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This one should be not too hard ...

Below crafted octagon features my PSE profile image.

PSE profile image

It is designed using eight identical grey scaled squares.

This 2D image, which I once incidentally drew, appears, with some fantasy, to have some 3D effects.

The white vertices of all these squares have integer coordinates.

Question: What is the (minimal) area of this octagon.

Pure numeric answer is appreciated but some explanation is welcome.

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    $\begingroup$ Regarding the "spoiler": reveal spoiler It's not 3D, it's a tesseract! $\endgroup$ Commented Jan 5 at 22:11
  • $\begingroup$ @AxiomaticSystem I also believe see some reveal spoiler Esher effect. $\endgroup$ Commented Jan 6 at 9:34
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    $\begingroup$ @AxiomaticSystem, that is what I saw initially. I really had to step back and see the octagon. $\endgroup$ Commented Jan 6 at 18:29

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The minimal area of the octagon is

$ 48 = 8 \times 8 - 4 \times \left( 1 \times 1 \right) - 8 \times \frac{3 \times 1}{2} $, if the white vertices of the squares have integer coordinates in the coordinate system with axes parallel to sides of the rectangular picture. The vertices of the octagon have coordinates $\left( \pm 4, 0 \right)$, $\left( 0, \pm 4 \right)$, $\left( \pm 3, \pm 3 \right)$.
image1

Explanation:

The only smaller octagon with integer coordinates has vertices $\left( \pm 3, 0 \right)$, $\left( 0, \pm 3 \right)$, $\left( \pm 2, \pm 2 \right)$ (other possibilities give a non-convex octagon or a square) and area $ 24 = 6 \times 6 - 4 \times \left( 1 \times 1 \right) - 8 \times \frac{2 \times 1}{2} $. However, in this case the eight internal vertices form a square with sides parallel to the coordinate axes, while the picture in the puzzle shows internal vertices forming a square with diagonals parallel to the sides of the picture. image2

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    $\begingroup$ I wonder if the coordinate system is allowed to be at an angle relative to the image. $\endgroup$ Commented Jan 5 at 22:01
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    $\begingroup$ I think the picture can be rotated in any way so as to minimize the area, but the OP should confirm. $\endgroup$ Commented Jan 5 at 22:02
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    $\begingroup$ @FirstNameLastName Isn't the second image in this answer a rotated improvement? $\endgroup$ Commented Jan 5 at 22:54
  • $\begingroup$ Second image, indeed, is a rotated one, similar to first, and, also has integer square vertices coordinates. But the sides of the second octagon are not parallel to the sides of the OP octagon. $\endgroup$ Commented Jan 6 at 9:33
  • $\begingroup$ Welcome to PSE: Note that there are also next bigger solutions (from -N to +N), but this OP one (N=4), perhaps, looks a bit more visual interesting. $\endgroup$ Commented Jan 10 at 0:11

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