"Triangles", tetragon, 4-sided pyramid, these all describe the same thing: the simplest brush you have avaialable to build stuff with, most commonly describes as "triangles" in the forums.
NOTE: sometimes it is only necessary to clip a rectangualr brush into two triangles to make an "invalid solid" error and others go away, but sometimes if you have avery difficult shape, you are better off using purely tetrahedrons or "triangles".
It is nearly impossible to get errors VMing with only these brushes. (clipping is another story of course).
To avoid the dreaded "leaf saw" errors on your during VIS Compile, construct your complex detail items and props in hammer entirely out of 4-sided polygons.
Included are a cube cut into 6, 4-sided polygons, and a partially hollow cube created out of 4, 4-sided polygons, slightly contorted.
I would recommend using only the first mentioned, 6-piece cube for your construction, as the weirdly hollow, 4-piece cube is more of a novelty, just showing you it's possible. However, if you can find a use for the 4-piece cube, go ahead!
= )
Hope this helps some people endeavoring to create fancy brushwork in Valve Hammer Editor! Again, DO NOT make your entire map out of these stuctures! Doing so would be highly impractical/unnecessary, and boost your compile time to a unmanageable level very quickly..
= )
Unless you're high.
Using tetrahedrons eliminate the appearance of invalid brushes. It's what i use to make all my crazy creations.
=)
Also, the leaf saw into leafs is not an error, its just a warning and it can be ignored, as it won't make you map unplayable.
Just make sure your brush work is grid-safe (IE: vertices on the grid) and compile with full vis for the final compile. Its also recommended to use cliptype precise or cliptype simple.
It is virtually impossible to make leaf-saw errors using these structures, that's the whole point, again, not for your whole map, just for crazy detail items and props!
A tetragon is something entirely different, it's a 2D shape with four edges (tetra gon - four angles). Squares and rectangles are examples of tetragons.