Swirl Ball
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A few months back, I saw a 2D illustration in Mathographics that was begging to be expanded into 3D.
A spiral from Robert Dixon’s Mathographics. It wanted to be 3D.
Here’s my “spin” on that expansion. First, I started with a simple circle.
Only two shapes matter: triangles and circles.
/usr/lib/ruby/vendor_ruby/rubygems/dependency.rb:315:in `to_specs': Could not find 'coderay' (>= 0) among 94 total gem(s) (Gem::MissingSpecError) Checked in 'GEM_PATH=/var/www/.local/share/gem/ruby/3.2.0:/var/lib/gems/3.2.0:/usr/local/lib/ruby/gems/3.2.0:/usr/lib/ruby/gems/3.2.0:/usr/lib/x86_64-linux-gnu/ruby/gems/3.2.0:/usr/share/rubygems-integration/3.2.0:/usr/share/rubygems-integration/all:/usr/lib/x86_64-linux-gnu/rubygems-integration/3.2.0:/home/johnch/.gems' , execute `gem env` for more information from /usr/lib/ruby/vendor_ruby/rubygems/dependency.rb:327:in `to_spec' from /usr/lib/ruby/vendor_ruby/rubygems/core_ext/kernel_gem.rb:57:in `gem' from ./coderay:24:in `'
I anticipated doing a lot of plotting of polar coordinates, so I factored out a helper function.
/usr/lib/ruby/vendor_ruby/rubygems/dependency.rb:315:in `to_specs': Could not find 'coderay' (>= 0) among 94 total gem(s) (Gem::MissingSpecError) Checked in 'GEM_PATH=/var/www/.local/share/gem/ruby/3.2.0:/var/lib/gems/3.2.0:/usr/local/lib/ruby/gems/3.2.0:/usr/lib/ruby/gems/3.2.0:/usr/lib/x86_64-linux-gnu/ruby/gems/3.2.0:/usr/share/rubygems-integration/3.2.0:/usr/share/rubygems-integration/all:/usr/lib/x86_64-linux-gnu/rubygems-integration/3.2.0:/home/johnch/.gems' , execute `gem env` for more information from /usr/lib/ruby/vendor_ruby/rubygems/dependency.rb:327:in `to_spec' from /usr/lib/ruby/vendor_ruby/rubygems/core_ext/kernel_gem.rb:57:in `gem' from ./coderay:24:in `'
Next up was shaping the circle into a lobed, flowery shape. I fed the angle into the sine function to get an undulating radius as I traversed the circle’s perimeter.
The circle bloomed into a flower. It was a sine.
/usr/lib/ruby/vendor_ruby/rubygems/dependency.rb:315:in `to_specs': Could not find 'coderay' (>= 0) among 94 total gem(s) (Gem::MissingSpecError) Checked in 'GEM_PATH=/var/www/.local/share/gem/ruby/3.2.0:/var/lib/gems/3.2.0:/usr/local/lib/ruby/gems/3.2.0:/usr/lib/ruby/gems/3.2.0:/usr/lib/x86_64-linux-gnu/ruby/gems/3.2.0:/usr/share/rubygems-integration/3.2.0:/usr/share/rubygems-integration/all:/usr/lib/x86_64-linux-gnu/rubygems-integration/3.2.0:/home/johnch/.gems' , execute `gem env` for more information from /usr/lib/ruby/vendor_ruby/rubygems/dependency.rb:327:in `to_spec' from /usr/lib/ruby/vendor_ruby/rubygems/core_ext/kernel_gem.rb:57:in `gem' from ./coderay:24:in `'
Next I wanted a stack of flowers. However, I like abstraction a lot, so I made a flower function first.
/usr/lib/ruby/vendor_ruby/rubygems/dependency.rb:315:in `to_specs': Could not find 'coderay' (>= 0) among 94 total gem(s) (Gem::MissingSpecError) Checked in 'GEM_PATH=/var/www/.local/share/gem/ruby/3.2.0:/var/lib/gems/3.2.0:/usr/local/lib/ruby/gems/3.2.0:/usr/lib/ruby/gems/3.2.0:/usr/lib/x86_64-linux-gnu/ruby/gems/3.2.0:/usr/share/rubygems-integration/3.2.0:/usr/share/rubygems-integration/all:/usr/lib/x86_64-linux-gnu/rubygems-integration/3.2.0:/home/johnch/.gems' , execute `gem env` for more information from /usr/lib/ruby/vendor_ruby/rubygems/dependency.rb:327:in `to_spec' from /usr/lib/ruby/vendor_ruby/rubygems/core_ext/kernel_gem.rb:57:in `gem' from ./coderay:24:in `'
I wanted the stack of flowers eventually to shape themselves into a ball, so it made sense to me to walk the altitudes from -90 to 90. At the extremes -90 and 90, I wanted more flowers because the radius changes faster at the ends and I needed to capture the detail. So, I fed the azimuth angle into another sine function to calculate a flower slice’s z-coordinate.
A stack of flowers, with more detail at the ends.
/usr/lib/ruby/vendor_ruby/rubygems/dependency.rb:315:in `to_specs': Could not find 'coderay' (>= 0) among 94 total gem(s) (Gem::MissingSpecError) Checked in 'GEM_PATH=/var/www/.local/share/gem/ruby/3.2.0:/var/lib/gems/3.2.0:/usr/local/lib/ruby/gems/3.2.0:/usr/lib/ruby/gems/3.2.0:/usr/lib/x86_64-linux-gnu/ruby/gems/3.2.0:/usr/share/rubygems-integration/3.2.0:/usr/share/rubygems-integration/all:/usr/lib/x86_64-linux-gnu/rubygems-integration/3.2.0:/home/johnch/.gems' , execute `gem env` for more information from /usr/lib/ruby/vendor_ruby/rubygems/dependency.rb:327:in `to_spec' from /usr/lib/ruby/vendor_ruby/rubygems/core_ext/kernel_gem.rb:57:in `gem' from ./coderay:24:in `'
Now I shrunk the flowers according to the altitude. At the extremes -90 and 90, I wanted the flowers to be small. At 0, I wanted the flower to be its full size. Cosine to the rescue!
The flower stack, with its slices shrunk according to their altitude.
/usr/lib/ruby/vendor_ruby/rubygems/dependency.rb:315:in `to_specs': Could not find 'coderay' (>= 0) among 94 total gem(s) (Gem::MissingSpecError) Checked in 'GEM_PATH=/var/www/.local/share/gem/ruby/3.2.0:/var/lib/gems/3.2.0:/usr/local/lib/ruby/gems/3.2.0:/usr/lib/ruby/gems/3.2.0:/usr/lib/x86_64-linux-gnu/ruby/gems/3.2.0:/usr/share/rubygems-integration/3.2.0:/usr/share/rubygems-integration/all:/usr/lib/x86_64-linux-gnu/rubygems-integration/3.2.0:/home/johnch/.gems' , execute `gem env` for more information from /usr/lib/ruby/vendor_ruby/rubygems/dependency.rb:327:in `to_spec' from /usr/lib/ruby/vendor_ruby/rubygems/core_ext/kernel_gem.rb:57:in `gem' from ./coderay:24:in `'
Then it was time make a solid. I added a surface solidifier. I stopped interDelta + 1 times going through the azimuth angles and intraDelta + 1 times going through the altitudinal angles.
The flower stack made into a solid.
/usr/lib/ruby/vendor_ruby/rubygems/dependency.rb:315:in `to_specs': Could not find 'coderay' (>= 0) among 94 total gem(s) (Gem::MissingSpecError) Checked in 'GEM_PATH=/var/www/.local/share/gem/ruby/3.2.0:/var/lib/gems/3.2.0:/usr/local/lib/ruby/gems/3.2.0:/usr/lib/ruby/gems/3.2.0:/usr/lib/x86_64-linux-gnu/ruby/gems/3.2.0:/usr/share/rubygems-integration/3.2.0:/usr/share/rubygems-integration/all:/usr/lib/x86_64-linux-gnu/rubygems-integration/3.2.0:/home/johnch/.gems' , execute `gem env` for more information from /usr/lib/ruby/vendor_ruby/rubygems/dependency.rb:327:in `to_spec' from /usr/lib/ruby/vendor_ruby/rubygems/core_ext/kernel_gem.rb:57:in `gem' from ./coderay:24:in `'
Finally, I added a little twist between flower slices.
A twisted flower stack. Dream accomplished.
/usr/lib/ruby/vendor_ruby/rubygems/dependency.rb:315:in `to_specs': Could not find 'coderay' (>= 0) among 94 total gem(s) (Gem::MissingSpecError) Checked in 'GEM_PATH=/var/www/.local/share/gem/ruby/3.2.0:/var/lib/gems/3.2.0:/usr/local/lib/ruby/gems/3.2.0:/usr/lib/ruby/gems/3.2.0:/usr/lib/x86_64-linux-gnu/ruby/gems/3.2.0:/usr/share/rubygems-integration/3.2.0:/usr/share/rubygems-integration/all:/usr/lib/x86_64-linux-gnu/rubygems-integration/3.2.0:/home/johnch/.gems' , execute `gem env` for more information from /usr/lib/ruby/vendor_ruby/rubygems/dependency.rb:327:in `to_spec' from /usr/lib/ruby/vendor_ruby/rubygems/core_ext/kernel_gem.rb:57:in `gem' from ./coderay:24:in `'
Missing from this process are the steps where I made a bunch of mistakes.
I think cornflower blue is pretty.
