Windmills of Your Mind Spirals/Circles

Ben Sparks, Feb 8, 2018

Some of the resources used in Ben's talk "Windmills of Your Mind" The Sunflower Spirals is also used in a Numberphile video on the Golden Ratio. Surd Spiral Mandelbrot Orbits Sunflower spirals

Table of Contents

  1. Intro
    1. Circles within circles - Trigonometry
  2. Surd Spiral (Theodorus)
    1. Surd Spiral (Spiral of Theodorus)
  3. Complex Spiral (Mandelbrot)
    1. Mandelbrot Painter
    2. Mandelbrot Iteration Orbits
  4. Sunflower Spiral (Phi)
    1. Flower (dark)

1.

Intro

  • 1. Circles within circles - Trigonometry

1.1.

Circles within circles - Trigonometry

What do the projections of a point on a circle look like?[br]Try:[br]Adjust the slider to '2 points' - the inner points follow sine and cosine waves.[br]Adjust the slider to '3 points'.[br]Go higher.[br]Experiment by turning the paths and outer circles on or off
Try:[br]Adjust the slider to '2 points' - the inner points follow sine and cosine waves.[br]Adjust the slider to '3 points'.[br]Go higher.[br]Experiment by turning the paths and outer circles on or off
– Ben Sparks

2.

Surd Spiral (Theodorus)

  • 1. Surd Spiral (Spiral of Theodorus)

2.1.

Surd Spiral (Spiral of Theodorus)

Spiral of Theodorus
– Ben Sparks

3.

Complex Spiral (Mandelbrot)

  • 1. Mandelbrot Painter
  • 2. Mandelbrot Iteration Orbits

3.1.

Mandelbrot Painter

Drag point 'C' around, to 'paint' a picture of the mandelbrot set - black points are where the iteration is stable, dark blue where unstable. [br](It's a bit like doing a 'brass rubbing')[br]You can see the true picture by clicking 'Show Picture'
Mandelbrot Painter
– Ben Sparks

3.2.

–

4.

Sunflower Spiral (Phi)

  • 1. Flower (dark)

4.1.

Flower (dark)

– Ben Sparks

Information