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Fourier Transform 🤝 Spherical Harmonics = Viral Budding Geometry Viral Budding of an Infected Cell

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Mathelirium

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11,817 views • 4 months ago •via X (Twitter)

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Spherical Harmonics 🤝 Fourier Transform I’m modeling the cell surface as a time-varying field u(θ,φ). Budding is energy shifting into higher angular frequencies. Decompose u into spherical harmonics (the Fourier basis on a sphere), evolve the coefficients aₗₘ(t) where growth/smoothing is clean, then inverse-transform back into a breathing, budding 3D surface. #SphericalHarmonics #FourierTransform #HarmonicAnalysis #SpectralMethods #SignalProcessing #AppliedMath #MathAnimation #ScientificVisualization #Biophysics
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Spherical Harmonics 🤝 Fourier Transform I’m modeling the cell surface as a time-varying field u(θ,φ). Budding is energy shifting into higher angular frequencies. Decompose u into spherical harmonics (the Fourier basis on a sphere), evolve the coefficients aₗₘ(t) where growth/smoothing is clean, then inverse-transform back into a breathing, budding 3D surface. #SphericalHarmonics #FourierTransform #HarmonicAnalysis #SpectralMethods #SignalProcessing #AppliedMath #MathAnimation #ScientificVisualization #Biophysics

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This video presents a recent breakthrough called the Sparse Fourier Transform (SFT). This algorithm yields an exponential speed-up over the celebrated Fast Fourier Transform (FFT) when asked to extract a small number of dominant Fourier coefficients.
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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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