The impossible triangle and other impossible figures are optical illusions that have inspired much artwork since their discovery by Oscar Reutersvärd in the 1930s. This paper shows how I use the impossible triangle and its variations to create a series of sculptures woven with seed beads and thread, using a bead weaving technique known as “cubic right angle weave” (CRAW). The flexibility of the beadwork eliminates the paradox of the optical illusion by substituting curves for the otherwise straight beams. The resulting “highly unlikely” beaded art objects generate surfaces that twist like Möbius bands around the objects, often making several distinct paths, which make interesting colorings possible. The many beaded examples include triangles, squares and frames. The beading techniques were then applied to tetrahedra and a dodecahedron that generate no corresponding optical illusion.