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Home > Gwen L. Fisher :: Woven Beads

"Coordinate Axis, Highly Unlikely Square and Highly Unlikely Triangle," by Gwen Fisher (

Materials: seed beads, Nymo nylon thread

These three pieces are woven from box stitch (also known as 3D right angle weave), which essentially takes the Cartesian tiling of 3-space with cubes, and places one bead on each edge of some subset of the tiling. With box stitch, rows and columns of cubes (attached face to face) can be woven into any continuous arrangement. Then, I add extra beads at the vertices of each cube to give the object more structure and decoration. The Coordinate Axis shows the basic structure of box stitch, and is also suitable for a game of children's Jacks. The Highly Unlikely Square and Triangle are beaded versions of the Impossible Triangle of Roger Penrose that was made famous by M.C. Escher. Compared with a regular square frame or triangular frame like you might hang on your wall, these frames have one quarter turn on each side. To see the effect of these twists, imagine painting a regular square frame with four colors to identify four paths: inside, outside, front and back. A similar coloring on the Highly Unlikely Square identifies four paths or faces, one of which is outlined with gold seed beads. Starting at the corner closest to the camera traveling clockwise, the golden face is outside, back, inside, front. In fact, all four faces are congruent. The effect of the quarter turns on the Highly Unlikely Triangle is different; there is only one face that travels around the triangle four times. -- Gwen Fisher (

Group.jpg Mobius2.jpg Mobius1.jpg OctaCluster.jpg sierp-5.jpg

American Mathematical Society