Nvidia GeForce 4090, Chaos Theory, Lorenz Strange Attractor, 50 Million Dots
172024-10-27. Chaos theory. Solution in the Lorenz attractor using: ρ (rho) = 28, σ (sigma) = 10, and β (beta) = 8/3. 50,000,000 time step iterations, each drawn as a dot, with a time step of 0.0000125.
CHAOS THEORY
Edward Lorenz attempted to recreate a simulation of weather patterns by modeling 12 variables (temperature, wind speed, etc.). He failed to recreate the simulation due to rounding off values for the input. He discovered chaos theory: a small change in an input can have drastic changes in the outcome.
LORENZ SYSTEM
The Lorenz system is a system of differential equations. It has chaotic solutions for some parameter values and initial conditions. Tiny changes in initial conditions evolve to completely different trajectories.
LORENZ EQUATIONS
The model is a system of three ordinary differential equations now known as the Lorenz equations. The equations describe the rate of change of three quantities over time, in a two-dimensional fluid layer uniformly warmed from below and cooled from above.
dx/dt = σ (y - x)
dy/dt = x (ρ - z) - y
dz/dt = xy - βz
x is proportional to the rate of convection
y is proportional to the horizontal temperature variation
z is proportional to the vertical temperature variation.
LORENZ ATTRACTOR
The Lorenz attractor is a set of chaotic solutions of the Lorenz system. σ, ρ, and β are constant system parameters proportional to the Prandtl number, Rayleigh number, and physical dimensions of the fluid layer.
When ρ = 28, σ = 10, and β = 8/3, the Lorenz system has chaotic solutions, though not all solutions are chaotic. Almost all initial points will tend to an invariant set: the Lorenz attractor, which is a strange attractor, a fractal, and a self-excited attractor with respect to all three equilibria. An attractor is called "strange" if it has a fractal structure. Something is a fractal if it has non-integer Hausdorff dimension. Fractals are named after their fractional (non-integer dimension). Consider the coastline paradox as an example.
COLORING
Hues of primary and secondary colors are used to indicate rate of change in time for combined x, y, z using Pythagoras distance:
Red = FAST rate of change in time
Yellow
Green
Cyan = MEDIUM rate of change in time
Blue
Magenta
Red = SLOW rate of change in time
RENDERING
Dots are drawn as 2x2 dots at 1920 x 1080. Then scaled up to 4K using nearest neighbour to help battle video compression artifacts.
Vertical FOV = 70.000°.
Horizontal FOV = 102.448°
Aspect Ratio = 16:9
CAMERA
Camera is manually controlled with an Xbox 360 controller using the two proportional thumbsticks, two proportional triggers, and two bumpers, to control the 6 Degrees of Freedom camera. Yes, it's difficult to keep all 6 inputs under control, with smooth motion, and keep the rotational frame (roll) appropriate.
I've considered recording my input, then post-processing it to be smoother. However, there is something nice about the input coming directly my manual manipulation, in real-time, to catch the intricacies of the chaotic solution, fractal, etc.
HARDWARE
Alienware m18 R2 + 4090 GPU. (Purchased first day Dell made it available: January 23, 2024.)
NVIDIA GeForce RTX 4090 16GB GDDR6 (laptop version)
14th Gen Intel Core i9 14900HX (24-Core, 36MB L3 Cache, up to 5.8GHz Max Turbo Frequency) (16 E-Cores, and 8 P-Cores hyperthreaded = 32 V-Cores total)
64 GB: 2 x 32 GB, DDR5, 5200 MT/s, non-ECC, dual-channel
18" FHD+ (1920 x 1200) 480Hz, 3ms, ComfortView Plus, NVIDIA G-SYNC + DDS, 100% DCI-P3, FHD IR Camera
AlienFX RGB backlit Alienware CherryMX ultra low-profile mechanical keyboard
4 TB, M.2, PCIe NVMe, SSD
PLAYLIST
Xona System 8: • Game Dev Engine #1. A New Beginning.
Voxel: • Voxel Engine #8 - Correct Peripheral ...
Ray Cast: • 3D Ray Casting Engine: #4 (Global Gam...
Graph-All: • #1 Graph-All Equation Renderer - Mult...
Scroll Shmup: • The First Pixel: Devlog #9 - Enemies,...
Road: • Pseudo 3D Road #8 - infinite vegetati...
Arena Shmup: • Arena Shmup Demo #3 - Object Pool Opt...
Wave Function: • A.I. Texture Generation, High Res (Wa...
3D Polygon: • 3D Polygon Engine #9 - Multiple Mater...
GW-BASIC: • GW-BASIC - Escape From Monster Cavern...
GPU: • Nvidia GeForce 4090, Icosahedron Crys...
WEBSITE
GitHub: https://github.com/JDoucette
Blog: http://thefirstpixel.com/
Studio: http://xona.com/