[color=#999999]This activity belongs to the GeoGebra [i][url=https://www.geogebra.org/m/r2cexbgp]Road Runner (beep, beep)[/url][/i] book. [/color][br][br]In this construction, we can travel along a [i]Hamiltonian path[/i] [url=https://www.geogebra.org/m/yybrap57#material/bn8y4sz5][img]https://www.geogebra.org/resource/scjbyz2p/0tuzuVw455vxurEw/material-scjbyz2p.png[/img][/url] of the edges of a dodecahedron. To facilitate our orientation, we have colored three pairs of opposite faces differently.[br][br]In the case of straight segments, such as the edges of a polyhedron, the curvature is zero, so the curvature vector at point C remains undefined. Therefore, we have chosen [b][color=#6aa84f]N[/color][/b], the normal vector to the curve, as the vector:[br][center][b][color=#cc0000][b][color=#6aa84f]N[/color][/b][/color][/b] = [b][color=#cc0000]T[/color][/b] ⊗ UnitVector(C)[/center]Since the dodecahedron is centered at the origin O, the position vector of C has the direction OC. Thus, the binormal vector [b][color=#0000ff]B[/color][/b] of the Frenet frame (defined as [b][color=#cc0000]T[/color][/b] ⊗ [b][color=#cc0000][b][color=#6aa84f]N[/color][/b][/color][/b]) will coincide with the direction OC, making the "vertical" of our point of view precisely that.[br][br]Note that the projection performed when the point C is kept fixed (above, "Point view") may appear incomplete at times. This is not the case. It simply happens that, in the projection, some of the edges coincide with others (meaning the Hamiltonian path continues along these coinciding edges when projected), giving the impression that the path abruptly stops at some points.[br][br][color=#cc0000]Note: To improve execution agility, it is recommended to download the GGB file. Remember that, in this case, it may be necessary to readjust the height of the 3D view to reposition the point in the desired position. For this adjustment, the magenta slider that appears once downloaded may be helpful (you may need to temporarily hide the 2D graphics view to see it).[/color]

[color=#999999][color=#999999]Author of the construction of GeoGebra: [color=#999999][url=https://www.geogebra.org/u/rafael]Rafael Losada[/url][/color][/color][/color]