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"Green Squares (from series, the Mathematical Basis of the Arts) (ca. 1934)" by Joseph Schillinger is a meticulously structured composition that exemplifies the intersection of mathematical precision and artistic expression. This piece is part of Schillinger’s broader exploration into the geometrical and mathematical underpinnings of art. The painting features an arrangement of green squares that are expertly aligned and proportioned, forming a visual rhythm that draws the viewer into its systematic yet dynamic order.

The painting primarily consists of green squares of various shades and sizes, arranged in a pattern that suggests both movement and stability. The central motif uses a precise grid system to create a harmonious and balanced composition. The different tones of green range from deep forest hues to lighter, almost lime shades, creating a sense of depth and complexity. The artist uses subtle variations in color intensity and square size to guide the eye through the composition, ensuring that each element feels both individually significant and integral to the overall design.

1. Location: The original painting "Green Squares" (ca. 1934) by Joseph Schillinger is part of the collection at the Smithsonian American Art Museum in Washington, D.C. This prestigious location ensures the preservation and accessibility of Schillinger’s influential work.

2. Artist Background: Joseph Schillinger (1895-1943) was a Ukrainian-American artist, composer, music theorist, and mathematician. His interdisciplinary preoccupations made him a unique figure in both the art and music worlds. Schillinger's most notable contribution is the Schillinger System of Musical Composition, which applied mathematical principles to music theory.

3. Artistic Movement and Career: Schillinger's work, including "Green Squares," fits into the wider context of early 20th-century movements that sought to meld science and art, such as Constructivism, De Stijl, and the Bauhaus movement. His paintings often reflect a fascination with order, symmetry, and the application of mathematical rules to create aesthetically pleasing compositions. This painting is a clear representation of Schillinger's belief in a rational basis for art, aligning closely with his theories on the Mathematical Basis of the Arts.

4. Noteworthy Details:
- Technique: Schillinger's methodical approach is evident in the precise execution of "Green Squares." Each square is carefully placed to conform to the overall mathematical logic of the piece.
- Conceptual Framework: The painting exemplifies Schillinger's broader theory that all forms of art can be understood and created through mathematical laws. This is not just a visual artwork but also an intellectual proposition about the nature of creativity.
- Legacy: While Schillinger is often more recognized for his contributions to music theory, his visual artworks are crucial in understanding his interdisciplinary approach. "Green Squares" serves as a visual testament to his belief in the unity of artistic and mathematical principles.

In summary, "Green Squares" (ca. 1934) is a significant work by Joseph Schillinger that showcases his commitment to finding the mathematical basis of artistic expression. It stands as a historical testament to the early 20th century's avant-garde movements and is preserved as part of the Smithsonian American Art Museum's collection.