Origami and the Catastrophe Theory   http://www.paperfolding.com/chengchit/3d.php  

The origami Perimeter Problem and it's Solution:  
http://hverrill.net/pages~helena/origami/perimeter/

Edward Crankshaw: http://home.hiwaay.net/~ejcranks/arth193b.html   Scroll down to read about origami and math...
Sometimes an overview is good.   Just how and what is Origami to Education?  http://www.users.waitrose.com/~pureland/japanpaper.htm

If you are an elementary school teacher & have not heard of Math in Motion I really recomment it!
Origami and Math:  So much can be said and has for the benefits of using origami to teach math concepts and more.  Because they have said is so well from elementary lesson plans, math in Motion, to complicated models, illustrations, lessons on geometry, & discussions of really advanced math  --  I will list links for you here to start your learnng about origami and math!

Elementary School Art:  Bird in Thirds Lesson for second graders.. check this out!
http://art-smart.ci.manchester.ct.us/how_to/birds_in_thirds.html

Made from squares:  Simple lesson on the whale. Well laid out, benefits listed, good for elementary and small beginner folders!   http://www.mathinmotion.com/whalefld.html   Simple math ....
 
Math In Motion: lists it this way. Quickly easy to see:
http://www.mathinmotion.com/mimedbens.htm
           "As teachers, we get what we ask for.If we ask only for simple numerical answers, children will value onlyprocedures and computational tasks. But, if we ask for discussion, explanation, and elaboration; and if we reward these kinds of answers, then children will value understanding and meaning."--John L. Higgins, Arithmetic Teacher, January 1988 Mathematics 
Develop Shape, Size, and Color Recognition
Develop Geometric Fundamentals
Develop Math Concepts and Vocabulary Develop Symmetry - Congruence - Angles
Develop Fractions - Ratio - Proportion - Measurement
Develop Problem Solving, Analytical and Critical Thinking Skills Investigate 3-Dimensional Objects - Spatial Relationships Explore Patterns and Make Connections

Jim Plank's Origami: Modular - http://www.cs.utk.edu/~plank/plank/pics/origami/origami.html
Diagrams and gallery of many geometric models. 

Geometry Junkyard: Origami - http://www.ics.uci.edu/~eppstein/junkyard/origami.html
Resource listing of links for information about the relationship between origami and geometry. 


Paper Folding - http://www.lwcd.com/paper-folding/
Details of a book which teaches math through origami. 

Teaching Mathematical Thinking Through Origami - http://csis.pace.edu/~meyer/origami/
Suggestions for using origami to teach concepts in mathematics, with diagrams of models.

Wayne's Paper Creations:  http://home.istar.ca/~wko/education_and_origami.html    He writes   "First, there are the obvious geometrical connections such as symmetry, angles, intersections and so on.  There are also other less obvious conceptual connections as well;  I'll try to illustrate with a few simple examples:        For example,  if you fold a paper in half once, the result will be two rectangles when you open it back up  (2^1).   If you fold the paper in half twice and open it up, you end up with 4 rectangles (2^2).  If you fold the paper in half three times you get 8 rectangles (2^3) and so on.  We therefore see a simple power relationship.            However, this also produces an intuitive understanding of the concept that any non-zero base to the power of zero is one!  In our situation, it can be illustrated as how many rectangles will you get if you do not fold the paper in half at all - one rectangle or 2^0=1.          [The other examples are omitted since they require diagrams, or an actual model.  The omitted examples include: some area type puzzles;  using the special 1, 2 and square root of 3 triangle to fold a 60 degree angle; a simple visualization of the Pythagorean Theorem; a simple box involving calculating the volume and then generalizing the volume using algebra and variables]      In summary,  origami can be an extremely useful tool in education and mathematics in particular.  He writes more but I wanted to quote him here.

Meenakshi's Modular Mania - http://www.geocities.com/mmukhopadhyay/origami.html
Image Galleries of original modular models as well as picture links to other designs. 

Modular Origami - http://www.geocities.com/golics/index_en_sum.htm
Instructions, diagrams and pictures of various modular models in English, German, and Russian. 

Paper Models of Polyhedra - http://www.geocities.com/SoHo/Exhibit/5901/
Gallery, diagrams, and VRML of the models. 

Tom's Origami Gallery - http://web.merrimack.edu/~thull/gallery/modgallery.html
Personal gallery of modular and tessellation models. 

Origami Mathematics - http://web.merrimack.edu/~thull/OrigamiMath.html
Mathematics of paper folding; includes a bibliography of articles and journals, Combinatorial geometry syllabus, and a tutorial on geometric constructions. Photo gallery of completed modular, geometric, and tessellation models. 

The Origami Junkyard: http://www.ics.uci.edu/~eppstein/junkyard/origami.html is the listing of names, sites, books and more you will find listed below here under more.  But incase you want to see it for yourself...
More!  Origami
The Japanese art of paper folding is obviously geometrical in nature. Some origami masters have looked at constructing geometric figures such as regular polyhedra from paper. In the other direction, some people have begun using computers to help fold more traditional origami designs. This idea works best for tree-like structures, which can be formed by laying out the tree onto a paper square so that the vertices are well separated from each other, allowing room to fold up the remaining paper away from the tree. Bern and Hayes (SODA 1996) asked, given a pattern of creases on a square piece of paper, whether one can find a way of folding the paper along those creases to form a flat origami shape; they showed this to be NP-complete. Related theoretical questions include how many different ways a given pattern of creases can be folded, whether folding a flat polygon from a square always decreases the perimeter, and whether it is always possible to fold a square piece of paper so that it forms (a small copy of) a given flat polygon. Cranes, planes, and cuckoo clocks. Announcement for a talk on mathematical origami by Robert Lang. Crumpling paper: states of an inextensible sheet. Cut-the-knot logo. With a proof of the origami-folklore that this folded-flat overhand knot forms a regular pentagon.
·          Folding geometry. Wheaton college course project on modular origami.
·          Geometric paper folding. David Huffman.
·          Rona Gurkewitz' Modular Origami Polyhedra Systems Page. With many nice images from two modular origami books by Gurkewitz, Simon, and Arnstein.
·          Knotology. How to form regular polyhedra from folded strips of paper?
·          The Margulis Napkin Problem. Jim Propp asked for a proof that the perimeter of a flat origami figure must be at most that of the original starting square. Gregory Sorkin provides a simple example showing that on the contrary, the perimeter can be arbitrarily large.
·          Mathematical origami, Helena Verrill. Includes constructions of a shape with greater perimeter than the original square, tessellations, hyperbolic paraboloids, and more.
·          A mathematical theory of origami. R. Alperin defines fields of numbers constructible by origami folds.
·          Mostly modular origami. Valerie Vann makes polyhedra out of folded paper.
·          Number patterns, curves, and topology, J. Britton. Includes sections on the golden ratio, conics, Moiré patterns, Reuleaux triangles, spirograph curves, fractals, and flexagons.
·          Origami: a study in symmetry. M. Johnson and B. Beug, Capital H.S.
·          The origami game. Rick Nordal challenges folders to make a sequence of geometric shapes with a single sheet of origami paper as quickly as possible.
·          Origami & math, Eric Andersen.
·          Origami mathematics, Tom Hull, Merrimack.
·          Origami Menger Sponge built from Sonobe modules by K. & W. Burczyk.
·          Origami polyhedra. Jim Plank makes geometric constructions by folding paper squares.
·          Origami tessellations, Alex Bateman.
·          Ozzigami tessellations, papercraft, unfolded peel-n-stick glitter Platonic solids, and more.
·          Paper folding a 30-60-90 triangle. From the geometry.puzzles archives.
·          Paperfolding and the dragon curve. David Wright discusses the connections between the dragon fractal, symbolic dynamics, folded pieces of paper, and trigonometric sums.
·          Paper models of polyhedra.
·          Polyhedra plaited with paper strips, H. B. Meyer. See also Jim Blowers' collection of plaited polyhedra.
·          Rabbit style object on geometrical solid. Complete and detailed instructions for this origami construction, in 3 easy steps and one difficult step.
·          The RUG FTP origami archive contains several papers on mathematical origami.
·          Spidron, a triangulated double spiral shape tiles the plane and various other surfaces. With photos of related paperfolding experiments. Daniel Erdely.  the inventor of the Spidron System (Thanks to Mr. Erdely for alerting us to additions to this entry!) 
·          Spring into action. Dynamic origami. Ben Trumbore, based on a model by Jeff Beynon from Tomoko Fuse's book Spirals.
·          The tea bag problem. How big a volume can you enclose by two square sheets of paper joined at the edges? See also Andrew Kepert's teabag problem page.
·          Unfolding polyhedra. A common way of making models of polyhedra is to unfold the faces into a planar pattern, cut the pattern out of paper, and fold it back up. Is this always possible?
·          
IF you landed here at Origami Designs, & this is your first stop in cyber space seeking data...   Here goes:  Origami  &  Math!!.  There is a load of information here so please scroll down to see it all: including books!
Origami and Math:  So much can be said and has for the benefits of using origami to teach math concepts and more.  Because they have said is so well from elementary lesson plans, math in Motion, to complicated models, illustrations, lessons on geometry, & discussions of really advanced math  --  I will list links for you here to start your learnng about origami and math!

Elementary School Art:  Bird in Thirds Lesson for second graders.. check this out!
http://art-smart.ci.manchester.ct.us/how_to/birds_in_thirds.html

Made from squares:  Simple lesson on the whale. Well laid out, benefits listed, good for elementary and small beginner folders!   http://www.mathinmotion.com/whalefld.html   Simple math ....
 
Math In Motion: lists it this way. Quickly easy to see:
http://www.mathinmotion.com/mimedbens.htm
           "As teachers, we get what we ask for.If we ask only for simple numerical answers, children will value onlyprocedures and computational tasks. But, if we ask for discussion, explanation, and elaboration; and if we reward these kinds of answers, then children will value understanding and meaning."--John L. Higgins, Arithmetic Teacher, January 1988 Mathematics 
Develop Shape, Size, and Color Recognition
Develop Geometric Fundamentals
Develop Math Concepts and Vocabulary Develop Symmetry - Congruence - Angles
Develop Fractions - Ratio - Proportion - Measurement
Develop Problem Solving, Analytical and Critical Thinking Skills Investigate 3-Dimensional Objects - Spatial Relationships Explore Patterns and Make Connections

Jim Plank's Origami: Modular - http://www.cs.utk.edu/~plank/plank/pics/origami/origami.html
Diagrams and gallery of many geometric models. 

Geometry Junkyard: Origami - http://www.ics.uci.edu/~eppstein/junkyard/origami.html
Resource listing of links for information about the relationship between origami and geometry. 


Paper Folding - http://www.lwcd.com/paper-folding/
Details of a book which teaches math through origami. 

Teaching Mathematical Thinking Through Origami - http://csis.pace.edu/~meyer/origami/
Suggestions for using origami to teach concepts in mathematics, with diagrams of models.

Wayne's Paper Creations:  http://home.istar.ca/~wko/education_and_origami.html    He writes   "First, there are the obvious geometrical connections such as symmetry, angles, intersections and so on.  There are also other less obvious conceptual connections as well;  I'll try to illustrate with a few simple examples:        For example,  if you fold a paper in half once, the result will be two rectangles when you open it back up  (2^1).   If you fold the paper in half twice and open it up, you end up with 4 rectangles (2^2).  If you fold the paper in half three times you get 8 rectangles (2^3) and so on.  We therefore see a simple power relationship.            However, this also produces an intuitive understanding of the concept that any non-zero base to the power of zero is one!  In our situation, it can be illustrated as how many rectangles will you get if you do not fold the paper in half at all - one rectangle or 2^0=1.          [The other examples are omitted since they require diagrams, or an actual model.  The omitted examples include: some area type puzzles;  using the special 1, 2 and square root of 3 triangle to fold a 60 degree angle; a simple visualization of the Pythagorean Theorem; a simple box involving calculating the volume and then generalizing the volume using algebra and variables]      In summary,  origami can be an extremely useful tool in education and mathematics in particular.  He writes more but I wanted to quote him here.

Meenakshi's Modular Mania - http://www.geocities.com/mmukhopadhyay/origami.html
Image Galleries of original modular models as well as picture links to other designs. 

Modular Origami - http://www.geocities.com/golics/index_en_sum.htm
Instructions, diagrams and pictures of various modular models in English, German, and Russian. 

Paper Models of Polyhedra - http://www.geocities.com/SoHo/Exhibit/5901/
Gallery, diagrams, and VRML of the models. 

Tom's Origami Gallery - http://web.merrimack.edu/~thull/gallery/modgallery.html
Personal gallery of modular and tessellation models. 

Origami Mathematics - http://web.merrimack.edu/~thull/OrigamiMath.html
Mathematics of paper folding; includes a bibliography of articles and journals, Combinatorial geometry syllabus, and a tutorial on geometric constructions. Photo gallery of completed modular, geometric, and tessellation models. 

The Origami Junkyard: http://www.ics.uci.edu/~eppstein/junkyard/origami.html is the listing of names, sites, books and more you will find listed below here under more.  But incase you want to see it for yourself...
More!  Origami
The Japanese art of paper folding is obviously geometrical in nature. Some origami masters have looked at constructing geometric figures such as regular polyhedra from paper. In the other direction, some people have begun using computers to help fold more traditional origami designs. This idea works best for tree-like structures, which can be formed by laying out the tree onto a paper square so that the vertices are well separated from each other, allowing room to fold up the remaining paper away from the tree. Bern and Hayes (SODA 1996) asked, given a pattern of creases on a square piece of paper, whether one can find a way of folding the paper along those creases to form a flat origami shape; they showed this to be NP-complete. Related theoretical questions include how many different ways a given pattern of creases can be folded, whether folding a flat polygon from a square always decreases the perimeter, and whether it is always possible to fold a square piece of paper so that it forms (a small copy of) a given flat polygon. Cranes, planes, and cuckoo clocks. Announcement for a talk on mathematical origami by Robert Lang. Crumpling paper: states of an inextensible sheet. Cut-the-knot logo. With a proof of the origami-folklore that this folded-flat overhand knot forms a regular pentagon.
·          Folding geometry. Wheaton college course project on modular origami.
·          Geometric paper folding. David Huffman.
·          Rona Gurkewitz' Modular Origami Polyhedra Systems Page. With many nice images from two modular origami books by Gurkewitz, Simon, and Arnstein.
·          Knotology. How to form regular polyhedra from folded strips of paper?
·          The Margulis Napkin Problem. Jim Propp asked for a proof that the perimeter of a flat origami figure must be at most that of the original starting square. Gregory Sorkin provides a simple example showing that on the contrary, the perimeter can be arbitrarily large.
·          Mathematical origami, Helena Verrill. Includes constructions of a shape with greater perimeter than the original square, tessellations, hyperbolic paraboloids, and more.
·          A mathematical theory of origami. R. Alperin defines fields of numbers constructible by origami folds.
·          Mostly modular origami. Valerie Vann makes polyhedra out of folded paper.
·          Number patterns, curves, and topology, J. Britton. Includes sections on the golden ratio, conics, Moiré patterns, Reuleaux triangles, spirograph curves, fractals, and flexagons.
·          Origami: a study in symmetry. M. Johnson and B. Beug, Capital H.S.
·          The origami game. Rick Nordal challenges folders to make a sequence of geometric shapes with a single sheet of origami paper as quickly as possible.
·          Origami & math, Eric Andersen.
·          Origami mathematics, Tom Hull, Merrimack.
·          Origami Menger Sponge built from Sonobe modules by K. & W. Burczyk.
·          Origami polyhedra. Jim Plank makes geometric constructions by folding paper squares.
·          Origami tessellations, Alex Bateman.
·          Ozzigami tessellations, papercraft, unfolded peel-n-stick glitter Platonic solids, and more.
·          Paper folding a 30-60-90 triangle. From the geometry.puzzles archives.
·          Paperfolding and the dragon curve. David Wright discusses the connections between the dragon fractal, symbolic dynamics, folded pieces of paper, and trigonometric sums.
·          Paper models of polyhedra.
·          Polyhedra plaited with paper strips, H. B. Meyer. See also Jim Blowers' collection of plaited polyhedra.
·          Rabbit style object on geometrical solid. Complete and detailed instructions for this origami construction, in 3 easy steps and one difficult step.
·          The RUG FTP origami archive contains several papers on mathematical origami.
·          Spidron, a triangulated double spiral shape tiles the plane and various other surfaces. With photos of related paperfolding experiments. Daniel Erdely.  the inventor of the Spidron System (Thanks to Mr. Erdely for alerting us to additions to this entry!) 
·          Spring into action. Dynamic origami. Ben Trumbore, based on a model by Jeff Beynon from Tomoko Fuse's book Spirals.
·          The tea bag problem. How big a volume can you enclose by two square sheets of paper joined at the edges? See also Andrew Kepert's teabag problem page.
·          Unfolding polyhedra. A common way of making models of polyhedra is to unfold the faces into a planar pattern, cut the pattern out of paper, and fold it back up. Is this always possible?
·          
" One fold, two folds, three folds, & four,
  Halves, and thirds, and then fold some more,
  Paper and creases, intersecting lines
  Origami  & math can be a fine time!

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                             All rights reserved.
 
Want to know more than is offered above?   See below: 
Sometimes an overview is good.   Just how and what is Origami to Education?  http://www.users.waitrose.com/~pureland/japanpaper.htm

If you are an elementary school teacher & have not heard of Math in Motion I really recomment it!
Origami and the Catastrophe Theory   http://www.paperfolding.com/chengchit/3d.php  

The origami Perimeter Problem and it's Solution:  
http://hverrill.net/pages~helena/origami/perimeter/

Edward Crankshaw: http://home.hiwaay.net/~ejcranks/arth193b.html   Scroll down to read about origami and math...
43 Mathmatically based models