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The Shakers, whose motto was "Hands to work and hearts to God", began weaving seat bottoms over 100 years ago, in the 1890's, with narrow, hand-woven cotton fabrics (which we now call "Shaker Tape"). A sturdy fabric, available in a variety of delightful colors, is usually woven in a checkerboard or herringbone design, using complementary or contrasting colors for more interest. The seat is typically padded with either a foam cushion or cotton batting for added comfort.
Shaker Tape is an excellent choice for children's chairs, stools, rockers, and ladder back chairs.
The first 3 photos are of a Shaker stool woven in Cedar Red and Beige. (The colors are much more vibrant than these photos would seem to indicate!)
The next 2 photos are of a chair woven in Dark Blue and Veridian Green.