# Articles for keyword: math

Diagrams for a simple modular from 2:1 rectangles, released in open access to celebrate the World Origami Days 2016.
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by Jon Tucker
This articles shows how to construct arbitrary square grids via folding alone, using Haga's First Theorem.
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This is a followup of my previous article, Pentakis Dodecahedron (Issue 35), featuring variation patterns. Mono paper such as copy paper or Tant is a must.
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by Mukul Achawal
Diagrams for a 30 unit modular dodecahedron with color change by Mukul Achawal of India.
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Diagrams for a simple modular from squares 4" or smaller. For larger constructions though, use paper of proportion $$5:3\sqrt{3}$$, i.e., $$1:1.039$$.
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Photo diagrams for a 30 unit modular with subtle curves by Aldos Marcell of Nicaragua. Assemblies with other number of units possible as well.
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Diagrams for a simple Sonobe type modular made from approximately 1:5 rectangles. This design is great for any leftover strips you may have amassed when sizing paper for other projects.
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Diagrams for a 12 or 30 unit modular with color change. The starting paper size ratio for an unit is 1:3.
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by Troels Højer Jensen and Hans Dybkjær
In origami we frequently need to find an $$n$$th of a paper, often in order to divide it into an $$n \times n$$ grid. This article generalizes a common technique for finding references and provides some insight into the geometric properties of paper.
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Diagrams for a 30-unit Sonobe type design with color change. Other assemblies such as 3, 6, 12 or larger number of units are possible as well.
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Diagrams for a 30 unit modular design.
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by Hans Dybkjær
This article tells the tale of the higher spheres of oranges and apples: How I got there, and how to make them.
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Building Block Units (BBU) are a new family of modular origami units, with over one hundred different interlocking module designs.
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by Meenakshi Mukerji & Ittai Hacohen
The 3-unit Sonobe hexahedron (Toshie's Jewel) and the 12-unit Sonobe octahedral assembly are well known Sonobe constructions. But did you know that you can also construct the former with double the number of units, and the latter with half the number of units, i.e., both shapes from 6 units?
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A discussion of an alternate Sonobe Unit assembly which produces a surprisingly different result than the conventional one. Diagrams included.
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Narong Krined
Diagrams for a beautiful 12 or 30 unit Sonobe Variation.
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Diagrams for a beautiful 12 or 30 unit modular.
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Diagrams for the solid version of the Compound of 5 Tetrahedra aka the 47th Stellation of the Icosahedron, similar to the very popular frame version by Tom Hull/Francis Ow, known as Five Intersecting Tetrahedra or FIT. Some mathematics has been discussed as well.
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Diagrams for a 30-unit modular design by Ekaterina Lukasheva.
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A quick and easy method of folding a heptagon by Jacques Justin and some related discussions. Francesco Mancini found the method in a pile of letters and notes that he inherited from Roberto Morassi's origami archive.
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Diagrams for a 30-unit modular design by Ekaterina Lukasheva of Russia.
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Tridecagon, also known as the triskaidecagon, is a 13-sided polygon. There are several origami methods already available for folding the tridecagon but the simplicity of my approach may be of interest to people. You may use the tridecagon to transpose origami designs based on other regular polygons.
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Diagrams for a 30 piece modular design by Natalia Romanenko of Moldova.
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Diagrams for a 30-unit modular design by Ekaterina Lukasheva of Russia.
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Toyoaki Kawai’s method of making a pentagon from a square is a widely used one. This article demonstrates how to extend his method to a decagon and shows examples of transpositions of well known designs to pentagons and decagons.
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by Irina Reutskaya
Diagrams for a 30-piece modular design by Irina Reutskaya of Russia.
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by Daniel Reutsky
Diagrams for a 30-piece modular design by Daniel Reutsky of Russia.
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Diagrams for a 30-piece modular design by Maria Vakhrusheva of Russia.
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Curved-crease origami can be designed by considering the properties of ruling lines, lines on the crease pattern that remain straight in the 3D folded form. This technique was developed by David Huffman, who identified conic section curves has being particularly suitable for curved-crease designs. Two examples using ellipses are given as crease patterns.
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Diagrams for a 30-piece modular design by Valentina Minayeva of Ukraine.
by Sjaak Adriaanse
Paper hoarders will appreciate this nifty tool for cutting leftover pieces of paper into common size ratios like 4 by 3, Golden and Silver rectangles, or the ratio of the dollar bill.
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by Thomas E. Cooper
Diagrams for a simple pyramid model with variations, used for to help teach geometry.
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by Arnold Tubis
Arnold Tubis offers yet another dollar bill fold of George Washington, framed on both sides of the model using two dollar bills.
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by Thomas Hull
Several variations on a hexagon-based, iso-area, geometric collapse method are shown. Some of these were taught at the 2013 OrigamiUSA Annual Convention in New York City.
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by Arnold Tubis
A folding method for closed masu boxes from a single square, generalized to masu-like structures with regular polygonal bases.
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by Sy Chen
A generalized folding construction to divide the sides of triangles (and other non-square shapes) into rational divisions.
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A rotationally symmetric solid curved fold, folded from a regular hexagon.
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by Arnold Tubis, John Andrisan, and Christopher Pooley
Part two in a series examining the mathematics behind the golden ratio in some geometric boxes.
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by Arnold Tubis and Carmen Sprung
Tubis and Sprung show that the same starting shapes used previously to create generalized masu boxes [Tubis and Pooley 2012] can be used to produce $$n$$-pointed 3D stars.
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The analysis of a geometric construction for 1/3, and other fractions.
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Learn to fold a family of modular units that can create a wide variety of deltahedra, polyhedra whose faces are equilateral triangles.
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by Bennett Arnstein
Proof of Lewis Simon's construction for the trisection of the side of a square or the short side of a rectangle.
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by Arnold Tubis and Christopher Pooley
Tubis and Pooley explore $$n$$-sided generalizations of the masu and one of its many decorative-lids. Detailed video instructions are provided at the Origami Player site.
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Meenakshi Mukerji
30 pieces of paper are folded to make a modular version of the Compound of Five Octahedra model.
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Thomas Hull
This wave model is fun to fold and has a lot of math in it!
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A pleated cone sliced by multiple planes creates this geometric model reminiscent of a breaking wave.
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by Arnold Tubis, John Andrisan, and Christopher Pooley
Paper folding exercises involving the golden section of a line, the golden rectangle, and the golden triangle provide interesting geometry-teaching supplements.
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by Tom Hull
Learn some of the history of origami geometry, as well as the story of Margherita Liazzolla Beloch, the first origami mathematician!
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A rotationally symmetric geometric shape, folded from a hexagon, based on Jeannine Mosely's "Bud".
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by Tom Hull
A summary of Bern & Hayes' proof that flat-foldability in origami is computationally hard!
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Design and folding instructions for one of the 54 polypolyhedra using dollar bills.
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by Arnold Tubis and Crystal E. Mills
A method for making four-compartment side–to–side or corner–to–corner divider inserts for prism-shape containers with square faces is generalized so as to produce n equal compartments of specified height for a container with an n–sided regular-polygon face.
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Arnold Tubis
Two conundrums concerning the Betsy Ross Five-Pointed Star: the provenance of the Pattern–for-Stars artifact and the surprising incompleteness of fold and one-cut descriptions for making the star.
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Thomas Hull
The names Maekawa and Kawasaki are known to origamists as great origami creators. But did you know they have Theorems named after them too? And so does the French paper folder Jacques Justin. See what these Theorems are all about. Warning: Math ahead!
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By cmorrow [at] mtholyoke.edu (Charlene Morrow)
An expanded version of a 5OSME convention commentary that appears in the Winter 2011 issue of The Paper, pp 18-19. Many more interesting experiences and color photos that could not be included in The Paper version due to limited space.
Course information for an MIT graduate course in Geometric Folding Algorithms.
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Tom Hull
Every time you fold paper, your fingers are doing calculus. Read more to learn how smart your fingers are!
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by Tom Hull
The origami wind spinner is a traditional, if somewhat obscure model of repeated pleat folds. We ask ourselves, "What kind of shapes can paper form with these simple pleats?" and, "How much can we make a square piece of paper rotate with this pleating scheme?" The answers are surprising and fun!
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by Tom Hull
Learn how to fold Molly Kahn's 3-unit modular Hexahedron and marvel at the multitude of math manifested around this model!
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At the recent 5OSME convention, an impromptu challenge involved incorporating a "nonexistent" fold, the origami hypar, into definitely existent origami models.