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)
|
A diamond that is a rhombus with diagonals 
and 
, where 
, has inradius
 |  |  |
(2)
|
 |  |  |
(3)
|
Writing the diamond as an algebraic curve gives the quartic curve
 |
(4)
|
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 are integer lattice points satisfying
the inequality 
, for nonnegative
integer 
.
There are then 
squares in the order-
diamond, which is precisely the centered
square number of order 
.
For 
, 1, ..., 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.
