how can I wrap a sphere without any overlaps or creases?

Question

I WANT TO WRAP A PIECE OF PAPER ROUND A BALL WITHOUT ANY OVERLAPS, WHAT SHAPE DO I NEED TO MAKE THE PAPER TO ACHIEVE THIS?

Answer 1

I can answer this question. However, it is difficult to answer.

Have you ever seen a world map with Mercator Projection.

The World is considered round. Mercator Projection Unravels the world into a Flat plane.

See the following link for the explanation and mathematics:

http://en.wikipedia.org/wiki/Mercator_projection

Make a solution of Paper and water. Then add Elmer's glue to adhere the Paper to the Ball or sphere. You have now covered the ball in paper, Any other attempt would be too complex, unless you use the mathematics from the site i listed.

Good Luck.

Answer 2

you guys arn't reading the question right. how can he get this "piece of paper around a ball"

It is impossible my friend with out creasing it, the only thing i can think of is using several cut out "cone shapes" but then you risk overlapping.
I'm thinking 8 cone shape pieces of paper that are each half the diameter long. Kinda like flatening the ball and cutting it into 4 pie shapes.
Then have 4 connected by the points at the top and 4 connected from the bottom. they will al be able to wrap around the ball without overlapping but you would have a lot of tape work to do.

Answer 3

That's a damn good question in 2D planes meeting 3D objects. Could I recommend asking it in Mathematics? My own maths isnt up to the challenge at all. As a practical point though I would imagine you would want there to be some overlap just so you can seal it up properly. There is the good suggestion of putting it in a box though, any weird shaped present will give major clues to what it is. Kids might spend hours trying to figure it out based on the shape. I know I used to! regards.

Answer 4

I'll give it a shot. We are looking for a volume that is a half sphere with radius r, from which a portion of a sphere of radius 1 has scooped out a portion. The volume of the smaller hemisphere of radius r is: V = 2πr³/3 If we consider a sphere of radius one. The distance from the center of the sphere where a plane could cut the sphere to get a circle of radius one--use the Pythagorean Theorem. The distance is √(1 - r²). Integrate from y = √(1 - r²) to y = 1 V = ∫(πx²)dy = π∫(r² - y²)dy = π(r²y - y³/3) | [Evaluated from y = √(1 - r²) to y = 1] = π(r² - 1/3) - π(r²√(1 - r²) - (1 - r²)^(3/2)/3) = (π/3)(3r² - 1) - (π/3)√(1 - r²)[3r² - (1 - r²)] = (π/3)(3r² - 1) - (π/3)√(1 - r²) (4r² - 1) = (π/3)[(3r² - 1) - √(1 - r²) (4r² - 1)] So the volume in question is: V = 2πr³/3 - (π/3)[(3r² - 1) - √(1 - r²) (4r² - 1)] = (π/3)[2r³ - 3r² + 1] + (π/3)[√(1 - r²) (4r² - 1)] Find r to maximize the volume. Take the first derivative and set equal to zero to find the critical points. dV/dr = (π/3)[6r² - 6r] + (π/3)[-r(4r² - 1)/√(1 - r²) + 8r√(1 - r²)] = 0 6r(r² - 1) - r(4r² - 1)/√(1 - r²) + 8r√(1 - r²) = 0 6r(r² - 1)^(3/2) - r(4r² - 1) + 8r(1 - r²) = 0 6r(r² - 1)^(3/2) + r(-4r² + 1 + 8 - 8r²) = 0 6r(r² - 1)^(3/2) + 3r(3 - 4r²) = 0 2r(r² - 1)^(3/2) + r(3 - 4r²) = 0 Hmmm...

Answer 5

If this is not for practical purposes, i.e. wrapping. Cut paper into four segments (picture an orange cut into 1/4), you'll need to know the dimensions of the ball at numerous points on the sphere to calculate the taper from widest to narrowest point. Difficult.....

Answer 6

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Answer 7

Think of the outside of the segments of a Terry's orange.
You will have to cut the paper to that shape and stick it on.

Or look at an old map of the earth which is cut into a shape.

Answer 8

Given that the termination each end produces a width that must be replicated, determine a width that equates with the end circumferences such that the the sum of each equals the whole.
Measure the largest diameter, the middle, and draw an arc where both sides are equal. Now you have one segment.
Duplicate to match your chosen number and you are finished.

Answer 9

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Answer 10

Ok, I'm no mathematician but this seems obvious to me. The one place where you see the surface of a ball on a piece of paper is in an atlas of the world. That is the shape you want but you will have to enlarge it or shrink it to suit the size of your ball.