Zankapfel
03-18-2011, 01:31 AM
Impossible objects are a type of optical illusion involving ambiguous visual descriptions of figures that cannot physically exist.
It is shown by way of example that such objects can be further developed using standard fractal techniques to create new and more complex designs that retain the perceptual illusion, sometimes allowing additional illusions to emerge from the process.
The balanced Pythagorean tree is used to efficiently render impossible fractals that display the perceptual effect across decreasing levels of scale.
So, combining three of our favourite things -geometry, maths and art- here are some very fun "Impossible Fractals".
http://www.cameronius.com/graphics/impossible-fractals-figures/impossible-fractals-fig-1.jpg
The tri-bar, the Koch snowflake and the Sierpinski gasket.
http://www.cameronius.com/graphics/impossible-fractals-figures/impossible-fractals-fig-2.jpg
Two iterations of an impossible snowflake (with acute and obtuse generators shown).
http://www.cameronius.com/graphics/impossible-fractals-figures/impossible-fractals-fig-3.jpg
An alternative snowflake design that emphasizes the perceptual effect (with generator shown).
http://www.cameronius.com/graphics/impossible-fractals-figures/impossible-fractals-fig-4.jpg
Impossible gaskets are more troublesome.
http://www.cameronius.com/graphics/impossible-fractals-figures/impossible-fractals-fig-5.jpg
The Devil's fork and the Cantor set.
http://www.cameronius.com/graphics/impossible-fractals-figures/impossible-fractals-fig-6.jpg
The Devil's gatling gun (2-level), timber offcuts (2-level) and comb (3-level).
http://www.cameronius.com/graphics/impossible-fractals-figures/impossible-fractals-fig-7.jpg
4-bar designs (with sharp and truncated generators) applied to the square Sierpinski curve.
http://www.cameronius.com/graphics/impossible-fractals-figures/impossible-fractals-fig-11.jpg
An impossible multibar Peano-Gosper curve.
http://www.cameronius.com/graphics/impossible-fractals-figures/impossible-fractals-fig-12.jpg
Reutersvärd's "Meander" and a Hilbert meander.
http://www.cameronius.com/graphics/impossible-fractals-figures/impossible-fractals-fig-15.jpg
An impossible fern (balanced 30° Pythagorean tree).
http://www.cameronius.com/graphics/impossible-fractals-figures/impossible-fractals-fig-16.jpg
A spiral tri-bar and hexagonally bound isometric spirals.
Credit: Cameron Bolitho Browne, Bachelor of Arts in Computer Science.
It is shown by way of example that such objects can be further developed using standard fractal techniques to create new and more complex designs that retain the perceptual illusion, sometimes allowing additional illusions to emerge from the process.
The balanced Pythagorean tree is used to efficiently render impossible fractals that display the perceptual effect across decreasing levels of scale.
So, combining three of our favourite things -geometry, maths and art- here are some very fun "Impossible Fractals".
http://www.cameronius.com/graphics/impossible-fractals-figures/impossible-fractals-fig-1.jpg
The tri-bar, the Koch snowflake and the Sierpinski gasket.
http://www.cameronius.com/graphics/impossible-fractals-figures/impossible-fractals-fig-2.jpg
Two iterations of an impossible snowflake (with acute and obtuse generators shown).
http://www.cameronius.com/graphics/impossible-fractals-figures/impossible-fractals-fig-3.jpg
An alternative snowflake design that emphasizes the perceptual effect (with generator shown).
http://www.cameronius.com/graphics/impossible-fractals-figures/impossible-fractals-fig-4.jpg
Impossible gaskets are more troublesome.
http://www.cameronius.com/graphics/impossible-fractals-figures/impossible-fractals-fig-5.jpg
The Devil's fork and the Cantor set.
http://www.cameronius.com/graphics/impossible-fractals-figures/impossible-fractals-fig-6.jpg
The Devil's gatling gun (2-level), timber offcuts (2-level) and comb (3-level).
http://www.cameronius.com/graphics/impossible-fractals-figures/impossible-fractals-fig-7.jpg
4-bar designs (with sharp and truncated generators) applied to the square Sierpinski curve.
http://www.cameronius.com/graphics/impossible-fractals-figures/impossible-fractals-fig-11.jpg
An impossible multibar Peano-Gosper curve.
http://www.cameronius.com/graphics/impossible-fractals-figures/impossible-fractals-fig-12.jpg
Reutersvärd's "Meander" and a Hilbert meander.
http://www.cameronius.com/graphics/impossible-fractals-figures/impossible-fractals-fig-15.jpg
An impossible fern (balanced 30° Pythagorean tree).
http://www.cameronius.com/graphics/impossible-fractals-figures/impossible-fractals-fig-16.jpg
A spiral tri-bar and hexagonally bound isometric spirals.
Credit: Cameron Bolitho Browne, Bachelor of Arts in Computer Science.