monotile notebook

doc/examples/monotile.md

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+---
+jupytext:
+  formats: ipynb,md:myst
+  text_representation:
+    extension: .md
+    format_name: myst
+    format_version: 0.13
+    jupytext_version: 1.14.6
+kernelspec:
+  display_name: SageMath 9.7
+  language: sage
+  name: sagemath
+---
+
+# Playing with the monotile
+
+The monotile is a polygon that tiles the plane by rotation and reflections. Interestingly
+one can build two translation surfaces out of it.
+
+The monotile is the following polygon.
+```{code-cell}
+from flatsurf import Polygon, MutableOrientedSimilaritySurface
+
+K = QuadraticField(3)
+a = K.gen()
+
+# build the vectors
+l = [(1, 0), (1, 2), (a, 11), (a, 1), (1, 4), (1, 6), (a, 3),
+     (a, 5), (1, 8), (1, 6), (a, 9), (a, 7), (1, 10), (1, 0)]
+vecs = []
+for m, e in l:
+    v = vector(K, [m * cos(2*pi*e/12), m * sin(2*pi*e/12)])
+    vecs.append(v)
+p = Polygon(edges=vecs)
+```
+
+One can build translation surfaces by gluing parallel edges. There is an
+ambiguity in doing so because of the horizontal segments. We a surface
+`Sbase` where non-ambiguous gluings are performed.
+```{code-cell}
+from collections import defaultdict
+d = defaultdict(list)
+for i, e in enumerate(p.edges()):
+    e.set_immutable()
+    d[e].append(i)
+```
+
+```{code-cell}
+Sbase = MutableOrientedSimilaritySurface(K)
+_ = Sbase.add_polygon(p)
+for v in list(d):
+    if v in d:
+        indices = d[v]
+        v_op = -v
+        v_op.set_immutable()
+        opposite_indices = d[v_op]
+        assert len(indices) == len(opposite_indices), (len(indices), len(opposite_indices))
+        if len(indices) == 1:
+            del d[v]
+            del d[v_op]
+            Sbase.glue((0, indices[0]), (0, opposite_indices[0]))
+```
+
+Next we recover the ambiguous edges and build the two possible remaining gluings.
+```{code-cell}
+assert len(d) == 2
+(i0, j0), (i1, j1) = d.values()
+```
+
+```{code-cell}
+S1 = MutableOrientedSimilaritySurface.from_surface(Sbase)
+S1.glue((0, i0), (0, i1))
+S1.glue((0, j0), (0, j1))
+S1.set_immutable()
+```
+
+```{code-cell}
+S2 = MutableOrientedSimilaritySurface.from_surface(Sbase)
+S2.glue((0, i0), (0, j1))
+S2.glue((0, j0), (0, i1))
+S2.set_immutable()
+```
+
+We indeed obtain translation surfaces
+```{code-cell}
+print(S1.category())
+print(S2.category())
+```
+
+And one can compute their genera
+```{code-cell}
+print(S1.genus(), S2.genus())
+```
+
+and strata
+```{code-cell}
+print(S1.stratum(), S2.stratum())
+```
+
+TODO: failure
+```{code-cell}
+S1.triangulate()
+```
+
+TODO: failure
+```{code-cell}
+S2.triangulate()
+```

doc/news/monotile.rst

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+**Added:**
+
+* monotile notebook example