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 ….

7 thoughts on “4D Metamorphosis puzzle

  1. Hi Ark,
    I had to think over the problem with the value of m. Here are my thoughts.

    In the case of pendulum when the energy is high, m < 1, the pendulum makes whole circles, movement is full, movement is elliptic.
    In the case of pendulum when the energy is low, m > 1, movement is not full, movement is hyperbolic.

    In the case of non-symetric top, either when energy is low or high, the range of angular momentum components is not full, so with both energies the true value of m is grater than 1 but it is converted to be less then 1 in the formulas.
    Therefore in the case of non-symetric top the value of m is not a gauge. The value of energy is the gauge.
    Angular momentum of non-symetric top changes hyperbolicly.

    1. Testing…

      Works again.
      I was playing with a plugin that would enable to see all comments, as Ronan requaested. But it seems to be incompatible with the edit time window.

  2. And what about my comment from 22:48 ?

    Was it 4 000 001 and not 400 001
    was it -200 200 and not -2000 2000
    was the step 0.01 and not 0.001 ?

    1. “was the step 0.01 and not 0.001 ?”

      Yes. Fixed. 400,000 takes about 10 minutes to generate. Now I want to make HD movie of the metamorphosis. But first I will redo the animation of yesterday using these points. That will be for today’s post. Also, in the animation of yesterday I was taking time from 0 to 2000. This produces asymmetry. Taking time from -2000 to 2000 produces a nice closed formation. Deserves a name….

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