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Vectors and matrices Experience it live.

From coordinates to the inverse: vectors, dot and cross products, linear equations and matrices in pictures.

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Coordinates after Descartes: geometry becomes algebra (3D)

A curve as a picture and as an equation at once: line, circle, parabola and ellipse on a 3D coordinate board, the equation as an animated transformation. Point test with P (on, inside, outside), intersection points with a line by substitution, discriminant and the quadratic formula, oblique axes as in Descartes (mixed term with cos ω) and the step into space: without z a cylinder and a plane, with z a sphere, ellipsoid and paraboloid.

Analytic geometryTry it

Right-handed systems in 2D and 3D

What makes (a, b) or (a, b, c) right-handed? A 3D hand fits itself to the vectors: thumb a, index finger b, middle finger c; in the plane the fingers curl from a to b. det > 0 as the clean definition, signed parallelepiped and area. Rotating keeps the orientation, swapping and reflecting reverse it, with a quiz.

Analytic geometryTry it

Vector workshop

Vectors as arrows and as number pairs: shift arrows in parallel (same vector), a + b along both ways around the parallelogram, b − a and closed vector chains, multiples r·a flipping for r < 0, normalising, coordinates via e₁ and e₂ with the magnitude as Pythagoras — |a + b| ≤ |a| + |b| live.

Linear algebraTry it

Vectors in ℝⁿ

Vectors beyond arrows: monthly sales as a vector in ℝ¹², position + velocity as a state in ℝ⁶. a + b, r·a and a·b component-wise, the dot product as a running sum, magnitude and cos φ as the similarity of two curves (b = 2·a ⇒ cos φ = 1), time step position + Δt·v in 3D.

Linear algebraTry it

Linear combination & linear independence (3D)

r·a + s·b + t·c as a vector chain: the span is a line, a plane or all of space. Moving c into the plane of a and b collapses the space grid to a plane, det = 0 and a non-trivial zero chain appears. Decomposition mode F = r·a + s·b, graphically via the parallelogram and as a linear system.

Linear algebraTry it

Phone vectors: three vectors, one plane

The keypad rows 1-2-3, 4-5-6, 7-8-9 as vectors in space: the game of the closed vector chain finds a − 2b + c = 0, so they are linearly dependent and lie in one plane. Seen edge-on, all three arrows lie on one line. For comparison c = (7, 8, 10) sticks out, the chain no longer closes, det ≠ 0.

Linear algebraTry it

Horse and rail: the dot product as work

A horse pulls a wagon at an angle α: the force splits into a part along the rail that does work and a perpendicular part that is lost. W = |F| · |s| · cos α; at 90° the wagon stands still. This gives the dot product F · s.

Linear algebraTry it

Dot product: coordinate formula from the definition

Why a·b = a1b1 + a2b2 follows from a·b = |a||b|cos φ: decompose into unit vectors, insert, multiply out, evaluate the boxes e_i·e_j with cos 0° = 1 and cos 90° = 0 — the zero terms drop out. In 2D (4 terms) and 3D (9 terms), optionally with a numerical example.

Analytic geometryTry it

Shoelace method: the cross product

The cross product a × b by the school rule: two columns, a copy below, cross out the first and last row, lace three times crosswise (down minus back). The laces draw themselves, the calculations stand next to them, the result gets brackets. Check with the dot product using a, b or c = (7, 8, 9), |a × b| as the area of the parallelogram.

Analytic geometryTry it

Equations as scales

Equations as beam balances: add two scales and knock out an unknown with Gauss – coloured stones against metal weights.

Linear algebraTry it

Matrizen rechnen: Produkt, Inverse, Transponierte

Matrizenrechnung Schritt für Schritt animiert: A·B als Zeile mal Spalte, wahlweise im Falk-Schema, mit Tausch A·B ↔ B·A; die Inverse über (A | E) mit Gauß-Jordan und exakten Brüchen; die Transponierte als Spiegelung an der Hauptdiagonale.

Linear algebraTry it

Matrices like numbers: identity, inverse, A·x = b

Who plays the role of 0 and 1 for matrices? The zero matrix for plus, the identity for times, not the matrix full of ones: (2 3; 5 1) times the all-ones matrix gives (5 5; 6 6). Opposite matrix and inverse as reciprocal, A·x = b unfolds into three equations, x = A⁻¹·b, Gauss-Jordan on (A | I) and the phone matrix 1 to 9, which has no inverse (0 = 2).

Linear algebraTry it

Collections

More collections. Curated by us.

Project management

Planning and steering projects: the iron triangle, stakeholders, network diagrams, earned value, schedule risk, team size, Kanban and waterfall against Scrum.

8 appsOpen collection

Explizit oder rekursiv: die Folgentreppe

Financial mathematics

From sequences to loans: arithmetic and geometric growth, savings plans, annuities, compounding within the year and perpetuities.

11 appsOpen collection

Archaeology

Reading the traces: burial mounds in section, radiocarbon dating, settlements in the landscape and the tomb of Tutankhamun.

5 appsOpen collection

Warum ist ein Roboterarm so schwer zu steuern? (3D)

Roboterhand im Raum — sechs Achsen, acht Lösungen (3D)

Roboterarm mit Gewicht — Haltemoment, Regelung, Kopplung (3D)

Robotics

Why robot arms are heavy, how six axes guide a hand and how control holds the load.

4 appsOpen collection

Relationales Modell: ein Topf voller Tabellen

Schlüssel zur Wohnung: Primär- und Fremdschlüssel

SQL-Fließband: Die Maschine liest anders

SQL and databases

Tables, keys, joins, groups and subqueries you can touch: how a database really reads.

7 appsOpen collection

Zauberkasten KI: ein Funke, sechs Kräfte

Das Wortorakel: wie ein Sprachmodell schreibt

Die schöne Fassade: Warum KI-Antworten so überzeugend wirken

Zauberspruch-Werkstatt: gut mit KI sprechen

Understanding AI

Generative AI in strong images: what it can do, how it writes, why it convinces, how to use it well and where to take care.

7 appsOpen collection

Automotive technology

Engine, brakes, high voltage, CAN bus and chassis: technology and diagnostics of the modern vehicle.

9 appsOpen collection

Platz für Bäume: Altstadt Reutlingen (3D)

Asphalt raus: Entsiegelung Innenstadt (3D)

Urban climate

Heat in the city centre: heat islands, shade, trees and unsealing, using the old town of Reutlingen as an example.

5 appsOpen collection

Dark-Patterns-Detektor: Abofallen im Online-Shop

Verständlichkeits-Werkstatt: ein Verbraucherbrief in fünf Schritten

Consumer knowledge

Reading the small print, spotting subscription traps, placing risk figures and making texts understandable.

4 appsOpen collection

Differential equations

From equation to curve: initial values, linear equations, substitution and the worlds in which they apply.

7 appsOpen collection

Schwerpunkt-Brett: der Mittelwert als Gleichgewicht

Statistics from the start

Mean, median, mode, box plot and levels of measurement: descriptive statistics to try out.

9 appsOpen collection

Vier Wege zu fertigen: Baustelle, Werkstatt, Reihe, Fluss (3D)

Verschwendung finden: sieben Arten in der Fertigung (3D)

Los oder One-Piece-Flow: zehn Stück, zwei Wege (3D)

Wertstromkarte TIM Corp.: vom Ist- zum Soll-Zustand

Factory and production

Production principles, waste, one-piece flow, value streams, bills of materials and routing: how a factory is organised.

6 appsOpen collection

Cyber Resilience Act: Betroffenheitsprüfung

Cyber Resilience Act

What the CRA requires of manufacturers: checking whether it applies, meeting reporting deadlines, keeping software bills of materials.

5 appsOpen collection

Brüche multiplizieren mit kreuzweisem Kürzen

Brüche dividieren: Wie oft passt es hinein?

Fractions

Reducing, common denominators, multiplying, dividing and mixed numbers: fractions step by step.

7 appsOpen collection

Zagiers Windmühlen: Primzahlen als Summe zweier Quadrate

Museumswächter: Warum n/3 Wachen reichen

From Erdős' Book

Proofs of special beauty to touch: Zagier's windmills, the museum guards and Buffon's needle.

3 appsOpen collection

Gezupfte Saite: Wo man zupft, bestimmt den Klang

Ein Radio aus Mathematik: Koeffizienten herausfiltern

Fourier to hear and see

How oscillations break down: coefficients, symmetry, phasor chains, the Gibbs phenomenon and the plucked string.

7 appsOpen collection

Understanding programming

What happens inside the machine: objects, methods, operators, slicing and sorting in Python, visible step by step.

9 appsOpen collection

Benz Patent-Motorwagen 1886 — der Viertakt im Schnitt (3D)

Erster Dieselmotor 1897 — Zündung durch Verdichtung (3D)

Masterpieces of engineering

From the Patent Motor Car to the Z3: famous machines and experiments cut open in 3D.

11 appsOpen collection

Understanding and playing music

Keyboard, intervals, scales, chords, the circle of fifths and groove: music theory you hear and play.

9 appsOpen collection

Klanglabor: Synthesizer, Sampler, Sounddesign

Sound & sound design

How sound is made and shaped: synthesizers and samplers, envelopes and overtones, waveforms and timbre, voice and orchestra, room acoustics and sound scenography — all audible in your browser.

16 appsOpen collection

Scenography

Staging space, sight, light, sound and paths: visitor flows, showcase lighting, room acoustics, sound islands, projection mapping and the pavilions of world expos to try out.

15 appsOpen collection

Die Ebene ist nicht fünffärbbar (Hadwiger–Nelson)

Gaußscher Graben — kein Weg ins Unendliche

Open Problems, Solved by AI

Mathematical questions open for decades, for which an AI model proposed solutions in 2026, as interactive apps: understand the problem, experience the result.

32 appsOpen collection

AI and Machine Learning

From neurons and gradients to transformers, RLHF and agents: machine learning you can touch.

44 appsOpen collection

Molecular Biology

DNA, replication, transcription, translation and cell division as explorable 3D models.

24 appsOpen collection

Art and Architecture

Perspective, ornament, structure and space: how art and architecture work with mathematics.

21 appsOpen collection

Autonome Verkehre in logistischen Systemen (3D)

Modern Logistics

Hyperloop, autonomous vehicles, warehouses and conveyors: logistics as interactive simulation.

17 appsOpen collection

Medicine and Anatomy

Heart, eye, voice, skeleton and nerves: the human body in 3D, layer by layer.

10 appsOpen collection

Statistical Traps

Monty Hall, Simpson, Berkson and base rates: where intuition fails us with numbers.

14 appsOpen collection

Schwarzes Loch — Gravitationslinse und Schatten (3D)

Astronomy and Cosmos

Planets, stars, black holes and space probes: the cosmos to explore.

10 appsOpen collection

Kanalrhodopsin: ein Lichtschalter für Nervenzellen

Optogenetik: Zelltypen, Maus und Sehhilfe

Neutrinos: Geisterteilchen, die durch die Erde fliegen

Nobel Prizes 2026

Light switches for nerve cells, a telescope made of ice, mirror-image chemistry and interest rates: the 2026 Nobel Prizes to try out.

8 appsOpen collection

Asch-Linien: Die Mehrheit irrt mit Absicht

Zuschauer-Effekt: Alle warten, keiner hilft

Deceived in the Lab

Planted participants, smoke in the room, one dollar for a lie and experimenters who are themselves the subjects: famous social psychology experiments to try yourself.

5 appsOpen collection