# MARVELOUS MODULAR ORIGAMI PDF

Download as PDF, TXT or read online from Scribd. Flag for inappropriate content . Marvelous Modular Origami Meenakshi Mukerji A K Peters. Natick. in such a way as to create magnificent three-dimensional structures without the use Modular origami isn't limited to polyhedra—flat stars, coasters, tessellated . The Sonobe unit is one of the foundations of modular origami. There are many variations . Reference: Book: Marvelous Modular Origami by Meenakshi Mukerji.

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Marvelous Modular cittadelmonte.info 18/13/ PM Jasmine Dodecahedron 1 (top) and 3 (bottom). (Se. Marvelous Modular Origami by Meenakshi Mukerji. Clockwise: Module Decoration Box from 2xls adapted from dollar bills. Module Dodecahedron from degree module from 4x3s. Module Decoration.

This richly illustrated book provides step-by-step instructions for the construction of over 30 different modular origami structures. The diagrams are clear, crisp, and easy to follow, and are accompanied by inspiring color photographs. Additional tips encourage the reader to design their own original creations. They range from simple Sonobe to floral and geometrical constructions. All are eye-catching and satisfying to fold, and the finished constructions are pleasing to behold. Also included are short sections on the mathematics behind the shapes and optimum color choices.

Edition 1st Edition. First Published DOI https: Pages 92 pages. Export Citation. Get Citation. Mukerji, M. The folding tasks will enrich geometry teaching and create interesting learning environments at secondary school level.

The directions, accompanied by quality illustrations, are easy to follow. This would be an excellent reference for members of a mathematics club who are looking for projects. Check out the link above for more details. Meenakshi Mukerji Adhikari was introduced to origami in her early childhood by her uncle Bireshwar Mukhopadhyay.

She rediscovered origami in its modular form as an adult, quite by chance in , when she was living in Pittsburgh, PA. A friend took her to a class taught by Doug Philips, and ever since she has been folding modular origami and displaying it on her very popular website www.

She has many designs to her own credit. Meenakshi was born and raised in Kolkata, India. She obtained her BS in electrical engineering at the prestigious Indian Institute of Technology, Kharagpur, and then came to the United States to pursue a master s in computer science at Portland State University in Oregon. She worked in the software industry for more than a decade but is now at home in California with her husband and two sons to enrich their lives and to create her own origami designs.

Would you like to tell us about a lower price? If you are a seller for this product, would you like to suggest updates through seller support? This richly illustrated book provides step-by-step instructions for the construction of over 30 different modular origami structures. The diagrams are clear, crisp, and easy to follow, and are accompanied by inspiring color photographs. Additional tips encourage the reader to design their own original creations. Read more Read less.

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Page 1 of 1 Start over Page 1 of 1. Exploring 3D Geometric Designs. Meenakshi Mukerji. Animal Origami for the Enthusiast: John Montroll. Exquisite Modular Origami. Ekaterina Lukasheva. Origami Omnibus: Paper Folding for Everybody.

## Marvelous Modular Origami: Meenakshi Mukerji: cittadelmonte.info: Books

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Top Reviews Most recent Top Reviews. There was a problem filtering reviews right now. Please try again later. Paperback Verified Purchase. Pinch flaps halfway. Repeat Steps 4—7 at the back.

Tuck corner back along pre-creased line. Open mountain fold from Step 3. Lock two units together by tucking back along crease made in Step 5. Form one face of the dodecahedron by assembling five units in a ring in the order shown.

The last two units will be difficult to lock; the last one may be left unlocked. Assemble sixth unit as shown to form a vertex. Continue making faces and vertices to complete the dodecahedron. Start with Step 9 of the previous model and tuck marked flap under. Pull corner out from behind. Turn over and repeat Steps 1—3. Pocket Tab Assemble units as in previous model to arrive at the finished model.

Note that locking gets difficult in this model. Tab Assemble units as in previous dodecahedron to arrive at the finished model below. Use paper of the same size for the template and the units.

Four-inch squares will yield finished models about 4" in height. The template will also be cre- ated using origami methods. The use of a template will not only expedite the folding process but it will also reduce unwanted creases on a unit, making the finished model look neater.

Making a Template 2 2. Valley fold top layer only. Pull bottom flap out to the top. Fold into thirds following the number sequence.

Mountain fold at center. Turn over and repeat. Open mountain fold from Step 5. Unfold crease from Step 6 and repeat at the back. Mountain fold along edge of back flap. Valley fold to match dots. Valley fold tip. Unfold Steps 13 and Finished Template Unit Repeat Steps 12—14 at the back and then open to Step 5. Then, fold 30 units following the diagrams below. Use same size paper for the template and the model.

Lay on template and make the mountain creases. Fan fold into thirds following number sequence. The portion above CD helps lock units together. Continue assembling faces and vertices as in Daisy Dodecahedron to arrive at the finished model. Crease and open center fold. Then valley fold into thirds and open.

Repeat on DC. Mountain fold to re-crease portions of the flap shown.

The lock of unit 2 will lie above line BC of unit 1. Crease center fold. Valley fold flaps as shown. Mountain fold as shown. Mountain fold to re -crease the portions of AB and CD shown.

Snip with scissors along the dotted lines, approximately less than half way from A to B. The lock of unit 2 will lie above line BC. Continue assembling as in Daisy Dodecahedron to arrive at finished model.

Valley fold each edge twice. Valley fold existing creases. Mountain fold to re-crease the portions of AB and CD shown. Assemble as in Daisy Dodecahedron to arrive at the finished model.

Assemblies shown in regular and reverse colorings. Mountain fold corners and tuck under flap. Valley fold corners, top layer only. C Pocket Finished Unit Assemblies shown in regular and reverse colorings. Do Steps 1 through 7 of Swirl Dodecahedron 1. Valley fold to match dots and repeat on the back.

## Marvelous Modular Origami

Re-crease mountain fold shown. Open mountain fold from Step 8. Perform Steps 10 and 11 of Swirl Dodecahedron 1 to form locks. The Lightning Bolt and the Star Windows are essentially Sonobe-type models, but, unlike Sonobe models, they both have openings or windows. It was only when someone asked me if I was inspired by the Curler Units that it first came to my attention. Although the models seem similar at first glance, they are really quite different, as you will see.

The Curler Units are far more versatile than my Twirl Octahedron units. With the former you can make virtually any polyhedral assembly, but with the latter you can only make an octahedral assembly.

Start with A 4 paper. Pinch ends of centerline, then crease and open the diagonal shown. Pinch halfway points as marked. Fold to align with diagonal crease from Step 1. Fold to align with existing diagonal crease from Step 1. Fold to bisect marked angles.

Open last three creases. Repeat Step 11 on the back. Unlike Sonobe models, pyramid tips will be open.

You can also make models out of 3, 6 , 12, or more units. Use same size paper for both Twirl and Frame Units. Twirl Units 3.

Valley fold and open top flap only. Fold one diagonal. Mountain fold rear flap. Open last two folds. Make curl as tight as possible. You may keep curls held with miniature clothespins overnight for tighter curls. Fold small portions of the two corners shown.

Crease and open like waterbomb base. Finished Frame Unit 3. Collapse like a waterbomb base. Note that Step 2 is optional: Making one face of the octahedron. To form one face of the octahedron, connect three Frame Units and three Twirl Units as shown above.

Continue forming all eight faces to complete the octahedron. Pinch ends of center fold and then cupboard fold. Crease and open mountain and valley folds as shown.

Reverse fold corners. Open gently, turn over, and orient like the finished unit. Collapse center like waterbomb base. Continue assembling in a dodecahedral manner to complete model.

Although traditional origami begins with a square, many modern models use rectangles. The following is organized in ascending order of aspect ratio. Obtain paper sizes as below, and use them as templates to cut papers for your actual units. While many models look nice made with a single color, there are many other models that look astounding with the use of multiple colors.

Sometimes random coloring works, but symmetry lovers would definitely prefer a homogenous color tiling. Determining a solution such that no two units of the same color are adjacent to each other is quite a pleasantly challenging puzzle. For those who do not have the time or patience, or do not Three-color tiling of an octahedron Every face has three distinct colors.

Three-color tiling of an icosahedron Every face has three distinct colors. In the following figures each edge of a polyhedron represents one unit, dashed edges are invisible from the point of view. It is obvious that for a homogeneous color tiling the number of colors you choose should be a sub-multiple or factor of the number of edges or units in your model.

For example, for a unit model, you can use three, five, six or ten colors. Four-color tiling of an octahedron Every vertex has four distinct colors, and every face has three distinct colors. Three-color tiling of a dodecahedron Every vertex has three distinct colors. Six-color tiling of an icosahedron Every vertex has five distinct colors, and every face has three distinct colors.

Six-color tiling of a dodecahedron Every face has five distinct colors, and every vertex has three distinct colors. Hence, it is not surprising that so many mathematicians, scientists, and engineers have shown a keen interest in this field—some have even taken a break from their regular careers to delve deep into the depths of exploring the connection between origami and mathematics.

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So strong is the bond that a separate international origami conference, Origami Science, Math, and Education OSME has been dedicated to its cause since Professor Kazuo Haga of University of Tsukuba, Japan, rightly proposed the term origamics in to refer to this genre of origami that is heavily related to science and mathematics.

Background The mathematics of origami has been extensively studied not only by origami enthusiasts around the world but also by mathematicians, scientists, and engineers, as well as artists. These interests date all the way back to the late nineteenth century when Tandalam Sundara Row of India wrote the book Geometric Exercises in Paper Folding in He used novel methods to teach concepts in Euclidean geometry simply by using scraps of paper and a penknife.

The geometric results were so easily attainable that the book became very attractive to teachers as well as students and is still in print to this day. It also inspired quite a few mathematicians to investigate the geometry of paper folding. Around the s, Italian mathematician Margherita Beloch found out that origami can do more than straightedge-and-compass geometric constructions and discovered something similar to one of the Huzita-Hatori Axioms explained next.

All geometric origami constructions involving single-fold steps that can be serialized can be performed using some or all of these seven axioms. Together known as the Huzita-Hatori Axioms, they are listed below: While a lot of studies relating origami to mathematics were going on, nobody actually formalized any of them into axioms or theorems until when Japanese mathematician Humiaki Huzita and Italian mathematician Benedetto Scimemi laid out a list of six axioms to define the algebra and geometry of origami.

Later, in , origami enthusiast Hatori Koshiro added a seventh axiom. Physicist, engineer, and leading origami artist Dr. Robert Lang has proved that these seven axioms are complete, i. Given two lines L1 and L2, a line can be folded placing L1 onto L2. Given two points P1 and P2, a line can be folded placing P1 onto P2.

Given two points P1 and P2, a line can be folded passing through both P1 and P2. Given a point P and a line L, a line can be folded passing through P and perpendicular to L.

This can be demonstrated by geometry students playing around with some paper. Discoveries similar to the Kawasaki Theorem have also been made independently by mathematician Jacques Justin. Physicist Jun Maekawa discovered another fundamental origami theorem. It states that the difference between the number of mountain creases M and the number of valley creases V surrounding a vertex in the crease pattern of a flat origami is always 2. There are some more fundamental laws and theorems of mathematical origami, including the laws of layer ordering by Jacques Justin, which we will not enumerated here.

Other notable contributions to the foundations of mathematical origami have been made by Thomas Hull, Shuji Fujimoto, K. Husimi, M. The use of computational mathematics in the design of origami models has been in practice for many years now.

Some of the first known algorithms for such designs were developed by computer scientists Ron Resch and David Huffman in the sixties and seventies. In more recent years Jun Maekawa, Toshiyuki Meguro, Fumiaki Kawahata, and Robert Lang have done extensive work in this area and took it to a new level by developing computer algorithms to design very complex origami models.

Particularly notable is Dr. Due to the fact that paper is an inexpensive and readily available resource, origami has become extremely useful for the purposes of modeling and experimenting. There are a large number of educators who are using origami to teach various concepts in a classroom. Origami is already being used to model polyhedra in mathematics classes, viruses in biology classes, molecular structures in chemistry classes, geodesic domes in architecture classes, DNA in genetics classes, and crystals in crystallography classes.

Origami has its applications in technology as well, some examples being automobile airbag folding, solar panel folding in space satellites, design of some optical systems, and even gigantic foldable telescopes for use in the geostationary orbit of the Earth. Professor Koryo Miura and Dr. Lang have made major contributions in these areas. Modular Origami and Mathematics As explained in the preface, modular origami almost always means polyhedral or geometric modular origami, although there are some modulars that have nothing to do with geometry or polyhedra.

For the purposes of this essay, we will assume that modular origami refers to polyhedral modular origami. A polyhedron is a three-dimensional solid that is bound by polygonal faces. A polygon in turn is a two-dimensional figure bound by straight lines. Aside from the real polyhedra themselves, modular origami involves the construction of a host of other objects that are based on polyhedra. But, in fact, for every model there is an underlying polyhedron.

The most referenced polyhedra for origami constructions are undoubtedly the five Platonic solids. First, one must determine whether a unit is a face unit, an edge unit or a vertex unit, i. Face units are the easiest to identify. Most modular model units and all models presented in this book except the Twirl Octahedron use edge units. For edge units there is a second step in- volved—one must identify which part of the unit which is far from looking like an edge actually associates with the edge of a polyhedron.

Unfortunately, most of the modular origami creators do not bother to specify whether a unit is of face type, edge type, or vertex type because it is generally perceived to be quite intuitive. But, to a beginner, it may not be so intuitive. On closer observation and with some amount of trial and error, though, one may find that it is not so difficult after all. Once the identifications are made and the folder can see through the maze of superficial designs and perceive the unit as a face, an edge, or a vertex, assembly becomes much simpler.

Now it is just a matter of following the structure of the polyhedron to put the units in place. With enough practice, even the polyhedron chart need not be consulted anymore.

The Design Process How flat pieces of paper can be transformed into such aesthetically pleasing three-dimensional models simply by folding is always a thing to ponder. Whether it is a one-piece model or an interlocked system of modules made out of several sheets of paper, it is equally mind-boggling.

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