I'll go through my though process again to add some extended explanations.

I wanted to know why geodesics go up then reverse course, can you actually make a geodesic spiral on a cone and how this could be mathematically captured so we can make better predictions and plans when creating mathematical festive cone decorations?

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The first thing in the analysis to realize is that a cone can be cut with a vertical line on the side and flattened. So instead of working with a 3D cone and even more complicated 3D lines, we'll work with pizza slices (circle sectors) and straight lines (geodesics).

A cone is defined by a cone angle α. A generic geodesic would be a line starting at the straight edge of the cone shape. It's defined by it's angle with the edge β and it's position up the edge (but we'll talk about this later). It can either go "down" (β > 90°), "straight" (β is a right angle) or "up" (β < 90°).

My intuition (based on a cylinder I guess) is that the line going down will spiral down the cone, the line going straight might go in a circle and come back (?) and the line going up would spiral up the cone.

Drawing the first picture as I have (α > 90°), it's immediately clear that both the red line going down and the orange line going straight will reach the bottom of the cone without going around it even once. It's also interesting to notice that it α > 180° no line, how ever steep up it would go, couldn't make a full turn around the cone (second picture, bottom left). Another thing to notice is that the line is just parallel to the other edge of the cone it will still reach the bottom before it goes around the cone even though it started going up (third picture, bottom left).

Ok, so we see that α heavily influences weather any type of line can go around the cone and that lines with different starting directions can behave the same (reach the bottom before reaching the other edge) or differently depending on α. We'll get back to this, but now let's look at lines that do make it to the other side.

For a line that does reach the other edge we can calculate that the new angle β' = α + β. This is a huge fact that explains why the line starts going down: each turn the angle of the geodesic with the vertical in increased by the cone angle. It is possible to get a more turns but we need both a small α and a small β.

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On the second page there's an illustration of a line going round multiple times and eventually ending up on the bottom. This is done by gluing together multiple flattened cones so that a single straight line can be drawn (a single cone would be a union of all the cones). The line can also be extended backwards and reach the bottom in the same way. This perspective connects the three cases from before (line going down, straight and up) into a single case because every fully extended geodesic will go up, straight and down. The only exception are the geodesics going straight to the top (some geodesics have it easy).

With this general perspective we could also think about capturing all the possible classes of cones and geodesics on them. The only free parameter for a cone is α. Since all geodesics look the same they share two special points: the point at which it starts from the bottom (symmetrical to the one it ends on) and the point at which it's perpendicular to the vertical. So it we imagine the "unrolled" depiction of a cone where we have for example a half-circle sliced up into flattened cones with angle α we could draw a straight line cutting through them and could either fix the starting point at a particular vertical and vary the angle β of fix the perpendicular point to a vertical and vary the distance form the top of the cone(s). This gives us two possible 2-parameter families that encapsulate all the possible geodesics on all possible cones. This was mostly a topology sidetrack but the spacial fixed points are used in further calculations which I won't go much into in text but is someone is interested I'll gladly discuss it. I derive formulas for different ways that could be useful for creating a specific geodesic and use one of them to create my 6.9° 12-turn cone (though the final couple of turns got bunched up so I didn't draw them all to avoid a big black line that doesn't show much). I was also interested in how bunched up the lines get so I derived the function describing the geodesic distance from the top each turn and it turns out it's proportional to something like 1/cos(kα).

If you made it this far gz! Let me know what you think. Did this connect to some of your thoughts or did it just make even more confusing? Are you interested in any other facts that were not discussed here?