Hi -

I would like to prove, that on an equilateral triangular grid such as this one but extended to infinity

it is not possible to mark 5 points of triangles that form a perfect pentagon.

I would think it is also not possible to resize the orange triangle so that all three points are vertexes of the triangular grid.

Can anyone help me to prove or disprove this?

It seems to me that if the grid is infinite it must be possible to come arbitrarily close to achieving it by increasing the size ?

Oh and of course the next questions would be - if it is true that this cannot be done ...

is it also true for a tetrahedral/octahedral honeycomb in 3D ?

And the equivalent in all higher dimensions ?

Thanks in advance ~ DV