Is the diamond shape (rhombus) necessarily different from the square (sides not perpendicular, different lengths of the diagonals), or is the square a special case of a diamond?
Sometimes I see a square (with right angles) that is called a diamond just because of being rotated by 45 degrees from the position where two of the sides were horizontal and the other two vertical, but I doubt that just this rotation (which is an isometric transformation, without any deformation) is sufficient for the square to become another shape.
The standard terminology in plane geometry (if the English language even has one) is that a square is a rhombus which is also a rectangle; equivalently, in a more exhotic and minimalistic way, a kite which is also a rectangle. Of course plane geometry is invariant by the action of the group of plane isometries, and therefore it isn't concerned with the graphical rendering of a square in an "organized" space where one or more directions are more important than the others.
Answered by Gae. S. on November 21, 2020
Mathematically speaking, something is called a rhombus, when it has four courners and all edges have the same length.
Something is called a square, when it has four courners, all edges have the same length and the four interior angles are $90°$.
So yes, a square is a special case of a rhombus.
Answered by Nurator on November 21, 2020
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