Infill pattern comparison I'm relatively new to 3d printing, and wanted to get a few things understood.
Firstly, I am unclear on how Hexagonal infill is stronger than, say, diamond pattern. 

Can anyone explain how the different shape causes the structure to be stronger? I saw a few places that hex is stronger; usually, more vertices means 'weaker' shape (i.e. a triangle is stronger than a square), so how does that work with hex vs diamond?

Also, in small objects, where the printer makes only a single dot as the infill (a dot instead of a line in larger objects), does the infill strengthen the object at all?

EDIT: I am trying to understand the effect of the infill pattern on the strength of the print, regardless of print time.
Hex grids are used for different reasons than triangular grids (such as you often see on bridges and roof systems). Triangles are especially good at being rigid, while hex grids are very material-efficient for a given strength. The second reason ($) is typically more important for 3D printing.

Triangles do have fewer vertices than squares, but it's not always true that "fewer vertices" means stronger. Vertices are one kind of weak point. But in a triangle, vertex "angle-holding" failures simply don't matter. You can fasten 3 bars together with hinges or other joints that have little resistance to changing angle, and the triangle is still rigid.

In contrast, rectangular grids can (and do -- https://www.youtube.com/watch?v=5t9MpNTSbYg) completely collapse if their vertices aren't rigid enough. That combines badly with the fact that vertices concentrate forces, so have to be much stronger than sides in comparable settings.

A triangle cannot change without changing multiple things -- at least 2 angles and a side, or all three sides. Intuitively, the sides and vertices of a triangle work together for strength. This advantage of triangles doesn't transfer to hexagons, but hexagons have two other advantages: First, if you want to fill a space with a repeating shape, hexagons use less material than other shapes. And second, hexagons keep all the individual "walls" shorter compared to others shapes, which makes them less prone to bend.

The material efficiency was proven by Thomas Hales in 1998, according to http://www.slate.com/articles/health_and_science/science/2015/07/hexagons_are_the_most_scientifically_efficient_packing_shape_as_bee_honeycomb.html. His paper "The Honeycomb Structure" is available at https://www.communitycommons.org/wp-content/uploads/bp-attachments/14268/honey.pdf