The mathematical approach to change over time. Most dynamical systems are nonlinear and generally unsolvable, and though deterministic are often unpredictable.

Logistic map

logistic map image

Clifford attractor

clifford vectors image

Grid map

Grid map image

Pendulum phase space



Trajectories of any dynamical equation may stay bounded or else diverge towards infinity. The borders between bounded and unbounded trajectories can take on spectacular fractal geometries.

Polynomial roots I


Polynomial roots II


Julia sets

julia set1

Mandelbrot set

disappearing complex mandelbrot

Henon map


Clifford map


Logistic map

logistic map image


Primes are unpredictable

\[\lnot \exists n, m : (g_n, g_{n+1}, g_{n+2}, ... , g_{n + m - 1}) \\ = (g_{n+m}, g_{n+m+1}, g_{n+m+2}, ..., g_{n + 2m - 1}) \\ = (g_{n+2m}, g_{n+2m+1}, g_{n+2m+2}, ..., g_{n + 3m - 1}) \\ \; \; \vdots\]

Aperiodicity implies sensitivity to initial conditions

\[f(x) : f^n(x(0)) \neq f^{n+k}(x(0)) \forall k \implies \\ \forall x_1, x_2 : \lvert x_1 - x_2 \rvert < \epsilon, \; \\ \exists n \; : \lvert f^n(x_1) - f^n(x_2) \rvert > \epsilon\]

Aperiodic maps, irrational numbers, and solvable problems

\[\Bbb R - \Bbb Q \sim \{f(x) : f^n(x(0)) \neq f^k(x(0))\} \\ \text{given} \; n, k \in \Bbb N \; \text{and} \; k \neq n \\\]

Irrational numbers on the real line

\[\Bbb R \neq \{ ... x \in \Bbb Q, \; y \in \Bbb I, \; z \in \Bbb Q ... \}\]

Discontinuous aperiodic maps

\[\{ f_{continuous} \} \sim \Bbb R \\ \{ f \} \sim 2^{\Bbb R}\]

Poincaré-Bendixson and dimension

\[D=2 \implies \\ \forall f\in \{f_c\} \; \exists n, k: f^n(x) = f^k(x) \; if \; n \neq k\]

Computability and Periodicity I: the Church-Turing thesis

\[\\ \{i_0 \to O_0, i_1 \to O_1, i_2 \to O_2 ...\}\]

Computability and Periodicity II

\[x_{n+1} = 4x_n(1-x_n) \implies \\ x_n = \sin^2(\pi 2^n \theta)\]

Nonlinearity and dimension


Reversibility and periodicity

\[x_{n+1} = rx_n(1-x_n) \\ \; \\ x_{n} = \frac{r \pm \sqrt{r^2-4rx_{n+1}}}{2r}\]

Additive transformations

random fractal

Fractal Geometry



As for any natural science, an attempt to explain observations and predict future ones using hypothetical statements called theories. Unlike the case for axiomatic mathematics, such theories are never proven because some future observation may be more accurately accounted for by a different theory. As many different theories can accurately describe or predict any given set of observations, it is customary to favor the simplest as a result of Occam’s razor.

Three Body Problem I

3 body image

Three Body Problem II: Parallelized Computation with CUDA

3 body image



Quantum Mechanics

\[P_{12} \neq P_1 + P_2 \\ P_{12} = P_1 + P_2 + 2\sqrt{P_1P_2}cos \delta\]


The study of life, observations of which display many of the features of nonlinear mathematical systems: an attractive state resistant to perturbation, lack of exact repeats, and simple instructions giving rise to intricate shapes and movements.

Genetic Information Problem

coral image


lake image

Deep Learning

Machine learning with layered representations. Originally inspired by efforts to model the animalian nervous system, much work today is of somewhat dubious biological relevance but is extraordinarily potent for a wide range of applications. For some of these pages and more as academic papers, see here.

Image Classification

neural network architecture

Input Attribution and Adversarial Examples

neural network architecture

Input Generation I: Classifiers

generated badger

Input Generation II: Vectorization and Latent Space Embedding

wordnet recovered from imagenet

Input Generation III: Input Representations

resnet googlenet transformation

Input Representation I: Depth and Representation Accuracy

layer autoencoding

Input Representation II: Vision Transformers

vision transformer layer representations

Language Representation I: Spatial Information

vision transformer layer representations

Language Representation II: Sense and Nonsense

\[\mathtt{This \; is \; a \; prompt \; sentence} \\ \mathtt{channelAvailability \; is \; a \; prompt \; sentence} \\ \mathtt{channelAvailability \; millenn \; a \; prompt \; sentence} \\ \dots \\ \mathtt{redessenal \; millenn-+-+DragonMagazine}\]

Language Representation III: Noisy Communication on a Discrete Channel

\[a = \mathtt{The \; sky \; is \; blue.} \\ a_g = \mathtt{The \; sky \; is \; blue \lt s \gt}\] The wipers on the bus go swish swish
Predicted next token: sw

Language Features

\[O_f = [:, \; :, \; 2000-2004] \\ a_g = \mathtt{called \; called \; called \; called \; called} \\ \mathtt{ItemItemItemItemItem} \\ \mathtt{urauraurauraura} \\ \mathtt{vecvecvecvecvec} \\ \mathtt{emeemeemeemeeme} \\\]

Feature Visualization I

features 2

Feature Visualization II: Deep Dream


Feature Visualization III: Transformers and Mixers



autoencoding of landscapes

Diffusion Inversion


Generative Adversarial Networks

network architecture

Normalization and Gradient Stability

network architecture

Small Language Models for Abstract Sequences

network architecture

Interpreting Sequence Models

deep learning attributions

Training Memory

Limitations of Neural Networks

discontinous proof

Small Projects

Game puzzles


Programs to compute things

\[\; \\ \begin{vmatrix} a_{00} & a_{01} & a_{02} & \cdots & a_{0n} \\ a_{10} & a_{11} & a_{12} & \cdots & a_{1n} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ a_{n0} & a_{n1} & a_{n2} & \cdots & a_{nn} \\ \end{vmatrix} \; \\\] \[\; \\ 5!_{10} = 12\mathbf{0} \to 1 \\ 20!_{10} = 243290200817664\mathbf{0000} \to 4 \\ n!_k \to ? \; \\\]

High voltage

High voltage engineering projects: follow the links for more on arcs and plasma.

Tesla coil

tesla coil arcs


fusor image

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