Trying to divide a spherical shell into equal-volume "wedges

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Mechatherium
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Joined: Thu Jul 03, 2014 1:44 am UTC

Trying to divide a spherical shell into equal-volume "wedges

Postby Mechatherium » Tue Feb 02, 2016 3:44 am UTC

What I think I need is a method of dividing a sphere into an arbitrary number of pieces of equal surface area, especially if the number is odd. The reason I'm asking for a way to divide by surface area, not volume is because the spherical shells I want to divide have a "thickness" of 1 and an outer radius of from 2 to 16 where outer-radius - inner-radius =1, as shown in the attached image:

shell sketch.png


Here's a little table showing the radii of each shell and the number of parts. Unbelievably, I did the math (in Excel) and each part in each concentric shell ought to have the same volume as the innermost sphere (radius=1).

Code: Select all

Radius    Parts
1         none (undivided)
2         7
3         19
4         37
5         61
6         91
7         127
8         169
9         217
10        271
11        331
12        397
13        469
14        547
15        631
16        721

Total parts = 4096

If anyone knows how to divide up a sphere as I'm asking please let me know--and please, use LOTS of pictures! :)

>-)
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Re: Trying to divide a spherical shell into equal-volume "we

Postby >-) » Tue Feb 02, 2016 6:21 am UTC

Pick any axis of the sphere which passes through the center. Then, to split the sphere into n regions, simply create n slices from pole to pole, each with angle 2pi/n in width.

This sliced orange is an example where n = 8 (i think -- the image is too blurry to tell), and the width of each slice is pi/4
Of course, you're slicing a shell here, not the whole sphere, so pretend you can only see the orange peel in this image.

Spoiler:
Image

If you want to try something more challenging, I found this interesting looking paper on slicing a sphere into sphere into equal-area regions each with a small diameter. (whereas the sphere slicing method above creates long, thin slices with large diameter)
http://maths-people.anu.edu.au/~leopard ... e-talk.pdf

Lothario O'Leary
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Re: Trying to divide a spherical shell into equal-volume "we

Postby Lothario O'Leary » Fri Feb 05, 2016 11:37 pm UTC

IIRC, if you divide the diameter of a sphere into N equal parts, and cut the sphere perpendicular to it, the resulting slices will surprisingly have equal area (the long ones will be thin, and the short ones will be wide, and it happens to compensate exactly).
Then you can join up a few neighboring slices and cut them up in wedges instead. Which is basically what Macquarie does, as it happens, but I figured that out before I looked at the Macquarie article :D

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Goemon
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Re: Trying to divide a spherical shell into equal-volume "we

Postby Goemon » Wed Feb 10, 2016 3:04 am UTC

Your pattern of "number of slices in each shell" is actually kind of interesting because you (not coincidentally) get the same pattern in two dimensions if you do this (you can demonstrate to yourself with a collection of pennies or similar coins, or examine a hexagonal game board):

Start with one hexagon (or penny) (total = 1)
Arrange a ring of hexagons around the original hexagon, forming a single larger hexagon with 2 small hexagons on each side (TOTAL small hexagons = 7)
Add another ring of hexagons around the 7. This requires 12 new hexagons to form a new large hexagon, now with three small hexagons per side (TOTAL small hexes = 19)
Add another ring to form one large hexagon with four small hexagons on each side (total small hexes = 37)

etc.

So you could form a similar pattern with your pieces, (though you won't be able to do it with hexagons) with the caveat that each piece in each shell is half the volume of the inner sphere. Just take one hemisphere of say the innermost shell, draw a circle on it such that the circle consumes 1/7 of the total volume, and then divide the outer ring into six equal pieces. Repeat with the other hemisphere to create a total of 14 equal volume "slices" (rather than the 7 you actually wanted). On the next outer shell, you'll have two circles on each hemisphere: one inner piece at the pole, six pieces in the "middle" ring, and 12 more around the outside for a total of 19 on each hemisphere.

The end result would have the drawbacks that (a) there are twice as many pieces in each shell as you were aiming for, and (b) all the pieces from different "rings" and different shells would be different sizes and shapes, but on the other hand, they'd all be fairly similar and all exactly half the volume of the central sphere.
Life is mostly plan "B"

Mechatherium
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Joined: Thu Jul 03, 2014 1:44 am UTC

Re: Trying to divide a spherical shell into equal-volume "we

Postby Mechatherium » Tue Feb 23, 2016 4:05 am UTC

to begin--thanks everybody!

Sorry I didn't get back sooner--other stuff I had to deal with!

I read the Macquarie-sphere talk pdf before. I have to admit much of the math was over my head. I think Lothario O'Leary was trying to explain it in simpler terms but I didn't quite get what he was after (math dummy--need pictures!)

There is a walk through of the Feige-Schechtman algorithm in the PDF.

If I'm not mistaken, my hollow outer spheres ought to be divisible into a series of plug-shaped segments each with the volume of the unit sphere (r = 1)

1) The first step is to determine how big the spherical cap should be. The area of the cap ought to be (4*pi*r^2)/n where n is the number of pieces I need to divide the sphere into. I have some inkling how I would do that but I'd like some more detail if possible.

2) Next, I figure out the optimal packing arrangement of the n caps over the sphere. Two will be at the poles. Again, the math is provided but it's hard to visualize what I need to do..

The last three steps are frankly beyond my limited math experience.

If anyone would like to help walk me through at least r=2 and r=3 I'd be grateful.

Tony

s23bog
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Joined: Mon Aug 12, 2019 1:31 am UTC

Re: Trying to divide a spherical shell into equal-volume "wedges

Postby s23bog » Mon Aug 12, 2019 1:43 am UTC

I don't know if you've found an answer, I didn't really bother to check. But I would like to start a discussion about this very thing. I have a structure with which I am experimenting. I am currently using it as an experimental greenhouse. It is an ellipsoid, or sorts. But, I have a 3D computer that is publicly available, if you are interested. I don't know how many equal parts you ar looking to divide an sphere into, but I think this particular arrangement has a very good chance of being perfectly divided.

The shape is difficult to describe in words, but I certainly can. Anyhow, the polyhedron at the center, which probably can be fitted to the outer 18 points of the spheroid is called a tetrakis cuboctahedron. If your interested, I'd be more than happy to share with you a couple of links. :idea:


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