Saturday. It was warm and sunny. Went for a bicycle raid. Working on generating the exceptional geodesic for d=0.5.
But, until that is ready, I have made another geodesic line on the three-sphere, this time for d=0.4999999 and 180 degrees rotation in 4D. Click on the image to open animation
Update: I have produced the geodesic for d=1/2. Here is the result for t going from -2000 to 2000:
It is a great surprise for me. Is it really one closed line? Or, under microscope, it will split into many-many lines, very close, but with a subtle structure?
Update at 22:12
Progress. Here is one point on the trajectory (I created animation to see how it is moving)
And here is zoom on the line when this point was:
The line is not a line. These are many many lines, but very close to each other.
I can’t help it. I am obsessed with geodesics.
Quoting from Encyclopedia of Mathematics:
The notion of a geodesic line (also: geodesic) is a geometric concept which is a generalization of the concept of a straight line (or a segment of a straight line) in Euclidean geometry to spaces of a more general type. The definitions of geodesic lines in various spaces depend on the particular structure (metric, line element, linear connection) on which the geometry of the particular space is based. In the geometry of spaces in which the metric is considered to be specified in advance, geodesic lines are defined as locally shortest. In spaces with a connection a geodesic line is defined as a curve for which the tangent vector field is parallel along this curve. In Riemannian and Finsler geometries, where the line element is given in advance (in other words, a metric in the tangent space at each point of the considered manifold is given), while the lengths of lines are obtained by subsequent integration, geodesic lines are defined as extremals of the length functional.
Geodesic lines were first studied by J. Bernoulli and L. Euler, who attempted to find the shortest lines on regular surfaces in Euclidean space.
The play takes place at Mr. Jourdain’s house in Paris. Jourdain is a middle-aged “bourgeois” whose father grew rich as a cloth merchant. The foolish Jourdain now has one aim in life, which is to rise above this middle-class background and be accepted as an aristocrat. To this end, he orders splendid new clothes and is very happy when the tailor’s boy mockingly addresses him as “my Lord”. He applies himself to learning the gentlemanly arts of fencing, dancing, music and philosophy, despite his age; in doing so he continually manages to make a fool of himself, to the disgust of his hired teachers. His philosophy lesson becomes a basic lesson on language in which he is surprised and delighted to learn that he has been speaking prose all his life without knowing it.
We are speaking geodesics all the time without knowing it.
On the regular two-dimensional sphere geodesics are parts of great circles
A smart ant would travel from A to B along path c, from A to C along b, and from B to C along a. These are lines as straight as only possible, that are on the surface. Geodesics on the ideal sphere are all of finite length, the maximal ones are closed, they are great circles.
But if the sphere is not so ideal, if it is a three-axial ellipsoid, even if a little bit deformed from being ideally spherical, then everything changes. Geodesics do not close, they can behave in a strange way. Again, on a small scale, they are as straight as possible, but going always “straight” we can, after long enough time, find ourselves almost everywhere. This is a difficult subject – see Geodesics on an ellipsoid.
We are not on an ellipsoid, but we are on something similar in spirit: we are on three-dimensional sphere endowed with three-axial geometry. It is the three moments of inertia that determine geometry of our After all is the whole universe of the rigid body that is rotating about its center of mass. Three parameters determine rotation matrix. Unit quaternions which we use do the same. But not all rotations are equal. Some are for the body “harder” than other, some are “easier”. This depends whether the instantaneous rotation axis is close to the one with the greatest moments of inertia, or the smallest one. Therefore also “distance” between two rotations that somehow measures “effort” exerted by the body, will depend on the direction in
I know, we will have to learn all that. But, on the other hand, it is something that should be expected. For some reason that we do not fully understand, lot fundamental equations of dynamics can be derived from “least action principle”. But least action can be translated into “shortest way”- with the appropriate definition of the distance. Therefore, perhaps, all physics can be converted into geometry, and the Nature is simply choosing always the shortest path (or, sometimes, the longest).
So, here is again the same animation, but there are more details.
You will have to click on the image to open animation in a new window. There are 91 frames. I am rotating the observer every one degree from 0 to 90 degrees. Something rather strange happens around 84 degrees. That is the extracted frame above. The animation represents one geodesic line. I mean part of it. Time goes from -2000 to 2000 (of whatever units we use). It can go from minus infinity to infinity. I have no idea how the picture would change with the extension of the length of time. Probably it would be getting denser and denser. The phenomenon here may be similar to the one exhibited by the famous chaotic Lorentz attractor.
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!
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 onto the three-dimensional Euclidean space 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 using the function defined in Eq. (8) of Quaternions – If they can’t see you, they can’t eat you. I take I generate 400 001 unit quaternions taking from to with step I do stereographic projection, and obtain the following image:
Then I take the quaternion that describes rotation about -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 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 ….
You do not have to know that these are geodesic lines. In 
endowed with three-axial geometry. It is the three moments of inertia 
that determine geometry of our 
That was the subject of the last post: 
defined in Eq. (8) of 
I generate 400 001 unit quaternions taking 
from 
to 
with step 
I do stereographic projection, and obtain the following image:
that describes 
rotation about 
-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:
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.
