**Contents:**

- Isometries
- How isometries combine
- the dihedral group

- The braid patterns
- Plane patterns & symmetries
- The 17 plane patterns
- More plane truth
- Archimedean tilings • Coxeter graphs • Wythoff's construction

- Matrix and vector algebra
- Isometries in 3-space
- Quaternions and rotations
- complex numbers • quaternion in-betweening • Bezier curves

- Fractals and nature
- snowflakes • self-similarity • topological dimension • Cantor • Sierpinski

- Basic topology
- metric spaces • subspaces and product spaces • limits and closed sets • continuity • quotient spaces

- Compact sets, connected sets, holes and homeomorphisms
- The existence and uniqueness of fractals
- Hausdorff distance • Fixed Point theorem • winding numbers

- Iterated function systems
- affine maps • factorising rotations into shears • finding an IFS • condensation • attractors

- Addresses, measures, and the random iteration algorithm
- Hausdorff dimension

- Julia, Mandelbrot, and beyond
- dynamics of iterated complex functions