Can an equilateral triangle have vertices at integral lattice points?
Integral lattice points are such points as (101, 254) or (3453, 12), but not points such as (123.4, 1) or (√2, 5)
If you can't find a solution in the 2D Cartesian plane, can you find one in a 3 (or more) dimensional space?
Let's assume that there is such an equilateral triangle with integral lattice points. Call it triangle ABC, and set point A as the origin.
Extend BC to double its length to point P. APB is a 30-60-90 triangle with integral lattice points. Note that AP=sqrt(3)*AB.
P
|\
| \
| \
| C
| / \
| / \
|/ \
A_______B
Let's call the coordinates of B (X,Y). P's coordinates must be (ħsqrt(3)*-Y,ħsqrt(3)*X) because the direction (vector?) from A is rotated 90 degrees and the magnitude is sqrt(3) times more. Since X and Y are integers, P cannot be an integral lattice point. This leads to a contradiction, so therefore, there is no such triangle.
As for part 2, (1,0,0), (0,1,0), (0,0,1) is the smallest such equilateral triangle, but there are many more oriented in different ways.
|
|
Posted by Tristan
on 2004-06-20 15:52:09 |
Proof
