diff --git a/PyAPD/apds.py b/PyAPD/apds.py
index a0dcf79..f8635c8 100644
--- a/PyAPD/apds.py
+++ b/PyAPD/apds.py
@@ -42,6 +42,7 @@ def __init__(
         error_tolerance=0.01,
         pixel_size_prefactor=2,
         seed=-1,
+        periodic=False,
     ):
         """
         Construct an anisotropic power diagram system.
@@ -86,6 +87,11 @@ def __init__(
             Multiplier applied to the computed pixel count. Default: 2.
         seed : int
             Manual random seed (< 0 means no seed is set). Default: -1.
+        periodic : bool
+            If True, distances between seeds and points use the minimum-image
+            convention across the (rectilinear) domain, so the power diagram
+            is periodic and grains may wrap across the domain boundary.
+            Default: False (fully backward-compatible).
         """
 
         self.N = N
@@ -103,6 +109,15 @@ def __init__(
 
         self.set_domain(domain)
 
+        self.periodic = bool(periodic)
+        # Per-dimension edge lengths of the (rectilinear) domain. Kept on the
+        # same device/dtype as the domain so KeOps arithmetic stays homogeneous.
+        edge = (self.domain[:, 1] - self.domain[:, 0]).to(
+            device=self.device, dtype=self.dt
+        )
+        self._L_tensor = edge  # shape (D,), plain torch
+        self._L = LazyTensor(edge.view(1, 1, self.D))  # broadcast over (N, M, D)
+
         self.set_X(X)
 
         self.set_As(As)
@@ -387,6 +402,22 @@ def mask_pixels(self, mask):
             self.W = initial_guess_heuristic(self.As, self.target_masses, self.D)
             self.w = LazyTensor(self.W.view(self.N, 1, 1))
 
+    def _displacement(self, y, x):
+        """Return ``y - x`` as a KeOps LazyTensor, wrapped to the minimum
+        image across the periodic domain when ``self.periodic`` is True.
+
+        Uses the KeOps-compatible expression ``dy - L * round(dy / L)``,
+        which rounds ``dy / L`` to the nearest integer per component.
+        ``LazyTensor.floor()`` is unavailable on some KeOps builds and the
+        Python ``%`` operator is not defined between LazyTensors, so the
+        ``.round()`` form is used.
+        """
+        dy = y - x
+        if self.periodic:
+            shift = (dy / self._L).round()
+            dy = dy - self._L * shift
+        return dy
+
     def assemble_apd(
         self, record_time=False, verbose=False, color_by=None, backend="auto"
     ):
@@ -396,7 +427,8 @@ def assemble_apd(
         if self.Y is None:
             self.assemble_pixels()
         start = time.time()
-        D_ij = ((self.y - self.x) | self.a.matvecmult(self.y - self.x)) - self.w
+        dy = self._displacement(self.y, self.x)
+        D_ij = (dy | self.a.matvecmult(dy)) - self.w
         # Find which grain each pixel belongs to
         grain_indices = D_ij.argmin(dim=0, backend=backend).ravel()
         time_taken = time.time() - start
@@ -410,6 +442,34 @@ def assemble_apd(
         else:
             return grain_indices
 
+    def grain_of(self, points, backend="auto"):
+        """Return the grain index that owns each point in ``points``.
+
+        Parameters
+        ----------
+        points : torch.Tensor of shape (M, D)
+            Query points, in the same coordinate frame as ``self.domain``.
+            When ``self.periodic`` is True, query points are compared to each
+            seed using the minimum-image distance across the domain.
+        backend : str
+            KeOps reduction backend (forwarded to argmin).
+
+        Returns
+        -------
+        torch.Tensor of shape (M,), dtype int64
+            For each row of ``points``, the grain index ``i`` minimising
+            ``(y - x_i) . A_i . (y - x_i) - w_i`` under the active metric.
+        """
+        pts = points.to(device=self.device, dtype=self.dt)
+        if pts.ndim != 2 or pts.shape[1] != self.D:
+            raise ValueError(
+                f"grain_of: expected (M, {self.D}), got {tuple(pts.shape)}"
+            )
+        y = LazyTensor(pts.view(1, pts.shape[0], self.D))
+        dy = self._displacement(y, self.x)
+        D_ij = (dy | self.a.matvecmult(dy)) - self.w
+        return D_ij.argmin(dim=0, backend=backend).view(-1)
+
     def plot_apd(
         self, color_by=None, mode="auto", alpha=None, ps_scale=False, marker_scale=20.0
     ):
@@ -603,7 +663,8 @@ def OT_dual_function(self, W, backend="auto"):
         self.W = W
         self.w = LazyTensor(self.W.view(self.N, 1, 1))
 
-        D_ij = ((self.y - self.x) | self.a.matvecmult(self.y - self.x)) - self.w
+        dy = self._displacement(self.y, self.x)
+        D_ij = (dy | self.a.matvecmult(dy)) - self.w
         idx = D_ij.argmin(dim=0, backend=backend).view(-1)
 
         ind_select = torch.index_select(self.X, 0, idx) - self.Y
@@ -623,7 +684,8 @@ def check_optimality(
         if self.Y is None:
             self.assemble_pixels()
 
-        D_ij = ((self.y - self.x) | self.a.matvecmult(self.y - self.x)) - self.w
+        dy = self._displacement(self.y, self.x)
+        D_ij = (dy | self.a.matvecmult(dy)) - self.w
 
         grain_indices = D_ij.argmin(dim=0, backend=backend).ravel()
         volumes = torch.bincount(grain_indices, self.PS, minlength=self.N)
@@ -707,7 +769,8 @@ def adjust_X(self, backend="auto"):
         if not self.optimality:
             print("Find optimal W first!")
         else:
-            D_ij = ((self.y - self.x) | self.a.matvecmult(self.y - self.x)) - self.w
+            dy = self._displacement(self.y, self.x)
+            D_ij = (dy | self.a.matvecmult(dy)) - self.w
             grain_indices = D_ij.argmin(dim=0, backend=backend).ravel()
             normalisation = torch.bincount(grain_indices, self.PS, minlength=self.N)
             new_X0 = (
@@ -730,6 +793,12 @@ def adjust_X(self, backend="auto"):
             else:
                 self.X = torch.stack([new_X0, new_X1], dim=1)
 
+            if self.periodic:
+                # Centroids may fall outside the box when a grain straddles the
+                # periodic seam; wrap them back into [domain[:, 0], domain[:, 1)).
+                origin = self.domain[:, 0]
+                self.X = origin + torch.remainder(self.X - origin, self._L_tensor)
+
             self.optimality = False
             self.x = LazyTensor(self.X.view(self.N, 1, self.D))
 
diff --git a/tests/test_apds_periodic.py b/tests/test_apds_periodic.py
new file mode 100644
index 0000000..231025b
--- /dev/null
+++ b/tests/test_apds_periodic.py
@@ -0,0 +1,117 @@
+"""Tests for the periodic (minimum-image) distance support in apd_system."""
+
+import pytest
+import torch
+
+from PyAPD import apd_system
+
+
+def test_periodic_flag_accepted():
+    """apd_system must accept periodic=True without raising."""
+    apd = apd_system(N=4, D=3, periodic=True, seed=0, device="cpu")
+    assert apd.periodic is True
+
+
+def test_non_periodic_is_default():
+    """periodic defaults to False (backward-compatible)."""
+    apd = apd_system(N=4, D=2, seed=0, device="cpu")
+    assert apd.periodic is False
+
+
+def test_periodic_single_seed_uniform_labels():
+    """A single seed in a periodic box must label every voxel as grain 0,
+    regardless of where the seed sits (corner, centre, edge)."""
+    L = 1.0
+    for seed_pos in [[0.0, 0.0, 0.0], [0.5, 0.5, 0.5], [0.99, 0.01, 0.5]]:
+        X = torch.tensor([seed_pos])
+        apd = apd_system(
+            X=X,
+            D=3,
+            domain=torch.tensor([[0.0, L], [0.0, L], [0.0, L]]),
+            periodic=True,
+            pixel_params=(16, 16, 16),
+            device="cpu",
+        )
+        labels = apd.assemble_apd().cpu()
+        assert torch.all(labels == 0), (
+            f"seed at {seed_pos} produced labels {labels.unique()}"
+        )
+
+
+def test_periodic_grain_wraps_across_seam():
+    """Two seeds at x = 0.0 and x = 0.5 (y, z = 0.5). Under periodicity,
+    grain 0 (seed at the seam) owns x in [0, 0.25] u [0.75, 1.0] and grain 1
+    owns x in [0.25, 0.75]. So the columns at x=0 and x=L (column 0 and column
+    M-1) must BOTH belong to grain 0 -- the grain wraps across the periodic
+    seam. Without periodicity, column M-1 would be closer to seed 1 (x=0.5)
+    than to seed 0 (x=0.0), so it would belong to grain 1."""
+    L = 1.0
+    M = 32
+    X = torch.tensor([[0.0, 0.5, 0.5], [0.5, 0.5, 0.5]])
+    apd = apd_system(
+        X=X,
+        D=3,
+        domain=torch.tensor([[0.0, L], [0.0, L], [0.0, L]]),
+        periodic=True,
+        pixel_params=(M, 16, 16),
+        device="cpu",
+    )
+    labels = apd.assemble_apd().reshape(M, 16, 16).cpu()
+    # column 0 and column M-1 (both ~0.016 from the seam at x=0) must be in
+    # the same grain under periodicity, and that grain is grain 0.
+    assert torch.all(labels[0] == 0)
+    assert torch.all(labels[-1] == 0)
+    # and the middle column (x ~ 0.5) must be in grain 1
+    assert torch.all(labels[M // 2] == 1)
+
+
+def test_grain_of_matches_assemble_apd_on_voxel_centres():
+    """grain_of() on the pixel centres must reproduce assemble_apd()'s output
+    exactly -- same metric, same data, same result."""
+    L = 1.0
+    apd = apd_system(
+        N=5,
+        D=3,
+        domain=torch.tensor([[0.0, L], [0.0, L], [0.0, L]]),
+        periodic=True,
+        pixel_params=(16, 16, 16),
+        seed=42,
+        device="cpu",
+    )
+    apd.find_optimal_W(verbose=False)
+    voxel_labels = apd.assemble_apd().cpu()
+    # apd.Y holds the voxel centres (filled by assemble_pixels via assemble_apd).
+    point_labels = apd.grain_of(apd.Y).cpu()
+    assert torch.equal(voxel_labels, point_labels)
+
+
+def test_grain_of_continuous_points():
+    """grain_of must accept arbitrary (M, 3) torch tensors in the box."""
+    L = 1.0
+    apd = apd_system(
+        N=4,
+        D=3,
+        domain=torch.tensor([[0.0, L], [0.0, L], [0.0, L]]),
+        periodic=True,
+        seed=7,
+        device="cpu",
+    )
+    apd.find_optimal_W(verbose=False)
+    pts = torch.tensor(
+        [
+            [0.5, 0.5, 0.5],
+            [0.0, 0.0, 0.0],
+            [0.99, 0.01, 0.5],
+            [0.1, 0.9, 0.7],
+        ]
+    )
+    labels = apd.grain_of(pts)
+    assert labels.shape == (4,)
+    assert int(labels.min()) >= 0 and int(labels.max()) < 4
+
+
+def test_grain_of_rejects_wrong_shape():
+    """grain_of must validate the (M, D) shape of its input."""
+    apd = apd_system(N=4, D=3, periodic=True, seed=1, device="cpu")
+    with pytest.raises(ValueError):
+        apd.grain_of(torch.zeros(5, 2))  # D=2 points into a D=3 system