The triaugmented triangular prism has two types of [[closed geodesic]]s. of a polyhedronThese are the paths on theits surface avoidingthat the vertices andare locally lookstraight: likethey theavoid shortestvertices path.of Inthe other wordspolyhedron, these paths follow straight line segments across eachthe facefaces that intersectthey cross, and createform [[complementary angles]] on the two incident faces of theeach edge asthat they cross. In the caseOne of athe triaugmentedtwo triangular prism, and with the resulttypes of closed geodesic'slengths,runs itparallel hasto twothe typessquare base of closeda geodesics:pyramid, through the firsteight geodesicfaces crossessurrounding the edgespyramid. ofFor twoa equilateralpolyhedron squarewith pyramidsunit-length and a triangular prismsides, anthis [[equator]]geodesic of the solid, withhas length of <math>4</math>;. theThe secondother type of closed geodesic crosses ten of the edgespolyhedron offaces, perpendicularly to an threeedge equilateralof squareeach pyramids,face; withit has length of <math>\sqrt{19}\approx 4.36</math>.{{r|lptw}}
== Fritsch graph ==
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| last3 = Traub | first3 = Cynthia M.
| last4 = Weyhaupt | first4 = Adam G.
| doi = 10.12732/ijpam.v89i2.1 | doi-access = free
| issue = 2
| journal = International Journal of Pure and Applied Mathematics
| pages = 123–139
| title = Coloring graphs to classify simple closed geodesics on convex deltahedra.
