If you know the notation for the Rubik's Cube, the Pyraminx notation will be no problem. Each of the four corners is labeled as shown on the right. One-layer turns only affect the trivial tips, the little pyramidal pieces on the corners, so we won't use them for algorithms, but a one-layer turn of the R corner, for example, would be labeled as r (if it is a clockwise turn) or r' (if it was counterclockwise). Similarly, a two-layer turn of the R corner would be labeled as R or R', and all of the other corners follow the same notation. There are two important types of pieces on a Pyraminx: the four corners (the pieces touching the trivial tips) and the six edges (the other pieces).
The image on the left shows what we want to accomplish: we want to correctly orient all of the corners, solve all of the trivial tips, and also solve two edges. This seems like a difficult task, but it can actually be done very quickly.
(solved!)
R U' R' U' R U' R'
R' U' R U' R' U' R
R U R' U R U' R'
R' U R U R' U R
U L U' R U' R' U L'
L U' R U R' U L' U'
R' L R L' U L' U' L
L R' L' R U' R U R'
R' U L' U L U' R U'
L U' R U' R' U L' U
L U R U' R' L'
L' B' U' B U L
R' L' U' L U R
R U B U' B' R'
B' R' U' R U B
B U L U' L' B'
L R U R' U' L'
L' U' B' U B L
R' U' L' U L R
R B U B' U' R'
B' U' R' U R B
B L U L' U' B'
R L' U L U' L R' L'
R U' R' U L' U L U'
U' R U' R' U L' U L
U L' U L U' R U' R'
L' R U' R' U R' L R
L' U L U' R U' R' U
U L' U L R U' B U' B' R'
R B U B' U R' L' U' L U'
U' L' B' U B' L' B' L'
U' R' B' R' B' U B' R'
U L B L B U' B L
U R B U' B R B R
R U' R' L' U' L U'
U' R U' R' L' U' L
U R U' R' L' U' L U
U' L' U L R U R' U'
U L' U L R U R'
L' U L R U R' U
