You can fold a simple unit from a standard U.S. business card that can be combined with other units to create a tetrahedron, octahedron, or icosahedron. The tabs on the business cards hold the other cards in place. It is really quite elegant.
You need a supply of business cards with dimensions that have a ratio of 7 to 4. This corresponds to the standard 2 inch by 3.5 inch business cards in the U.S. If you have business cards of other dimensions, you can trim some off of to make it skinnier. For example, if you have credit card-sized cards 85.6 x 53.98 mm, you can trim 5 mm. to make it narrower. If your cards are 85 x 55 mm, trim off 6 mm. You can also simply cut some card stock.
Fold a business card in half so that the bottom right-hand corner touches the upper left-hand corner.
Fold the flaps over the equilateral triangles that you just created. Make your creases sharp by pushing them flat with your thumbnail, a pen cap, or some other improvised tool.
Open the card slightly. This will be a "left-handed unit."
Create a "right-handed unit" by repeating the process above except that you start by folding the card so that bottom left-hand corner touches the upper right-hand corner. This will be a "right-handed unit."
Make as many of these units as you need to produce the polyhedron you want: one of each will make a tetrahedron, two of each will make an octahedron and five of each will produce an icosahedron.
Slip the units together so the flaps hook over and hold the other units in place. You don't have to use glue. However the icosahedron is quite tricky to make without glue because the model doesn't say together until the last unit is in place.
It is only a coincidence that the simple fold shown here produces equilateral triangles. The arctangent of (4/7) is almost 30° so the fold gives you a pretty good approximation of a 60° angle.
You can make other polyhedral models with these units. Try creating the deltahedra.
This model was independently invented by Jeannine Mosely and Kenneth Kawamura.
This site has big selection of other business card models you can make. I can imagine getting hooked on these for the sheer simplicity of the concept and the broad range of models you can make. Isn't the internet great?