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10 November 1997 Vol. 2, No. 45 THE MATH FORUM INTERNET NEWS Origami & Tessellations | Erich's Packing Center | Swim & Walk THE MATHEMATICS OF PAPER-FOLDING: ORIGAMI AND TESSELLATIONS ORIGAMI MATH: THOMAS HULL http://www.math.uri.edu/~hull/OrigamiMath.html When paper is folded, an origami geometry is at work. Tom Hull provides information on investigations into the mathematics of origami as carried out by mathematicians, scientists, and artists. Contents include: - a tutorial on origami geometric constructions that presents Humiaki Huzita's origami axiom list and compares it to traditional straightedge and compass constructions, with instructions for trisecting angles and doubling cubes - a model: five intersecting tetrahedra - an origami math bibliography - a listing of upcoming origami math events Hull's bibliography on origami geometry and education includes articles from such widely available publications as "Mathematics Teacher" and "Mathematical Intelligencer." http://www.math.uri.edu/~hull/Geombib.html \|/ ALEX BATEMAN'S ORIGAMI PAGE http://www.sanger.ac.uk/~agb/Origami/origami.html Directions for folding Alex Bateman's "square dance," "honeycomb," and "linked circles," with postscript files to be downloaded. Other models by Dino Andreozzi, Nick Robinson, and Edwin Corrie are also provided. HELENA'S ORIGAMI - H. A. VERRILL http://www.mast.queensu.ca/~helena/origami/ An extensive illustrated collection of modular, tessellating, and mathematical origami: tessellation techniques, twists, and origami weave tessellations; perimeter problem, hyperbolic parabola, and spherical origami. PAPER FOLDING BY CHRIS PALMER http://www.cea.edu/sarah/chris/ Palmer displays original folded paper patterns, including double twist grafts and ring twist octagons, and crease patterns such as flower towers and collapse octagons. \|/ For other ORIGAMI sites, use the Math Forum's Quick Search: http://mathforum.org/dumpgrepform.html -|-\-/-|-\-/-|-\-/-|-\-/-|-\-/-|-\-/-|-\-/-|-\-/-|-\-/-|-\-/-|- ERICH'S PACKING CENTER - ERICH FRIEDMAN http://www.stetson.edu/~efriedma/packing.html Erich enjoys packing geometric shapes into other geometric shapes: triangles in squares, circles in triangles, squares in circles, and more, with illustrations, equations, and indications of packings that have been proved optimal. Also: - animations and 3-dimensional packings rectangles in squares circles in rectangles cubes into cubes spheres into cubes packing polyominoes - other packing problems harmonic series of polygons geometric series of polygons tiling squares with squares packing Pythagorean L's - combinatorial geometry problems minimizing maximum rectangles tree planting problems maximizing squares -|-\-/-|-\-/-|-\-/-|-\-/-|-\-/-|-\-/-|-\-/-|-\-/-|-\-/-|-\-/-|- SWIM AND WALK http://www.emsl.pnl.gov:2080/docs/mathexpl/swimwalk.html You are on one bank of a river and want to get to a point on the other bank exactly opposite. You plan to do this first by swimming across (while being swept downstream by the current), then by walking back upstream. You think you might save time by swimming upstream; it will take longer to cross the river, but it should take less time to walk back upstream and you hope to come out ahead overall. Does this actually work? If it does, how much time can you save and under what conditions? Hanford's step-by-step approach spans a wide range of mathematics, starting with time = distance/rate and progressing through the Pythagorean Theorem, graphing, numerical optimization, curve-fitting, and calculus. This problem is presented by the MATH EXPLORATION QUILT, Innovative Mathematics Education from Hanford School, WA. http://www.emsl.pnl.gov:2080/docs/mathexpl/ -|-\-/-|-\-/-|-\-/-|-\-/-|-\-/-|-\-/-|-\-/-|-\-/-|-\-/-|-\-/-|- CHECK OUT OUR WEB SITE: The Math Forum http://mathforum.org/ Ask Dr. Math http://mathforum.org/dr.math/ Problem of the Week http://mathforum.org/geopow/ Internet Resources http://mathforum.org/~steve/ Join the Math Forum http://mathforum.org/join.forum.html Send comments to the Math Forum Internet Newsletter editors _o \o_ __| \ / |__ o _ o/ \o/ __|- __/ \__/o \o | o/ o/__/ /\ /| | \ \ / \ / \ /o\ / \ / \ / | / \ / \ |

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