Can you explain why it is impossible to construct this triangle?

Recreating the designs in this challenge requires you to break a problem down into manageable chunks and use the relationships between triangles and hexagons. An exercise in detail and elegance.

Can you find a general rule for finding the areas of equilateral triangles drawn on an isometric grid?

Using LOGO, can you construct elegant procedures that will draw this family of 'floor coverings'?

Make an equilateral triangle by folding paper and use it to make patterns of your own.

If the yellow equilateral triangle is taken as the unit for area, what size is the hole ?

Using the interactivity, can you make a regular hexagon from yellow triangles the same size as a regular hexagon made from green triangles ?