reality: Imaginary and Complex numbers represent rotations in a plane. They can equally be used in classical physics. A wave on an ocean can be described by its height and vertical velocity as a Complex number wave equation.  

Myth: Quantum physics is weird because you square a complex number to get a probability

Reality: While there isn't a single killer explanation for the Born rule in any interpretation of quantum physics, there are several proposed reasons within Many Worlds, they just aren't bullet proof. Whatever the final proof, it seems very sensible that the number (or weight) of child "worlds" would be in proportion to the wave intensity or energy density of the wave at each point. This energy integrates the "effort" in reaching a particular height or velocity linearly from 0, which is half the square norm of the complex number. The half is not observable so can be ignored. The weight for each point on the wave is the size of the multiverse state space following the sensor interaction. And we divide by the total weight (state space size) because the total size is also not observable, giving the Born rule.

Each "world" sees a different measurement outcome, and repeated experiments appear to be give stochastic results due to the inability to predict or control which branch of the branching worlds will be aware in.

Myth: Quantum physics uses spinors, which are square roots of vectors, a mysterious space

Reality: spinors are not the square root of vectors, they are closer to the 'square root' of sections of vector bundles. So they represent transformations in a connected space. Spinors are used to represent classical light polarisation equations too, the 'spin up' and 'spin down' are the orthogonal transverse waves and superpositions describe diagonal and circularly polarised waves.

Myth: particles have to turn twice to return to the same orientation, it is mysterious

Reality: this spin 1/2 property (represented in spinors) can be seen in real life whenever the object remains connected, such as the belt and plate trick, and most clearly seen on anti-twister videos.

so spin 1/2 most likely indicates that the particle is connected to the space around it. 

Myth: Spinors are only ever observed as up or down no matter the angle

Reality: Such particles are a superposition of both in any continuous mixture. The 'up' and 'down' refer to the two components of the spinor, which represent two types of rotation in the above animation, or the two types of rotation on the Clifford torus (a stereographic projection of the spinor's 4 components)

Those two types of rotation are independent parts of the wave function, so can be physically split by a magnet leading to the binary results of the Stern–Gerlach experiment.  

Myth: None of the maths is real until you make an observation

Reality: If the probability of observation results from that maths then that is as real as any other mathematical law of nature. The wave function is a very real and accurate description of what is happening, and in the Many Worlds Interpretation it is real since there is no distinction between the quantum and the classical.

Myth: The uncertainty principle says that some states of particles cannot be known together

Reality: This is the wrong way to look at it. Particles are really waves and a fact about all waves (including waves on a beach) is that a sharp wave is built from almost all wavelengths (making its momentum poorly determined) and a long sine wave's momentum is well determined but doesn't have a well determined location. That is all the uncertainty principle is telling us. It is not that position and momentum are the fundamental quantities that we can't access, it is that they are a poor chocie of quantities to represent a wave. It is an inadequacy of our choice of summarising parameters, not mysteriously inaccessible knowledge.

Myth: The uncertainty principle is because you affect the object by measuring it

Reality: This is an old attempt at explaining the uncertainty principle, but is not the cause of it.

Myth: Quantum physics has action at a distance

Reality: Nothing communicates faster than the speed of light. In Many Worlds it should be clear that if two particles are 'entangled' (e.g. splitting a particle into a spin up and spin down version and moving one up to the moon) then the 'world splitting' when someone observes the particle on Earth will be interdependent with the world splitting when someone else observes the particle on the moon. It is only considered unusual behaviour if you are expecting the events to be independent. The Bell inequalities are a problem if your physics needs to have non-locality, realism and separability. Many Worlds has non-locality and realism, but not separability (independence). 

However the "world splitting" happens exactly it never propagates faster than light. And remember that "splitting" doesn't mean anything moving apart, just regions of the universal wave function becoming decoherent and therefore independent from each other. 

Myth: Quantum physics isn't just beyond what we know, it is beyond what we *can* know

Reality: There's no indication that this is true, or that the physicist who said it was qualified to make such a claim.

Summary

If we assume something similar to the Many Worlds Interpretation then we live in a deterministic and time reversible multiverse that obeys simple (linear) wave equations. It is classical wave physics in a bigger arena, and using some more modern maths tools. 

In fact it is the simplicity (linearity) of the physics that allows superpositions and therefore multiple "worlds" to exist over the same universal wave function. 

General Relativity is also a field equation but is nonlinear (since it acts on the arena itself), which may explain why it does not easily fit into the quantum picture. 

Spinors

I started to learn a bit about spinors, so this is my early take on them. Firstly 

Pauli spinors:

The equation for rotating a vector v in 3D is

for unit quaternion q. Noting that v is placed in the i,j,k positions with zero w.

The equivalent in spinor notation is on 2x2 complex-valued matrices:

where the dagger symbol is the conjugate transpose. 

U is an SU(2) matrix:

and V is a 'Pauli vector' which is derived from the vector v=(x,y,z):

 is a direct equivalent of because SU(2) matrices are isomorphic to unit quaternions. There is a direct translation and multiplication works the same.

 

Spinors fit in when we factor V into the outer product of two complex length-2 vectors, to give:

Each of those vectors is a two-component spinor, as it has two complex components, labelled with the xi letter for standard Pauli spinors.

 Now unlike with the quaternion formula, we can now throw away the right half of the formula and work on just the left hand side, where a single spinor is oriented using an SU(2) matrix. We only need to add the conjugate transpose right hand side when we want to turn the result into a pure 3D rotation.

So what exactly are U and the Pauli spinor in this single product?

The spinor can be thought of as a quaternion -- it maps spinor (a+bi, c+di) to quaternion (a,b,c,d) -- but only a quaternion as a special isoclinic rotation, not as a rotator, it is the 2x2 SU(2) matrix which acts like a quaternion rotator, i.e. to rotate other objects. 

The rotational axis of the spinor is just the ratio of the two complex components mapped onto the Riemann sphere (i.e. mapped stereographically).

I said 'special' isoclinic rotation as a quaternion (or an SU(2) matrix) does not rotate in an arbitrary pair of orthogonal planes, but in a pair such that one of the planes intersects the quaternion's w axis. For quaternions you are able to achieve an arbitrary 2-axis rotation of a 4D vector using:

The same is possible with the spinor notation. Allowing spinors to set the phases for any 2-axis 4D (or complex 2D) rotation.

So as with quaternions, there are three ways you can use these 2-component Pauli spinors:

1. for 3D rotations, using the formulae above.

2. for general 4D rotations (equivalently 2-axis complex 2D rotations)

3. natively to represent special isoclinic rotations in 4D (or complex 2D), as in: 

In this native 4D space we can picture it in 3D as a vortex, with the spinor components giving the phase offset in the two toroidal rotation planes. 

So 2-component spinors are useful whenever there are two orthogonal planes which are rotated equally (isoclinic). That includes light polarisation (phase space for up and right transverse directions is 4D isoclinic). This phenomenon is completely non-quantum but uses spinors to represent the light 2D phase.