An infinite triangular lattice
Geometry
Level
3
You have an infinite triangular lattice.
Is it possible to pick two points on the lattice such that no point on the lattice forms the third point of an equilateral triangle?
Yes
This is an unsolvable question
No
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1 solution
The triangular lattice points are of the form: z = m + n/2 + i×n×sqrt(3)/2, where i is the imaginary unit and m, n are integers. WLOG assume one of the points is (0,0) and the other is (m+n/2, n×sqrt(3)/2). If we multiply z by e^(i×π/3) we rotate our line segment by π/3. After this transformation we get an equilateral triangle. Now we need to make sure our new point is of the form like z. After some multiplication we obtain that indeed it is [ just switching (m, n) -> (-n, m+n) ]. Thus for every two triangular lattice points we can construct a point which makes an equilateral triangle with them.