That means this:

...will become this:

Do this for All four of the corner pieces. After doing this, there is no reason whatsoever to turn the outer corners again for the rest of the solution.

If you're trying to solve a tetraminx, you can ignore everything I just said.

The following segments are called the intermediate corners. These are represented by the shaded in areas of this picture...

Twist the pyraminx/tetraminx so that the intermediate corners line up on each side. This should make something like a Mitsubishi logo on each side like this.

Mitsubishi!

Tip:

Do you find it hard to determine what color a side will end up? (i.e. you line up 2 blue intermediate corners and you try to align the third corner but it doesn't have any blue on it) Look at the corner piece on the opposite side. There are 3 stickers on it. The color of the opposite side will be the color that is missing among those 3 stickers. Look at the picture above for example. The corner in the middle of the picture has blue, red, and yellow on it. The missing color is green. Therefore the opposite side is green.

The next 2 steps will focus on positioning the pieces between the intermediate corners from the previous step. In the future, I will call these the 'edge pieces' or 'edges'.

This step will solve the three edges on the bottom. In this picture I have selected blue and I am holding it so that blue is on the bottom.

Now we're going to solve the purple area.

Look for an edge piece in the orange region pictured above that has your base color on it. (In my case, blue) The probability of not being able to find a base colored edge on the top row is 1 out of 20.

After you find it, take note of the color of the other sticker on this edge. Let's say for example you find a blue edge piece that also has a yellow sticker on it. Hold the pyraminx so that the yellow-face-to-be is in front and the blue face is still on the bottom. Now turn the top so that the blue/yellow edge is in front and the blue sticker is facing you. Like in this picture...

...or it could possibly look like this...

If while looking at the yellow face, the blue/yellow edge is on the left, do this...

(Illustrations show front and right faces)

If while looking at the yellow face, the blue/yellow edge is on the right, do this...

(Illustrations show front and right faces)

Continue this until all the base colored pieces on the top row are on the bottom row. If a piece you need is already on the bottom row but in the wrong place or the wrong orientation, put a random piece from the top row in its place with the method above. The piece you need will now be available on the top row.

You should have something that looks like this...

And then do this...

In this case do this (colors don't have to match the example)...

And the other possible direction:

In this case do this (colors don't have to match the example)...

And that's all there is to it.