The connection between mathematics and
art goes back thousands of years. Mathematics has been
used in the design of Gothic cathedrals, Rose windows,
oriental rugs, mosaics and tilings. Geometric forms were
fundamental to the cubists and many abstract expressionists,
and award-winning sculptors have used topology as the
basis for their pieces. Dutch artist M.C. Escher represented
infinity, Möbius ands, tessellations, deformations,
reflections, Platonic solids, spirals, symmetry, and
the hyperbolic plane in his works.

Mathematicians and artists continue to
create stunning works in all media and to explore the
visualization of mathematics--origami, computer-generated
landscapes, tesselations, fractals, anamorphic art, and
more.

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"Sierpinski Tetrahedron (View I)" in glass bugle beads, size 11/0 and 8/0 seed beads, Fireline thread, by Gwen L. Fisher, California Polytechnic State University, San Luis Obispo and beAd Infinitum There are several ways to build a polyhedron with beads. One technique that will always work is to align the hole of a bead along each edge of the polyhedron. Then, the thread connects the beads at the vertices of the polyhedron. The most stable polyhedron is the tetrahedron because it is made of all triangles. In a beaded tetrahedron, there are three sets of beads in each loop, like the three sides of a triangle. Any regular tetrahedral beaded bead will naturally require six identical sets of beads, one set for each of the six edges of the tetrahedron. In this case, a set is three beads: a short, a long and a short. Rather than give an example of the simplest tetrahedron, I have used a more complex design based on the structure resulting from the third iteration in the construction of the "Sierpinski Tetrahedron" with its 64 little tetrahedrons. Adding a bead at each interior vertex is necessary to stabilize the structure and make it more rigid. --- Gwen L. Fisher (www.beadinfinitum.com)

"Sierpinski Tetrahedron (View II)" in glass bugle beads, size 11/0 and 8/0 seed beads, Fireline thread, by Gwen L. Fisher, California Polytechnic State University, San Luis Obispo and beAd Infinitum There are several ways to build a polyhedron with beads. One technique that will always work is to align the hole of a bead along each edge of the polyhedron. Then, the thread connects the beads at the vertices of the polyhedron. The most stable polyhedron is the tetrahedron because it is made of all triangles. In a beaded tetrahedron, there are three sets of beads in each loop, like the three sides of a triangle. Any regular tetrahedral beaded bead will naturally require six identical sets of beads, one set for each of the six edges of the tetrahedron. In this case, a set is three beads: a short, a long and a short. Rather than give an example of the simplest tetrahedron, I have used a more complex design based on the structure resulting from the third iteration in the construction of the "Sierpinski Tetrahedron" with its 64 little tetrahedrons. Adding a bead at each interior vertex is necessary to stabilize the structure and make it more rigid. --- Gwen L. Fisher (www.beadinfinitum.com)

"Octahedral Cluster" in white opalite glass, seed beads, Nymo nylon thread, by Gwen L. Fisher, California Polytechnic State University, San Luis Obispo, and beAd Infinitum. Copyright 2005 by Gwen L. Fisher.The regular octahedron has 8 triangular faces, 6 vertices each of valence 4, and 12 edges. These 12 edges correspond to the 12 largest (white) beads in the Octahedral Cluster beaded bead. The 6 vertices of the octahedron appear as 6 stars with 4 points each. The 8 triangular faces correspond to where the points of the stars meet. This beaded bead is hollow, yet structurally stable. The stability comes from the way the small beads fit snugly into the spaces between the larger beads. The beaded bead shows virtually no thread; only beads are visible. It is springy between the fingertips and reforms its shape remarkably well. When free from compression, it is round from every angle.
-- Gwen L. Fisher (www.beadinfinitum.com)

"Mobius Frame with 2 Holes (View II)" in seed beads, Nymo nylon thread, by Gwen L. Fisher, California Polytechnic State University, San Luis Obispo, and beAd InfinitumThis Mobius Frame is 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. This Mobius Frame represents two distinct mathematical objects. First, one can view this object as assembled from cube (or cube-like) shapes. We might be tempted to try to build such an object from wood using three long beams and two short beams. However, like the Impossible Triangle, this Mobius Frame cannot be built in 3D using all straight lines and right angles. In connecting the beams at their ends, the sides of the beams need to twist. The flexibility of the thread connecting the beads allows the beaded frame to twist to accommodate the necessary turns to build this object in 3D. The second way to view this object is to see it as a patch of an infinite surface with no thickness and two holes. Since the surface has no thickness, ignore the layer of purple beads in the middle. The blue and green coloring of the largest faces shows that this surface has two distinct faces. --- Gwen L. Fisher (www.beadinfinitum.com)

"Mobius Frame with 2 Holes (View I)" in seed beads, Nymo nylon thread, by Gwen L. Fisher, California Polytechnic State University, San Luis Obispo, and beAd InfinitumThis Mobius Frame is 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. This Mobius Frame represents two distinct mathematical objects. First, one can view this object as assembled from cube (or cube-like) shapes. We might be tempted to try to build such an object from wood using three long beams and two short beams. However, like the Impossible Triangle, this Mobius Frame cannot be built in 3D using all straight lines and right angles. In connecting the beams at their ends, the sides of the beams need to twist. The flexibility of the thread connecting the beads allows the beaded frame to twist to accommodate the necessary turns to build this object in 3D. The second way to view this object is to see it as a patch of an infinite surface with no thickness and two holes. Since the surface has no thickness, ignore the layer of purple beads in the middle. The blue and green coloring of the largest faces shows that this surface has two distinct faces. --- Gwen L. Fisher (www.beadinfinitum.com)

"Coordinate Axis, Highly Unlikely Square and Highly Unlikely triangle" in seed beads, Nymo nylon thread, by Gwen L. Fisher, California Polytechnic State University, San Luis Obispo, and beAd InfinitumThese 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 L. Fisher (www.beadinfinitum.com)

"Sierpinski Tetrahedron (View III)" in glass bugle beads, size 11/0 and 8/0 seed beads, Fireline thread, by Gwen L. Fisher, California Polytechnic State University, San Luis Obispo and beAd Infinitum There are several ways to build a polyhedron with beads. One technique that will always work is to align the hole of a bead along each edge of the polyhedron. Then, the thread connects the beads at the vertices of the polyhedron. The most stable polyhedron is the tetrahedron because it is made of all triangles. In a beaded tetrahedron, there are three sets of beads in each loop, like the three sides of a triangle. Any regular tetrahedral beaded bead will naturally require six identical sets of beads, one set for each of the six edges of the tetrahedron. In this case, a set is three beads: a short, a long and a short. Rather than give an example of the simplest tetrahedron, I have used a more complex design based on the structure resulting from the third iteration in the construction of the "Sierpinski Tetrahedron" with its 64 little tetrahedrons. Adding a bead at each interior vertex is necessary to stabilize the structure and make it more rigid. --- Gwen L. Fisher (www.beadfinitum.com)

"Sierpinski Tetrahedron (View IV)" in glass bugle beads, size 11/0 and 8/0 seed beads, Fireline thread, by Gwen L. Fisher, California Polytechnic State University, San Luis Obispo and beAd Infinitum There are several ways to build a polyhedron with beads. One technique that will always work is to align the hole of a bead along each edge of the polyhedron. Then, the thread connects the beads at the vertices of the polyhedron. The most stable polyhedron is the tetrahedron because it is made of all triangles. In a beaded tetrahedron, there are three sets of beads in each loop, like the three sides of a triangle. Any regular tetrahedral beaded bead will naturally require six identical sets of beads, one set for each of the six edges of the tetrahedron. In this case, a set is three beads: a short, a long and a short. Rather than give an example of the simplest tetrahedron, I have used a more complex design based on the structure resulting from the third iteration in the construction of the "Sierpinski Tetrahedron" with its 64 little tetrahedrons. Adding a bead at each interior vertex is necessary to stabilize the structure and make it more rigid. --- Gwen L. Fisher (www.beadfinitum.com)

"Sierpinski Tetrahedron (View V)" in glass bugle beads, size 11/0 and 8/0 seed beads, Fireline thread, by Gwen L. Fisher, California Polytechnic State University, San Luis Obispo and beAd Infinitum There are several ways to build a polyhedron with beads. One technique that will always work is to align the hole of a bead along each edge of the polyhedron. Then, the thread connects the beads at the vertices of the polyhedron. The most stable polyhedron is the tetrahedron because it is made of all triangles. In a beaded tetrahedron, there are three sets of beads in each loop, like the three sides of a triangle. Any regular tetrahedral beaded bead will naturally require six identical sets of beads, one set for each of the six edges of the tetrahedron. In this case, a set is three beads: a short, a long and a short. Rather than give an example of the simplest tetrahedron, I have used a more complex design based on the structure resulting from the third iteration in the construction of the "Sierpinski Tetrahedron" with its 64 little tetrahedrons. Adding a bead at each interior vertex is necessary to stabilize the structure and make it more rigid. --- Gwen L. Fisher (www.beadfinitum.com)