Geodesic
Closed geodesics on the surface of a simplex
Sbornik. Mathematics, Tome 198 (2007) no. 2, pp. 243-260

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The closed non-self-intersecting geodesics on the surface of a three-dimensional simplex are studied. It is proved that every geodesic on an arbitrary simplex can be realized on a regular simplex. This enables us to obtain a complete classification of all geodesics and describe their structure. Conditions for the existence of geodesics are obtained for an arbitrary simplex. It is proved that a simplex has infinitely many essentially different geodesics if and only if it is isohedral. Estimates for the number of geodesics are obtained for other simplexes. Bibliography: 13 titles. 
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     author = {V. Yu. Protasov},
     title = {Closed geodesics on the surface of a simplex},
     journal = {Sbornik. Mathematics},
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     volume = {198},
     number = {2},
     year = {2007},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2007_198_2_a4/}
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V. Yu. Protasov. Closed geodesics on the surface of a simplex. Sbornik. Mathematics, Tome 198 (2007) no. 2, pp. 243-260. http://geodesic.mathdoc.fr/item/SM_2007_198_2_a4/