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Volume 2, Number 45

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10 November 1997                                Vol. 2, No. 45


Origami & Tessellations | Erich's Packing Center | Swim & Walk


                   ORIGAMI MATH: THOMAS HULL

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."


                   ALEX BATEMAN'S ORIGAMI PAGE

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

An extensive illustrated collection of modular, tessellating,
and mathematical origami: tessellation techniques, twists,
and origami weave tessellations; perimeter problem,
hyperbolic parabola, and spherical origami.


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:



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

    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.



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