Can a Christmas Tree have evenly spaced lights with no leftover string?

[ckirmser]

Got a question:

I've got a Christmas tree that needs lights strung on it, but I want to know how to hang the lights so that the entire string is spaced evenly and it ends at the top with nothing left over.

I figure there's a formula to derive this, but I don't know what it may be.

Any ideas?

Thanx!

 

[ckirmser]

Optimally, it'd be nice just to have a general set of formulas. Maybe I could just plug in the tree dimensions - basic cone - and could figure out, say, what the light string length should be to get a particular angle for each loop of lights, things like that.

 

[jbriggs444]

It is not an easy problem. One difficulty is pinning down exactly what is meant by "spaced evenly". The picture above shows each loop of lights parallel to the one before and climbing the tree at the same angle. That is misleading. A little thought shows that either one condition or the other can apply. Not both. You can see that problem in action by looking through picture in #1 to imagine the angle made by the loops in the back. At the bottom of the tree, those hidden loops are nearly horizontal. At the top of the tree they are nearly vertical.

So we need to nail the problem down carefully. At the top of the tree for instance, does the string go nearly vertical? Must it stop short of the top of the tree so that it is not "too close" to itself on a path that goes around the tree horizontally?

 

[ckirmser]

Well, the lines are not supposed to be taken literally. They were included just to graphically indicate the value being sought. Ideally, the angle of the lights would be equal regardless of from what direction the tree is being observed.

I figure that if a cone is just a triangle rolled up, then the hypotenuse would be equal to the length of the string of lights and that there might be formulas to derive the dimensions I'm after. Something probably involving integrals or derivatives and my last calculus was in 1985, so I am woefully out of my depth

Or, rather than a cone, how about a spiral?

If a flexible wire, rigid enough to maintain its spiral, but flexible enough to be extended along the Z-axis, with the length of the wire when straight is the length of the light string and the radius of the spiral is the radius of the bottom of the tree, was grasped by the innermost end and that end lifted, lifting each loop of the spiral equally, to the height of the tree, might there be a way to derive the separation between each loop of the spiral? (yes, that sentence could probably use some periods)

'Course, I may be visualizing it all wrong, but is what I'm looking for possible, if one only takes the graphic figuratively?

 

[jbriggs444]

The problem is that constant separation and constant slope cannot both be attained. That's not a question of how it is drawn. That is a real physical problem.