From ca34cd97e62cc45a65f6a553cb688dea1c274001 Mon Sep 17 00:00:00 2001
From: janbridley <bradleyjenj@gmail.com>
Date: Wed, 3 Sep 2025 16:52:49 -0400
Subject: [PATCH 01/18] WIPOn symmetries

---
 rowan/mapping/symmetries.py | 143 ++++++++++++++++++++++++++++++++++++
 1 file changed, 143 insertions(+)
 create mode 100644 rowan/mapping/symmetries.py

diff --git a/rowan/mapping/symmetries.py b/rowan/mapping/symmetries.py
new file mode 100644
index 0000000..04ece2a
--- /dev/null
+++ b/rowan/mapping/symmetries.py
@@ -0,0 +1,143 @@
+"""."""
+
+from itertools import combinations, permutations, product
+from typing import Iterable
+
+import numpy as np
+from numpy.typing import ArrayLike
+
+from rowan import from_axis_angle
+
+
+def cyclic_permutations(a):
+    """Generate the cyclic permutations of an array a."""
+
+    def make_permutation_generator(a):
+        """Return a list of functions that roll an array by 0,1,2...n."""
+        n, sign = 1, len(a)
+        return [
+            lambda arr, k=j: arr[(sign * k) :] + arr[: (sign * k)] for j in range(n)
+        ]
+
+    return [op(a) for op in make_permutation_generator(a)]
+
+
+def sign_changes(even: bool = True, d: int = 3):
+    """Get all even (or odd) sign changes of a vector in ℝ3."""
+    all_changes = np.array([*product([1, -1], repeat=d)])
+    if even == "all":
+        return all_changes
+    return all_changes[np.prod(all_changes, axis=1) == (1 if even else -1)]
+
+
+def generate_tetrahedral_group():
+    """Generates the 24 quaternions of the chiral tetrahedral group <T>."""
+    quats = {
+        # 8 (4+4) quaternions from permutationsof (±1, 0, 0, 0)
+        (1, 0, 0, 0),
+        (0, 1, 0, 0),
+        (0, 0, 1, 0),
+        (0, 0, 0, 1),
+        (-1, 0, 0, 0),
+        (0, -1, 0, 0),
+        (0, 0, -1, 0),
+        (0, 0, 0, -1),
+        *product([-0.5, 0.5], repeat=4),  # 16 quaternions from 1/2 * (±1, ±1, ±1, ±1)
+    }  # TODO: test leaves tetrahedron unchanged?
+    return np.array(sorted(quats))
+
+
+def generate_octahedral_group():
+    """Generates the 48 quaternions of the octahedral group."""
+    c = 1 / np.sqrt(2)
+
+    quats = {
+        # 24 elements of the tetrahedral group
+        *(tuple(x) for x in generate_tetrahedral_group()),
+        # 24 quaternions from permutations of 1/√2 * (±1, ±1, 0, 0)
+        *permutations([c, c, 0, 0]),
+        *permutations([c, -c, 0, 0]),
+        *permutations([-c, -c, 0, 0]),
+    }
+
+    return np.array(sorted(quats))
+
+
+def generate_cyclic_group(n: int, axis: ArrayLike = (0, 0, 1)):
+    """Generates the 2n quaternions of the cyclic group <C_n>.
+
+    Args:
+        n (int): The index of the cyclic group C_n. The order of the
+                 resulting group will be 2n.
+        axis (np.ndarray): The axis of rotation for the group.
+
+    Returns:
+        np.ndarray: A (2n, 4) array of quaternions.
+    """
+    # return np.array([from_axis_angle(axis, 2 * np.pi * k / n) for k in range(n)])
+    return from_axis_angle(axis, np.linspace(0, 2 * np.pi, n, endpoint=False)).round(15)
+
+
+def generate_dicyclic_group(n: int, axis: ArrayLike = (0, 0, 1)):
+    """Generates the 4n quaternions of the dicyclic group <D_n>.
+
+    Args:
+        n (int): The index of the cyclic group D_n. The order of the
+                 resulting group will be 4n.
+        axis (np.ndarray): The axis of rotation for the group.
+
+    Returns:
+        np.ndarray: A (4n, 4) array of quaternions.
+    """
+    return np.concatenate(
+        (
+            # 2n quaternions from the cyclic group
+            generate_cyclic_group(n, axis),
+            #     hetas = np.linspace(0, np.pi, n, endpoint=False)
+            # rv = np.pi * np.vstack([np.zeros(n), np.cos(thetas), np.sin(thetas)]).T
+            # g2 = np.roll(rv, axis, axis=1)
+            # from_axis_angle(axis, np.linspace(0, np.pi, n, endpoint=False)) # TODO
+        )
+    )
+
+
+print(len(generate_tetrahedral_group()))
+
+print(generate_cyclic_group(4))
+# print((generate_binary_octahedral_group()))
+
+
+def even_permutations(it: Iterable):
+    """Return permutations containing an even number of transpositions from an iterable.
+
+    Args:
+        it (typing.Iterable): The iterable to permute.
+
+    Returns:
+        typing.Generator: An (N!/2, len(it)) generator of permutations.
+    """
+    return (
+        p for p in permutations(it) if not sum(a > b for a, b in combinations(p, 2)) % 2
+    )
+    # if sum(a > b for (i, a) in enumerate(p) for b in p[i + 1 :]) % 2 == 0
+
+
+def generate_binary_icosahedral_group():
+    """Generates the 48 quaternions of the octahedral group."""
+    φ = (1 + np.sqrt(5)) / 2
+
+    quats = {
+        # 24 elements of the tetrahedral group
+        *(tuple(x) for x in generate_tetrahedral_group()),
+        # 96 quaternions from even permutations of 1/2 * (0, ±1, ±1/φ, ±φ)
+        *(
+            q
+            for a, b, c in product([1 / 2, -1 / 2], repeat=3)
+            for q in even_permutations([0, a, b / φ, c * φ])
+        ),
+    }
+    return np.array(sorted(quats))
+
+
+print(len(generate_binary_icosahedral_group()))
+# TODO: test against Scipy, and make sure objects are invariant

From 865eff27bf0fa2e15341da65cb390300561e3f57 Mon Sep 17 00:00:00 2001
From: janbridley <bradleyjenj@gmail.com>
Date: Wed, 3 Sep 2025 16:54:25 -0400
Subject: [PATCH 02/18] Clean up

---
 rowan/mapping/symmetries.py | 47 ++++++++++++++++---------------------
 1 file changed, 20 insertions(+), 27 deletions(-)

diff --git a/rowan/mapping/symmetries.py b/rowan/mapping/symmetries.py
index 04ece2a..a7356d3 100644
--- a/rowan/mapping/symmetries.py
+++ b/rowan/mapping/symmetries.py
@@ -43,7 +43,7 @@ def generate_tetrahedral_group():
         (0, 0, -1, 0),
         (0, 0, 0, -1),
         *product([-0.5, 0.5], repeat=4),  # 16 quaternions from 1/2 * (±1, ±1, ±1, ±1)
-    }  # TODO: test leaves tetrahedron unchanged?
+    }
     return np.array(sorted(quats))
 
 
@@ -78,33 +78,27 @@ def generate_cyclic_group(n: int, axis: ArrayLike = (0, 0, 1)):
     return from_axis_angle(axis, np.linspace(0, 2 * np.pi, n, endpoint=False)).round(15)
 
 
-def generate_dicyclic_group(n: int, axis: ArrayLike = (0, 0, 1)):
-    """Generates the 4n quaternions of the dicyclic group <D_n>.
-
-    Args:
-        n (int): The index of the cyclic group D_n. The order of the
-                 resulting group will be 4n.
-        axis (np.ndarray): The axis of rotation for the group.
-
-    Returns:
-        np.ndarray: A (4n, 4) array of quaternions.
-    """
-    return np.concatenate(
-        (
-            # 2n quaternions from the cyclic group
-            generate_cyclic_group(n, axis),
-            #     hetas = np.linspace(0, np.pi, n, endpoint=False)
-            # rv = np.pi * np.vstack([np.zeros(n), np.cos(thetas), np.sin(thetas)]).T
-            # g2 = np.roll(rv, axis, axis=1)
-            # from_axis_angle(axis, np.linspace(0, np.pi, n, endpoint=False)) # TODO
-        )
-    )
-
+# def generate_dicyclic_group(n: int, axis: ArrayLike = (0, 0, 1)):
+#     """Generates the 4n quaternions of the dicyclic group <D_n>.
 
-print(len(generate_tetrahedral_group()))
+#     Args:
+#         n (int): The index of the cyclic group D_n. The order of the
+#                  resulting group will be 4n.
+#         axis (np.ndarray): The axis of rotation for the group.
 
-print(generate_cyclic_group(4))
-# print((generate_binary_octahedral_group()))
+#     Returns:
+#         np.ndarray: A (4n, 4) array of quaternions.
+#     """
+#     return np.concatenate(
+#         (
+#             # 2n quaternions from the cyclic group
+#             generate_cyclic_group(n, axis),
+#             #     hetas = np.linspace(0, np.pi, n, endpoint=False)
+#             # rv = np.pi * np.vstack([np.zeros(n), np.cos(thetas), np.sin(thetas)]).T
+#             # g2 = np.roll(rv, axis, axis=1)
+#             # from_axis_angle(axis, np.linspace(0, np.pi, n, endpoint=False)) # TODO
+#         )
+#     )
 
 
 def even_permutations(it: Iterable):
@@ -119,7 +113,6 @@ def even_permutations(it: Iterable):
     return (
         p for p in permutations(it) if not sum(a > b for a, b in combinations(p, 2)) % 2
     )
-    # if sum(a > b for (i, a) in enumerate(p) for b in p[i + 1 :]) % 2 == 0
 
 
 def generate_binary_icosahedral_group():

From c87df3fbb2c8c9296504482288a5babd93a609b6 Mon Sep 17 00:00:00 2001
From: janbridley <bradleyjenj@gmail.com>
Date: Thu, 4 Sep 2025 09:39:02 -0400
Subject: [PATCH 03/18] Add more_itertools dependency

---
 .../workflows/environments/requirements-build.txt  |  2 +-
 .../workflows/environments/requirements-doc.txt    |  8 ++++----
 .../environments/requirements-test-3.10.txt        | 14 +++++++++-----
 .../environments/requirements-test-3.11.txt        | 10 ++++++----
 .../environments/requirements-test-3.12.txt        | 10 ++++++----
 .../environments/requirements-test-3.13.txt        | 10 ++++++----
 .../environments/requirements-test-3.9.txt         | 10 +++++++---
 pyproject.toml                                     |  1 +
 8 files changed, 40 insertions(+), 25 deletions(-)

diff --git a/.github/workflows/environments/requirements-build.txt b/.github/workflows/environments/requirements-build.txt
index 54d30bd..a0f5df7 100644
--- a/.github/workflows/environments/requirements-build.txt
+++ b/.github/workflows/environments/requirements-build.txt
@@ -1,6 +1,6 @@
 # This file was autogenerated by uv via the following command:
 #    uv pip compile --python-version 3.13 --python-platform linux requirements-build.in
-build==1.2.2.post1
+build==1.3.0
     # via -r requirements-build.in
 packaging==25.0
     # via build
diff --git a/.github/workflows/environments/requirements-doc.txt b/.github/workflows/environments/requirements-doc.txt
index 2e1c595..ee8d021 100644
--- a/.github/workflows/environments/requirements-doc.txt
+++ b/.github/workflows/environments/requirements-doc.txt
@@ -1,12 +1,12 @@
 # This file was autogenerated by uv via the following command:
-#    uv pip compile --python-version 3.13 --python-platform linux requirements-doc.in
+#    uv pip compile --python-version 3.12 --python-platform linux requirements-doc.in
 alabaster==1.0.0
     # via sphinx
 babel==2.17.0
     # via sphinx
-certifi==2025.6.15
+certifi==2025.8.3
     # via requests
-charset-normalizer==3.4.2
+charset-normalizer==3.4.3
     # via requests
 docutils==0.21.2
     # via
@@ -24,7 +24,7 @@ packaging==25.0
     # via sphinx
 pygments==2.19.2
     # via sphinx
-requests==2.32.4
+requests==2.32.5
     # via sphinx
 roman-numerals-py==3.1.0
     # via sphinx
diff --git a/.github/workflows/environments/requirements-test-3.10.txt b/.github/workflows/environments/requirements-test-3.10.txt
index 72c07fa..009977d 100644
--- a/.github/workflows/environments/requirements-test-3.10.txt
+++ b/.github/workflows/environments/requirements-test-3.10.txt
@@ -1,20 +1,24 @@
 # This file was autogenerated by uv via the following command:
 #    uv pip compile --python-version 3.10 --python-platform linux requirements-test.in
-exceptiongroup==1.2.2
+exceptiongroup==1.3.0
     # via pytest
 iniconfig==2.1.0
     # via pytest
-numpy==2.2.5
+numpy==2.2.6
     # via
     #   -r requirements-test.in
     #   scipy
 packaging==25.0
     # via pytest
-pluggy==1.5.0
+pluggy==1.6.0
     # via pytest
-pytest==8.3.5
+pygments==2.19.2
+    # via pytest
+pytest==8.4.1
     # via -r requirements-test.in
-scipy==1.15.2
+scipy==1.15.3
     # via -r requirements-test.in
 tomli==2.2.1
     # via pytest
+typing-extensions==4.15.0
+    # via exceptiongroup
diff --git a/.github/workflows/environments/requirements-test-3.11.txt b/.github/workflows/environments/requirements-test-3.11.txt
index 9df1bae..e16644a 100644
--- a/.github/workflows/environments/requirements-test-3.11.txt
+++ b/.github/workflows/environments/requirements-test-3.11.txt
@@ -2,15 +2,17 @@
 #    uv pip compile --python-version 3.11 --python-platform linux requirements-test.in
 iniconfig==2.1.0
     # via pytest
-numpy==2.2.5
+numpy==2.3.2
     # via
     #   -r requirements-test.in
     #   scipy
 packaging==25.0
     # via pytest
-pluggy==1.5.0
+pluggy==1.6.0
     # via pytest
-pytest==8.3.5
+pygments==2.19.2
+    # via pytest
+pytest==8.4.1
     # via -r requirements-test.in
-scipy==1.15.2
+scipy==1.16.1
     # via -r requirements-test.in
diff --git a/.github/workflows/environments/requirements-test-3.12.txt b/.github/workflows/environments/requirements-test-3.12.txt
index 084a80a..7b59bb5 100644
--- a/.github/workflows/environments/requirements-test-3.12.txt
+++ b/.github/workflows/environments/requirements-test-3.12.txt
@@ -2,15 +2,17 @@
 #    uv pip compile --python-version 3.12 --python-platform linux requirements-test.in
 iniconfig==2.1.0
     # via pytest
-numpy==2.2.5
+numpy==2.3.2
     # via
     #   -r requirements-test.in
     #   scipy
 packaging==25.0
     # via pytest
-pluggy==1.5.0
+pluggy==1.6.0
     # via pytest
-pytest==8.3.5
+pygments==2.19.2
+    # via pytest
+pytest==8.4.1
     # via -r requirements-test.in
-scipy==1.15.2
+scipy==1.16.1
     # via -r requirements-test.in
diff --git a/.github/workflows/environments/requirements-test-3.13.txt b/.github/workflows/environments/requirements-test-3.13.txt
index 0c81039..165e829 100644
--- a/.github/workflows/environments/requirements-test-3.13.txt
+++ b/.github/workflows/environments/requirements-test-3.13.txt
@@ -2,15 +2,17 @@
 #    uv pip compile --python-version 3.13 --python-platform linux requirements-test.in
 iniconfig==2.1.0
     # via pytest
-numpy==2.2.5
+numpy==2.3.2
     # via
     #   -r requirements-test.in
     #   scipy
 packaging==25.0
     # via pytest
-pluggy==1.5.0
+pluggy==1.6.0
     # via pytest
-pytest==8.3.5
+pygments==2.19.2
+    # via pytest
+pytest==8.4.1
     # via -r requirements-test.in
-scipy==1.15.2
+scipy==1.16.1
     # via -r requirements-test.in
diff --git a/.github/workflows/environments/requirements-test-3.9.txt b/.github/workflows/environments/requirements-test-3.9.txt
index 2a921c5..6148d75 100644
--- a/.github/workflows/environments/requirements-test-3.9.txt
+++ b/.github/workflows/environments/requirements-test-3.9.txt
@@ -1,6 +1,6 @@
 # This file was autogenerated by uv via the following command:
 #    uv pip compile --python-version 3.9 --python-platform linux requirements-test.in
-exceptiongroup==1.2.2
+exceptiongroup==1.3.0
     # via pytest
 iniconfig==2.1.0
     # via pytest
@@ -10,11 +10,15 @@ numpy==2.0.2
     #   scipy
 packaging==25.0
     # via pytest
-pluggy==1.5.0
+pluggy==1.6.0
     # via pytest
-pytest==8.3.5
+pygments==2.19.2
+    # via pytest
+pytest==8.4.1
     # via -r requirements-test.in
 scipy==1.13.1
     # via -r requirements-test.in
 tomli==2.2.1
     # via pytest
+typing-extensions==4.15.0
+    # via exceptiongroup
diff --git a/pyproject.toml b/pyproject.toml
index e5a10ab..a05ad3f 100644
--- a/pyproject.toml
+++ b/pyproject.toml
@@ -24,6 +24,7 @@ classifiers = [
 ]
 dependencies = [
     "numpy>=1.21",
+    "more_itertools"
 ]
 
 [project.optional-dependencies]

From 338b02b41b3d50e7cf1811cce99333aa276d2a16 Mon Sep 17 00:00:00 2001
From: janbridley <bradleyjenj@gmail.com>
Date: Thu, 4 Sep 2025 10:18:31 -0400
Subject: [PATCH 04/18] WIP on tests and more efficient generation

---
 rowan/mapping/__init__.py   |  8 ++++
 rowan/mapping/symmetries.py | 74 +++++++++++++++++++------------------
 tests/test_mapping.py       | 53 ++++++++++++++++++++++++++
 3 files changed, 99 insertions(+), 36 deletions(-)

diff --git a/rowan/mapping/__init__.py b/rowan/mapping/__init__.py
index 4de0f71..abde5a2 100644
--- a/rowan/mapping/__init__.py
+++ b/rowan/mapping/__init__.py
@@ -58,6 +58,14 @@
 
 from ..functions import from_matrix, rotate, to_matrix
 from ..geometry import angle
+from .symmetries import _cyclic_permutations
+from .symmetries import generate_icosahedral_group as _generate_icosahedral_group
+from .symmetries import (
+    generate_octahedral_group as _generate_octahedral_group,
+)
+from .symmetries import (
+    generate_tetrahedral_group as _generate_tetrahedral_group,
+)
 
 __all__ = ["kabsch", "davenport", "procrustes", "icp"]
 
diff --git a/rowan/mapping/symmetries.py b/rowan/mapping/symmetries.py
index a7356d3..ea05584 100644
--- a/rowan/mapping/symmetries.py
+++ b/rowan/mapping/symmetries.py
@@ -4,17 +4,17 @@
 from typing import Iterable
 
 import numpy as np
+from more_itertools import distinct_permutations
 from numpy.typing import ArrayLike
 
-from rowan import from_axis_angle
+from ..functions import from_axis_angle
 
 
-def cyclic_permutations(a):
-    """Generate the cyclic permutations of an array a."""
-
-    def make_permutation_generator(a):
-        """Return a list of functions that roll an array by 0,1,2...n."""
-        n, sign = 1, len(a)
+def _cyclic_permutations(a):
+    def make_permutation_generator(a, forward=True):
+        """Return a list of functions that roll an array by 0,1,2...n"""
+        n = len(a)
+        sign = -1 if forward else 1
         return [
             lambda arr, k=j: arr[(sign * k) :] + arr[: (sign * k)] for j in range(n)
         ]
@@ -32,35 +32,35 @@ def sign_changes(even: bool = True, d: int = 3):
 
 def generate_tetrahedral_group():
     """Generates the 24 quaternions of the chiral tetrahedral group <T>."""
-    quats = {
+    quats = [
         # 8 (4+4) quaternions from permutationsof (±1, 0, 0, 0)
         (1, 0, 0, 0),
-        (0, 1, 0, 0),
-        (0, 0, 1, 0),
-        (0, 0, 0, 1),
         (-1, 0, 0, 0),
+        (0, 1, 0, 0),
         (0, -1, 0, 0),
+        (0, 0, 1, 0),
         (0, 0, -1, 0),
+        (0, 0, 0, 1),
         (0, 0, 0, -1),
         *product([-0.5, 0.5], repeat=4),  # 16 quaternions from 1/2 * (±1, ±1, ±1, ±1)
-    }
-    return np.array(sorted(quats))
+    ]
+    return np.array(quats)
 
 
 def generate_octahedral_group():
     """Generates the 48 quaternions of the octahedral group."""
     c = 1 / np.sqrt(2)
 
-    quats = {
-        # 24 elements of the tetrahedral group
-        *(tuple(x) for x in generate_tetrahedral_group()),
-        # 24 quaternions from permutations of 1/√2 * (±1, ±1, 0, 0)
-        *permutations([c, c, 0, 0]),
-        *permutations([c, -c, 0, 0]),
-        *permutations([-c, -c, 0, 0]),
-    }
-
-    return np.array(sorted(quats))
+    return np.array(
+        [
+            # 24 elements of the tetrahedral group
+            *generate_tetrahedral_group(),
+            # 24 quaternions from distinct permutations of 1/√2 * (±1, ±1, 0, 0)
+            *distinct_permutations([c, c, 0, 0]),
+            *distinct_permutations([c, -c, 0, 0]),
+            *distinct_permutations([-c, -c, 0, 0]),
+        ]
+    )
 
 
 def generate_cyclic_group(n: int, axis: ArrayLike = (0, 0, 1)):
@@ -115,22 +115,24 @@ def even_permutations(it: Iterable):
     )
 
 
-def generate_binary_icosahedral_group():
+def generate_icosahedral_group():
     """Generates the 48 quaternions of the octahedral group."""
     φ = (1 + np.sqrt(5)) / 2
 
-    quats = {
-        # 24 elements of the tetrahedral group
-        *(tuple(x) for x in generate_tetrahedral_group()),
-        # 96 quaternions from even permutations of 1/2 * (0, ±1, ±1/φ, ±φ)
-        *(
-            q
-            for a, b, c in product([1 / 2, -1 / 2], repeat=3)
-            for q in even_permutations([0, a, b / φ, c * φ])
-        ),
-    }
-    return np.array(sorted(quats))
+    return np.array(
+        [
+            # 24 elements of the tetrahedral group
+            *generate_tetrahedral_group(),
+            # 96 quaternions from even permutations of 1/2 * (0, ±1, ±1/φ, ±φ)
+            *(
+                q
+                for a, b, c in product([1 / 2, -1 / 2], repeat=3)
+                for q in even_permutations([0, a, b / φ, c * φ])
+            ),
+        ]
+    )
 
 
-print(len(generate_binary_icosahedral_group()))
+print(len(generate_icosahedral_group()))
+# print(generate_binary_icosahedral_group())
 # TODO: test against Scipy, and make sure objects are invariant
diff --git a/tests/test_mapping.py b/tests/test_mapping.py
index 81687f9..5f3a49f 100644
--- a/tests/test_mapping.py
+++ b/tests/test_mapping.py
@@ -1,8 +1,11 @@
 """Test algorithms for point-cloud mapping."""
 
 import unittest
+from itertools import product
 
+import matplotlib.pyplot as plt
 import numpy as np
+from coxeter.shapes import ConvexPolyhedron
 
 from rowan import from_axis_angle, from_matrix, mapping, random, rotate
 
@@ -191,6 +194,56 @@ def test_icp_mismatched(self):
             # a heuristic since we can't guarantee exact matches
             assert np.mean(norms) < 0.5
 
+    def test_symmetric_quaternions_tetrahedral(self):
+        """Verify our symmetrically equivalent quaternions are correct."""
+        tet = [
+            (-0.5, -0.5, 0.5),
+            (-0.5, 0.5, -0.5),
+            (0.5, -0.5, -0.5),
+            (0.5, 0.5, 0.5),
+        ]  # Verts in sorted order
+
+        quats = mapping._generate_tetrahedral_group()
+        for quat in quats:
+            res = sorted(rotate(quat, tet).tolist())
+            np.testing.assert_array_equal(res, tet, err_msg=f"q={quat}")
+
+    def test_symmetric_quaternions_octahedral(self):
+        """Verify our symmetrically equivalent quaternions are correct."""
+        cube = [*product([-0.5, 0.5], repeat=3)]  # Verts in sorted order
+
+        quats = mapping._generate_octahedral_group()
+        for quat in quats:
+            res = sorted(rotate(quat, cube).tolist())
+            np.testing.assert_allclose(res, cube, err_msg=f"q={quat}", atol=4e-16)
+
+    def test_symmetric_quaternions_icosahedral(self):
+        """Verify our symmetrically equivalent quaternions are correct."""
+        φ = (1 + np.sqrt(5)) / 2
+        icos = sorted(
+            [
+                *mapping._cyclic_permutations([0, -1, -φ]),
+                *mapping._cyclic_permutations([0, -1, φ]),
+                *mapping._cyclic_permutations([0, 1, -φ]),
+                *mapping._cyclic_permutations([0, 1, φ]),
+            ]
+        )
+
+        fix, ax = plt.subplots(subplot_kw={"projection": "3d"})
+        ConvexPolyhedron(icos).plot(ax=ax)
+
+        quats = mapping._generate_icosahedral_group()
+        for quat in quats:
+            res = sorted(rotate(quat, icos).tolist())
+            poly = ConvexPolyhedron(res)
+            print(poly.volume)
+            try:
+                np.testing.assert_allclose(res, icos, err_msg=f"q={quat}", atol=4e-16)
+            except AssertionError as e:
+                poly.plot(ax=ax, label_verts=True)
+                plt.show()
+                raise AssertionError(e) from e
+
 
 if __name__ == "__main__":
     unittest.main()

From 164384897489811fe3c620965335711d68385520 Mon Sep 17 00:00:00 2001
From: janbridley <bradleyjenj@gmail.com>
Date: Thu, 4 Sep 2025 11:07:51 -0400
Subject: [PATCH 05/18] Clean up and test

---
 tests/test_mapping.py | 74 +++++++++++++++++++++++++++++++++----------
 1 file changed, 57 insertions(+), 17 deletions(-)

diff --git a/tests/test_mapping.py b/tests/test_mapping.py
index 5f3a49f..44b9686 100644
--- a/tests/test_mapping.py
+++ b/tests/test_mapping.py
@@ -3,9 +3,8 @@
 import unittest
 from itertools import product
 
-import matplotlib.pyplot as plt
 import numpy as np
-from coxeter.shapes import ConvexPolyhedron
+import scipy
 
 from rowan import from_axis_angle, from_matrix, mapping, random, rotate
 
@@ -204,45 +203,86 @@ def test_symmetric_quaternions_tetrahedral(self):
         ]  # Verts in sorted order
 
         quats = mapping._generate_tetrahedral_group()
+        assert len(quats) == 24
         for quat in quats:
             res = sorted(rotate(quat, tet).tolist())
             np.testing.assert_array_equal(res, tet, err_msg=f"q={quat}")
 
+        # Compare our array of generated quaternions to Scipy
+        np.testing.assert_array_equal(
+            sorted(quats.tolist()),
+            sorted(
+                [
+                    *scipy.spatial.transform.Rotation.create_group("T")
+                    .as_quat()
+                    .tolist(),
+                    *(
+                        -scipy.spatial.transform.Rotation.create_group("T").as_quat()
+                    ).tolist(),
+                ]
+            ),
+        )
+
     def test_symmetric_quaternions_octahedral(self):
         """Verify our symmetrically equivalent quaternions are correct."""
         cube = [*product([-0.5, 0.5], repeat=3)]  # Verts in sorted order
 
         quats = mapping._generate_octahedral_group()
+        assert len(quats) == 48
         for quat in quats:
             res = sorted(rotate(quat, cube).tolist())
             np.testing.assert_allclose(res, cube, err_msg=f"q={quat}", atol=4e-16)
 
+        # Compare our array of generated quaternions to Scipy
+        np.testing.assert_allclose(
+            sorted(quats.tolist()),
+            sorted(
+                [
+                    *scipy.spatial.transform.Rotation.create_group("O")
+                    .as_quat()
+                    .tolist(),
+                    *(
+                        -scipy.spatial.transform.Rotation.create_group("O").as_quat()
+                    ).tolist(),
+                ]
+            ),
+            atol=4e-16,
+        )
+
     def test_symmetric_quaternions_icosahedral(self):
         """Verify our symmetrically equivalent quaternions are correct."""
         φ = (1 + np.sqrt(5)) / 2
-        icos = sorted(
+        icos = np.sort(
             [
                 *mapping._cyclic_permutations([0, -1, -φ]),
                 *mapping._cyclic_permutations([0, -1, φ]),
                 *mapping._cyclic_permutations([0, 1, -φ]),
                 *mapping._cyclic_permutations([0, 1, φ]),
-            ]
+            ],
+            axis=0,
         )
 
-        fix, ax = plt.subplots(subplot_kw={"projection": "3d"})
-        ConvexPolyhedron(icos).plot(ax=ax)
-
         quats = mapping._generate_icosahedral_group()
-        for quat in quats:
-            res = sorted(rotate(quat, icos).tolist())
-            poly = ConvexPolyhedron(res)
-            print(poly.volume)
-            try:
-                np.testing.assert_allclose(res, icos, err_msg=f"q={quat}", atol=4e-16)
-            except AssertionError as e:
-                poly.plot(ax=ax, label_verts=True)
-                plt.show()
-                raise AssertionError(e) from e
+        assert len(quats) == 120
+
+        # TODO: we get the correct vertices, but the ordering is difficult to ensure.
+        # Is there a better way to test for equivalence here?
+        # for i, quat in enumerate(quats):
+        #     res = np.sort(rotate(quat, icos).round(14), axis=0)
+        #     np.testing.assert_allclose(res, icos, err_msg=f"q{i}={quat}", atol=4e-16)
+
+        # Compare our array of generated quaternions to Scipy
+        np.testing.assert_allclose(
+            np.sort(quats, axis=0),
+            np.sort(
+                [
+                    *scipy.spatial.transform.Rotation.create_group("I").as_quat(),
+                    *(-scipy.spatial.transform.Rotation.create_group("I").as_quat()),
+                ],
+                axis=0,
+            ),
+            atol=4e-16,
+        )
 
 
 if __name__ == "__main__":

From a9cc0d773eecad5036a50ca7f8a03d41ccc425f2 Mon Sep 17 00:00:00 2001
From: janbridley <bradleyjenj@gmail.com>
Date: Thu, 4 Sep 2025 13:31:00 -0400
Subject: [PATCH 06/18] Lint and add class

---
 rowan/mapping/__init__.py   | 19 +++++++------
 rowan/mapping/symmetries.py | 53 ++++++++++++++++++++++++++-----------
 tests/test_mapping.py       | 40 ++++++++++++++++++----------
 3 files changed, 72 insertions(+), 40 deletions(-)

diff --git a/rowan/mapping/__init__.py b/rowan/mapping/__init__.py
index abde5a2..1932fcc 100644
--- a/rowan/mapping/__init__.py
+++ b/rowan/mapping/__init__.py
@@ -58,16 +58,15 @@
 
 from ..functions import from_matrix, rotate, to_matrix
 from ..geometry import angle
-from .symmetries import _cyclic_permutations
-from .symmetries import generate_icosahedral_group as _generate_icosahedral_group
-from .symmetries import (
-    generate_octahedral_group as _generate_octahedral_group,
-)
-from .symmetries import (
-    generate_tetrahedral_group as _generate_tetrahedral_group,
-)
-
-__all__ = ["kabsch", "davenport", "procrustes", "icp"]
+from .symmetries import SymmetricallyEquivalentQuaternions
+
+__all__ = [
+    "kabsch",
+    "davenport",
+    "procrustes",
+    "icp",
+    "SymmetricallyEquivalentQuaternions",
+]
 
 
 def kabsch(X, Y, require_rotation=True):
diff --git a/rowan/mapping/symmetries.py b/rowan/mapping/symmetries.py
index ea05584..c5b0edb 100644
--- a/rowan/mapping/symmetries.py
+++ b/rowan/mapping/symmetries.py
@@ -7,19 +7,8 @@
 from more_itertools import distinct_permutations
 from numpy.typing import ArrayLike
 
-from ..functions import from_axis_angle
-
-
-def _cyclic_permutations(a):
-    def make_permutation_generator(a, forward=True):
-        """Return a list of functions that roll an array by 0,1,2...n"""
-        n = len(a)
-        sign = -1 if forward else 1
-        return [
-            lambda arr, k=j: arr[(sign * k) :] + arr[: (sign * k)] for j in range(n)
-        ]
-
-    return [op(a) for op in make_permutation_generator(a)]
+# from ..functions import from_axis_angle
+from rowan.functions import from_axis_angle
 
 
 def sign_changes(even: bool = True, d: int = 3):
@@ -133,6 +122,38 @@ def generate_icosahedral_group():
     )
 
 
-print(len(generate_icosahedral_group()))
-# print(generate_binary_icosahedral_group())
-# TODO: test against Scipy, and make sure objects are invariant
+class SymmetricallyEquivalentQuaternions(np.ndarray):
+    """Equivalent quaternions for a particular point group."""
+
+    def __new__(cls, data, group):  # noqa: D102
+        obj = np.asarray(data).view(cls)
+        obj.flags.writeable = False
+        obj._group = group  # noqa: SLF001
+        return obj
+
+    def __array_finalize__(self, obj):  # noqa: D105
+        if obj is None:
+            return
+        self._group = getattr(obj, "_group", None)
+
+    @classmethod
+    def __class_getitem__(cls, group: str):
+        """Create the symmetrically equivalent orientations for the provided group."""
+        if group == "T":
+            return cls(data=generate_tetrahedral_group(), group=group)
+        if group == "O":
+            return cls(data=generate_octahedral_group(), group=group)
+        if group == "I":
+            return cls(data=generate_icosahedral_group(), group=group)
+        msg = f"Unknown group '{group}' does not match valid options {{'T', 'O', 'I'}}"
+        raise ValueError(msg)
+
+    def __str__(self):  # noqa: D105
+        arrstr = np.array2string(self, floatmode="maxprec", threshold=24, precision=9)
+        return (
+            f"SymmetricallyEquivalentQuaternions['{self._group}'] "
+            f"(order {len(self)}):\n{arrstr}"
+        )
+
+    def __repr__(self):  # noqa: D105
+        return self.__str__()
diff --git a/tests/test_mapping.py b/tests/test_mapping.py
index 44b9686..792fee7 100644
--- a/tests/test_mapping.py
+++ b/tests/test_mapping.py
@@ -13,6 +13,18 @@
 half = np.array([0.5, 0.5, 0.5, 0.5])
 
 
+def _cyclic_permutations(a):
+    def make_permutation_generator(a, forward=True):
+        """Return a list of functions that roll an array by 0,1,2...n."""
+        n = len(a)
+        sign = -1 if forward else 1
+        return [
+            lambda arr, k=j: arr[(sign * k) :] + arr[: (sign * k)] for j in range(n)
+        ]
+
+    return [op(a) for op in make_permutation_generator(a)]
+
+
 class TestMapping(unittest.TestCase):
     """Test mapping functions."""
 
@@ -202,7 +214,7 @@ def test_symmetric_quaternions_tetrahedral(self):
             (0.5, 0.5, 0.5),
         ]  # Verts in sorted order
 
-        quats = mapping._generate_tetrahedral_group()
+        quats = mapping.SymmetricallyEquivalentQuaternions["T"]
         assert len(quats) == 24
         for quat in quats:
             res = sorted(rotate(quat, tet).tolist())
@@ -227,7 +239,7 @@ def test_symmetric_quaternions_octahedral(self):
         """Verify our symmetrically equivalent quaternions are correct."""
         cube = [*product([-0.5, 0.5], repeat=3)]  # Verts in sorted order
 
-        quats = mapping._generate_octahedral_group()
+        quats = mapping.SymmetricallyEquivalentQuaternions["O"]
         assert len(quats) == 48
         for quat in quats:
             res = sorted(rotate(quat, cube).tolist())
@@ -251,18 +263,18 @@ def test_symmetric_quaternions_octahedral(self):
 
     def test_symmetric_quaternions_icosahedral(self):
         """Verify our symmetrically equivalent quaternions are correct."""
-        φ = (1 + np.sqrt(5)) / 2
-        icos = np.sort(
-            [
-                *mapping._cyclic_permutations([0, -1, -φ]),
-                *mapping._cyclic_permutations([0, -1, φ]),
-                *mapping._cyclic_permutations([0, 1, -φ]),
-                *mapping._cyclic_permutations([0, 1, φ]),
-            ],
-            axis=0,
-        )
-
-        quats = mapping._generate_icosahedral_group()
+        # φ = (1 + np.sqrt(5)) / 2
+        # icos = np.sort(
+        #     [
+        #         *_cyclic_permutations([0, -1, -φ]),
+        #         *_cyclic_permutations([0, -1, φ]),
+        #         *_cyclic_permutations([0, 1, -φ]),
+        #         *_cyclic_permutations([0, 1, φ]),
+        #     ],
+        #     axis=0,
+        # )
+
+        quats = mapping.SymmetricallyEquivalentQuaternions["I"]
         assert len(quats) == 120
 
         # TODO: we get the correct vertices, but the ordering is difficult to ensure.

From 89a75a6f9ef7827f406e234d06aed0ed825f06c2 Mon Sep 17 00:00:00 2001
From: janbridley <bradleyjenj@gmail.com>
Date: Thu, 4 Sep 2025 13:33:59 -0400
Subject: [PATCH 07/18] Swap class getitem for create_group

---
 rowan/mapping/symmetries.py | 2 +-
 tests/test_mapping.py       | 6 +++---
 2 files changed, 4 insertions(+), 4 deletions(-)

diff --git a/rowan/mapping/symmetries.py b/rowan/mapping/symmetries.py
index c5b0edb..bae6e9e 100644
--- a/rowan/mapping/symmetries.py
+++ b/rowan/mapping/symmetries.py
@@ -137,7 +137,7 @@ def __array_finalize__(self, obj):  # noqa: D105
         self._group = getattr(obj, "_group", None)
 
     @classmethod
-    def __class_getitem__(cls, group: str):
+    def create_group(cls, group: str):
         """Create the symmetrically equivalent orientations for the provided group."""
         if group == "T":
             return cls(data=generate_tetrahedral_group(), group=group)
diff --git a/tests/test_mapping.py b/tests/test_mapping.py
index 792fee7..2647fda 100644
--- a/tests/test_mapping.py
+++ b/tests/test_mapping.py
@@ -214,7 +214,7 @@ def test_symmetric_quaternions_tetrahedral(self):
             (0.5, 0.5, 0.5),
         ]  # Verts in sorted order
 
-        quats = mapping.SymmetricallyEquivalentQuaternions["T"]
+        quats = mapping.SymmetricallyEquivalentQuaternions.create_group("T")
         assert len(quats) == 24
         for quat in quats:
             res = sorted(rotate(quat, tet).tolist())
@@ -239,7 +239,7 @@ def test_symmetric_quaternions_octahedral(self):
         """Verify our symmetrically equivalent quaternions are correct."""
         cube = [*product([-0.5, 0.5], repeat=3)]  # Verts in sorted order
 
-        quats = mapping.SymmetricallyEquivalentQuaternions["O"]
+        quats = mapping.SymmetricallyEquivalentQuaternions.create_group("O")
         assert len(quats) == 48
         for quat in quats:
             res = sorted(rotate(quat, cube).tolist())
@@ -274,7 +274,7 @@ def test_symmetric_quaternions_icosahedral(self):
         #     axis=0,
         # )
 
-        quats = mapping.SymmetricallyEquivalentQuaternions["I"]
+        quats = mapping.SymmetricallyEquivalentQuaternions.create_group("I")
         assert len(quats) == 120
 
         # TODO: we get the correct vertices, but the ordering is difficult to ensure.

From 51988ce9cbba78f0840c32fd656469018a277a61 Mon Sep 17 00:00:00 2001
From: janbridley <bradleyjenj@gmail.com>
Date: Thu, 4 Sep 2025 13:38:51 -0400
Subject: [PATCH 08/18] Update docs

---
 rowan/mapping/symmetries.py | 39 ++++++++++++++++++++++++++++++++++---
 1 file changed, 36 insertions(+), 3 deletions(-)

diff --git a/rowan/mapping/symmetries.py b/rowan/mapping/symmetries.py
index bae6e9e..f7cdc79 100644
--- a/rowan/mapping/symmetries.py
+++ b/rowan/mapping/symmetries.py
@@ -123,7 +123,12 @@ def generate_icosahedral_group():
 
 
 class SymmetricallyEquivalentQuaternions(np.ndarray):
-    """Equivalent quaternions for a particular point group."""
+    """Equivalent quaternions for a particular point group.
+
+    This class is a read-only subclass of `np.ndarray` that holds the set of
+    quaternions representing the symmetry operations of a specific point group
+    ('T', 'O', or 'I').
+    """
 
     def __new__(cls, data, group):  # noqa: D102
         obj = np.asarray(data).view(cls)
@@ -131,14 +136,42 @@ def __new__(cls, data, group):  # noqa: D102
         obj._group = group  # noqa: SLF001
         return obj
 
-    def __array_finalize__(self, obj):  # noqa: D105
+    def __array_finalize__(self, obj):
+        """Finalize the array, ensuring attributes are passed on."""
         if obj is None:
             return
         self._group = getattr(obj, "_group", None)
 
     @classmethod
     def create_group(cls, group: str):
-        """Create the symmetrically equivalent orientations for the provided group."""
+        """Create the set of symmetrically equivalent quaternions.
+
+        Parameters
+        ----------
+        group : {'T', 'O', 'I'}
+            The name of the point group. Must be one of {'T', 'O', 'I'}.
+
+        Returns:
+        --------
+        SymmetricallyEquivalentQuaternions
+            An instance containing the quaternions for the specified group.
+
+        Raises:
+        -------
+        ValueError
+            If the `group` string is not one of the valid options.
+
+        Examples:
+        ---------
+        >>> tetrahedral_quats = SymmetricallyEquivalentQuaternions.create_group("T")
+        >>> print(tetrahedral_quats._group)
+        T
+        >>> print(len(tetrahedral_quats))
+        24
+
+        >>> assert np.array_equal(tetrahedral_quats[0], [1,0,0,0])
+
+        """
         if group == "T":
             return cls(data=generate_tetrahedral_group(), group=group)
         if group == "O":

From 616eccc07914b71c6c402d26721d739bbb7d0b0d Mon Sep 17 00:00:00 2001
From: janbridley <bradleyjenj@gmail.com>
Date: Thu, 4 Sep 2025 13:40:49 -0400
Subject: [PATCH 09/18] Update lockfiles

---
 .github/workflows/environments/requirements-test-3.10.txt | 4 +++-
 .github/workflows/environments/requirements-test-3.11.txt | 4 +++-
 .github/workflows/environments/requirements-test-3.12.txt | 4 +++-
 .github/workflows/environments/requirements-test-3.13.txt | 4 +++-
 .github/workflows/environments/requirements-test-3.9.txt  | 4 +++-
 .github/workflows/environments/requirements-test.in       | 1 +
 6 files changed, 16 insertions(+), 5 deletions(-)

diff --git a/.github/workflows/environments/requirements-test-3.10.txt b/.github/workflows/environments/requirements-test-3.10.txt
index 009977d..224d952 100644
--- a/.github/workflows/environments/requirements-test-3.10.txt
+++ b/.github/workflows/environments/requirements-test-3.10.txt
@@ -4,6 +4,8 @@ exceptiongroup==1.3.0
     # via pytest
 iniconfig==2.1.0
     # via pytest
+more-itertools==10.8.0
+    # via -r requirements-test.in
 numpy==2.2.6
     # via
     #   -r requirements-test.in
@@ -14,7 +16,7 @@ pluggy==1.6.0
     # via pytest
 pygments==2.19.2
     # via pytest
-pytest==8.4.1
+pytest==8.4.2
     # via -r requirements-test.in
 scipy==1.15.3
     # via -r requirements-test.in
diff --git a/.github/workflows/environments/requirements-test-3.11.txt b/.github/workflows/environments/requirements-test-3.11.txt
index e16644a..96b2032 100644
--- a/.github/workflows/environments/requirements-test-3.11.txt
+++ b/.github/workflows/environments/requirements-test-3.11.txt
@@ -2,6 +2,8 @@
 #    uv pip compile --python-version 3.11 --python-platform linux requirements-test.in
 iniconfig==2.1.0
     # via pytest
+more-itertools==10.8.0
+    # via -r requirements-test.in
 numpy==2.3.2
     # via
     #   -r requirements-test.in
@@ -12,7 +14,7 @@ pluggy==1.6.0
     # via pytest
 pygments==2.19.2
     # via pytest
-pytest==8.4.1
+pytest==8.4.2
     # via -r requirements-test.in
 scipy==1.16.1
     # via -r requirements-test.in
diff --git a/.github/workflows/environments/requirements-test-3.12.txt b/.github/workflows/environments/requirements-test-3.12.txt
index 7b59bb5..d95a4cf 100644
--- a/.github/workflows/environments/requirements-test-3.12.txt
+++ b/.github/workflows/environments/requirements-test-3.12.txt
@@ -2,6 +2,8 @@
 #    uv pip compile --python-version 3.12 --python-platform linux requirements-test.in
 iniconfig==2.1.0
     # via pytest
+more-itertools==10.8.0
+    # via -r requirements-test.in
 numpy==2.3.2
     # via
     #   -r requirements-test.in
@@ -12,7 +14,7 @@ pluggy==1.6.0
     # via pytest
 pygments==2.19.2
     # via pytest
-pytest==8.4.1
+pytest==8.4.2
     # via -r requirements-test.in
 scipy==1.16.1
     # via -r requirements-test.in
diff --git a/.github/workflows/environments/requirements-test-3.13.txt b/.github/workflows/environments/requirements-test-3.13.txt
index 165e829..3133622 100644
--- a/.github/workflows/environments/requirements-test-3.13.txt
+++ b/.github/workflows/environments/requirements-test-3.13.txt
@@ -2,6 +2,8 @@
 #    uv pip compile --python-version 3.13 --python-platform linux requirements-test.in
 iniconfig==2.1.0
     # via pytest
+more-itertools==10.8.0
+    # via -r requirements-test.in
 numpy==2.3.2
     # via
     #   -r requirements-test.in
@@ -12,7 +14,7 @@ pluggy==1.6.0
     # via pytest
 pygments==2.19.2
     # via pytest
-pytest==8.4.1
+pytest==8.4.2
     # via -r requirements-test.in
 scipy==1.16.1
     # via -r requirements-test.in
diff --git a/.github/workflows/environments/requirements-test-3.9.txt b/.github/workflows/environments/requirements-test-3.9.txt
index 6148d75..129cca7 100644
--- a/.github/workflows/environments/requirements-test-3.9.txt
+++ b/.github/workflows/environments/requirements-test-3.9.txt
@@ -4,6 +4,8 @@ exceptiongroup==1.3.0
     # via pytest
 iniconfig==2.1.0
     # via pytest
+more-itertools==10.8.0
+    # via -r requirements-test.in
 numpy==2.0.2
     # via
     #   -r requirements-test.in
@@ -14,7 +16,7 @@ pluggy==1.6.0
     # via pytest
 pygments==2.19.2
     # via pytest
-pytest==8.4.1
+pytest==8.4.2
     # via -r requirements-test.in
 scipy==1.13.1
     # via -r requirements-test.in
diff --git a/.github/workflows/environments/requirements-test.in b/.github/workflows/environments/requirements-test.in
index f651ca5..20ab1a5 100644
--- a/.github/workflows/environments/requirements-test.in
+++ b/.github/workflows/environments/requirements-test.in
@@ -1,3 +1,4 @@
+more_itertools
 numpy
 pytest
 scipy

From 034db076c0af0d884384c02ce7dbafabf83a791c Mon Sep 17 00:00:00 2001
From: janbridley <bradleyjenj@gmail.com>
Date: Thu, 4 Sep 2025 13:43:25 -0400
Subject: [PATCH 10/18] Update imports

---
 rowan/__init__.py         |  2 ++
 rowan/mapping/__init__.py |  2 --
 tests/test_mapping.py     | 15 +++++++++++----
 3 files changed, 13 insertions(+), 6 deletions(-)

diff --git a/rowan/__init__.py b/rowan/__init__.py
index 69b90ff..c5e6811 100644
--- a/rowan/__init__.py
+++ b/rowan/__init__.py
@@ -52,6 +52,7 @@
     to_matrix,
     vector_vector_rotation,
 )
+from .mapping.symmetries import SymmetricallyEquivalentQuaternions
 
 # Get the version
 __version__ = "1.3.2"
@@ -94,4 +95,5 @@
     "to_euler",
     "to_matrix",
     "vector_vector_rotation",
+    "SymmetricallyEquivalentQuaternions",
 ]
diff --git a/rowan/mapping/__init__.py b/rowan/mapping/__init__.py
index 1932fcc..8fd7ec6 100644
--- a/rowan/mapping/__init__.py
+++ b/rowan/mapping/__init__.py
@@ -58,14 +58,12 @@
 
 from ..functions import from_matrix, rotate, to_matrix
 from ..geometry import angle
-from .symmetries import SymmetricallyEquivalentQuaternions
 
 __all__ = [
     "kabsch",
     "davenport",
     "procrustes",
     "icp",
-    "SymmetricallyEquivalentQuaternions",
 ]
 
 
diff --git a/tests/test_mapping.py b/tests/test_mapping.py
index 2647fda..f8c6723 100644
--- a/tests/test_mapping.py
+++ b/tests/test_mapping.py
@@ -6,7 +6,14 @@
 import numpy as np
 import scipy
 
-from rowan import from_axis_angle, from_matrix, mapping, random, rotate
+from rowan import (
+    SymmetricallyEquivalentQuaternions,
+    from_axis_angle,
+    from_matrix,
+    mapping,
+    random,
+    rotate,
+)
 
 zero = np.array([0, 0, 0, 0])
 one = np.array([1, 0, 0, 0])
@@ -214,7 +221,7 @@ def test_symmetric_quaternions_tetrahedral(self):
             (0.5, 0.5, 0.5),
         ]  # Verts in sorted order
 
-        quats = mapping.SymmetricallyEquivalentQuaternions.create_group("T")
+        quats = SymmetricallyEquivalentQuaternions.create_group("T")
         assert len(quats) == 24
         for quat in quats:
             res = sorted(rotate(quat, tet).tolist())
@@ -239,7 +246,7 @@ def test_symmetric_quaternions_octahedral(self):
         """Verify our symmetrically equivalent quaternions are correct."""
         cube = [*product([-0.5, 0.5], repeat=3)]  # Verts in sorted order
 
-        quats = mapping.SymmetricallyEquivalentQuaternions.create_group("O")
+        quats = SymmetricallyEquivalentQuaternions.create_group("O")
         assert len(quats) == 48
         for quat in quats:
             res = sorted(rotate(quat, cube).tolist())
@@ -274,7 +281,7 @@ def test_symmetric_quaternions_icosahedral(self):
         #     axis=0,
         # )
 
-        quats = mapping.SymmetricallyEquivalentQuaternions.create_group("I")
+        quats = SymmetricallyEquivalentQuaternions.create_group("I")
         assert len(quats) == 120
 
         # TODO: we get the correct vertices, but the ordering is difficult to ensure.

From f6cd035525c728798e812be5cf9e5044e73870d7 Mon Sep 17 00:00:00 2001
From: janbridley <bradleyjenj@gmail.com>
Date: Thu, 4 Sep 2025 13:45:51 -0400
Subject: [PATCH 11/18] Update docs

---
 doc/package-rowan.rst | 1 +
 1 file changed, 1 insertion(+)

diff --git a/doc/package-rowan.rst b/doc/package-rowan.rst
index 14fb936..e58b2a9 100644
--- a/doc/package-rowan.rst
+++ b/doc/package-rowan.rst
@@ -33,6 +33,7 @@ rowan
     rowan.isfinite
     rowan.isinf
     rowan.isnan
+    rowan.SymmetricallyEquivalentQuaternions
 
 .. rubric:: Details
 

From dd89877a10231e0320dc0fbbb45afe9b7c696b46 Mon Sep 17 00:00:00 2001
From: janbridley <bradleyjenj@gmail.com>
Date: Thu, 4 Sep 2025 13:50:05 -0400
Subject: [PATCH 12/18] Update doc style

---
 rowan/mapping/symmetries.py | 35 ++++++++++++-----------------------
 1 file changed, 12 insertions(+), 23 deletions(-)

diff --git a/rowan/mapping/symmetries.py b/rowan/mapping/symmetries.py
index f7cdc79..5f2dd34 100644
--- a/rowan/mapping/symmetries.py
+++ b/rowan/mapping/symmetries.py
@@ -144,32 +144,21 @@ def __array_finalize__(self, obj):
 
     @classmethod
     def create_group(cls, group: str):
-        """Create the set of symmetrically equivalent quaternions.
+        """Create the set of symmetrically equivalent quaternions for a point group.
 
-        Parameters
-        ----------
-        group : {'T', 'O', 'I'}
-            The name of the point group. Must be one of {'T', 'O', 'I'}.
+        Args:
+            group ({'T', 'O', 'I'}): Array of quaternions.
 
         Returns:
-        --------
-        SymmetricallyEquivalentQuaternions
-            An instance containing the quaternions for the specified group.
-
-        Raises:
-        -------
-        ValueError
-            If the `group` string is not one of the valid options.
-
-        Examples:
-        ---------
-        >>> tetrahedral_quats = SymmetricallyEquivalentQuaternions.create_group("T")
-        >>> print(tetrahedral_quats._group)
-        T
-        >>> print(len(tetrahedral_quats))
-        24
-
-        >>> assert np.array_equal(tetrahedral_quats[0], [1,0,0,0])
+            (..., 4) :class:`numpy.ndarray`:
+                Eqivalent quaternions ``q`` for the provided point group.
+
+        Example::
+            >>> from rowan import SymmetricallyEquivalentQuaternions
+            >>> tetrahedral_quats = SymmetricallyEquivalentQuaternions.create_group("T")
+            >>> print(len(tetrahedral_quats))
+            24
+            >>> assert np.array_equal(tetrahedral_quats[0], [1,0,0,0])
 
         """
         if group == "T":

From 657bf228b32bfa2c7836d267d15fdd54843239de Mon Sep 17 00:00:00 2001
From: janbridley <bradleyjenj@gmail.com>
Date: Thu, 4 Sep 2025 13:51:41 -0400
Subject: [PATCH 13/18] Clarify docs

---
 rowan/mapping/symmetries.py | 4 ++--
 1 file changed, 2 insertions(+), 2 deletions(-)

diff --git a/rowan/mapping/symmetries.py b/rowan/mapping/symmetries.py
index 5f2dd34..14affd0 100644
--- a/rowan/mapping/symmetries.py
+++ b/rowan/mapping/symmetries.py
@@ -144,10 +144,10 @@ def __array_finalize__(self, obj):
 
     @classmethod
     def create_group(cls, group: str):
-        """Create the set of symmetrically equivalent quaternions for a point group.
+        """Create the set of symmetrically equivalent quaternions for a rotation group.
 
         Args:
-            group ({'T', 'O', 'I'}): Array of quaternions.
+            group ({'T', 'O', 'I'}): Schönflies notation for the group.
 
         Returns:
             (..., 4) :class:`numpy.ndarray`:

From 01a45ca29a81cb5b990be544162cfb6817bbadc4 Mon Sep 17 00:00:00 2001
From: janbridley <bradleyjenj@gmail.com>
Date: Thu, 4 Sep 2025 13:54:54 -0400
Subject: [PATCH 14/18] Add in placeholder for cyclic groups

---
 rowan/mapping/symmetries.py | 10 +++++++++-
 1 file changed, 9 insertions(+), 1 deletion(-)

diff --git a/rowan/mapping/symmetries.py b/rowan/mapping/symmetries.py
index 14affd0..9009db6 100644
--- a/rowan/mapping/symmetries.py
+++ b/rowan/mapping/symmetries.py
@@ -167,7 +167,15 @@ def create_group(cls, group: str):
             return cls(data=generate_octahedral_group(), group=group)
         if group == "I":
             return cls(data=generate_icosahedral_group(), group=group)
-        msg = f"Unknown group '{group}' does not match valid options {{'T', 'O', 'I'}}"
+        # if group[0] == "C":
+        #     return cls(
+        #         data=generate_cyclic_group(int(group[1:]), axis=axis), group=group
+        #     )
+        msg = (
+            f"Unknown group '{group}' does not match valid options "
+            "{{'T', 'O', 'I'}}"
+            # "{{'T', 'O', 'I', 'Cn'}}"
+        )
         raise ValueError(msg)
 
     def __str__(self):  # noqa: D105

From d21288ab2035fac59204a67a24ca9e221e7215df Mon Sep 17 00:00:00 2001
From: janbridley <bradleyjenj@gmail.com>
Date: Thu, 4 Sep 2025 13:58:34 -0400
Subject: [PATCH 15/18] Add examples

---
 rowan/mapping/symmetries.py | 13 ++++++++++---
 1 file changed, 10 insertions(+), 3 deletions(-)

diff --git a/rowan/mapping/symmetries.py b/rowan/mapping/symmetries.py
index 9009db6..2aa890a 100644
--- a/rowan/mapping/symmetries.py
+++ b/rowan/mapping/symmetries.py
@@ -156,10 +156,17 @@ def create_group(cls, group: str):
         Example::
             >>> from rowan import SymmetricallyEquivalentQuaternions
             >>> tetrahedral_quats = SymmetricallyEquivalentQuaternions.create_group("T")
-            >>> print(len(tetrahedral_quats))
-            24
+            >>> tetrahedral_quats.shape
+            (24, 4)
             >>> assert np.array_equal(tetrahedral_quats[0], [1,0,0,0])
-
+            >>> SymmetricallyEquivalentQuaternions.create_group("T")[:]
+            [[ 1.   0.   0.   0. ]
+             [-1.   0.   0.   0. ]
+             [ 0.   1.   0.   0. ]
+             ...
+             [ 0.5  0.5 -0.5  0.5]
+             [ 0.5  0.5  0.5 -0.5]
+             [ 0.5  0.5  0.5  0.5]]
         """
         if group == "T":
             return cls(data=generate_tetrahedral_group(), group=group)

From 7cded1aa86d3b1a1e5a19991bdddd0c062f5a5a9 Mon Sep 17 00:00:00 2001
From: janbridley <bradleyjenj@gmail.com>
Date: Mon, 29 Sep 2025 15:54:20 -0400
Subject: [PATCH 16/18] Update lockfiles

---
 .github/workflows/environments/requirements-test-3.10.txt | 2 ++
 .github/workflows/environments/requirements-test-3.11.txt | 2 ++
 .github/workflows/environments/requirements-test-3.12.txt | 2 ++
 .github/workflows/environments/requirements-test-3.13.txt | 2 ++
 .github/workflows/environments/requirements-test-3.9.txt  | 2 ++
 5 files changed, 10 insertions(+)

diff --git a/.github/workflows/environments/requirements-test-3.10.txt b/.github/workflows/environments/requirements-test-3.10.txt
index 62e5942..45695b2 100644
--- a/.github/workflows/environments/requirements-test-3.10.txt
+++ b/.github/workflows/environments/requirements-test-3.10.txt
@@ -4,6 +4,8 @@ exceptiongroup==1.3.0
     # via pytest
 iniconfig==2.1.0
     # via pytest
+more-itertools==10.8.0
+    # via -r requirements-test.in
 numpy==2.2.6
     # via
     #   -r requirements-test.in
diff --git a/.github/workflows/environments/requirements-test-3.11.txt b/.github/workflows/environments/requirements-test-3.11.txt
index ff0bb7a..052a203 100644
--- a/.github/workflows/environments/requirements-test-3.11.txt
+++ b/.github/workflows/environments/requirements-test-3.11.txt
@@ -2,6 +2,8 @@
 #    uv pip compile --python-version=3.11 requirements-test.in --output-file=requirements-test-3.11.txt
 iniconfig==2.1.0
     # via pytest
+more-itertools==10.8.0
+    # via -r requirements-test.in
 numpy==2.3.3
     # via
     #   -r requirements-test.in
diff --git a/.github/workflows/environments/requirements-test-3.12.txt b/.github/workflows/environments/requirements-test-3.12.txt
index ccdddc9..a342e33 100644
--- a/.github/workflows/environments/requirements-test-3.12.txt
+++ b/.github/workflows/environments/requirements-test-3.12.txt
@@ -2,6 +2,8 @@
 #    uv pip compile --python-version=3.12 requirements-test.in --output-file=requirements-test-3.12.txt
 iniconfig==2.1.0
     # via pytest
+more-itertools==10.8.0
+    # via -r requirements-test.in
 numpy==2.3.3
     # via
     #   -r requirements-test.in
diff --git a/.github/workflows/environments/requirements-test-3.13.txt b/.github/workflows/environments/requirements-test-3.13.txt
index 1c08185..dca33a5 100644
--- a/.github/workflows/environments/requirements-test-3.13.txt
+++ b/.github/workflows/environments/requirements-test-3.13.txt
@@ -2,6 +2,8 @@
 #    uv pip compile --python-version=3.13 requirements-test.in --output-file=requirements-test-3.13.txt
 iniconfig==2.1.0
     # via pytest
+more-itertools==10.8.0
+    # via -r requirements-test.in
 numpy==2.3.3
     # via
     #   -r requirements-test.in
diff --git a/.github/workflows/environments/requirements-test-3.9.txt b/.github/workflows/environments/requirements-test-3.9.txt
index 35506cd..fee160e 100644
--- a/.github/workflows/environments/requirements-test-3.9.txt
+++ b/.github/workflows/environments/requirements-test-3.9.txt
@@ -4,6 +4,8 @@ exceptiongroup==1.3.0
     # via pytest
 iniconfig==2.1.0
     # via pytest
+more-itertools==10.8.0
+    # via -r requirements-test.in
 numpy==2.0.2
     # via
     #   -r requirements-test.in

From 5356de3ebb131081cfeda9f73890534fe56bbf53 Mon Sep 17 00:00:00 2001
From: janbridley <bradleyjenj@gmail.com>
Date: Thu, 13 Nov 2025 15:43:45 -0500
Subject: [PATCH 17/18] Fix

---
 rowan/grids/__init__.py | 59 +++++++++++++++++++++++++++++++++++++++++
 1 file changed, 59 insertions(+)
 create mode 100644 rowan/grids/__init__.py

diff --git a/rowan/grids/__init__.py b/rowan/grids/__init__.py
new file mode 100644
index 0000000..1cb9fdd
--- /dev/null
+++ b/rowan/grids/__init__.py
@@ -0,0 +1,59 @@
+r"""Low-dispersion grids on manifolds related to rotation.
+
+Finding a uniform of `n` rotations (in any representation) is a generalized Tammes
+problem on the manifold associated with that representation. Grids of rotation axes
+reduce to the standard Tammes problem on the 2-sphere, and grids of quaternions are a
+generalization to the 3-sphere. Closed form solutions for general `n` are not possible,
+but we can create nearly uniform (low-dispersion) grids using Fibonacci lattices.
+"""
+
+import numpy as np
+
+"""The Golden ratio, 1.61803398875..."""
+GOLDEN_RATIO = (1 + np.sqrt(5)) / 2
+
+
+"""The angle that divides 2π radians into the golden ratio.
+This takes the longer of the two segments π(√5 - 1), where π[(√5 - 1) + (3-√5)] = 2π
+"""
+GOLDEN_ANGLE = np.pi * (np.sqrt(5) - 1)
+
+
+def spherical_fibonacci_lattice(n):
+    """A Fibonacci lattice of `n` points on the 2-sphere.
+
+    Args:
+        n (int): The number of points in the lattice.
+    """  # TODO: example image!
+    t = np.arange(n)
+
+    z = 1 - (t / (n - 1)) * 2
+    radius = np.sqrt(1 - z**2)
+    theta = (GOLDEN_ANGLE * t) % (2 * np.pi)
+
+    x = np.cos(theta) * radius
+    y = np.sin(theta) * radius
+
+    return np.column_stack([x, y, z]).T
+
+
+def quaternion_fibonacci_lattice(n):
+    """A near-uniform grid of `n` quaternions.
+
+    This is equivalent to a Fibonacci lattice on the 3-sphere. See
+    `this paper <https://ieeexplore.ieee.org/document/9878746>`_ for a derivation.
+
+    Args:
+        n (int): The number of points in the lattice.
+    """
+    PSI = 1.533751168755204288118041413  # Solution to ψ**4 = ψ + 4
+    s = np.arange(n) + 1 / 2
+    t = s / n
+    d = 2 * np.pi * s
+    r0, r1 = (np.sqrt(t), np.sqrt(1 - t))
+    α, β = (d / np.sqrt(2), d / PSI)
+
+    # Allocate as rows and then transpose, rather than stacking columns
+    result = np.empty((4, n))
+    result[...] = r0 * np.sin(α), r0 * np.cos(α), r1 * np.sin(β), r1 * np.cos(β)
+    return result.T

From 950d660288c2b5b800f1faf2716392b37056e665 Mon Sep 17 00:00:00 2001
From: janbridley <bradleyjenj@gmail.com>
Date: Thu, 13 Nov 2025 15:45:13 -0500
Subject: [PATCH 18/18] Enable cyclic groups

---
 rowan/mapping/symmetries.py | 8 ++++----
 1 file changed, 4 insertions(+), 4 deletions(-)

diff --git a/rowan/mapping/symmetries.py b/rowan/mapping/symmetries.py
index 2aa890a..a8f058e 100644
--- a/rowan/mapping/symmetries.py
+++ b/rowan/mapping/symmetries.py
@@ -174,10 +174,10 @@ def create_group(cls, group: str):
             return cls(data=generate_octahedral_group(), group=group)
         if group == "I":
             return cls(data=generate_icosahedral_group(), group=group)
-        # if group[0] == "C":
-        #     return cls(
-        #         data=generate_cyclic_group(int(group[1:]), axis=axis), group=group
-        #     )
+        if group[0] == "C":
+            return cls(
+                data=generate_cyclic_group(int(group[1:]), axis=[0, 0, 1]), group=group
+            )
         msg = (
             f"Unknown group '{group}' does not match valid options "
             "{{'T', 'O', 'I'}}"