A grid and a shape

[studiot]

Thank you for clarifying that it must not be placeable.

The nearest I have been able to come is to start with a 1 x 1 square, which obviously has area of 1 and is aligned to the grid blue points at its four corners.

 

If we now reduce one side only by an infinitesimal amount δ then the area now becomes 1(1-δ), which is strictly less than 1, by an infinitesimal amount.

This slants one side only away from its corner.

Lifting the square off the base two corners and shifting it sideways by  infinitesimal amounts leaves the square clear of all four blue grid points.

 

 

Edited May 6, 2023 by studiot

[Genady]

9 minutes ago, studiot said:

Thank you for clarifying that it must not be placeable.

The nearest I have been able to come is to start with a 1 x 1 square, which obviously has area of 1 and is aligned to the grid blue points at its four corners.

 

If we now reduce one side only by an infinitesimal amount δ then the area now becomes 1(1-δ), which is strictly less than 1, by an infinitesimal amount.

This slants one side only away from its corner.

Lifting the square off the base two corners and shifting it sideways by  infinitesimal amounts leaves the square clear of all four blue grid points.

 

 

I understand your description and there are no problems with it, but I am not clear about what it proves. I have tried a few posts above to reformulate the problem in a - hopefully - clearer way. Here I write it again:

You have a plane with the grid points on it as described in the OP. Somebody gives you a transparency with an inkblot on it. You position the transparency on top of your plane in such a way that the inkblot does not cover or touch any of the grid points. 

Prove that such a positioning exists for inkblot of any shape as long as its area is less than unity.

[studiot]

Pick's Theorem

 

Pick's theorem - Wikipedia

[Commander]

A shape like this 

[Genady]

4 hours ago, Commander said:

A shape like this 

If you shift the shape up, or the grid down as below, the shape does not intersect with the grid dots: