Also known, for obvious reasons, as the Sierpenski Triangle, the Sierpenski Gasket was first described by Waclaw Sierpenski (1882-1920) in 1902. It is related to the so-called Monster Curves because it has finite area but infinite perimeter. For years, mathematicians considered these shapes to be annoying curiosities but eventually they became a sort of grandfather to all fractal geometry. The shape includes delicately interspersed triangles-within-triangles, and the repeating (iterative) patterns continues “downward” as though your point-of-view could spiral down to infinity. With each spiral, more detail is revealed. Sierpenski Gaskets are formed as seen below.
They are also very similar to the four-sided Sierpenski Carpet and the 3-dimensional Sierpenski Pyramid.
This piece is based on variations of the basic Sierpenski Gasket as sketched by myself and others. The most important sketch is represented on the cover (computer graphics by Yuedong Merritt). The form of this piece represents a trip downward through the top triangle, with each repetition of material 1/3 smaller and tighter. In a sense, the piece is a Prelude, Toccata, & Rhapsody in 3 1/3 iterations. The first section (repeated 3 times) is a fantastical, chromatic bit of fireworks that uses super-fast scales, trills, and elaborate ornaments. The second section (repeated twice) is a finger-busting toccata that spans the keyboard. The final section is an homage to the pianism of the high romantic in which I attempt to “one-up” Liszt & Rachmaninoff. Each section is quite different, and strictly divided – with few attempts to blend or develop the materials together – as though they each formed one side of a triangle. Each repetition represents a new and smaller triangle, finally disappearing into infinity.
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