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A versatile modular origami unit. Good for making Platonic and Achimedean solids (wikipedia)

Two origami squares make one unit. These half units are based on the units of the triangle box.

origami-instructions.com easy modular triangle box

origami-fun.com origami triangle box

I cut squares, size 1-7/8 inch using this.

mcgillinc.com Stacking Squares

You could use a paper cutter or even scissors.

swingline.com ClassicCut Gulliotine Trimmer

In the photos I used cardstock.

Step 1: First Through Fourth Folds

1. Fold in half diagonally.

2. Pinch fold on long side of triangle. Fold point down as shown in photo.

3. Fold the other point symetrically.

4. Turn over. Fold two layers as shown in photo.

Step 2: Finish Two Half Modules

Half-open the last 3 folds, leaving the model with a 3d shape.

There are 2 corners of the paper together at bottom right. Fold one inward to the centre of the half module.

Make a second half module in the same way, except for the last step fold the other corner into the centre of the half module. Actually this is a reverse fold.

To join the 2 half modules, each unfolded corner goes into the other half module along side the inside folded corner.

The two half modules are held together tighter when the fold between the two is more closed.

Each module has four triangles which can fit into the pocket of a triangle in another module.

Step 3: Connecting Modules

Put two modules together by putting one triangle down into the pocket of another.

Creasing the triangles when one is fully inserted into the other locks them together.

Add a third module, making three coming together at a point. Three edges, one vertex. The folds of the already joined modules can be closed tighter to make it easier to insert the next module.

Add more locking folds as needed.

Step 4: Other Configurations of Modules

Four or five modules can also go around a vertex. The icosahedron model has 5 modules around each vertex. Three, four and more modules can go around to make faces: triangles, squares, etc.

Step 5: Another Example

I cut some more squares from printer paper. The new model is a truncated cube. It has six octagonal faces and eight triangular faces. It uses 36 modules. The triangular faces resemble the triangle box that is the basis of the module. I outlined one of the octagon faces in red (and pink where it is hidden behind other parts of the model). I outlined one of the triangle faces in blue. The problem with this model is that it pulls apart easily. For example, at the places with the arrows in the fourth picture. I had two ideas for how to keep it together better. but I did not want to build another copy of the model just to try both of them. So, I tried one method on one side, the other on the other side. The fifth picture shows locking creases added. The last picture shows that I reversed some of the modules. The red lines show the segments of the octagon that were facing into the model are now facing out. Oddly this works about as well as the locking creases. There is more, here,

truncated icosahedron

<p>I've never seen this before. How do you get from the step 4 to the finished product shown at the beginning?</p>
Each module corresponds to an edge of a polyhedron. You could start with a cube. It has 12 edges, so for 12 modules, you need 24 squares. Keep adding modules until all the edges of the cube are there. For an icosahedron or a dodecahedron you need 30 modules.

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