mathematical explainers, derived naturally
whenever τ appears in an equation, a circle will always emerge somewhere
MathTau publishes visual derivations of mathematical objects from first principles — not as definitions to memorize, but as inevitable consequences of simpler underlying structures.
Every article ships with interactive figures. Drag sliders, hit play, watch the geometry build from zero.
Whenever π appears in an equation, it should be replaced by τ.
Whenever τ appears in an equation, a circle will always emerge somewhere.
Why eiπ + 1 = 0 is not the most elegant equation — and eiτ = 1 is
Interactively derive eiθ = cos θ + i sin θ from the fold-and-add method. See the unit circle emerge from the series, and why the famous identity is just a half-frame of the full story.
Read in English → 阅读中文版
A visual journey from linear change to the hidden circle
Why does ∫ exp(−x²/2) dx = √τ? Four sections, from the most ordinary observation (constant acceleration) to a surprising consequence (a circle emerges from a one-dimensional integral). Each step is built interactively — drag, play, watch.
Read in English → 阅读中文版
Taylor, Cauchy, and Fourier are the same move
Three theorems taught in three different lecture rooms, with one mechanism underneath: spin an arrow and average it over a full turn — everything spinning cancels, only the still part survives. Taylor reads it at the center, Fourier on the rim, and Cauchy is the equals sign between them.
Read in English → 阅读中文版
A playable derivation of mathematics, one truth at a time
Before all numbers, a single point: O. From it, number, then circle, then calculus, rebuilt step by step as you play. Seal the Rift by drawing the Great Circle τ. The interactive companion to the MathTau book.
Begin your path → Survey the world map 🗺
A breathing circle that counts you down, one full turn at a time
Every breath draws one step around the circle; every full turn is one τ. Counting sheep, doubling, Fibonacci, set to rain and waves. Close your eyes and let the numbers spin.
Open the lullaby →