Drilled Holes in a Sphere

why 17?

Because 18 would be too easy of a problem that could be done with no calculation, and thus wouldn't test the student's knowledge?

You'll need something like an index table and some way of holding the ball... a machinist vise might work. Google both of those to get ideas. The rest is just math/geometry to figure out the angles etc.

Teach yourself a bit of spherical geometry. Learn all about steradians. Then think up an algorithm for dividing the sphere into N (maybe 16 or 17) equal solid angles. Not too hard if you've got a bit of time (which I haven't). Good luck.

(PS - try it in a plane (ie with a circle) first - it'll point you in the right directions).

why does 21 degrees come to mind??

Hi Fredski, I used 17 as an example, not actually my preferred number of holes. Starting with a circle is straightforward (360/17=21.2) When I throw in that 3rd dimension, I get a little dizzy. Not a student but always learning. Thanks for your comments.

Bill

your problem isn't that tough but we don't do others homework here. I bet if you explained your project you'd get more assistance

All, to help understand this request a little more, I'll outline the idea. I'm working on an application that uses balls from 1 to 6" diameter. These are machined from round rod slugs and are usually made in a dual chuck lathe (turn 2/3rds of the surface, grab it with the other chuck and finish). In an effort to lighten these balls, it's been suggested that we drill blind holes in them to a consistent depth but still maintain the maximum spherical surface area. The holes need to be distributed across the spherical surface area so that they are equidistant from their neighboring holes as much as possible. This is a bit of the golf ball dimple question, right? I'd rather not start looking at the various diameters of holes (dimples) as I hope to have less than 10% of the surface area as holes. A simplification of the problem could be just accepting 6 or 14 holes. I can lay that out on my tennis ball model. Thanks for your comments.

Bill

Golf ball dimples are not to lighten the ball.

If the object is light weight, can you use a foam ball?

What will you do with the balls.

If you want light I'd go with hollow spheres....cut them in half, machine them out, and weld them back together.....or just buy some...

http://www.sharpeproducts.com/store/metal-spheres-hollow-balls

Excellent explanation, but a small cosmetic imperfection.

Area of an triangle is : basis times height divided by 2

For an equilateral triangle with s the length of one side,

the height = (s√3)/2 and the basis is s/2 so the area = (s²√3)/8

No it's not..

The area of a right angle triangle can be found by:- A= Base X Altitude divided by 2 as you have said. ie A = (B X A)/2

An equilateral triangle has 3 sides at 60° to each other and so we can look at it as two right angle triangles back to back with the base of each being half the length of one side of the equilateral triangle.

The sine of 60° is √3/2, therefore the height is (√3/2)s. s being the length of side.

So to find the area of one half of the equilateral triangle we take the base - in this case half of one side (s/2) and multiply it by the height (√3/2)s and then divide the lot by 2

So Area of half the equilateral triangle A = ((s/2) x ((√3/2)s)/2)

So area of the whole triangle is A = (((s/2) x ((√3/2)s)/2)x2)

The last pair of 2s cancel each other so the area of the whole triangle can now be seen as A = ((s/2) x ((√3/2)s))

Simplifying that becomes A=(s²√3)/(2x2) or A=(s²√3)/4

Google "area of equilateral triangle" to confirm this if you have further doubts.

You are right,

shame on me....

That is the formula for the area of an equilateral triangle on a plane surface, not on a sphere.

What are the steel balls being used for and why do you need to make them lighter?

Can you use a different material?

Do you have access to a CNC mill or machine center?

GA. (The voting button doesn't work on my current browser.) The Archimedean polyhedra (and prisma and antiprosms) would also give equal hole-to-hole distances, but the areas within the polygons would differ.

I was thinking more of a circle and a poly. ga for you

Small correction:5 platonic solids (polyhedra): tetrahedron, octahedron, cube, isosahedron and dodecahedron. They have 4, 6, 8, 12, and 20 vertices, all equally spaced. icosahedron has 20 faces, and the dodecahedron has 12 in fact.

just nitpicking...

But no 17 equlateral corners. Seems like prime numbers is not a subset of corners for a a sphere.

You do realize that there are only certain numbers of points that can be evenly spaced on a sphere?

https://www.maths.unsw.edu.au/about/distributing-points-sphere

With an indexer. You have to decide how many holes. And figure how many degrees to turn it. Drill the first hole to a size you can force fit a pin into it. This hole has to be drilled straight and in the center. Not so tight of a fit that you can't get back out. But tight enough to hold while you drill the other holes. Put the pin in the hole. The ball with pin into the indexer collet. You have to set the indexer so the next hole drilled is centered. Drill the holes around the ball turning how ever many degrees you choose. After you have done the first row. Remove the ball, take out the pin and put it in another hole. The put it in the indexer and drill again. Doing until all the holes are drilled.

I hope this is a one time deal and not a production run.

that would work: a reference pin

He said they do a 2/3 rds turn in the CNC lathe. Why don't they add the hole drilling to that operation too?

That's if the machine has that capability?

Hi OM, the CNC lathe does have that capability with live tooling.

All, I appreciate the comments and insight you've offered here. It's complex but when we get one of these made, I'll try to post a photo. The guys who make golf balls have won my appreciation of something I've taken for granted. Cheers!

golf-balls are made in a computer......then spit out by the millions

It's late and I don't have time for an extensive example, but here are some thoughts:

Laser cutters attached to 6 axis robotic arms perform these kinds of operations regularly. Aiming the beam or tool normal to the surface to perform cutting is a bread and butter operation.

That said, from an actual programming perspective, think about projecting the hole axis centerlines equally spaced off the center point of the sphere out to the surface, not from the surface in. This, using an appropriate algorithm, will define the proper pierce points. I'm guessing you can easily visualize 6 centerlines without programming or even breaking a sweat.

So, in a nutshell, don't try to determine the hole locations on the surface, determine the centerlines from the centerpoint out.

Once the pierce vectors are determined then it's just a matter of programmatically defining the diameter or radius of the sphere and specifying the cutting depth.

Lastly, this whole operation could also be performed on a 5 axis mill with two rotational axes. I just happen to think of robotic programming first when dealing with spherical surface machining.

Hope this is clear. It's late and I'm tired.

Nite nite.

Hooker

Fit a volume with 17 equilateral corners

There a a very limited number of options to make the holes exactly equidistant. What is your interpretation of approximately equidistant? Any two holes drilled at random into a six inch diameter sphere are approximately equidistant if you allow a tolerance of nineteen inches.

Setting a question to which you as the question setter do not know the answer is more than stretching your students. If one of them comes up with a more elegant solution than any suggested here, how will you know that it is a more elegant solution?

OMG!! Do you think this is perhaps a homework question ....???

Hi Hilton, this is no homework question. I work for a light metals company and we produce the balls today on CNC lathes. The customer wants to take weight out of the ball but the spherical surface seals on a seat. It doesn't need to be a perfect seal so drilling blind holes is an option. Drilling blind holes that are evenly dispersed on the balls is the request and while I can ask the machinist to do it, I'd rather know how it should be done before asking them. There are some very insightful responses here and I appreciate the help. Thanks!

That's a relief! I also agree with your wanting to know how it's done before giving the work out to someone else. I operate in much the same way. However, sometimes you have no choice, and your CNC operator can probably explain how, after the software tells/ shows him exactly !

Good luck,

H.

I agree with Hilton. While you may choose a method that you think is fine, the machinist may already know how to perform the task of drilling the holes.

He might not take too kindly to having a non-machinist try to tell him how to do his job, either.

I'd probably approach him with the task and ask him what he thinks is the best way to do the job. You might pose a question along the lines of, "What do you think about drilling the holes (then propose your suggestion)" and see what he says.

You might try plunging the tool into the ball and then removing material by wobbling the ball around the cutter without enlarging the hole.

The machinist may be able to remove enough material using a much smaller number of holes this way.

Have you considered getting the balls hollow cast? That way the weight is taken care of without compromising the integrity of the surface. 3D printing is another alternative

Original_Macgyver has already asked, but can you use another material? Aluminium, for example, is only about 38% as dense as steel; you'd have to drill an awful lot of holes to lose 62% of your material!

Don't planetariums use spheres with lights in them for creating the "stars" on the spherical ceiling? How do they locate and hold the spheres for making the holes. I don't know, but however it is done may provide a technique for you after you determine where the evenly spaced holes should be. I think there is a spherical geometry; does that branch of mathematics provide any methods?