Equilateral Flat Unit Folding Instructions

Modular Origami Earth Day April 22nd 2010

Face-Based Modules:

Equilateral Triangle and Square Flat Units

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Table of Contents Equilateral Triangle Flat Unit.......................3 Square Flat Unit .............................................7 Joining Tabs..................................................10 What Can You Make With These? ............11 Sample Models..............................................12 Notes...............................................................13 Resources and Links.....................................14

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Equilateral Triangle Flat Unit Before

Kasahara, Kunihiko. Origami Omnibus. Japan Publications, 1998. After Step 1: Turn the paper so its colored side is down and fold in half from right to left.

0 Step 2: Unfold. Fold the lower right corner (A) upwards so it falls along the center crease made in step 1 (at B) and so that your crease goes through the lower left corner (C). Make sure the crease through the lower left corner is clean and sharp!

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Before

After Step 3: Unfold the crease you made in step 2. Fold the upper left corner (D) so it falls along the crease made in the previous step (E). The new crease should once again cleanly and sharply go through the lower left corner (C). Point C is one of the corners of the triangle. 0 Step 4: 0 Fold the point where the crease from step 3 intersects the right side (F), until it reaches the center crease from Step 1 (G). Your new crease will go through the lowerright corner (A).

Step 5: Flip the paper over. After this flip, the narrow end (H) should be pointing toward you.

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Before

After Step 6: Fold H to meet I. Fold the bottom vertex of the triangle (H) to the middle of the top edge so that it lies directly 0 above (I).

Step 7: Flip the paper over again. Fold the corner of the trapezoid which lies on the opposite side of the small white tab (A) to the corner of the trapezoid which is adjacent to the tab (J). Make sure that all the edges line up. Step 8: Fold the corner of the small white tab (K) over the triangle edge (L) so that it is out of the way. 0 Step 9: Fold and unfold. Fold the right corner of the rhombus (C) on top of the opposite corner (M) so that all of 0 the triangular edges match. Make a sharp crease through the middle, then unfold again.

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Before

After Step 10:

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Tuck both the open flap (C) and the paper behind it into the pocket beneath the other half of the triangle (M). Ensure that the triangular flap is inserted all the way inside the pocket and that the short flap behind it is completely tucked away. If the paper refuses to go in all the way, keep trying! The completed unit (flipped over.) Though both sides of the unit are functionally identical, the side of the triangle that was facing you during Step 9 is really the "back" of the unit. Models in which the "fronts" of all triangles are facing outward will tend to look nicer. Inspect your unit and pick the cleanest side as the “front,” or outward facing, side.

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Square Flat Unit Before

Fuse, Tomoko. Unit Polyhedron Origami . Japan Publications Trading, 2006. After Step 1: With the colored side down, fold the left edge over as far as you wish.

Step 2: Fold the right edge so that it lines up with the left edge.

Step 3: Fold the bottom up towards the center as far up as you wish.

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Square Flat Unit Fuse, Tomoko. Unit Polyhedron Origami . Japan Publications Trading, 2006. BEFORE AFTER Step 4: Fold top edge to down to align with the bottom edge.

Now you have completed one unit, but that is only half of the square!

Repeat steps 1-4 to make the other half. Step 5: Assemble the two halves to make a Square Flat Unit. Open one unit horizontally like a book. Open the other unit vertically so that its bottom crease (the dotted line) aligns with the top edge of the first unit. Close the 'book' by folding the first unit's flaps inward again. You may need to use your fingers to flatten the edges out inside the other half.

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Square Flat Unit Fuse, Tomoko. Unit Polyhedron Origami . Japan Publications Trading, 2006. BEFORE AFTER Step 6: Fold the top flap down over the bottom unit.

Step 7: Tuck the last flap inside the pocket area created by the first unit. You may need to use your fingers inside to flatten out the edges in between the halves.

Congratulations! You have completed the Square Flat Unit. Notice it has one deep pocket on each edge. Coincidentally, the four edges on the square unit are the same length as those of the triangle unit: exactly half the length of the original sheet of paper. This allows us to connect the Flat Square Units and Equilateral Triangle Units along their edges, but in order to do so, we will need to make joining tabs !

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Joining Tabs for Square & Triangle Flat Units Fuse, Tomoko. Unit Polyhedron Origami . Japan Publications Trading, 2006. Before After Step 1: Fold one full-sized sheet into quarters. Then unfold and cut or cleanly tear along the creases to make four squares of equal size .

Step 2: With each quarter-size square, fold in half both ways, then fold all four corners to meet at the center. You now have a joining tab. Each full sheet of paper will make four joining tabs.

Congratulations! You have completed four joining tabs. Remember that the joining tab edges have the same length as the edges for the Square Flat Unit and Equilateral Triangle Unit. This means that we can insert these tabs into the pockets of both the square and triangle units in order to “join” their edges together. Tip: to save time, Post-It Notes can be used as joining tabs. Most Post-It Notes are 7.5 x 7.5 cm², which is exactly ¼ the size of a regular sheet of origami paper.

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What Can You Make With These? Now that you have learned how to fold two different types of flat unit modules, it is time to assemble them using the joining tabs (and a bit of glue!)

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Sample Models Try to use what you have learned to assemble other models. For inspiration, we visited MathWorld and looked at their list of Johnson Solids (http://mathworld.wolfram.com/JohnsonSolid.html).

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Notes

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Resources and Links Interested in learning more? Books: Fuse, Tomoko. Unit Polyhedron Origami. Japan Publications Trading, 2006. Fuse, Tomoko. Unit Origami: Multidimensional Transformations. Japan Publications, 1990. Kasahara, Kunihiko. Origami Omnibus. Japan Publications, 1998. Links: Wolfram MathWorld , http://mathworld.wolfram.com : “ The web's most extensive mathematics resource.” Features interactive 3D virtual models using the LiveGraphics browser plug-in, plus lots of information on the mathematical properties of just about every named polyhedron out there. This is where we found the information on polyhedron nets. Alexander's Polyhedra Pages , http://polyhedra.doskey.com : Contains a treasure trove of technical information about Stewart Toroids (polyhedra made from other polyhedra) and other recent musings in the world of geometry. If you have a VRML plug-in like the MediaMachines FluxPlayer , all of the WRL file links will be interactive. ◦ JovoToys, http://www.jovotoys.com : From the same person as above, Alex Doskey. Contains hundreds of plastic and VRML models of geometrical shapes and figures. Just about anything on the website which is made of squares and triangles can be made with the modules in this booklet. This is a fantastic resource for inspiration. Wikipedia, http://en.wikipedia.org : The Free Encyclopedia. In addition to having information on just about every subject, there are plenty of articles here on Archimedian , Platonic , and Johnson solids, including spinning animations, presented in a format that is easy to navigate. This is a good complement to MathWorld, and often links back to it. GEOMAG Constructions, http://textodigital.com/P/GG/bindex.ph p: During his life, Rafael Millàn was a prolific Geomag model builder. This website is an archive of his work. Since Geomag units have many of the same physical limitations as origami modules, a lot of what he made, you can make as well. Geomag Wiki, http://geomag.wikia.com : A wiki full of real-world Geomag models ranging from the simple to the complex, most of which can also be constructed using modular origami. For an interesting challenge, search for the “meta-dodecahedron.”

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Stores: Marukai & Marukai SuperM Stores, http://www.marukai.com : A nice place to buy origami paper on the cheap. Shop online or visit one of several South Bay locations. ◦ Marukai Forum, Gardena: 1740 WEST ARTESIA BLVD., GARDENA, CA 90248 310660-6300 ▪ Members-only store. Purchase a one-day pass for $. This has the largest selection and variety of origami paper that we know of, and it is the only local location that we have found which carries 500-sheet packs! ◦ Marukai 98¢ Plus, Gardena: 1360 WEST ARTESIA BLVD., GARDENA, CA 90248 ▪ Japanese daiso, or $1.00 store. No membership required . A good selection of origami paper. ◦ Marukai 98¢ Plus, Torrance: 3832 SEPULVEDA BLVD. TORRANCE, CA 90505 ▪ Japanese daiso or $1.00 store. No membership required . A good selection of origami paper. Mitsuwa Market: Japanese Supermarket: 21515 Western Avenue, Torrance, CA ◦ Inside the market, there is a bookstore which stocks a limited supply of origami books and paper. There is also a stationary store inside which carries single-color packages. Although Mitsuwa is a little higher-priced, it offers a good variety of paper.

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Created by: U. Akotaobi & C. McCann April 22nd 2010

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