Math · Symmetry, tilings and fractals
Symmetry & Fractals
Reflect, rotate and tile like Escher, fold paper into snowflakes, then zoom into a fractal that never runs out of detail.
The course
7 worlds · 28 lessons
World 1 · Mirror, Turn, Slide
FREEReflection, rotation, translation, glide reflection, and what happens when you combine them- 01Lines of symmetry: the mirror testReflection symmetry and counting axes · 10 min
- 02Rotation symmetry and orderTurning a shape and getting the same shape back · 11 min
- 03Slide and glide: translation, glide reflection, friezesPatterns that repeat along a strip · 12 min
- 04Combining movesTwo reflections make a rotation or a slide, and the idea of a group · 13 min
World 2 · Tiling the Plane
PRORegular and semi-regular tilings, Escher-style tiles, the 17 wallpaper groups and aperiodic tiles- 01Regular tilings: why only three?The 360° rule at every corner · 12 minPRO
- 02Mixing shapes: semi-regular tilingsCorner configurations such as 3.6.3.6 and 4.8.8 · 12 minPRO
- 03Escher-style tilingsReshape a tile with an edge rule and it still tiles · 12 minPRO
- 04The 17 wallpaper groups (and Penrose)Every repeating pattern in the plane belongs to one of 17 types · 14 minPRO
World 3 · Symmetry of Solids
PROThe five Platonic solids, counting every rotation of a cube, and symmetry and handedness in molecules- 01The five Platonic solidsWhy there are exactly five, and Euler’s V − E + F = 2 · 12 minPRO
- 02All the rotations of a cubeCounting the 24 ways a cube can be turned onto itself · 12 minPRO
- 03Tetrahedron, octahedron, dodecahedron, icosahedronHow many symmetries each solid has, and the football · 13 minPRO
- 04Symmetry in molecules and chiralityPoint groups, and why a molecule can have a left and a right hand · 12 minPRO
World 4 · Snowflakes and Kaleidoscopes
PROWhy snow crystals have six arms, how weather shapes them, and how mirrors and folds copy a pattern- 01Why six arms?Hexagonal ice and the six-fold symmetry of snow crystals · 11 minPRO
- 02Temperature, humidity and shapeThe Nakaya diagram, and “no two snowflakes are alike” · 12 minPRO
- 03Kaleidoscopes: mirrors at an angleHow the angle between two mirrors sets the pattern · 12 minPRO
- 04Fold and cut: paper snowflakesFolding makes mirror lines, and mirror lines copy your cut · 12 minPRO
World 5 · The Koch Snowflake
PROSelf-similarity, a perimeter that runs away, an area that stays finite, and Sierpinski’s vanishing triangle- 01Self-similarity and iterationBuild the Koch snowflake from one simple rule · 11 minPRO
- 02The perimeter that runs awayA geometric sequence that grows without limit · 11 minPRO
- 03The area that stays finiteAdding a geometric series: 8/5 of the original · 13 minPRO
- 04Sierpinski: when volume vanishesTriangles with no area, the chaos game, and a pyramid with no volume · 14 minPRO
World 6 · Fractal Dimension and Fractals in Nature
PROMeasuring roughness with a number, why a coastline has no single length, and what nature really does- 01What does dimension mean?Scaling and the formula D = log N ÷ log s · 11 minPRO
- 02Koch, Sierpinski, Menger: non-integer dimensionsThe dimension table · 12 minPRO
- 03How long is a coastline?Box counting, rulers, and a famous paradox · 13 minPRO
- 04Fractals in nature, and the Fibonacci mythsWhat nature really does and what is only a story · 15 minPRO
World 7 · Fly into the Complex Plane
PROOrbits of z² + c, the Mandelbrot set, Julia sets and the 3D Mandelbulb- 01Iterating z → z² + cComplex numbers, orbits, and the escape rule |z| > 2 · 13 minPRO
- 02The Mandelbrot setEscape time, the cardioid and bulbs, and what “self-similar” really means here · 14 minPRO
- 03Julia setsFix c, vary the start, and read the Mandelbrot set as a map · 13 minPRO
- 04Fly into the MandelbulbA 3D cousin of the Mandelbrot set, and how ray marching draws it · 13 minPRO