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[Click here for another Shiny app on the Chaos Game]

[Click here for another Shiny app on the Chaos Game]

In the two dimensional version of the
Chaos Game
we start with a regular polygon and mark selected
points which will typically be the vertices. These points will be called
*endpoints*
and will be marked in red. The game begins by randomly choosing a
starting point and one of the endpoints.
Mark a new point at a fixed distance ratio from the starting point to the endpoint (e.g.,
halfway to the endpoint). Select another endpoint at random and,
with the most recently created point, repeat the process to generate the next point
and continue. By applying the right distance ratio the resulting set of points can converge
to a beautiful image known as a
*fractal*
. For each polygon the required
distance ratio to yield a fractal will be provided,
but try different settings to see what other patterns may arise!

For
**Triangle**
default value is 0.50

For
**Square**
default value is 0.67 (2/3)

For
**Pentagon**
default value is 0.63 (5/8)

For
**Hexagon**
default value is 0.67 (2/3)

For
**Tetrahedron**
default value is 0.50

For
**Cube**
default value is 0.33

For
**Dodecahedron**
default value is 0.38

Click
**Randomize**
to re-randomize outcomes based on current settings.

Shiny app by
Jimmy Doi

Base R code by
Jimmy Doi

Shiny source files:
GitHub Gist

This plot shows a step-by-step progression of the chaos game. At each step the randomly chosen red endpoint will be marked. Use the slider on the right and press the ▶ button to animate the plot. Advance the slider to n=100 before moving to the 'Extended Sequence' tab.

Press the
▶
button above to animate.

Advance slider manually at anytime.

Click**Randomize**
to re-randomize at current n.

Advance slider manually at anytime.

Click

Compare the plot above when n=100 to the plot when n=100 from the 'Initial Sequence' tab – they should be identical. This shows the plot shown here is simply a continuation of the chaos game. Advance the slider to n=1000 before moving to the 'Complete Sequence' tab.

Advance slider manually at anytime.

Click

Compare the plot above when n=1000 to the plot when n=1000 from the 'Extended Sequence' tab – they should be identical. This again shows the plot shown here is a continuation of the chaos game. Smaller plotting points are used to reveal the finer details of the completed plot.

Advance slider manually at anytime.

Click