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 bands, 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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"Octahedral Cluster," by Gwen Fisher (www.beadinfinitum.com) Copyright 2005 by Gwen L. Fisher.Materials: white opalite glass, seed beads, Nymo nylon thread

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 Fisher (www.beadinfinitum.com)Feb 19, 2016

"Sierpinski Tetrahedron (View V)," by Gwen Fisher (www.beadnfinitum.com) Materials: glass bugle beads, size 11/0 and 8/0 seed beads, Fireline thread

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 Fisher (www.beadfinitum.com)Feb 19, 2016

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 Fisher (www.beadfinitum.com) Feb 19, 2016

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 Fisher (www.beadfinitum.com) Feb 19, 2016

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 Fisher (www.beadinfinitum.com)Feb 19, 2016

"Sierpinski Tetrahedron (View I)," by Gwen Fisher (www.beadinfinitum.com) Materials: glass bugle beads, size 11/0 and 8/0 seed beads, Fireline thread

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 Fisher (www.beadinfinitum.com)Feb 19, 2016

"Flow of complex valued functions (4)," by Anne M. Burns, Long Island University, Brookville, NYInterpreting a complex valued function as a vector field over its domain provides a wealth of opportunities for producing visually appealing images. Starting with a plot of the vector field over a rectangular grid and assigning colors as functions of length and/or direction we easily discern the multiplicities of the zeros and poles of the function. Plotting the vector field of the function only along certain paths, inventing functions for assigning length and color to each vector produces some beautiful images. See more Flows of complex valued functions at http://myweb.cwpost.liu.edu/aburns/flows/flows.html. -- Anne M. BurnsOct 22, 2015

"Flow of complex valued functions (3)," by Anne M. Burns, Long Island University, Brookville, NYInterpreting a complex valued function as a vector field over its domain provides a wealth of opportunities for producing visually appealing images. Starting with a plot of the vector field over a rectangular grid and assigning colors as functions of length and/or direction we easily discern the multiplicities of the zeros and poles of the function. Plotting the vector field of the function only along certain paths, inventing functions for assigning length and color to each vector produces some beautiful images. See more Flows of complex valued functions at http://myweb.cwpost.liu.edu/aburns/flows/flows.html. -- Anne M. BurnsOct 22, 2015

"Flow of complex valued functions (2)," by Anne M. Burns, Long Island University, Brookville, NYInterpreting a complex valued function as a vector field over its domain provides a wealth of opportunities for producing visually appealing images. Starting with a plot of the vector field over a rectangular grid and assigning colors as functions of length and/or direction we easily discern the multiplicities of the zeros and poles of the function. Plotting the vector field of the function only along certain paths, inventing functions for assigning length and color to each vector produces some beautiful images. See more Flows of complex valued functions at http://myweb.cwpost.liu.edu/aburns/flows/flows.html. -- Anne M. BurnsOct 22, 2015

"Flow of complex valued functions (1)," by Anne M. Burns, Long Island University, Brookville, NYInterpreting a complex valued function as a vector field over its domain provides a wealth of opportunities for producing visually appealing images. Starting with a plot of the vector field over a rectangular grid and assigning colors as functions of length and/or direction we easily discern the multiplicities of the zeros and poles of the function. Plotting the vector field of the function only along certain paths, inventing functions for assigning length and color to each vector produces some beautiful images. See more Flows of complex valued functions at http://myweb.cwpost.liu.edu/aburns/flows/flows.html. -- Anne M. BurnsOct 22, 2015

"Circular Celtic Knot," by Gwen Fisher (www.beadinfinitum.com)Materials: size 11° and 15° seed beads and thread. 34 mm diameter.

Beaded knots are quite flexible. You can turn them inside out, and they can be worn as finger rings, of flatten them to make rosettes for pendants or earrings. This knot is a Brunnian link, meaning that if any one component is removed, then the remaining pieces are unlinked unknots. --- Gwen Fisher (www.beadfinitum.com) Oct 22, 2015

"Beaded Super Right Angle Weave Quilt for a Group of Order 18 (detail)," by Gwen Fisher (www.beadinfinitum.com)Materials: seed beads in sizes 8°, 11°, 15°, thread, silk and cotton fabric, and batting. 13 inches square.

This piece began as a study in color for what I call Super Right Angle Weave (SRAW), a bead weave based upon the regular tiling by squares. Each beaded patch is 6 square by 6 squares of the tiling. I weave loops of four beads in each square face and attach these loops across the edges of the tiling with a single bead between the loops. For this set, I use a coloring with three bead types (two types for the faces and one type for the edges). I chose three colors in each of three sizes, for a total of nine different bead types. The 18 patches are arranged in sets of three, where each row uses the same three bead types, but arranged differently. The patches of beadwork correspond to a group of order 18, and Tom Davis identified this group as the generalized dihedral group for E9. --- Gwen Fisher (www.beadfinitum.com)Oct 22, 2015

"Beaded Super Right Angle Weave Quilt for a Group of Order 18," by Gwen Fisher (www.beadinfinitum.com)Materials: seed beads in sizes 8°, 11°, 15°, thread, silk and cotton fabric, and batting. 13 inches square.

This piece began as a study in color for what I call Super Right Angle Weave (SRAW), a bead weave based upon the regular tiling by squares. Each beaded patch is 6 square by 6 squares of the tiling. I weave loops of four beads in each square face and attach these loops across the edges of the tiling with a single bead between the loops. For this set, I use a coloring with three bead types (two types for the faces and one type for the edges). I chose three colors in each of three sizes, for a total of nine different bead types. The 18 patches are arranged in sets of three, where each row uses the same three bead types, but arranged differently. The patches of beadwork correspond to a group of order 18, and Tom Davis identified this group as the generalized dihedral group for E9. --- Gwen Fisher (www.beadfinitum.com)Oct 22, 2015

"Hyperbolic Tiling," by Gwen Fisher (www.beadinfinitum.com)Materials: size 11° seed beads and thread. 63 mm diameter

This is a beaded version of the hyperbolic rhombitetrahexagonal tiling. This tiling is composed of squares and hexagons with three squares and one hexagon around every vertex. I made two of the types of squares green to emphasize the stripes in the tiling. The other type of square is purple, and the hexagons are pink. To make this tiling with bead weaving, I used an across-edge weave. In particular, for the squares, I weaved loops of four beads of the same color for each square, and loops of 6 beads for the hexagons. Then, I attached the loops with one bead per adjacent pair. So the holes of the beads that lie on the edges of the tiling are perpendicular to the the edges. --- Gwen Fisher (www.beadfinitum.com)Oct 22, 2015

"Unlikely Dodecahedron (View 2)," by Gwen Fisher (www.beadinfinitum.com)Materials: size 15° and 11° seed beads and thread. 23 mm on an edge, 58 mm diameter.

I applied the beading techniques I used to turn Impossible Triangles into beaded art objects to a dodecahedron. The Unlikely Dodecahedron generates no corresponding optical illusion. The faces form ten distinct paths that twist around the sculpture in unexpected ways. --- Gwen Fisher (www.beadfinitum.com)Oct 22, 2015