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You’re welcome.

The 600-cell has 120 vertices. The vertices are in the three-dimensional sphere. 60 of them on one hemisphere and the other 60 on the other hemisphere. When you rotate the three-sphere, the 600-cell that is inscribed inside the 3-sphere rotates with it as well. You can see that each hemisphere looks like a doughnut and the two hemisphere are linked together. Also, there’re 1200 faces in the 600-cell. For showing the inner faces we change the opacity of the faces randomly to let light rays get inside and reflect from them. What’s more, the edges that cross the equator of the 3-sphere have been removed so that you can see where one hemisphere ends and where the other begins. During the rotation of the three-sphere the equator rotates too and that’s why the interlinked doughnuts roll on each other.

@@iamazadi oh, I get it now. is there some interpolation as well? also, it's very confusing to say "3 dimensional sphere" to mean 3-sphere, as a 3 dimensional sphere is a 2-sphere.

@@ziggyzoggin the rotation of the three-sphere is done by first parameterizing points on a great circle of the 3-sphere using an angle, and then interpolating between an initial and a final angle. Multiplying the vertices of the 600-cell on either left or right by a parameterized point performs the rotation. One has to extend what we know about the geometry of the 2-sphere to develop tools for working with the 3-sphere, because spheres of different dimensions share geometric features.

One can prove the existence of connection one-forms using the same logic that Isaac Newton used to write his book: “The Mathematical Principles of Natural Philosophy”. So connection one-forms exist independent of us. However, in physics things have to come with the evidence of observation. The more the amount of evidence for something, the more likely that it exists. Connection-one forms appear in the Lagrangian of the standard model of particle physics. and very strong evidence supports its existence and role in the yang-Mills theory.

@@iamazadi I wasn't trying to debate a subject-object dichotomy of mathematical philosophy. We know mathematical conceptions change over time, and it's still up to debate as to whether or not anything exists independently of the subjective experience. However, I feel like you didn't answer the question. The point was what mechanism is used to generate this graph? Is the, "one-form," moving like an inch worm independently, or is it constructed as dependent upon the cartesian grid in which it is placed?

Good question! The connection 1-form depends on the metric that is defined on a principal bundle. You can check that both logically and numerically using Julia for example. As a definition, 1-forms take values from the tangent space at one point in the bundle and produce values in the Lie algebra of the fibers. But the metric also depends on the connection. So, we don’t know which one to start with in our computations, either way the two properties of a connection 1-form is the same and verifiable. Connection 1-forms are not unique and the way to verify whether an object is a connection 1-form or not would be to test two equalities. Let’s say that g is the fiber action in the bundle. First, the pullback of the connection by the right action of g equals the adjoint of the connection by g inverse. Second, vertical vectors in the tangent space are constant under the application of the connection. In other words, connection 1-forms eat horizontal vectors and produce zero. I’ve written a suite of unit test just to verify these two properties at the level of linear algebra. I’ve also unit tested the axioms of the linear algebra itself in a separate file. See github.com/iamazadi/Porta.jl/blob/master/test/hopfbundle_tests.jl and github.com/iamazadi/Porta.jl/blob/master/test/spacetime_tests/real4_tests.jl for example.

Thanks Mr. kamranifard

Thanks for the input! We made the plotter and hooked up the robot for real-time visualization: Visualizing the Acceleration Vector of Gravity ruclips.net/video/ib1llkPrtsM/видео.html

There should be a way to add the physical force that was exerted on the top of the robot to the control signal. Gathering data for a machine learning model is an objective.

It makes sense because smoothing the tilt angle readings slows the reaction time and consequently widens the dead band.

The Goal of Balancing is accomplished in this video: ruclips.net/video/sR8kfuUdOo8/видео.html

Thanks

Why does it need more struts?

Both

The Hopf Fibration Explained (A Math Explainer) ruclips.net/video/qZNTPCMM19s/видео.html

Fixed all of the broken links in the description. Now, you can visit the links again!

@@iamazadi All better now, thanks!

Try to find the books on the genesis library: libgen.is

Fixed all of the broken links in the description. Now, you can visit the links again!

Noted. Thanks for the feedback.

You’re welcome!

The Hopf Fibration Explained ruclips.net/video/qZNTPCMM19s/видео.html

Thanks!🙏

Two software packages: Makie.jl and Porta.jl. Source code: github.com/iamazadi/Porta.jl