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A realm of my own construct... engineered from math.

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Mathelirium

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579,450 görüntüleme • 7 ay önce •via X (Twitter)

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Much better visually. We now visualize the 3D version of our stunning Blaschke-Quotient Flow from our previous post.🤗🤩 What follows is a mesmerizing break of dawn in the some realm of complex space with zeros and poles still spinning under SU(1,1) Möbius symmetries on the Poincaré disk…but phase is now shaded by the Fubini-Study metric. 🤯🤯🤯
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Geometry 🤝 Topology What you're seeing here is called Hopf Fibration. In pure-math terms, S³ (the unit sphere in 4D) can be decomposed into circles called fibers. The Hopf map sends each of those circles to a single point on S². So you don’t fill S³ with random loops, you fill it with a perfectly organized family of circles, one circle sitting above each point of S². For the animation, we sample thousands of fibers on S³, apply a smooth SO(4) rotation (a real rotation in 4D, not a 3D cheat), then stereographically project into R³ so we can actually draw it. Pure mathematics is beautiful alz #HopfFibration #Topology #Geometry #StereographicProjection #PureMathematics

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Of all professions, electrical engineers are the ones that impress me the most. It’s not that they know more math. It’s how naturally they use it. A lot of ideas that sit in the pure-math neighborhood end up powering things like cryptography, coding theory, and information theory. I’d always known that in theory. What shocked me was seeing the same ideas running real systems: probability steering decisions in digital comms, optimisation shaping hardware, and information theory acting like a hard constraint on what’s even possible. Working with them on research was humbling. It made me feel like I knew nothing, in the best way. It also made me rethink what being good at math means. In my optimisation course, Space Mapping was one of the concepts that really stuck with me: you keep a computationally cheap coarse model f_c(x) that runs fast but lies, a brutally expensive fine model f_f(x) that tells the truth, and you iteratively adjust a mapping T so that f_c(T(x)) shadows f_f(x) where it matters. You do almost all the optimisation on the cheap side and call the fine model only sparingly. It’s a very engineer move: admit the model is wrong, then make it useful anyway. John Bandler, a Canadian engineer and professor, formalised this in the early 1990s and showed you could make full-wave electromagnetic optimisation practical rather than masochistic. He founded Optimization Systems Associates in 1983 to commercialise the idea, and in 1997 Hewlett-Packard bought the company and folded its tools into what became HP EEsof, then Agilent, now Keysight’s RF design stack. #SpaceMapping #ComputationalElectromagnetics #RFDesign #NonLinearOptimization #AntennaDesign

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