IntroductionLaying out the points on a sphere is pretty straight forward. But keep in mind that although it's very exact on a piece of paper, it's less so on wood. You'll be fudging things a bit here and there. And it's okay. Just determine points from more than one other point and mark the spot in the middle of the various possibilities.
Types of PointsThere are 4 basic types of points that you'll probably want to layout at some time. They are:
Ball with 12 primary points and 20 satellite points
Note: There are no hard and fast rules about which points get used for what. Many of the projects we discuss use the primary and satellite points or they use the main and clearance points as focus and secondary points. This doesn't have to be the case. You could mark out the 12 and 20 points and then make the 20 points the larger holes and the 12 points the smaller holes. The same can be said of the 6 main and 8 clearance points. I can't promise that you won't run into challenges, but once you figure out how this whole sphere thing works, let your imagination be your guide.
12 Primary Points
The 12 primay points on a sphere are vertices of an included icosahedron. That's a solid figure with 20 faces and 12 intersections. The points are 1.051 * radius from each other. As our balls have a 1-1/4" radius, our points will be 1-5/16" apart. Here's how to do it:
|1||Start out by marking the most obvious end grain point on the sphere. Number it as #1.|
|2||Set your compass to 1-5/16".|
|3||Mark a circle from point #1. We'll call this circle A.|
|4||Mark another circle from any point on circle A. We'll call this circle B. Circle B will cross point #1 and will cross circle A twice.|
|5||From each point where circle B crosses circle A, mark another circle. Repeat until you have 5 circles centered on circle A and 5 points on circle A that are each the intersection of 3 circles. These are the 5 primary points around point #1.|
|6||You should also now have 5 additional points that are each the intersection of 2 circles. These are the 5 points around what will be point #2, which is the opposite end grain point from point #1.|
|7||From each of the points that are the intersection of 2 circles, that is, the lower 5 points if you consider #1 to be the top point, mark an arc to determine the other end grain point. You can also refine the lower 5 points by marking an arch from each of them to the others while you have the compass set to find the lower end grain point.|
|8||Label the lower end grain point #2. Label each of the points around #1 as 3 through 7. Label each of the points around #2 as 8 through 12. This completes the layout of the 12 primary points.|
Notes - Once you get the hang of it, this goes very quickly. I draw a full circle from point #1 but then only mark arcs from each of the other points. It makes clean up a little faster. I find it best to work from one end grain point towards the other. That way the series of arcs that find point #2 shows me how round my ball is and how accurate my points are.
20 Satellite Points
The 20 satellite points, sometimes called constellation points, are easy to find once you have the 12 primary points marked. These points are optional for your balls, so I suggest creating balls with and without them to see what you like. You'll need to mark a circle around each of the primay points and find the points where they intersect. You can do it by trial and error, but it's easier to follow the steps below:
- Find and mark the 12 primay points on a 2-1/2" sphere.
- Erase any circles left from finding the 12 primay points, keeping the points you marked. This will make it a little easier.
- Set your compass to 13/16" (0.32 * 2-1/2").
- Mark a circle from each primary point.
- The intersections of these points are the 20 Constellation points. You'll need to fudge the points a bit as you did with the 12 primary points.
- Mark them, but numbering them isn't important.
6 Main Points
Projects using the 6 main points provide a whole new perspective on what's possible. They're often cut with a larger radius and may require a modified 'fence' on the cutter's holder so that its wide enough to cross the wider gap.
- Find the most obvious end grain point of your 2-1/2" sphere. Mark it as #1.
- Set your compass to 1-3/4". This is the radius of your sphere * 1.414.
- Mark a circle from point #1. This should be the equator of your sphere. Let's call this circle A.
- From any point on circle A, mark circle B. Circle B should cross circle A twice.
- Mark the 2 points where circle A and B cross as point #3 and point #4.
- From point #3 or #4, mark circle C.
- At the 2 points where circle C crosses circle A, mark point #5 and point #6.
- At the point where circle B crosses circle C, mark point #2.
- Note that points 1 and 2 are both end grain points and should probably be worked first.
8 Clearance Points
I think the orginal purpose of the 8 clearance points had to do with the reaching of the hollowing tool to the center of the point furthest from any 3 main points. That's certainly still a valid reason for using them, but don't let it constrain you.
- Find and mark the 6 main points on a 2-1/2" sphere.
- Erase any circles left from finding the main points, keeping the points you marked. This will make it a little easier.
- Set your compass to 15/16". This is the radius of your sphere * 0.765.
- Mark a circle from each main point.
- The intersections of these points are the 8 clearance points. You'll probably need to fudge them just a bit to get the pefect.
- Mark them, but numbering isn't important.