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← The Invariant — back to the explainer Companion · learning map

From Grade 8 to the frontier.

Every demonstration on the main page sits at the top of a ladder that starts with rotating a shape on graph paper. Here is the climb — five threads, five tiers, and what to learn at each rung to reach each idea. None of it requires genius; it requires order.

The five threads

Everything here is one of five paths up.

The site's concepts aren't a random pile — they fall into five strands that braid together. Linear algebra and calculus are the shared trunk most of them pass through.

Symmetry & groups
→ D₄ · orbit–stabilizer · representation theory
Complex & Fourier
→ the orbit (eiθ) · the DFT · Gabor vision · formants
Geometry & space
→ the Erlangen hierarchy · SO(2)/SO(3)
Calculus & physics
→ Noether's theorem
Discrete & topology
→ the Euler characteristic
The ladder

Five tiers, Grade 8 to graduate school.

Each rung lists what to learn; the colour on a topic shows which thread it feeds. Grey = shared trunk that several threads need.

0
Grade 8
The launchpad — you may already have all of this.
Negatives, fractions, ratiosarithmetic fluency
Solving linear equationsalgebra basics
Transformations on a gridtranslate · reflect · rotate
The coordinate plane(x, y)
Angles, area, Pythagorasplane geometry
Counting & simple graphsdiscrete seeds
1
Grades 9–10
Structure appears.
Functions & their graphsthe central object
Congruence & similaritytransformations, made precise
Right-triangle trigonometrysin · cos · tan
Exponents & logarithmsscaling laws
Vectors & systems of equationsintro
Polyhedra & Euler's V−E+F=2your first invariant
2
Grades 11–12
The toolkit. (Often called pre-calculus + calculus.)
Trig functions & identitiesperiodicity
Complex numberspolar form · eiθ · De Moivre
Matrices & 2D/3D transformslinear maps, concretely
Introductory calculuslimits · derivatives · integrals
Sequences, series, proof by inductionrigour
3
Early university
The machinery. This tier unlocks most of the site.
Linear algebrathe hub — spaces, eigenvalues, basis change
Abstract algebra Igroups · subgroups · cosets · Lagrange
Fourier analysisseries · transform · the DFT
Classical mechanicsNewton → first Lagrangian
Multivariable calculusgradients, fields
Discrete math & graph theorynetworks & counting
4
Advanced
The frontier — this is the main page.
Representation theorycharacters · irreducibles → §10
Lie / matrix groupsSO(2), SO(3) → §06
Affine & projective geometryKlein's Erlangen → §05
TopologyEuler characteristic, genus → §08/§11
Lagrangian mechanics + Noethervariations → §12
Signal processing / visionGabor energy model → §13
Acoustic phoneticsformants → §14
Per-concept chains

For each idea: the path, and how you know you're ready.

Pick a demonstration you want to truly understand and follow its chain. The green checkpoint is the test: if you can do it, that section will read as obvious rather than magical.

§04

The dihedral group D₄

T0 grid transformations → T1 congruence → T3 group axioms (closure, identity, inverses).
Ready whenYou can list the square's 8 symmetries and compose any two without drawing them.
§09

Orbit & stabilizer

§04 + T3 cosets & Lagrange's theorem.
Ready whenYou can prove and apply |G| = |orbit|·|stabilizer| on a new example.
§10

Representation theory

T3 linear algebra + abstract algebra + T3 Fourier (for the DFT link) → T4 characters & irreducibles.
Ready whenYou can read a character table and split a representation into irreducibles.
§03

The orbit & De Moivre

T1 trig → T2 complex numbers in polar form.
Ready whenYou can use eiθ = cos θ + i sin θ and De Moivre to take powers and roots.
§10 · §13

The DFT & Gabor magnitude

T2 complex numbers → T3 Fourier series & transform → the discrete & windowed cases.
Ready whenYou can compute a 4-point DFT by hand and explain why |coefficient| is shift-invariant.
§05

The Erlangen hierarchy

T2 matrices → T3 linear algebra → T4 affine & projective geometry (homogeneous coordinates).
Ready whenYou can apply a homography to a point and name which invariant each geometry preserves.
§06

SO(2) & SO(3)

T2 trig & vectors → T3 linear algebra (rotation matrices) → T4 matrix / Lie groups.
Ready whenYou can multiply two 3D rotation matrices and show the product depends on the order.
§12

Noether's theorem

T2 calculus → T3 mechanics + multivariable calc → T4 Lagrangian mechanics & the calculus of variations.
Ready whenYou can derive conservation of momentum from a translation-invariant Lagrangian.
§08 · §11

The Euler characteristic

T1 Euler's polyhedron formula → T3 graph theory → T4 topology (homeomorphism, genus).
Ready whenYou can compute V−E+F for any triangulated surface and relate it to genus (χ = 2 − 2g).
§13

Visual cortex & Gabor cells

T3 Fourier → T4 signal processing + a little neuroscience.
Ready whenYou can explain a Gabor filter (Gaussian × sinusoid) and why the complex-cell energy is phase-invariant.
§14

Phonemes & formants

T1 logarithms + waves → T3 Fourier → T4 acoustic phonetics.
Ready whenYou can read a spectrogram, find the formants, and explain why their ratios are speaker-invariant.
Where to learn it

Suggested courses & texts, by tier.

A curated, not exhaustive, list — leaning toward the clearest free option per topic, with a standard textbook alongside. Green = free online.

Tiers 0–2 · school math → toolkit
  • Khan Academy — full Grade 8 → calculus, free. free · video + practice
  • 3Blue1Brown — “Essence of Calculus”, complex numbers, and intuition for everything below. free · video
  • Paul's Online Math Notes — algebra → calculus reference. free
Tier 3 · linear algebra (the hub)
  • 3Blue1Brown — “Essence of Linear Algebra”. free · video
  • Gilbert Strang — Introduction to Linear Algebra + MIT 18.06. lectures free
  • Sheldon Axler — Linear Algebra Done Right. textbook
Tier 3–4 · groups → representations
  • Nathan Carter — Visual Group Theory. The closest book in spirit to this site. textbook
  • Charles Pinter — A Book of Abstract Algebra (gentle). textbook
  • Benjamin Steinberg — Representation Theory of Finite Groups. after algebra
Tier 3–4 · Fourier · vision · speech
Tier 4 · topology
  • David Richeson — Euler's Gem (popular, the χ story). accessible
  • James Munkres — Topology (point-set). textbook
  • Allen Hatcher — Algebraic Topology. free PDF
Tier 4 · Lie groups & geometry
  • John Stillwell — Naive Lie Theory (gentle SO(2)/SO(3)). textbook
  • Kristopher Tapp — Matrix Groups for Undergraduates. textbook
  • H. S. M. Coxeter — Projective Geometry (Erlangen). textbook
Tier 4 · mechanics & Noether
Vision & the brain
  • David Hubel — Eye, Brain, and Vision (Nobel-laureate, on V1). free online
  • Adelson & Bergen (1985) — the energy model behind §13. paper
  • Peterson & Barney (1952) — the vowel-formant data behind §14. paper

Titles and authors are given so you can find the current edition; only stable homepages are linked. There is no single “right” order — pick a thread that excites you and climb it; the others will start to make sense as you go.

§ Share · pass it on

From Grade 8 to the frontier — pass the map on.

Pick a hook — each is a different door into the same idea.