4D Metamorphosis puzzle

Metamorphosis in 3D is well known.

Metamorphosis is one of the most widely used life-history strategies of animals. The dramatic differences between larval and adult forms allow the stages to exploit different habitats and food sources, and also allow the extreme adaptation of one stage for a particular role, such as dispersal. In amphibians and many marine invertebrates, metamorphosis is an ancestral condition and its origins are buried deep in the evolution of these groups. In insects, however, the earliest forms showed direct development (were ametabolous) and the evolution of metamorphosis then fuelled their dramatic radiation …

James W. Truman & Lynn M. Riddiford
The origins of insect metamorphosis
Nature 401, 447-452 (30 September 1999) | doi:10.1038/46737

It is however well-known that life itself cannot be explained using 3D alone:

Perhaps life did not begin on Earth at all, but was brought here from elsewhere in space, a notion known as panspermia. For instance, rocks regularly get blasted off Mars by cosmic impacts, and a number of Martian meteorites have been found on Earth that some researchers have controversially suggested brought microbes over here, potentially making us all Martians originally. Other scientists have even suggested that life might have hitchhiked on comets from other star systems. However, even if this concept were true, the question of how life began on Earth would then only change to how life began elsewhere in space.

Oh, and if you thought all that was mysterious, consider this: Scientists admit they don’t even have a good definition of life!

7 Theories on the Origin of Life
By Charles Q. Choi, Live Science Contributor | March 24, 2016 06:46pm ET

When physics will take seriously the idea that space and time extend beyond 3+1 dimensions, we will see a tremendous jump in science and technology. I knew it long ago, that is why I have contributed to the international bestseller

Riemannian Geometry, Fibre Bundles, Kaluza-Klein Theories And All That (World Scientific Lecture Notes in Physics)
by Arkadiusz Jadczyk (Author), R Coquereaux (Author)

But, jokes aside, we do have metamorphosis in 4D, when we project stereographically three-dimensional sphere S^3 onto the three-dimensional Euclidean space \mathbf{R}^3. That was the subject of the last post: Quaternions – If they can’t see you, they can’t eat you.

Here are more details.
I am generating trajectory in S^3 using the function q(t) defined in Eq. (8) of Quaternions – If they can’t see you, they can’t eat you. I take I_1=1, I_2=2, I_3=3, d=0.34. I generate 400 001 unit quaternions taking t from t=-2000 to t=2000 with step 0.01. I do stereographic projection, and obtain the following image:

Then I take the quaternion
q=\cos \pi/4-\mathbf{j} \sin \pi/4 that describes -\pi/2 rotation about y-axis. I multiply all my 400 001 unit quaternions by this quaternion from the left. I obtain 400 001 new unit quaternions. I project them stereographically and plot:

That is supposed to be the same insect. The laboratory has been rotated by -90 degrees. The insect is the same. What we did was “group translation”, “left shift”. It should preserve the topological structure. The “shape” should be the “essentially the same”. And yet it does not seem so.
I am perplexed ….

Quaternions – If they can’t see you, they can’t eat you

I did not know quaternions can manage the art of camouflage – until today. Camouflage and metamorphosis, they come together. People are wearing masks, to disguise, to cheat, to the extent that they cheat themselves. Quaternions took me by surprise. As a kid I was told about metamorphosis.

I never witnessed it myself. Until, as I said, today, until I looked at these two pictures:

Quaternion path a’la van Zon and Schofield

Quaternion path a’la van Celledoni and Zanna (Kosenko)

In theory, if I am not making errors here, both images are supposed to represent the same story, except that it is being told by two different witnesses. It looks like a butterfly turning into caterpillar. How can it be? It is, after all, precise mathematics, not a psychology.
The first image represents the path in the group \mathrm{SU}(2) that we have discussed before in Introducing geodesics. Then, in Standing on the shoulders of giants – Reboot, examining the image obtained using my old code based on the paper of Van Zon and Schofield, I realized that:

We have a nice curly trajectory, but there are also strange straight line spikes. On animation they are, perhaps, harmless. But if something like this happens with the software controlling the flight of airplanes, space rockets or satellites – people may die. There is a BUG in the algorithm.

Bugs are no good. As I have written it elsewhere:

I believe that the Universe has Purpose, that it is much like a computer program of great complexity, and that “we” – the IGUS-es – have a role in its evolution. For a while our role can be described simply as “debugging units.” In short, my present answer to the question “why are we here?” reads: DEBUGGING THE UNIVERSE.

With the somewhat less ambitious task of removing the zigzag bug from the spinning quaternion history we decided to follow the giants and we have ended in the new algorithm based on the paper and the code by Celledoni and Zanna, the code described in The final answer for the Universe in which m<1. I will repeat this code here, with a small change that makes it more universal, as discussed in Attitude matrix for m<1.

The case m<1.
The solution \mathbf{L}(t) of the Euler’s equations has two trajectories. For one with L_3(t)>0 we take

(1)   \begin{eqnarray*} A_1&=&\sqrt{\frac{I_1 (d I_3-1)}{I_3-I_1}},\\ A_2&=&\sqrt{\frac{I_2 (d I_3-1)}{I_3-I_2}},\\ A_3&=&\sqrt{\frac{I_3 (1-d I_1)}{I_3-I_1}},\\ \end{eqnarray*}

while for the other one, with L_3(t)<0 we take

(2)   \begin{eqnarray*} A_1&=&-\sqrt{\frac{I_1 (d I_3-1)}{I_3-I_1}},\\ A_2&=&\sqrt{\frac{I_2 (d I_3-1)}{I_3-I_2}},\\ A_3&=&-\sqrt{\frac{I_3 (1-d I_1)}{I_3-I_1}}.\\ \end{eqnarray*}


(3)   \begin{eqnarray*} B&=&\sqrt{\frac{(1-d I_1) (I_3-I_2)}{I_1 I_2 I_3}},\\ m&=&\frac{(d I_3-1) (I_2-I_1)}{(1-d I_1) (I_3-I_2)}. \end{eqnarray*}

The solution \mathbf{L}(t) of the Euler’s equations is given by

(4)   \begin{eqnarray*} L_1(t)&=&A_1\, \cn (Bt,m),\\ L_2(t)&=&A_2\, \sn (Bt,m),\\ L_3(t)&=&A_3\, \dn (Bt,m), \end{eqnarray*}

With constants \alpha and \nu defined as

(5)   \begin{equation*} \alpha=\frac{I_3-I_1}{\sqrt{\frac{I_1(1-d I_1)(I_3-I_2)I_3}{I_2}}}, \end{equation*}

(6)   \begin{equation*} \nu=\frac{I_1-dI_1I_3}{I_3-dI_1I_3} \end{equation*}

we set the phase variable \psi(t) as

(7)   \begin{equation*} \psi(t)=\frac{t}I_1+\arctan\left((A_2/A_3)\mathrm{sd}(Bt,m)  \right)-\alpha \Pi(\nu,\am(Bt,m),m),\ \end{equation*}

where the Jacobi function \mathrm{sd} is defined as \mathrm{sd}(u,m)=\sn(u,m)/\dn(u,m).
Then the quaternionic attitude solution is given by q(t)=W(t)+\mathbf{i}X(t)+\mathbf{j}Y(t)+\mathbf{k}Z(t), with

(8)   \begin{eqnarray*} W(t)&=&\cos \frac{\psi(t)}{2}\sqrt{\frac{1+L_1(t)}{2}},\\ X(t)&=&\sin \frac{\psi(t)}{2}\sqrt{\frac{1+L_1(t)}{2}},\\ Y(t)&=&\frac{L_3(t)\cos \frac{\psi(t)}{2}+L_2(t)\sin \frac{\psi(t)}{2}}{\sqrt{2(1+L_1(t))}},\\ Z(t)&=&\frac{-L_2(t)\cos \frac{\psi(t)}{2}+L_3(t)\sin \frac{\psi(t)}{2}}{\sqrt{2(1+L_1(t))}} \end{eqnarray*}

Why is this better than the previous formulas that were producing BUGS? The secret is in the denominator!

In the formulas for Y(t) and Z(t) we have 1+L_1(t) in the denominator. That is 1+A_1\, \cn (Bt,m). The constant A_1
is given by

    \[A_1=\sqrt{\frac{I_1 (d I_3-1)}{I_3-I_1}}.\]

The smallest possible value of d is 1/I_3. then A_1=0.. The largest possible value of d, while m\leq 1, is 1/I_2, when m=1. Then, with I_1<I_2<I_3,


Therefore A_1<1, therefore 1+A_1\, \cn (Bt,m)>0 (Remember that \cn is a cosinus of the amplitude!). So, we never have a problem with the denominator! That is the advantage of the new algorithm! No infinities, all is finite, what a relieve!

So, we take our flagship example with I_1=1, I_2=2, I_3=3, we choose d=0.34, just a little bit over the minimal value of 0.33.., and we draw the stereographic projection of q(t) from S^3 to \mathbf{R}^3.
What we get is the second image.
No resemblance whatsoever to the first one.
And yet, they are supposed to represent the same reality! How can it be possible? They are like day and night …

So, I created the METAMORPHOSIS – a continuous group translation.


There is still something strange there…. And it needs to understood. Life is exciting.

Introducing geodesics

Why are geodesics important? Probably because they are very simple. They are, in a sense, the simplest possible paths. Quoting from Richard Buckminster Fuller Domes and archives, 1960, 1965

Fuller inspired by his observations of nature. The inventor applies the concept of the geodesic line (the shortest line joining two points on a surface) to construct the most balanced, lightweight and resistant structure possible. His domes are a synthesis of all of the inventor’s fundamental precepts, combining a reasoned and aesthetic use of technological progress with a holistic conception of man’s relationship to nature. Such was the reputation of the inventor in the scientific domain that a family of carbon-based molecules with a geodesic structure was named after him: Buckminster fullerenes, later changed to fullerenes. Many of these molecules have played a role in recent nanotechnology discoveries.

Quoting from Wikipedia – Geodesics on an ellipsoid :

Geodesics on an ellipsoid of revolution
There are several ways of defining geodesics (Hilbert & Cohn-Vossen 1952, pp. 220–221). A simple definition is as the shortest path between two points on a surface. However, it is frequently more useful to define them as paths with zero geodesic curvature—i.e., the analogue of straight lines on a curved surface. This definition encompasses geodesics traveling so far across the ellipsoid’s surface (somewhat more than half the circumference) that other distinct routes require less distance. Locally, these geodesics are still identical to the shortest distance between two points.

If the Earth is treated as a sphere, the geodesics are great circles (all of which are closed) and the problems reduce to ones in spherical trigonometry. However, Newton (1687) showed that the effect of the rotation of the Earth results in its resembling a slightly oblate ellipsoid and, in this case, the equator and the meridians are the only closed geodesics. Furthermore, the shortest path between two points on the equator does not necessarily run along the equator. Finally, if the ellipsoid is further perturbed to become a triaxial ellipsoid (with three distinct semi-axes), then only three geodesics are closed and one of these is unstable.
On a triaxial ellipsoid, there are only three simple closed geodesics, the three principal sections of the ellipsoid given by X = 0, Y = 0, and Z = 0.

I am not playing with geodesics on three axial ellipsoid. But studying unexpected flips of asymmetric spinning top in zero gravity brings us very close to this circle of ideas. We do have three axial ellipsoid, because we have three different moments of inertia I_1<I_2<I_3. And we do have geodesics, even if we have to wait a little bit to see them nicely introduced in this series of posts. Histories of rotations in the rotation group are geodesics. We have seen a bunch of them in two previous posts. These were closed. Nothing particularly interesting – who did not see a circle? True, we have seen a bunch of circles spanning a torus, but these were generated artificially by a rotating observer.

But no it is time to move beyond the safe mode. If you have a spinning top in free space, and if you struck it, like Peggy Whitson

it will start to nutate. And when all three moments of inertia are different, this nutation is non-periodic. Geodesics have infinite length. They are certainly not the shortest connections between points in the group, in any sense. One geodesic line is wandering through the three-dimensional sphere S^3 sometimes almost returning to the starting point, then traveling far away. Strange are these trajectories.

Below is a part of one such trajectory, for I_1=1,I_2=2,I_3=3 and d=0.4. I am rotating it in animation, so that you can have a better view of its 3D structure. Of course, as in previous posts, this is stereographic projection from S^3.

Geodesic on three sphere
Geodesic of the left-invariant metric in the group \mathrm{SU}(2)

Of course I will have to explain the details …