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Diamond


Diamond

The term diamond is another word for a rhombus. The term is also used to denote a square tilted at a 45 degrees angle.

DiamondCurve

The diamond shape is a special case of the superellipse with parameter r=1, giving it implicit Cartesian equation

 |x|+|y|=1.
(1)

Since the diamond is a rhombus with diagonals p=2a and q=2b, it has inradius

r=(pq)/(2sqrt(p^2+q^2))
(2)
=(ab)/(sqrt(a^2+b^2)).
(3)
DiamondLines

Writing as an algebraic curve gives the quartic curve

 ((y^2)/(b^2)-(x^2)/(a^2)-1)^2=(4x^2)/(a^2),
(4)

which is a diamond curve with the diamond edges extended to infinity.

DiamondPolyomino

When considered as a polyomino, the diamond of order n can be considered as the set of squares whose centers satisfy the inequality |x|+|y|<=n. There are then 2n(n+1)+1 squares in the order-n diamond, which is precisely the centered square number of order n. For n=1, 2, ..., the first few values are 1, 5, 13, 25, 41, 61, 85, 113, 145, ... (OEIS A001844).

DiamondPolyiamond

The diamond is also the name given to the unique 2-polyiamond.


See also

Aztec Diamond, Centered Square Number, Diamond Graph, Kite, Lozenge, Parallelogram, Polyiamond, Quadrilateral, Rhombus, Superellipse, von Neumann Neighborhood

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References

Sloane, N. J. A. Sequence A001844/M3826 in "The On-Line Encyclopedia of Integer Sequences."

Cite this as:

Weisstein, Eric W. "Diamond." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Diamond.html

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