You are given an equilateral triangle. What is the most number of such identical triangles you can place such that they do not overlap, but each one touches the original triangle?
I think you can fit
An Infinite #
Place a triangle next to the original one so one edge is fully adjacent
Place a second triangle along that same edge, fully adjacent, but lift the far vertex vertically off the plane a bit so the new triangle rotates in 3-D
Place a third triangle along that same edge, but between the first two
Continue indefinitely. You will have an effect like the pages of a book spread open in a fan.