diff --git a/README.md b/README.md
index a90ad62..4c89a2b 100644
--- a/README.md
+++ b/README.md
@@ -17,3 +17,12 @@ Implementation of Danijar's [latest iteration](https://arxiv.org/abs/2509.24527v
     url     = {https://arxiv.org/abs/2509.24527}, 
 }
 ```
+
+```bibtex
+@misc{xiong2025ndrope,
+    author = {Jerry Xiong},
+    title  = {On n-dimensional rotary positional embeddings},
+    year   = {2025},
+    url    = {https://jerryxio.ng/posts/nd-rope/}
+}
+```
diff --git a/dreamer4/dreamer4.py b/dreamer4/dreamer4.py
index eae8877..9198091 100644
--- a/dreamer4/dreamer4.py
+++ b/dreamer4/dreamer4.py
@@ -6,10 +6,17 @@ from functools import partial
 import torch
 import torch.nn.functional as F
 from torch.nn import Module, ModuleList, Parameter, Sequential, Linear, RMSNorm, Identity
-from torch import cat, stack, tensor, Tensor, is_tensor
+from torch import cat, stack, arange, tensor, Tensor, is_tensor
 
 # ein related
 
+# b - batch
+# n - sequence
+# h - attention heads
+# d - feature dimension
+# f - frequencies (rotary)
+# p - positions (3 for spacetime in this work)
+
 import einx
 from einops import einsum, rearrange, repeat, reduce
 from einops.layers.torch import Rearrange
@@ -38,12 +45,87 @@ def exists(v):
 def default(v, d):
     return v if exists(v) else d
 
+def divisible_by(num, den):
+    return (num % den) == 0
+
 def l2norm(t):
     return F.normalize(t, dim = -1, p = 2)
 
 def softclamp(t, value = 50.):
     return (t / value).tanh() * value
 
+# golden gate rotary - Jerry Xiong, PhD student at UIUC
+# https://jerryxio.ng/posts/nd-rope/
+
+def _phi(m):
+    x = 2.
+    for _ in range(10):
+        x = (1. + x) ** (1. / (m + 1.))
+    return x
+
+def make_directions(n, d):
+    g = _phi(d)
+    alpha = (1.0 / g) ** arange(1, d + 1, dtype = torch.float64)
+    i = arange(1, n + 1, dtype = torch.float64).unsqueeze(1)
+    z = torch.fmod(i * alpha, 1.0)
+    directions = torch.erfinv(2.0 * z - 1.0)
+    directions = l2norm(directions)
+    return directions.float()
+
+class GoldenGateRoPENd(Module):
+    def __init__(
+        self,
+        dim_pos,
+        heads,
+        dim_head,
+        rope_min_freq = 1.,
+        rope_max_freq = 10000.,
+        rope_p_zero_freqs = 0., # proportion of frequencies set to 0
+    ):
+        super().__init__()
+        assert divisible_by(dim_head, 2)
+
+        n_freqs = dim_head // 2
+        n_zero_freqs = round(rope_p_zero_freqs * n_freqs)
+
+        omega = cat((
+            torch.zeros(n_zero_freqs),
+            rope_min_freq * (rope_max_freq / rope_min_freq) ** torch.linspace(0, 1, n_freqs - n_zero_freqs),
+        ))
+
+        directions = make_directions(heads * n_freqs, dim_pos)
+        directions = rearrange(directions, '(h f) p -> h f p', h = heads)
+
+        omega_expanded = rearrange(omega, 'f -> f 1')
+        self.register_buffer('freqs', directions * omega_expanded)  # shape: (h, f, p)
+
+    def forward(
+        self,
+        x,    # (b h n d)
+        pos   # (b n p)
+    ):
+        dtype = x
+
+        x, y = x.float().chunk(2, dim = -1) # (b, h, n, f)
+
+        freqs = rearrange(self.freqs, 'h f p -> 1 h 1 f p')
+        positions = rearrange(pos.float(), 'b n p -> b 1 n 1 p')
+
+        # thetas for freqs and positions (batch, head, seq, freq)
+
+        theta = reduce(freqs * positions, 'b h n f p -> b h n f', 'sum')
+
+        # apply rotations
+
+        cos_theta = torch.cos(theta)
+        sin_theta = torch.sin(theta)
+
+        x_out = x * cos_theta - y * sin_theta
+        y_out = x * sin_theta + y * cos_theta
+
+        out = cat((x_out, y_out), dim = -1)
+        return out.type_as(dtype)
+
 # multi-head rmsnorm
 
 class MultiHeadRMSNorm(Module):
@@ -58,7 +140,7 @@ class MultiHeadRMSNorm(Module):
 
     def forward(
         self,
-        x
+        x # (b h n d)
     ):
         normed = l2norm(x)
         scale = (self.gamma + 1.) * self.scale
@@ -117,7 +199,7 @@ class Attention(Module):
 
     def forward(
         self,
-        tokens,
+        tokens, # (b n d)
         kv_cache = None,
         return_kv_cache = False
     ):