This instructable is for Kasahara's single-sheet tetrahedron. The original is in his masterful book Origami for the Connoisseur. I have made some wordy instructions that go with the drawings, but I'm presenting them separately to avoid clutter. For some reason the drawings didn't load too well, but you can enlarge them by clicking the little i thingy.
If interested I've just posted diagrams for Jun Maekawa's Dodecahedron Unit
Step 1: What You Will Need
All you will need is a square sheet of paper. It is best, if you are new to origami, to start with a piece of paper that is one color on one side and a different color on the other side, in case you find it easier to follow my instructions that way. Otherwise, copy or printer paper or paper of that weight makes a crisper cleaner model. You might want to start with a sheet approximately 6" square.
Step 2: Preliminary Creases
This step, (two pages) shows how to make the preliminary creases. You should already know the symbols for Mountain fold, Valley fold, etc; There is nothing very complex here. Just mountain and valley folds...If you need a refresher course on the symbols, follow this link: http://www.fishgoth.com/origami/basics1.html
1. Valley fold paper in half, making only a small crease at the bottom, then unfold..., then fold the right side in half making only a small crease about half-way up...unfold.
2. Valley fold as indicated.
3. Valley fold as indicated.
4. Valley fold as indicated.
5. Mountain fold flaps behind, then unfold model completely except for those.
6. Fold and unfold as indicated.
7. Fold and unfold as indicated.
8. Fold and unfold as indicated. Make sure you match up the bottom edge of the flap with the center mark each time.
Step 3: Folding the Tetrahedron
Here comes the fun part. Up til now we've been working on a flat sheet of paper. That's over. I'm hoping the drawings I've made aren't too confusing.
A. Valley fold bottom left hand corner to the crease intersection as indicated.
B. Sorry, B didn't load too well...the horizontal "center-line" is a mountain fold, and the bisecting crease should be a valley fold...Anyway, bring the point up to the tip of the triangle as indidated, and lay the edge down flat.
C. 1. Valley fold the point at 1. to the point at the tip of the arrow, then
2. fold down flap to lock the last fold in place.
D. 1. What you're holding in your hand now should be something like an ice cream cone. Push in on the valley folds at the right of the model, as indicated, and slide the resultant tip at 1. up and under the flap as indicated. What you have in your hand now should resemble a tetrahedron with an extra face. Take the remaining tip at 2. and pull it down to the point as indicated.Tuck the flap in.
You should have a tetrahedron in your hand now. Here is a cool link:
Psic8 made it!