The term diamond is another word for a rhombus. The term is also used to denote a square tilted at a angle.
The diamond shape is a special case of the superellipse with parameter , giving it implicit Cartesian equation
(1)
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Since the diamond is a rhombus with diagonals and , it has inradius
(2)
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(3)
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Writing as an algebraic curve gives the quartic curve
(4)
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which is a diamond curve with the diamond edges extended to infinity.
When considered as a polyomino, the diamond of order can be considered as the set of squares whose centers satisfy the inequality . There are then squares in the order- diamond, which is precisely the centered square number of order . For , 2, ..., the first few values are 1, 5, 13, 25, 41, 61, 85, 113, 145, ... (OEIS A001844).
The diamond is also the name given to the unique 2-polyiamond.