Calling all Math Wizards....HELP!

[James Burke]

I am trying to create a soccer ball using a 12 inch gazing ball and have run into a mathematical dilemma.

I’ve disassembled a soccer ball and have reverse engineered the pieces for size and shape, but when I lay the vinyl on the ball, my theory comes to a screeching halt.

I’m only good for right angle trig, and a little bit of sine and cosine law....but I fear spherical trig is my next stop.

Are there any mathematicians out there?

I’m using AutoCAD and I can produce a design right down to the “nth” degree if needed, but I am unsure of where to begin.

I’m thinking I need to make an orthographic drawing first, and then project the polygons onto the spherical radius. And then after that, project the polygons back onto a flat surface for sending to the plotter. Essentially, that will create polygons with dished in sides that will meet up perfectly when applied to the ball. Make sense.

Anyway, I’d like to hear some good tips from those with a little more gray matter than I have.

Take a good long look at a soccer ball some time. They’re absolutely a work of art...getting pentagons to match up perfectly with hexagons. There are 12 pentagons and 20 hexagons on a soccer ball.

After some close observation, it sure appears to be the same design principle for the geodesic dome.

 

[randya]

http://mathworld.wolfram.com/TruncatedIcosahedron.html

http://mathtricks.org/math-geometry/soccer-ball-math/

Maybe this might help

 

[CheapVehicleWrap]

Are you trying to wrap a sphere?

 

[GAC05]

This might do it:
http://www.korthalsaltes.com/model.php?name_en=truncated icosahedron

 

[James Burke]

Are you trying to wrap a sphere?

Essentially, I am going to cut out 20 vinyl hexagons to cover the gazing ball. The pentagons will be formed by the uncovered ball.

The hexagons will be cut in a manner so that there will be approx. 1/4" between them so it mimics the seams between the sections and looks realistic.

Thank you for the links. I found some about the paper cutout truncated icosahedron, but those might not work as planned. On the paper model, the hexes and pentagons are basically flat, and the shared sides are created by a fold. It is a ball made of "multiple flats", so to say.

On the gazing ball, I'm dealing with a true spherical surface where those shared sides (of the vinyl) must follow the contour and remain parallel with each other.

I'll dig through the links some more and let you know what I find.

 

[showcase 66]

One of the reasons you are having a problem, is that a soccer ball is not a sphere. It only becomes that way after you fill it with air and each pentagon and hexagon are expanded. Your pieces will have a concave on all the sides. I think this will work.

Also is the sphere you are working with, is it 12" diameter. That is what I based it on. If not you should be able to scale it up or down. My math may be off a little since I have been drinking a little bit, but it should be close.

 

[showcase 66]

It is only for half of the ball as well

 

[James Burke]

Thank you CSDD. That's where I was headed, but I was unable to determine just how concave to make the sides.

Could you explain the process?

 

[showcase 66]

I will have to try to explain it tomorrow. I am not that good explainin mathmatical processes with out actually showing it. Right now my mind is a little foggy. I will get back to you tomorrow.

 

[James Burke]

As for scaling, can I take a 10 inch ball, divide it by 12 to equal .8333333333, and then scale down the hexes and pents by 83.333% ?.

Thanks for the help.

 

[showcase 66]

Ok. I have been trying to write this out for you to understand I think I am making it harder than it really needs to be. The pdf I sent is close but not exact. I think I have it right for you on the 12" ball. you should be able to scale it down at the 83.33333%.

I only have 1 hexagon and 1 pentagon in this pdf.

The radius of the concave should be right around the radius of the sphere. So in your case the 10" ball would have a 5" radius concave. The hardest part is getting the right size of the first piece.

Hope this works for you. I have been popping on and off here during the day and I will start writing it and it sounds really confusing.

If it works. Keep the design just in case someone else needs to do it.

 

[James Burke]

The hardest part is getting the right size of the first piece.

Exactly. I won't know for sure until I try do a full layout.

They start out looking good, but then things start getting crowded or spaced too far apart.

The radius on the hex and pentagon is a piece of cake with AutoCAD. All I have to do is make the polygon, and then apply an arc with the specified radius.

I'll keep you posted.

 

[James Burke]

After doing some more thinking, I believe I need to scale the polygons according based on total surface area of the sphere.

I get different three totally different scale factors when I compare diameters, volumes and surface areas between two known spheres. I strongly believe that the surface area is what really counts here. Does that make sense to you?

My proportions posted above were based on diameter only.

 

[showcase 66]

a 12" diameter ball has approx 450 sqin each pentagon covers approx 10.67 sq in. 10.67 x 12 = 128 sqin.

Hexagon covers approx 16.11 sqin. 16.11 x 20 = 322 sqin. 322+128 = 450

I get different three totally different scale factors when I compare diameters, volumes and surface areas between two known spheres. I strongly believe that the surface area is what really counts here. Does that make sense to you?

Correct. The actual difference between a 12" and a 10" is actually an approx. 30% drop in surface area. so your scale factor is 70% from 12 to 10.

You will have to scale it down as a regular hexagon and pentagon and then add the radius for the new size. Otherwise the radius will be wrong for them.

Out of curiosity. What exactly are you going to be doing with the sphere? Is it part of a sign?

Also if you wanted to have the "seam" inbetween the pieces you will need to add an inside stroke or something like that on what I sent over. I would actually cut it with a 1/16 inside stroke and apply it without weeding this stroke part out. After the application is done, then I would pull those pieces. If that makes sense.

I used autocad as well to create the pieces. I use it a lot on complex designs.

Hope this helps you. Make sure you post pix of this when you get it completed.

 

[James Burke]

The ball is to be part of a large "trophy", and will be fairly impressive once I get it all figured out.

I've been tweaking the design a little and it appears that the spherical radius applied to the polygon side is still a little small since it still has a little gap at mid point between the pieces. I'm thinking of increasing the radius to be equal with that of the sphere's diameter to see if that helps create a more parallel joint between the pieces.

You're correct about creating an inline. Once my polygon is set to size, I create a polyline out of it and do an offset. Laying a piece of vinyl with a cutout inside is a tricky proposition so I just cut out some strips equal to the gap between the polygons and use that as a guide to lay the vinyl and it works fairly well.

I've resorted to mixing a little theory with practicality, however. I've scribed strait layout lines on the vinyl (applied to the sphere) and have then removed the vinyl and done some measuring for distortion. That will become the basis of my "tweak" factor.

 

[GAC05]

good luck with it.
I'm watching and would like to see the shapes you end up with and the final result.

wayne k
guam usa

 

[visualeyez]

Maybe try making a 12" circle in your graphics software and using that to overlap and do a "cut out" of the sides of your polygons? That seems like it should distort back out properly to straight lines in my opinion.

 

[heyskull]

I have a friend who is a monumental mason who has a file for this as he had to make something similair in stone!
I will see if he still has the file.

SC

 

[James Burke]

In the flat, the designs all work out perfectly. It's the 3-D world where things get a little funky here.

I think I've got it on the run and will chime in later.


Jim