The Gaussian Integral. [

The Beautiful Gaussian Integral.

Fourier Integral & 3D Gaussian Wave

The π in the normal distribution has a geometric... origin. It comes from a famous mathematical result known as the Gaussian integral. ∫₋∞^∞ e^(−x²) dx = √π The interesting part is that e^(−x²) has no elementary antiderivative. Yet its integral over the entire real line can be calculated exactly. The solution involves turning a one-dimensional integration problem into a two-dimensional one. The resulting function, e^(−(x²+y²)), has circular symmetry, and polar coordinates reveal where π enters the calculation. This result is fundamental to probability theory. The familiar factor 1/(σ√(2π)) in the normal distribution comes from the same integral. It ensures that the total probability across all possible values is exactly 1. But its applications extend well beyond statistics. In engineering, Gaussian distributions are commonly used to model measurement errors and noise. With independent, zero-mean Gaussian errors of equal variance in an additive-error model, maximum-likelihood estimation is equivalent to minimizing the sum of squared residuals. Gaussian functions also appear in heat diffusion, where they describe how an initially concentrated point source spreads through an unbounded, homogeneous medium. In Fourier analysis, the transform of a Gaussian is another Gaussian, up to scaling. This property is useful in signal processing and image filtering. The same mathematical function connects probability distributions, statistical estimation, heat transfer, and frequency analysis. And the √π in its integral ultimately comes from the geometry of the plane.show more

Gaussian Splashing: Dynamic Fluid Synthesis with Gaussian Splatting

The visual demonstration of (a+b)³.

Gaussian splashing

When you do the math correctly👀

Gaussian Painters imported into #b3d as ellipsoids (3D Gaussian... Splatting plugin for Blender - work in progress)show more

The visual demonstration of (a+b)³. Nice.

Volume is the integral of area

The Aizawa Attractor. A simple equation, but beautiful chaotic... pattern. [show more

2. 4D Gaussian Splatting example:

Gaussian splatting is getting crazy!

The Fibonacci elephant. [

yoooooo.... 3D gaussian splatting + simulated kinematics

Comparison of the Wasserstein, Hellinger, Kullback-Leibler and reverse KL... on the space of Gaussian distributions.show more

Gaussian splat of a fire bug.

They just rendered a living world with 800,000 Gaussian... splats, live in your browser on WebGPU. ARE GAUSSIAN SPLATS ABOUT TO REPLACE POLYGONS?show more

Apple just trained a 3D Gaussian head reconstruction model... on 10,000+ subjects. Feed-forward. No test-time optimization. New identity in, reconstructed Gaussian head out. The UV-parameterized Gaussian representation decouples the number of Gaussians from the number and resolution of input images, making it practical to train with many high resolution views. And the heads are not just static either: text-conditioned identity generation, plus blendshape-driven latent animation across identities. We've been building in the 3D Gaussian Splatting space for a while. The gap between "research demo" and "works on real people at scale" is closing fast.show more