Tiling A Sphere: A 3-D Geometry Question

[SaxonViolence]

Friends,

 

One way to tile a Sphere is to use the "Dumbbells"—don't know what the technical name is for these. They are also used fairly frequently in Quilting, to tile 2-Dimensional Surfaces.

 

For anyone who doesn't know what I mean, examine a Baseball or a Tennis Ball.

 

If I want to tile my Sphere with exactly two "Dumbbells" and I want both of them to be symmetrical—is there a single size, shape or confirmation that will work? Or can I have a number of possible sets of dimensions for the two Dumbbells?

 

I'm thinking that I'm going to be limited to just one possible Dumbbell conformation.

 

Okay, given the Circumference of my Sphere—which will also give Radius, Circumference, Area and Volume if I need it...

 

Can I calculate the Length of Each Dumbbell from Extreme End to Extreme End; Width across the Widest part of the "Bell" and Width across the Narrowest Part of the "Neck"? 

 

{Obviously, Extreme Length + Narrowest Neck = Circumference.}

 

#1.} Yes, if I had a Baseball or Tennis Ball on hand, I could get an answer close enough for Government Purposes. I don't and it will be awhile before I an afford to invest in a Ball for no other purpose than to satisfy my Idle Curiosity—the Pauper-Hood of living on Disability, don't you know?

 

#2.} Why do I want to Know?

 

Yeah, I know that most Right-Thinking People regard Writing SF and Phantasy as a Vicious and Perverted Vice...

 

And yes, I'm Guilty.

 

I wanted to Imagine a series of Artificial Worlds with Identical Geography—at least so far as basic shape of the single sea and lone continent—all 50% Land Mass and 50% Ocean and all of the Continents and Oceans shaped like half of a Baseball.

 

I can kinda get a Jackass Guesstimate but for things as basic as simple Geometry I don't Like to have Egregious Errors of fact in my Novels.

 

 

Thanks!

 

 

Saxon Violence

Edited November 20, 2015 by SaxonViolence

[sanctus]

This what you are after?:
https://en.wikipedia.org/wiki/Spherical_polyhedron

[CraigD]

If I want to tile my Sphere with exactly two "Dumbbells" and I want both of them to be symmetrical—is there a single size, shape or confirmation that will work? Or can I have a number of possible sets of dimensions for the two Dumbbells?

A dumbbell-shaped “baseball cover” tiling of a sphere can have (infinitely) many possible shapes.

 

I don’t know a neat way to enumerate them, but there seem to be a few math papers on the subject. Richard Thompsons 1998 “Designing a Baseball Cover”, cited at Wolfram Mathworld and a few other places, seems a pretty good one.