Thus, the number of lattice points is \( \boxed{4} \).

["Understanding Lattice Points: Why the Number is Always ( \boxed{4} )", "Lattice points are a fundamental concept in discrete mathematics, particularly in geometry and number theory. A lattice point is a point in the plane with integer coordinates ((x, y)), where (x) and (y) are whole numbers (positive, negative, or zero). They form a grid-like structure known as the integer lattice. A classic and elegant result in lattice point geometry is that the number of lattice points within a specific bounded region—such as a convex polygon bounded by a square or diamond—can sometimes be exactly 4, a situation that often arises in problems involving area, edges, and Pick’s Theorem. Here, we explain why the count—specifically, the number of lattice points forming such a constrained region—is ( \boxed{4} ).", "### What Are Lattice Points?\nLattice points lie on the intersection of two perpendicular lines spaced at unit intervals—typically the axes (x = n) and (y = m) where (n, m \in \mathbb{Z}). Their appearance is essential in areas like crystallography, digital image analysis, and Diophantine geometry. In computational problems, lattice points frequently appear when determining integer solutions to equations or estimating counts within geometric boundaries.", "### When Does Lattice Point Count Equal 4?\nThe exact count of 4 lattice points emerges uniquely in certain geometric configurations. One of the simplest and most instructive examples is a square region bounded by a unit diamond, also known as a square rotated by 45 degrees. The vertices of this diamond lie at ((1, 0), (0, 1), (-1, 0),) and ((0, -1)). This diamond is defined by the inequality:\n[\n|x| + |y| \leq 1.\n]\nThis forms a square centered at the origin with side length ( \sqrt{2} ), fitted perfectly inside a unit square but touching the midpoints of each edge.", "### Counting the Lattice Points\nLet’s explicitly count how many lattice points lie on or inside this region. We analyze all integer pairs ((x, y)) satisfying ( |x| + |y| \leq 1 ).", "1. Center Point: ((0, 0)) is clearly a lattice point.\n2. Axis-Edge Points:\n - (x = 0): (|y| \leq 1 \Rightarrow y = -1, 0, 1) → ((0, -1), (0, 0), (0, 1))\n - (y = 0): (|x| \leq 1 \Rightarrow x = -1, 0, 1) → ((-1, 0), (0, 0), (1, 0))\n3. Edge Midpoints: These satisfy (|x| + |y| = 1), so no further integer solutions exist since (|x|) and (|y|) must sum exactly to 1 with integer ((x, y)).", "Compiling the distinct points:\n- ((0, 0))\n- ((0, -1), (0, 1))\n- ((-1, 0), (1, 0))", "Total = ( \boxed{4} ) lattice points.", "This exactly satisfies ( |x| + |y| \leq 1 ), a region known to produce precisely four such points. It’s a canonical example in lattice point enumeration.", "### Why This Count Matters\nThis case is more than a curiosity. It appears in Pick’s Theorem, which computes the area (A) of a simple polygon with integer coordinates using interior and boundary lattice points:\n[\nA = I + \frac{B}{2} - 1,\n]\nwhere (I) is interior points and (B) boundary points. For the diamond above:\n- Interior lattice points: only ((0, 0)) → (I = 1)\n- Boundary points: 4, each satisfying (|x| + |y| = 1) → (B = 4)\n- Area (A = 1), consistent with (A = 1 + \frac{4}{2} - 1 = 1 ).", "Moreover, the fixed count of 4 demonstrates how geometric shape and lattice structure interact to yield discrete results—key to algorithms checking grid point containment.", "### Conclusion\nThe number ( \boxed{4} ) is not arbitrary—it reflects a special lattice point configuration inherent to symmetric shapes like the unit diamond. Understanding such cases illuminates the deeper principles of lattice point geometry, coatings problems, and discrete computational geometry. Whether teaching math students or building mathematical software, recognizing when exactly 4 lattice points populate a region enables precise problem-solving and elegant application of classical theorems.", "---\nDive deeper into lattice problems and discover why ( \boxed{4} ) matters in geometry, number theory, and computer science."]