Tekne · τέχνη · the craft of exact figures
Geometry you can hold.
Twenty-seven instruments on one drafting sheet, from Pythagoras to statistical mechanics. Drag a point and every length, angle, slope and area is recomputed from its coordinates, then checked against the theorem that says what it must be. The same right triangle turns up on every sheet.
Figures held still
T-1
Pythagoras
a² + b² = c², drawn with a square on every side, proved by rearranging four triangles, then carried into space.
T-2
Cartesian
Points as pairs of numbers. Distance, lines, triangles and their centres, Pick's theorem, and every transformation as a matrix.
T-3
Platonic
The five regular solids built from coordinates, turned in your hand, and checked against Euler, Descartes and their closed forms.
Figures set in motion
T-4
Trigonometry
The unit circle with all six functions as segments, exact values generated rather than looked up, waves as turning vectors, and any triangle.
T-5
Calculus
Secants becoming tangents, sums becoming areas, polynomials becoming functions, each limit measured against its closed form.
T-6
Linear algebra
A matrix as a motion of the plane: determinant, eigenvectors and singular values you can see, exact systems, and least squares.
T-7
Differential equations
Slope fields and phase planes solved exactly, three integrators with their orders, and an oscillator that shows which ones keep energy.
Figures spread through space
T-8
Multivariable calculus
Gradients across contour maps, Newton's method on critical points, Green's theorem two ways, and a surface with its tangent plane.
T-9
Partial differential equations
Heat, waves and Laplace: each solved exactly and on a grid, with the stability limits and convergence rates measured.
T-10
Differential geometry
Curvature of curves and surfaces, Gauss's remarkable theorem checked numerically, Gauss–Bonnet, and geodesics that obey Clairaut.
Figures made abstract
T-11
Imaginary numbers
Multiplying by i as a quarter turn, quadratics whose roots leave the real line, Bombelli's cubic, and Euler's e^(iπ) = −1 built two ways.
T-12
Complex numbers
Multiplication as rotation, Gaussian integers that make Pythagorean triples, exact roots of unity, and conformal maps.
T-13
Functional analysis
Functions as vectors: Fourier coefficients as coordinates, Parseval as Pythagoras, Legendre by Gram–Schmidt, and the norms that are not.
T-14
Probability
A Galton board against the exact binomial, π by random points and falling needles, and Bayes' theorem three ways.
T-15
Matrix mechanics
Quantum theory as Heisenberg wrote it: spin on the Bloch sphere, the oscillator as matrices, and energy levels by diagonalisation.
Physics of motion
T-16
Newton's laws
Projectiles integrated against the parabola and against drag, collisions in exact fractions, and the forces on a slope cut from a Pythagorean triple.
T-17
Orbits
An ellipse read off one position and one velocity, three integrators held to Kepler's laws, the planets on a line of slope 3/2, and a figure eight.
T-18
Oscillations
A pendulum's period by Gauss's arithmetic–geometric mean, the exact normal modes of a chain, and resonance with Pythagoras in the denominator.
T-19
Rigid bodies
Exact moments of a lattice plate and Mohr's circle, a race down a slope won by 1/(1 + k), and the tennis racket's flip on an elliptic clock.
T-20
Special relativity
Spacetime's Pythagoras with a minus sign, speeds from triples so that every γ is a fraction, the twins, and E² = p² + m².
Physics of fields
T-21
Electrostatics
Field lines and equipotentials crossing at right angles, Gauss's law as a flux summed round a circle, and image charges placed by Pythagoras.
T-22
Circuits
Kirchhoff in exact fractions, a ladder that climbs Fibonacci to the golden ratio, impedance triangles, and the half a capacitor always wastes.
T-23
Magnetism
A charge turned by Boris's Pythagorean rotations, a loop's field from straight sides against its closed form, Helmholtz coils and Ampère's law.
T-24
Light
Snell's law as Fermat's quickest path, reflection where nothing is lost and water's Brewster angle is 3-4-5, and slits as arrows head to tail.
Physics of quanta and chance
T-25
Wave mechanics
A box whose grid is the chain of springs, probabilities that add like squares, tunnelling with |r|² + |t|² = 1, and a packet that spreads as a hypotenuse.
T-26
The hydrogen atom
A spectrum of fractions with Balmer and the deuterium shift, orbitals whose averages are exact, and Bohr's orbits as the waves that fit.
T-27
Statistical mechanics
Einstein solids counted exactly, a gas of disks relaxing to the law of a hypotenuse, and the Ising magnet against Onsager and Yang.
How it stays honest
Exact first
On the lattice every squared length is a whole number, so lengths print as radicals in simplest form. Sines of multiples of 15° and the roots of unity are computed in ℚ(√2, √3); systems, Gram–Schmidt and probabilities are exact fractions.
Two routes to every number
An area by the shoelace sum and by Heron; a flow by Runge–Kutta and by the matrix exponential; a curvature from outside and from inside; a posterior by counting and by odds; a spin by its closed form and by integration.
The residual is shown
Where two routes meet, their difference is printed: 0 when the arithmetic is exact, a few parts in 10¹⁶ when it is floating point, and a measured order when a method converges.