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I particularly like this 104-vertex spherical tessellation with its nearly octahedral symmetry and 18 interwoven triangle strips.

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80 vertices in 2-fold dihedral symmetry has triangle strips of 4 different lengths.

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We can also get 80-vertex tetrahedral symmetry with a more "traditional" arrangement of 12 pentagons and the rest hexagons.

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Here is an 80-vertex sphere in tetrahedral symmetry with 24 valence-7 vertices.

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Tetrahedral symmetry requires that a general point be in a set of 12 -- on each of the 4 faces in each of 3 orientations. You can also add 4 points at the vertices, 4 at each face center, or 6 at each edge center. Combined, any even number of points >= 4 can be arranged with tetrahedral symmetry, albeit not always evenly.

Here is 50 points in tetrahedral symmetry which requires that some of them have valence 7.

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50 vertices arranged in D6 symmetry is interesting in that it forms two different but close in length triangle strips -- one following the longitudes and the other the latitudes.

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23- and 29-vertex spheres both arrange in D3 symmetry with a single triangle strip.

And 22 vertices can also arrange with 2-fold cylindrical symmetry that runs all the pentagons together into one long strip. It produces one long triangle strip and three short ones.

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22 vertices can also arrange with 2-fold dihedral symmetry with two strips of six pentagons each separated by a single loop of 10 hexagons. The triangulation has two long triangle strips and two short ones.

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40 vertices in tetrahedral symmetry gives a mix of the two with 4 strips that wrap twice and three that only wrap once.

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22 vertices can also have tetrahedral symmetry, but now only have four strips that each wrap the sphere twice.

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16 vertices gives a triangulation with tetrahedral symmetry. It has 7 triangle strips.

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In contrast, the 12-vertex icosahedron has 6 triangle strips.

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If you walk triangle strips on this 67 vertex sphere triangulation with 5-fold dihedral symmetry, it makes one complete circuit.

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