Origami Polyhedra Design

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Origami Polyhedra Design
Author John Montroll
Publisher A K Peters
Publication date
2009

Origami Polyhedra Design is a book on origami designs for constructing polyhedra. It was written by the origami artist and mathematician John Montroll, and it was published in 2009 by A K Peters.

Contents

Topics

There are two traditional methods for making polyhedra out of paper: polyhedral nets and modular origami. In the net method, the faces of the polyhedron are placed to form an irregular shape on a flat sheet of paper, with some of these faces connected to each other within this shape; it is cut out and folded into the shape of the polyhedron, and the remaining pairs of faces are attached together. In the modular origami method, many similarly-shaped "modules" are each folded from a single sheet of origami paper, and then assembled to form a polyhedron, with pairs of modules connected by the insertion of a flap from one module into a slot in another module. This book does neither of those two things. Instead, it provides designs for folding polyhedra, each out of a single uncut sheet of origami paper. [1]

After a brief introduction to the mathematics of polyhedra and the concepts used to design origami polyhedra, book presents designs for folding 72 different shapes, organized by their level of difficulty. These include the regular polygons and the Platonic solids, [1] Archimedean solids, and Catalan solids, [2] as well as less-symmetric convex polyhedra such as dipyramids [3] and non-convex shapes such as a "sunken octahedron" (a compound of three mutually-perpendicular squares). [2] An important constraint used in the designs was that the visible faces of each polyhedron should have few or no creases; additionally, the symmetries of the polyhedron should be reflected in the folding pattern, to the extent possible, and the resulting polyhedron should be large and stable. [2]

Audience and reception

Reviewer Tom Hagedorn writes that "The book is well designed and organized and makes you want to start folding polyhedra," and that its instructions are "clear and easy to understand"; he recommends it to anyone interested in origami, polyhedra, or both. [1] Reviewer Rachel Thomas recommends it to origami folders, to demonstrate to them the beauty of geometric forms, and to mathematicians, to show these forms in a new light and demonstrate the creativity of origami design. [2] The book can also be used as a source for mathematical school projects, and to provide hands-on experience with geometry concepts such as length, angles, surface area, and volume; some of its designs are suitable for students as young as middle school, although others require more experience as an origami folder. [3]

See also

Related Research Articles

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In geometry, an Archimedean solid is one of 13 convex polyhedra whose faces are regular polygons and whose vertices are all symmetric to each other. They were first enumerated by Archimedes. They belong to the class of convex uniform polyhedra, the convex polyhedra with regular faces and symmetric vertices, which is divided into the Archimedean solids, the five Platonic solids, and the two infinite families of prisms and antiprisms. The pseudorhombicuboctahedron is an extra polyhedron with regular faces and congruent vertices, but it is not generally counted as an Archimedean solid because it is not vertex-transitive. An even larger class than the convex uniform polyhedra is the Johnson solids, whose regular polygonal faces do not need to meet in identical vertices.

<span class="mw-page-title-main">Regular icosahedron</span> Convex polyhedron with 20 triangular faces

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In geometry, a polyhedron is a three-dimensional figure with flat polygonal faces, straight edges and sharp corners or vertices.

In geometry, a Platonic solid is a convex, regular polyhedron in three-dimensional Euclidean space. Being a regular polyhedron means that the faces are congruent regular polygons, and the same number of faces meet at each vertex. There are only five such polyhedra:

<span class="mw-page-title-main">Origami</span> Traditional Japanese art of paper folding

Origami is the Japanese art of paper folding. In modern usage, the word "origami" is often used as an inclusive term for all folding practices, regardless of their culture of origin. The goal is to transform a flat square sheet of paper into a finished sculpture through folding and sculpting techniques. Modern origami practitioners generally discourage the use of cuts, glue, or markings on the paper. Origami folders often use the Japanese word kirigami to refer to designs which use cuts.

<span class="mw-page-title-main">Modular origami</span>

Modular origami or unit origami is a multi-stage paper folding technique in which several, or sometimes many, sheets of paper are first folded into individual modules or units and then assembled into an integrated flat shape or three-dimensional structure, usually by inserting flaps into pockets created by the folding process. These insertions create tension or friction that holds the model together. Some assemblies can be somewhat unstable because adhesives or string are not used.

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Cauchy's theorem is a theorem in geometry, named after Augustin Cauchy. It states that convex polytopes in three dimensions with congruent corresponding faces must be congruent to each other. That is, any polyhedral net formed by unfolding the faces of the polyhedron onto a flat surface, together with gluing instructions describing which faces should be connected to each other, uniquely determines the shape of the original polyhedron. For instance, if six squares are connected in the pattern of a cube, then they must form a cube: there is no convex polyhedron with six square faces connected in the same way that does not have the same shape.

<span class="mw-page-title-main">Rigid origami</span>

Rigid origami is a branch of origami which is concerned with folding structures using flat rigid sheets joined by hinges. That is, unlike in traditional origami, the panels of the paper cannot be bent during the folding process; they must remain flat at all times, and the paper only folded along its hinges. A rigid origami model would still be foldable if it was made from glass sheets with hinges in place of its crease lines.

The Alexandrov uniqueness theorem is a rigidity theorem in mathematics, describing three-dimensional convex polyhedra in terms of the distances between points on their surfaces. It implies that convex polyhedra with distinct shapes from each other also have distinct metric spaces of surface distances, and it characterizes the metric spaces that come from the surface distances on polyhedra. It is named after Soviet mathematician Aleksandr Danilovich Aleksandrov, who published it in the 1940s.

Geometric Folding Algorithms: Linkages, Origami, Polyhedra is a monograph on the mathematics and computational geometry of mechanical linkages, paper folding, and polyhedral nets, by Erik Demaine and Joseph O'Rourke. It was published in 2007 by Cambridge University Press (ISBN 978-0-521-85757-4). A Japanese-language translation by Ryuhei Uehara was published in 2009 by the Modern Science Company (ISBN 978-4-7649-0377-7).

Descartes on Polyhedra: A Study of the "De solidorum elementis" is a book in the history of mathematics, concerning the work of René Descartes on polyhedra. Central to the book is the disputed priority for Euler's polyhedral formula between Leonhard Euler, who published an explicit version of the formula, and Descartes, whose De solidorum elementis includes a result from which the formula is easily derived.

Adventures Among the Toroids: A study of orientable polyhedra with regular faces is a book on toroidal polyhedra that have regular polygons as their faces. It was written, hand-lettered, and illustrated by mathematician Bonnie Stewart, and self-published under the imprint "Number One Tall Search Book" in 1970. Stewart put out a second edition, again hand-lettered and self-published, in 1980. Although out of print, the Basic Library List Committee of the Mathematical Association of America has recommended its inclusion in undergraduate mathematics libraries.

Convex Polyhedra is a book on the mathematics of convex polyhedra, written by Soviet mathematician Aleksandr Danilovich Aleksandrov, and originally published in Russian in 1950, under the title Выпуклые многогранники. It was translated into German by Wilhelm Süss as Konvexe Polyeder in 1958. An updated edition, translated into English by Nurlan S. Dairbekov, Semën Samsonovich Kutateladze and Alexei B. Sossinsky, with added material by Victor Zalgaller, L. A. Shor, and Yu. A. Volkov, was published as Convex Polyhedra by Springer-Verlag in 2005.

<span class="mw-page-title-main">Blooming (geometry)</span>

In the geometry of convex polyhedra, blooming or continuous blooming is a continuous three-dimensional motion of the surface of the polyhedron, cut to form a polyhedral net, from the polyhedron into a flat and non-self-overlapping placement of the net in a plane. As in rigid origami, the polygons of the net must remain individually flat throughout the motion, and are not allowed to intersect or cross through each other. A blooming, reversed to go from the flat net to a polyhedron, can be thought of intuitively as a way to fold the polyhedron from a paper net without bending the paper except at its designated creases.

References

  1. 1 2 3 Hagedorn, Thomas R. (April 2010), "Review of Origami Polyhedra Design", MAA Reviews, Mathematical Association of America
  2. 1 2 3 4 Thomas, Rachel (December 2009), "Review of Origami Polyhedra Design", Plus Magazine
  3. 1 2 Luck, Gary S. (March 2011), "Review of Origami Polyhedra Design", The Mathematics Teacher, 104 (7): 558, JSTOR   20876948