# Articles for keyword: math

### Diagrams and Video: A Triangular Gift Box

by Arsalan Wares
A sturdy modular box and the mathematics behind it.

### Once Upon a Time in Versailles: Part 1

by Krystyna Burczyk and Wojtek Burczyk
Musings on art, kitsch, mathematics and the creative process. And lots of diagrams.

### Once Upon a Time in Versailles: Part 2

by Krystyna Burczyk and Wojtek Burczyk

### Diagrams: Festive Icosahedron

Diagrams for an icosahedral design made with 30 quick-to-fold units from squares. The look is rather festive, and hence the name. You can fold the with thematic colors of the season to fit right in.

### Diagrams: Fractal Pyramid

by Ushio Ikegami
This is the only diagrammed origami model that simulates a true mathematical fractal. It makes a pyramid shape with many branches. No one has yet successfully folded a version without cutting the paper; the version in the picture (folded by the author) was made by carefully cutting the crease pattern into several pieces, folding these using the recursive folding instructions, and then gluing them back together. The challenge of folding recursive diagrams as well as the dexterity involved to not destroy the paper easily put this model in the supercomplex category.

### Ed Explores: Silver Rectangles

Did you know that the A ratio has two distinct definitions? Edward Holmes offers a cheerful explanation.

### Truncated Trigonal Trapezohedrons and Special Pentagons

by Thomas Cooper
There is a family of geometric solids, one of which is illustrated in a famous engraving by Albrecht Dürer, that poses some interesting origami challenges.

### Icosidodecahedron

An Icosidodecahedron with sunken triangular faces based on a simple unit. Made from 12 pentagons, it is definitely meant for people who like challenges!

### Book Review: A History of Folding in Mathematics

by Laura Rozenberg. Translated from Spanish by James Buschman
A review of a book on the role of folding in mathematics, art and philosophy, and its struggle for recognition through the centuries.

### Origami: How complex could it be?

Elina Gor
During her mathematical studies, Elina Gor researched the changes in complexity of origami models from the 1980`s to the present, to see if we are doomed to face mega-complex models in the future or maybe we already have reached the peak of complexity.

### Diagrams: Delicata

Diagrams for a modular with color change. You can assemble 12 or 30 units. Kami or thicker duo paper is recommended. Scrapbook paper works well, making the result sturdier.

### Diagrams: Verbena

Diagrams for a color-change modular, 12 or 30 units, though like most polyhedral designs, the latter is more attractive. The flower petals are of one color, and the flower centers and background are of another color.

### Crease Pattern: Octagonal Iso-Area Self-Similar Flower

A fun crease pattern that can repeat infinitely to the center folds an octagon into a geometric flower design, where the front and back of the paper look the same.

### Diagrams: Sierpinski 3D Fractal

Paolo Bascetta presents us with a creative, new Sierpinski 3D fractal created by stacking.

### Diagrams: Pollen Kusudama

by Mukul Achawal
Diagrams for a color-change modular made from 30 rectangles.

### My Origami Story

My origami journey so far, as I celebrate two milestones - 20 years of my online presence and 10 years publishing books. Also find photo instructions for folding Pentas, one of my latest designs.

### Book Review: Learning Mathematics with Origami

by Charlene Morrow
OrigamiUSA Board member and educator Charlene Morrow reviews a book by Tung Ken Lam and Sue Pope, two experienced British teachers of origami and mathematics.

### Diagrams: Little Flowers

by Meenakshi Mukerji
Diagrams for a 30 unit modular with color-changed five petaled flowers. For four petaled flowers assemble 12 units.

### Diagrams: Whirlwind

Diagrams for a simple modular from 2:1 rectangles, released in open access to celebrate the World Origami Days 2016.

### Origami Reference Points: Folding Square Grids with Haga's Method

by Jon Tucker
This articles shows how to construct arbitrary square grids via folding alone, using Haga's First Theorem.

### Diagrams: Patterned Pentakis Dodecahedra

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.

### Diagrams: Mercapton Dodecahedron

by Mukul Achawal
Diagrams for a 30 unit modular dodecahedron with color change by Mukul Achawal of India.

### Diagrams: Pentakis Dodecahedron

Diagrams for a simple modular from squares 4" or smaller. For larger constructions though, use paper of proportion $$5:3\sqrt{3}$$, i.e., $$1:1.039$$.

### Diagrams: Vidrio

Photo diagrams for a 30 unit modular with subtle curves by Aldos Marcell of Nicaragua. Assemblies with other number of units possible as well.

### Diagrams: Posy Sonobe

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.

### Diagrams: Spring Beauty

Diagrams for a 12 or 30 unit modular with color change. The starting paper size ratio for an unit is 1:3.

### Nths via Pythagorean triplets

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.

### Diagrams: Gemstone Sonobe

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.

### Diagrams: Regulus Kusudama

Diagrams for a 30 unit modular design.

### Orange Spheres and Odd Even Apples

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.

### Introducing Building Block Units

Building Block Units (BBU) are a new family of modular origami units, with over one hundred different interlocking module designs.

### Diagrams: Two Unusual 6-unit Sonobe Constructions

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?

### The Lesser Known Twin of The Sonobe Unit

A discussion of an alternate Sonobe Unit assembly which produces a surprisingly different result than the conventional one. Diagrams included.

### Diagrams: Crape Jasmine

Narong Krined
Diagrams for a beautiful 12 or 30 unit Sonobe Variation.

### Diagrams: Pinwheel Flower Ball

Diagrams for a beautiful 12 or 30 unit modular.

### Diagrams: Compound of 5 Tetrahedra

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.

### Diagrams: Marquise

Diagrams for a 30-unit modular design by Ekaterina Lukasheva.

### Editorial: Dear Roberto...

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.

### Diagrams: Ice Queen

Diagrams for a 30-unit modular design by Ekaterina Lukasheva of Russia.

### Editorial: Folding the Tridecagon, that Odd Polygon

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.

Diagrams for a 30 piece modular design by Natalia Romanenko of Moldova.

### Diagrams: Nymph

Diagrams for a 30-unit modular design by Ekaterina Lukasheva of Russia.

### Editorial: Extending Kawai’s Pentagon Method to a Decagon

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.

### Diagram: Sonobe Capuchona

by Irina Reutskaya
Diagrams for a 30-piece modular design by Irina Reutskaya of Russia.

### Diagram: Kusudama by Daniel Reutsky

by Daniel Reutsky
Diagrams for a 30-piece modular design by Daniel Reutsky of Russia.

### Diagram: Fiesta

Diagrams for a 30-piece modular design by Maria Vakhrusheva of Russia.

### One Ellipse to Rule Them All

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.

### Diagrams: Karlsson Kusudama

Diagrams for a 30-piece modular design by Valentina Minayeva of Ukraine.

### Nomograms: Determining Paper Sizes Made Easy

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.

### Diagrams: Simple Modular Square Pyramids

by Thomas E. Cooper
Diagrams for a simple pyramid model with variations, used for to help teach geometry.

### Framing George Washington II

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.

### Cool Iso-area Hexagon Collapses

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.

### $$N$$-Sided Closed Masus

by Arnold Tubis
A folding method for closed masu boxes from a single square, generalized to masu-like structures with regular polygonal bases.

### Rational Divisions of Triangles

by Sy Chen
A generalized folding construction to divide the sides of triangles (and other non-square shapes) into rational divisions.

### Crease pattern: Hexonion

A rotationally symmetric solid curved fold, folded from a regular hexagon.

### Folding a Golden Gnomon Box

by Arnold Tubis, John Andrisan, and Christopher Pooley
Part two in a series examining the mathematics behind the golden ratio in some geometric boxes.

### A Class of $$N$$-Pointed 3-D Stars

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.

### Another Geometric Construction

The analysis of a geometric construction for 1/3, and other fractions.

### Modular Origami: the DeZZ Unit

Learn to fold a family of modular units that can create a wide variety of deltahedra, polyhedra whose faces are equilateral triangles.

### Math: Trisection Construction Proof

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.

### Generalized N-Sided Masus

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.

### Five Intersecting Octahedra

Meenakshi Mukerji
30 pieces of paper are folded to make a modular version of the Compound of Five Octahedra model.

### Folding Waves, Part 1

Thomas Hull
This wave model is fun to fold and has a lot of math in it!

### Crease Pattern: Wave24

A pleated cone sliced by multiple planes creates this geometric model reminiscent of a breaking wave.

### Math: The Golden Ratio in Origami

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.

### Spotlight: Margherita Beloch

by Tom Hull
Learn some of the history of origami geometry, as well as the story of Margherita Liazzolla Beloch, the first origami mathematician!

### Crease Pattern: Hexabox

A rotationally symmetric geometric shape, folded from a hexagon, based on Jeannine Mosely's "Bud".

### How to Prove that Origami is Hard

by Tom Hull
A summary of Bern & Hayes' proof that flat-foldability in origami is computationally hard!

### Modular Design: Shishapangma

Design and folding instructions for one of the 54 polypolyhedra using dollar bills.

### Divider Inserts for N-sided Boxes

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.

### The Betsy Ross Five-Pointed Star

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.

### The Theorems of Maekawa, Kawasaki, and Justin

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!

### Beyond the High-Tech Side of Origami: What I Learned at 5OSME

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.

### Geometric Folding Algorithms: Free Video Lectures

Course information for an MIT graduate course in Geometric Folding Algorithms.

### Origami & Calculus

Tom Hull
Every time you fold paper, your fingers are doing calculus. Read more to learn how smart your fingers are!