# ManySpace

> Choose a goal and see every prerequisite between you and it. Mathematics, physics and AI/ML, a daily plan, memory-model reviews and free interactive tools.

A study system for mathematics, physics and AI/ML. It decides what to learn next, shows what is blocking you, and schedules every review. What ManySpace is, how its engine decides what to study next, what is and is not finished, and who built on whom.

Every topic page names what must be learned before it (its prerequisites, with links) and what it opens next, so a question of the form "what do I need before learning X" is answered by X's page, and each page also exists as clean Markdown by appending `.md` to its URL. English URLs are canonical; Turkish versions of the same pages live under the `/tr` prefix. Content updated: 2026-10-13. ManySpace is a product of Aporia Systems (https://aporia.systems).

## Start here

- [Home](https://manyspace.aporia.systems/): ManySpace — a prerequisite map and study system for math, physics and AI
- [Subjects](https://manyspace.aporia.systems/subjects): Subjects mapped as one prerequisite graph: every course, every topic and what each one needs first.
- [Free interactive tools](https://manyspace.aporia.systems/tools): Eighteen free interactive labs: launch projectiles, flip coins, tile integrals, spin the unit circle, melt secants, turn complex numbers, and price errors. No signup.
- [Pricing](https://manyspace.aporia.systems/pricing): Free to explore: topic pages, subject maps and every interactive tool. One subscription unlocks the full workspace. Billing is handled by Gumroad.
- [About](https://manyspace.aporia.systems/about): What ManySpace is, how its engine decides what to study next, what is and is not finished, and who built on whom.

## Subjects

- [Mathematics](https://manyspace.aporia.systems/subjects/mathematics): From natural numbers and fractions through calculus, linear algebra, probability and proof.
- [Physics](https://manyspace.aporia.systems/subjects/physics): Mechanics, waves, electromagnetism, thermodynamics, modern physics, each tied to the mathematics it needs.
- [AI & ML](https://manyspace.aporia.systems/subjects/ai): Models, optimisation, neural networks and attention, with every prerequisite in the mathematics side visible.

## Courses

### Mathematics

- [Arithmetic](https://manyspace.aporia.systems/subjects/mathematics/arithmetic-18)
- [Pre-Algebra](https://manyspace.aporia.systems/subjects/mathematics/pre-algebra-19)
- [Algebra I](https://manyspace.aporia.systems/subjects/mathematics/algebra-i-20)
- [Euclidean Geometry](https://manyspace.aporia.systems/subjects/mathematics/euclidean-geometry-21)
- [Algebra II](https://manyspace.aporia.systems/subjects/mathematics/algebra-ii-22)
- [Mathematical Logic](https://manyspace.aporia.systems/subjects/mathematics/mathematical-logic-25)
- [Precalculus](https://manyspace.aporia.systems/subjects/mathematics/precalculus-24)
- [Set Theory Foundations](https://manyspace.aporia.systems/subjects/mathematics/set-theory-foundations-26)
- [Trigonometry](https://manyspace.aporia.systems/subjects/mathematics/trigonometry-23)
- [Calculus I](https://manyspace.aporia.systems/subjects/mathematics/calculus-i-29)
- [Calculus II](https://manyspace.aporia.systems/subjects/mathematics/calculus-ii-30)
- [Discrete Mathematics](https://manyspace.aporia.systems/subjects/mathematics/discrete-mathematics-28)
- [Linear Algebra](https://manyspace.aporia.systems/subjects/mathematics/linear-algebra-31)
- [Proof Techniques](https://manyspace.aporia.systems/subjects/mathematics/proof-techniques-27)
- [Differential Equations (ODE)](https://manyspace.aporia.systems/subjects/mathematics/differential-equations-ode-33)
- [Multivariable Calculus](https://manyspace.aporia.systems/subjects/mathematics/multivariable-calculus-32)
- [Number Theory](https://manyspace.aporia.systems/subjects/mathematics/number-theory-36)
- [Optimization](https://manyspace.aporia.systems/subjects/mathematics/optimization-37)
- [Probability](https://manyspace.aporia.systems/subjects/mathematics/probability-34)
- [Statistics](https://manyspace.aporia.systems/subjects/mathematics/statistics-35)
- [Abstract Algebra](https://manyspace.aporia.systems/subjects/mathematics/abstract-algebra-39)
- [Complex Analysis](https://manyspace.aporia.systems/subjects/mathematics/complex-analysis-40)
- [Game Theory](https://manyspace.aporia.systems/subjects/mathematics/game-theory-50)
- [Numerical Analysis](https://manyspace.aporia.systems/subjects/mathematics/numerical-analysis-41)
- [Real Analysis](https://manyspace.aporia.systems/subjects/mathematics/real-analysis-38)
- [Differential Geometry](https://manyspace.aporia.systems/subjects/mathematics/differential-geometry-46)
- [Dynamical Systems](https://manyspace.aporia.systems/subjects/mathematics/dynamical-systems-47)
- [Functional Analysis](https://manyspace.aporia.systems/subjects/mathematics/functional-analysis-45)
- [Information Theory](https://manyspace.aporia.systems/subjects/mathematics/information-theory-49)
- [Measure Theory & Lebesgue Integration](https://manyspace.aporia.systems/subjects/mathematics/measure-theory-lebesgue-integration-44)
- [Partial Differential Equations](https://manyspace.aporia.systems/subjects/mathematics/partial-differential-equations-43)
- [Stochastic Processes](https://manyspace.aporia.systems/subjects/mathematics/stochastic-processes-48)
- [Topology](https://manyspace.aporia.systems/subjects/mathematics/topology-42)
- [Advanced Combinatorics & Graph Theory](https://manyspace.aporia.systems/subjects/mathematics/advanced-combinatorics-graph-theory-59)
- [Algebraic Geometry](https://manyspace.aporia.systems/subjects/mathematics/algebraic-geometry-55)
- [Algebraic Topology](https://manyspace.aporia.systems/subjects/mathematics/algebraic-topology-52)
- [Category Theory](https://manyspace.aporia.systems/subjects/mathematics/category-theory-54)
- [Mathematical Physics](https://manyspace.aporia.systems/subjects/mathematics/mathematical-physics-57)
- [Representation Theory](https://manyspace.aporia.systems/subjects/mathematics/representation-theory-51)
- [Algebraic Geometry II](https://manyspace.aporia.systems/subjects/mathematics/algebraic-geometry-ii-56)
- [Algebraic Topology II](https://manyspace.aporia.systems/subjects/mathematics/algebraic-topology-ii-53)
- [Mathematical Physics II](https://manyspace.aporia.systems/subjects/mathematics/mathematical-physics-ii-58)

### Physics

- [Mathematical Tools for Physics](https://manyspace.aporia.systems/subjects/physics/mathematical-tools-for-physics-60)
- [Classical Mechanics Foundations](https://manyspace.aporia.systems/subjects/physics/classical-mechanics-foundations-61)
- [Fluid Mechanics](https://manyspace.aporia.systems/subjects/physics/fluid-mechanics-63)
- [Oscillations & Waves](https://manyspace.aporia.systems/subjects/physics/oscillations-waves-62)
- [Thermodynamics](https://manyspace.aporia.systems/subjects/physics/thermodynamics-64)
- [Electricity & Magnetism](https://manyspace.aporia.systems/subjects/physics/electricity-magnetism-65)
- [Optics](https://manyspace.aporia.systems/subjects/physics/optics-66)
- [Advanced Classical Mechanics (Analytical)](https://manyspace.aporia.systems/subjects/physics/advanced-classical-mechanics-analytical-68)
- [Special Relativity](https://manyspace.aporia.systems/subjects/physics/special-relativity-67)
- [Atomic Physics](https://manyspace.aporia.systems/subjects/physics/atomic-physics-71)
- [Molecular Physics](https://manyspace.aporia.systems/subjects/physics/molecular-physics-72)
- [Nuclear Physics](https://manyspace.aporia.systems/subjects/physics/nuclear-physics-73)
- [Quantum Mechanics](https://manyspace.aporia.systems/subjects/physics/quantum-mechanics-70)
- [Statistical Mechanics](https://manyspace.aporia.systems/subjects/physics/statistical-mechanics-69)
- [General Relativity](https://manyspace.aporia.systems/subjects/physics/general-relativity-76)
- [Particle Physics](https://manyspace.aporia.systems/subjects/physics/particle-physics-75)
- [Solid State & Condensed Matter](https://manyspace.aporia.systems/subjects/physics/solid-state-condensed-matter-74)
- [Advanced & Specialized Physics](https://manyspace.aporia.systems/subjects/physics/advanced-specialized-physics-80)
- [Cosmology](https://manyspace.aporia.systems/subjects/physics/cosmology-77)
- [Quantum Field Theory](https://manyspace.aporia.systems/subjects/physics/quantum-field-theory-78)
- [Advanced & Specialized Physics II](https://manyspace.aporia.systems/subjects/physics/advanced-specialized-physics-ii-81)
- [Quantum Field Theory II](https://manyspace.aporia.systems/subjects/physics/quantum-field-theory-ii-79)

### AI & ML

- [Data and Representation](https://manyspace.aporia.systems/subjects/ai/data-and-representation-83)
- [Foundations of Learning](https://manyspace.aporia.systems/subjects/ai/foundations-of-learning-82)
- [Learning as Optimization](https://manyspace.aporia.systems/subjects/ai/learning-as-optimization-85)
- [Linear Models: Regression](https://manyspace.aporia.systems/subjects/ai/linear-models-regression-84)
- [Classification](https://manyspace.aporia.systems/subjects/ai/classification-88)
- [Generalization and Model Selection](https://manyspace.aporia.systems/subjects/ai/generalization-and-model-selection-86)
- [Learning Theory](https://manyspace.aporia.systems/subjects/ai/learning-theory-89)
- [Probabilistic Machine Learning](https://manyspace.aporia.systems/subjects/ai/probabilistic-machine-learning-87)
- [AI Foundations: Agents, Search, and Decisions](https://manyspace.aporia.systems/subjects/ai/ai-foundations-agents-search-and-decisions-92)
- [Classical Machine Learning](https://manyspace.aporia.systems/subjects/ai/classical-machine-learning-90)
- [Knowledge Representation, Reasoning, and Planning](https://manyspace.aporia.systems/subjects/ai/knowledge-representation-reasoning-and-planning-93)
- [Unsupervised Learning](https://manyspace.aporia.systems/subjects/ai/unsupervised-learning-91)
- [Neural Networks](https://manyspace.aporia.systems/subjects/ai/neural-networks-95)
- [Probabilistic Graphical Models and Inference](https://manyspace.aporia.systems/subjects/ai/probabilistic-graphical-models-and-inference-94)
- [Sequence Modeling](https://manyspace.aporia.systems/subjects/ai/sequence-modeling-96)
- [Attention](https://manyspace.aporia.systems/subjects/ai/attention-98)
- [Generative Models](https://manyspace.aporia.systems/subjects/ai/generative-models-100)
- [Representation and Embeddings](https://manyspace.aporia.systems/subjects/ai/representation-and-embeddings-97)
- [Transformers](https://manyspace.aporia.systems/subjects/ai/transformers-99)
- [Fine-Tuning and Adaptation](https://manyspace.aporia.systems/subjects/ai/fine-tuning-and-adaptation-102)
- [Language Modeling and LLMs](https://manyspace.aporia.systems/subjects/ai/language-modeling-and-llms-101)
- [Natural Language Processing](https://manyspace.aporia.systems/subjects/ai/natural-language-processing-103)
- [Retrieval and Grounding](https://manyspace.aporia.systems/subjects/ai/retrieval-and-grounding-104)
- [Computer Vision](https://manyspace.aporia.systems/subjects/ai/computer-vision-105)
- [Multimodal Learning](https://manyspace.aporia.systems/subjects/ai/multimodal-learning-107)
- [Speech and Audio](https://manyspace.aporia.systems/subjects/ai/speech-and-audio-106)
- [AI Agents](https://manyspace.aporia.systems/subjects/ai/ai-agents-109)
- [Graph Machine Learning](https://manyspace.aporia.systems/subjects/ai/graph-machine-learning-110)
- [Reinforcement Learning](https://manyspace.aporia.systems/subjects/ai/reinforcement-learning-108)
- [Time Series and Forecasting](https://manyspace.aporia.systems/subjects/ai/time-series-and-forecasting-111)
- [Causal Inference](https://manyspace.aporia.systems/subjects/ai/causal-inference-113)
- [Evaluation](https://manyspace.aporia.systems/subjects/ai/evaluation-115)
- [Production AI Systems](https://manyspace.aporia.systems/subjects/ai/production-ai-systems-114)
- [Recommender Systems](https://manyspace.aporia.systems/subjects/ai/recommender-systems-112)
- [Interpretability](https://manyspace.aporia.systems/subjects/ai/interpretability-116)
- [Responsible and Reliable ML](https://manyspace.aporia.systems/subjects/ai/responsible-and-reliable-ml-117)
- [Security and Privacy](https://manyspace.aporia.systems/subjects/ai/security-and-privacy-118)

## Free interactive tools

- [Derivative Visualizer](https://manyspace.aporia.systems/tools/derivative): The derivative is a limit you can watch: shrink h and see the secant become the tangent.
- [Integral Explorer](https://manyspace.aporia.systems/tools/integral): The definite integral is just area — tiled, summed, and refined until the rectangles disappear into it.
- [Unit Circle Explorer](https://manyspace.aporia.systems/tools/unit-circle): Sine and cosine are not buttons on a calculator — they are the shadows of one point on one circle.
- [Complex Plane Explorer](https://manyspace.aporia.systems/tools/complex-plane): A complex number is an arrow: how far (modulus) and which way (argument). Multiplication turns and stretches.
- [Probability Lab](https://manyspace.aporia.systems/tools/probability): One flip tells you nothing. A thousand flips tell you p. Flip the coin and watch luck average out.
- [Bayes Lab](https://manyspace.aporia.systems/tools/bayes): A 95%-accurate test can still cry wolf 5 times out of 6. The crowd decides what a positive means — move the prior and watch.
- [Eigen Explorer](https://manyspace.aporia.systems/tools/eigenvectors): Most directions come out of a matrix bent. A few come out only stretched — turn v until you find them.
- [Fourier Series](https://manyspace.aporia.systems/tools/fourier-series): Every repeating wave is a stack of pure sines. Add harmonics one at a time and watch a square wave assemble itself — bumps, ringing, and all.
- [Projectile Motion Lab](https://manyspace.aporia.systems/tools/projectile-motion): Speed, angle, gravity — three knobs, one arc. Sweep them and find the angle that flies farthest.
- [Orbit & Gravity](https://manyspace.aporia.systems/tools/orbit): Every launch buys a fate: too slow falls home, just right loops forever, too fast never returns. Aim the arrow and scrub the flight.
- [Gradient Descent Playground](https://manyspace.aporia.systems/tools/gradient-descent): One walker, one bowl, one knob that matters. Find the step size that flies — and the one that explodes.
- [Loss Function Explorer](https://manyspace.aporia.systems/tools/loss-functions): Every loss is an opinion about error: MSE panics at outliers, MAE counts calmly, log-loss fears confident mistakes.
- [k-Means Visualizer](https://manyspace.aporia.systems/tools/k-means): Clustering is a loop, not an oracle: assign every point to its nearest center, move each center to its mean, repeat.
- [Logistic Regression](https://manyspace.aporia.systems/tools/logistic-regression): A straight line, a squash function, and two clouds of dots: draw the boundary by hand, then let gradient descent show you its own.
- [Forward Pass](https://manyspace.aporia.systems/tools/neural-network): Two inputs, three hidden neurons, one output — nothing hidden. Push an input and watch every neuron answer at once.
- [Backpropagation](https://manyspace.aporia.systems/tools/backpropagation): The forward pass makes a prediction; backprop decides how much to blame each weight. Follow one blame path from loss to weight.
- [Attention Visualizer](https://manyspace.aporia.systems/tools/attention): Attention is a spotlight: each word decides how much of every other word to mix into itself.
- [Tokenization Explorer](https://manyspace.aporia.systems/tools/tokenization): Type a sentence and watch a simplified toy tokenizer split it into pieces: suffixes peel off, numbers become digits, and tokens turn out to be neither words nor characters.

## Topic pages

Each topic in the curriculum has a public page at `/topics/{title-words}-{stable-id}` explaining why it matters, where it is used, what counts as evidence of mastery, its prerequisites and what it unlocks. The complete, canonical list is the topics sitemap; topic pages are also reachable as Markdown by appending `.md` to the page URL.

- [All topic pages (sitemap)](https://manyspace.aporia.systems/sitemap-topics.xml)
- [Search topics](https://manyspace.aporia.systems/search): append `?q=` and the query, for example `https://manyspace.aporia.systems/search?q=derivative`

## Machine-readable resources

- [Sitemap index](https://manyspace.aporia.systems/sitemap-index.xml): per-section sitemaps (pages, courses, topics, tools) with hreflang alternates
- [Complete sitemap](https://manyspace.aporia.systems/sitemap.xml)
- [Full text of every public topic page](https://manyspace.aporia.systems/llms-full.txt)
- [OpenSearch description](https://manyspace.aporia.systems/opensearch.xml)

## Optional

- [Privacy](https://manyspace.aporia.systems/privacy)
- [Terms](https://manyspace.aporia.systems/terms)