Last additions 
"A cm Pattern from a Minneapolis Skyline, (vertical format)," by Frank A. Farris, Santa Clara University, CAInk jet on paper.
Fantastical samurai appeared when I used a photograph of the Minneapolis skyline on an autumn day in conjunction with wave functions adapted for cm patterns.  Frank A. Farris
Sep 18, 2013


"Land and the Sea," by Daina Taimina (Cornell University, Ithaca, NY), photo © Daina TaiminaInspired by William Thurston's paper creations back in the 1960s, I thought if something can be made out of paper, it can also be crocheted, so I made my first crocheted hyperbolic planes in June 1997 by increasing stitches in constant ratioafter every two stitches I did an increase by one stitch. The number of stitches in each row grew exponentially, so after finishing my first small, very ruffled one I realized that to explore the hyperbolic plane I have to change the ratio of increase. For classroom use the best is to use the ratio 12:13it means to increase one stitch after every 12 single crochet stitches. See more crochet examples on my blog, Daina Taimina Fiber Sculptures  Daina Taimina (Cornell University, Ithaca, NY)Jun 21, 2013


"Manifold II (Bird's Eye View) in memory of Bill Thurston," by Daina Taimina (Cornell University, Ithaca, NY), photo © Daina TaiminaInspired by William Thurston's paper creations back in the 1960s, I thought if something can be made out of paper, it can also be crocheted, so I made my first crocheted hyperbolic planes in June 1997 by increasing stitches in constant ratioafter every two stitches I did an increase by one stitch. The number of stitches in each row grew exponentially, so after finishing my first small, very ruffled one I realized that to explore the hyperbolic plane I have to change the ratio of increase. For classroom use the best is to use the ratio 12:13it means to increase one stitch after every 12 single crochet stitches. See more crochet examples on my blog, Daina Taimina Fiber Sculptures  Daina Taimina (Cornell University, Ithaca, NY)Jun 21, 2013


"Manifold 5," by Daina Taimina (Cornell University, Ithaca, NY), photo © Daina TaiminaInspired by William Thurston's paper creations back in the 1960s, I thought if something can be made out of paper, it can also be crocheted, so I made my first crocheted hyperbolic planes in June 1997 by increasing stitches in constant ratioafter every two stitches I did an increase by one stitch. The number of stitches in each row grew exponentially, so after finishing my first small, very ruffled one I realized that to explore the hyperbolic plane I have to change the ratio of increase. For classroom use the best is to use the ratio 12:13it means to increase one stitch after every 12 single crochet stitches. See more crochet examples on my blog, Daina Taimina Fiber Sculptures  Daina Taimina (Cornell University, Ithaca, NY)Jun 21, 2013


"Manifold II (in tree) in memory of Bill Thurston," by Daina Taimina (Cornell University, Ithaca, NY), photo © Daina TaiminaInspired by William Thurston's paper creations back in the 1960s, I thought if something can be made out of paper, it can also be crocheted, so I made my first crocheted hyperbolic planes in June 1997 by increasing stitches in constant ratioafter every two stitches I did an increase by one stitch. The number of stitches in each row grew exponentially, so after finishing my first small, very ruffled one I realized that to explore the hyperbolic plane I have to change the ratio of increase. For classroom use the best is to use the ratio 12:13it means to increase one stitch after every 12 single crochet stitches. See more crochet examples on my blog, Daina Taimina Fiber Sculptures  Daina Taimina (Cornell University, Ithaca, NY)Jun 21, 2013


Autumn (Thanksgiving), by Daina Taimina (Cornell University, Ithaca, NY), photo @Daina Taimina44x44x29 cm, 20062010, wool, silk
Inspired by William Thurston's paper creations back in the 1960s, I thought if something can be made out of paper, it can also be crocheted, so I made my first crocheted hyperbolic planes in June 1997 by increasing stitches in constant ratioafter every two stitches I did an increase by one stitch. The number of stitches in each row grew exponentially, so after finishing my first small, very ruffled one I realized that to explore the hyperbolic plane I have to change the ratio of increase. For classroom use the best is to use the ratio 12:13it means to increase one stitch after every 12 single crochet stitches. See more crochet examples on my blog, Daina Taimina Fiber Sculptures  Daina Taimina (Cornell University, Ithaca, NY)Jun 21, 2013


"Seven Shades of Purple," by Daina Taimina (Cornell University, Ithaca, NY), photo © Daina TaiminaInspired by William Thurston's paper creations back in the 1960s, I thought if something can be made out of paper, it can also be crocheted, so I made my first crocheted hyperbolic planes in June 1997 by increasing stitches in constant ratioafter every two stitches I did an increase by one stitch. The number of stitches in each row grew exponentially, so after finishing my first small, very ruffled one I realized that to explore the hyperbolic plane I have to change the ratio of increase. For classroom use the best is to use the ratio 12:13it means to increase one stitch after every 12 single crochet stitches. See more crochet examples on my blog, Daina Taimina Fiber Sculptures  Daina Taimina (Cornell University, Ithaca, NY)Jun 21, 2013


"Turtles with Local Symmetry," Frank A. Farris, Santa Clara University, CAInk jet on paper, 2012.
There is no mirror symmetry in this image, only 6fold rotational symmetry. And yet our eye cannot help seeing symmetry in the turtle shapes. This "local symmetry" is the topic of an article, "Local symmetry in wallpaper," in preparation.  Frank A. FarrisJun 21, 2013


"A cmm Pattern from Peppers on a Cutting Board," by Frank A. Farris, Santa Clara University, CAInk jet on paper.
The bright oranges in this cmm pattern come from a photograph of assorted chopped peppers, with collard greens and the glint of the knife as nice contrasts.  Frank A. FarrisJun 21, 2013


"Tiffany Glass from a Mountain Gentian and its Negative," by Frank A. Farris, Santa Clara University, CAInk jet on paper, 2012.
Using a composite photograph of a mountain gentian juxtaposed with its negative to produce an intense contrast of purple with the original green, I used just the right waves to make wallpaper with horizontal mirrors, vertical glides, and 2fold rotational symmetry. In the notation of the International Union of Crystallographers, the symmetry group of this pattern is pmg, but if we allow colorswapping transformations as generalized symmetries, the larger group is cmm, so this pattern type is called cmm/pmg. The "Seeing Symmetry" virtual exhibition at includes more works and details.  Frank A. Farris
Jun 21, 2013


"Turtle, opus 269," by Robert J. LangOne uncut square of Origamido paper, 5", composed 1992, folded 2002
The intersections between origami, mathematics, and science occur at many levels and include many fields of the latter. Origami, like music, also permits both composition and performance as expressions of the art. Over the past 40 years, I have developed nearly 600 original origami compositions. About a quarter of these have been published with folding instructions, which, in origami, serve the same purpose that a musical score does: it provides a guide to the performer (in origami, the folder) while allowing the performer to express his or her own personality through interpretation and variation.  Robert J. LangMay 22, 2013


"Thataway, opus 570," by Robert J. LangOne uncut dollar, 2", composed and folded 2010
The intersections between origami, mathematics, and science occur at many levels and include many fields of the latter. Origami, like music, also permits both composition and performance as expressions of the art. Over the past 40 years, I have developed nearly 600 original origami compositions. About a quarter of these have been published with folding instructions, which, in origami, serve the same purpose that a musical score does: it provides a guide to the performer (in origami, the folder) while allowing the performer to express his or her own personality through interpretation and variation.  Robert J. LangMay 22, 2013


"Rattlesnake, opus 539," by Robert J. LangOne uncut rectangle of Thai unryu paper, 8", composed and folded 2008
The intersections between origami, mathematics, and science occur at many levels and include many fields of the latter. Origami, like music, also permits both composition and performance as expressions of the art. Over the past 40 years, I have developed nearly 600 original origami compositions. About a quarter of these have been published with folding instructions, which, in origami, serve the same purpose that a musical score does: it provides a guide to the performer (in origami, the folder) while allowing the performer to express his or her own personality through interpretation and variation.  Robert J. Lang
May 22, 2013


"QuezadaPot13, opus 590," by Robert J. LangOne uncut triskadecagon of Mexican yucca paper, 8", composed and folded 2009
The intersections between origami, mathematics, and science occur at many levels and include many fields of the latter. Origami, like music, also permits both composition and performance as expressions of the art. Over the past 40 years, I have developed nearly 600 original origami compositions. About a quarter of these have been published with folding instructions, which, in origami, serve the same purpose that a musical score does: it provides a guide to the performer (in origami, the folder) while allowing the performer to express his or her own personality through interpretation and variation.  Robert J. Lang
May 22, 2013


"Praying Mantis, opus 416," by Robert J. LangOne uncut square of paper, 4", composed and folded 2002
The intersections between origami, mathematics, and science occur at many levels and include many fields of the latter. Origami, like music, also permits both composition and performance as expressions of the art. Over the past 40 years, I have developed nearly 600 original origami compositions. About a quarter of these have been published with folding instructions, which, in origami, serve the same purpose that a musical score does: it provides a guide to the performer (in origami, the folder) while allowing the performer to express his or her own personality through interpretation and variation.  Robert J. LangMay 22, 2013


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