Rope a ball

Roping tree branches proves difficult

Every once in a while, I have a need to tie up a ball. Usually, it's to do with trees. Most recently, I was trying to secure a rope swing to a high and sturdy branch in an oak tree in my front yard. To do so, I would tie a thin and long rope to a smallish ball, and then attempt to throw the ball over the branch in the desired spot. This took many attempts, and what frequently happened was the ball would slip easily out of the rope. No matter how tightly I tied a lasso around the ball, it would always seem to have a little gap underneath the rope. Was my tying inadequate somehow?

Keeping a tight lasso is harder than it seems

I reach an obstacle of approach when lassoing objects tightly. In my article about a rope around the world, I showed that a little slack will allow a comparatively large gap if pulled tautly upward on one side. Suppose my ball is 6 inches in diameter, and I tie it so tightly, that I leave only 132 of an inch of slack.

\[ \delta = \frac{1}{32}\text{in} \times \frac{1\text{ radius}}{3\text{ in}} = \frac{1}{96} \text{ radius} \tag{slack in rope} \]
\[ \theta \approx \sqrt[3]{\frac{3 \delta}{2} } = \sqrt[3]{\frac{3 \times 1}{2 \times 96}}= \sqrt[3]{\frac{1}{64}} = \frac{1}{4} \text{ radians} \]
\[ \epsilon \approx \frac{\theta^2}{2} = \frac{1}{32} \text{ radius} \times \frac{3 \text{ in}}{1 \text{ radius}} = \frac{3}{32}\text{ in} \tag{gap height at peak} \]
\[ 2 \theta = \frac{1}{2} \text{ radians} \times \frac{3 \text{ in}}{1 \text{ radians}} = 1.5 \text{ in} \tag{gap width on ball} \]
\[ 2 \theta + \delta = \frac{3}{2} + \frac{1}{32} = \frac{49}{32} \approx 1.53 \text{ in} \tag{gap width in rope} \]

So that little bit of slack allowed a gap between the ball and the rope that was thrice as high and almost fifty times as long. It's no wonder, that no matter how tight I try and pull the rope, that the gap remains comparatively large.

What could be better than a lasso?

I believe this inevitable gap is what creates the instability which makes keeping a lasso around a ball so impractical. At the gap, the rope isn't pushing down on the ball, or applying friction against the ball. This lack of contact causes an instability such that any slight move in the rope will degenerate its grip quickly by loss of contact, tension and balance. And while tension is applied to the rope any bump to one side of the ball or other will only accelerate this loss, and the ball will pop out.

So what I had done was to tie a single slip knot. The result is that it could slip tight, but that didn't prevent the ball from slipping out.

I guessed quickly that I may improve my chances of that rope holding onto the ball, if I have rope spanning more area of the ball. One such attempt was a slight improvement in this regard which was to tie the same size lasso, but not as a slip knot. Instead I use the bowline, and I try to tie it to just barely the circumference of the ball. Then I take the standing part and attach it via another bowline to the midpoint of the lasso. Once spread around the ball, I can see I have trisected the surface area of my ball into three equal pieces. Think of the rope as 3 lines of longitude at 0°, 60° and 120° which connect at north and south poles. This may be better, but once again I find the ball slipping out all too quickly as the equal spacing of rope tends towards bunching.

Next I try the double lasso, or package ribbon approach which is like 4 lines of longitude equally spaced. Even with 4 longitude lines, I need tension and contact with the ball in order to maintain the placement of the ropes. However, the tension and contact requirements are exactly the weakness which caused my original lasso to fail, and hence the ball is able to slip out of its trap, with little adversity.

From longitude to latitude to butterflies

Once having tried multiple lines of longitude, without luck my next thought is that of latitude. Certainly, adding some lines of latitude would create a sufficient trap for my ball. But in pondering this, the puzzle turns from geometry to one of path traversal. I want to wrap my rope longitudinally, and then turn to do one or more turns of latitude. Somehow in end result I need my rope in a pattern which is planned and stable against any roughness of jostle. Attempts become fruitless in my desire to directly add wraps of latitude, for the desire to avoid a large degree of knot-tying and rope doubling prohibits much of turning. So my rope around a ball puzzle defies creation of an advantageous pattern without employing a strategy.

After some time, it hits me that my thought to lasso or gift wrap the rope must absolutely be the wrong concept to the task. Thinking more on the idea of adding both latitudes and longitudes of rope causes me to realize that in fact what I need is a net, not a lasso.

How does a net help? In very concept, a net is successful if the holes in the webbing are smaller than the object being netted. If I judge my lasso attempts with respect to what makes a good net, the failure is much more abundantly clear. Each longitudinal traversal attempt failed despite adding more traversals, because in fact each path of escape for the ball admitted the entire circumference of the ball. What can happen, eventually does happen, so of course the ball escapes my trap. And that of adding latitude does stand this test for building a proper net. However, in my original execution I had failed to implement it correctly. A proper net requires that each opening is fixed, usually achieved by a no slip knot at each crossing. This is something I did not consistently execute in my early attempts.

A sufficient net

I love to think about geometry of shapes which have vertices that line up to the surface of a sphere or ball. This primarily includes, studying regular polyhedra and polyhedrons whose faces are regular polygons. My first thought of the geometry of a net that would capture a ball, was to construct a net in the shape of a cube. After all, nets are composed of square holes, and a cube is a simple polyhedron formed of squares.

After a little thought, I was discouraged from a cube design, for a cuboid net actually proved more complicated than it seemed. A cube has 8 vertices, each with 3 edges. I soon realized I would probably be driven to build such a net from many short ropes. However, I wanted my net formed of a single rope. Building such a net from a single rope requires that at least 3 of my edges would need to be doubled.

Okay, putting the cube aside, I thought of the simplest polyhedron, the tetrahedron. As with the cube, each vertex joins 3 edges, but I have half the number of vertices. I can build a tetrahedron net, with only 1 doubled edge. And so that's just what I did! And it worked perfectly. No more chasing down a wild pitch of an escaped ball. So I wielded a ball to my heart's content. To great effect, I was able to place my thin rope at just the right point, and proceed with that placement to pull my climbing rope over the branch. To my great delight, I finished my rope swing with the help of a snugly swaddled baby basketball.

Also, just as important, in my study of nets I learned that the bowline knot is the knot of choice for constructing a net, for it is simple and does not slip. At the same time, the bowline is relatively easy to untie or adjust. More often, the bowline knot would be tied as a sheet bend knot, for they are actually the same knot.

It worked, but then the question becomes, did I actually create a sufficient net? Let me restate. The net divides the sphere into four parts. If my sizing of each part is approximately equal, and approximately snug with the ball, would my net be likely to have no hole large enough to permit the ball to leave the net? Turns out, I needed to do some simplex calculations to find my answer. Specifically, I need the angle that is made between the ball's center and two of it's vertices (knots). This angle matches the arc length, which is the rope's length along an edge. By symmetry, first I calculate the hole size as a proportion of thrice that angle. Then I check that it is less than a 360° circumference by some comfortable margin.

What's simpler than a simplex net?

So the 3-simplex, more commonly called a tetrahedron, gave me the minimal sufficient net for capturing the ball whose surface it hugs. But parts of this solution were not so simple.

  1. I needed trigonometry to prove that a tetrahedron gave a net of sufficiently small holes to prohibit a ball from escaping.
  2. I had to double an edge of the tetrahedron in order to tie the net with a single rope.

Upon studying the tetrahedron net and pausing to think, I noticed that I could pass the standing part of the rope between the ropes of the doubled edge, and once pulled taut, I could see a new net structure form which has 1 more vertex and 2 more faces. The new vertex came at the midpoint of the doubled edge. One new face comes by pulling the doubled edge open into a triangle, and the other comes by splitting an existing face with the standing part running to the doubled edge. The best part is that this new geometry has no doubled edges.

Triangular bipyramid

The structure of this new net was in the shape of a triangular bipyramid. I completely untied the net from my ball and carefully re-tied this shape. This new shape has 5 vertices, 6 faces and 9 edges. Also, this shape can be thought of as 3 lines of longitude, and 1 equatorial line of latitude. Furthermore, the end of the rope is at the south pole, and the standing part exits from the north pole. No edges have a double rope. And best of all, I don't need any fancy math to figure out if this net is sufficient. Each face is a spherical isosceles triangle with one edge longer than the other two. The longest edge is 13 the circumference of the ball. The other edges are 14 the circumference.

\[ \frac{1}{3} + \frac{1}{4} + \frac{1}{4} = \frac{5}{6} \]

With each hole in my net being less than a circumference, my net is sufficient, and having no doubled edges, it is clean. The net hole size went down an extra 112 from a barely sufficient 1112 circumference hole size. So in a way, the triangular bipyramid is a beautiful and bi-sufficient net for my controlled toss of a ball. While I managed a double improvement, it only cost me one extra knot and an extra 23 circumference of rope. In total, that's 5 sheet bends plus 2.5 circumferences of rope and one happy tosser. Anyone up for a game of tetherball?