lew@ihuxr.UUCP (08/15/83)
It is often said that a sphere has the smallest ratio of surface area to volume among possible shapes. (See net.sf-lovers) ... Well, here's proof to the contrary! For the sphere: ratio = ( 4*pi*r^2 ) / ( 4/3*pi*r^3 ) = 3/r For the cylinder circumscribed on the sphere: ratio = (height * circumference + 2 * base) / ( height * base ) = ( 2*r * 2*pi*r + 2 * pi*r^2 ) / (2*r * pi*r^2) = 3/r The same as for the sphere!!! Lew Mammel, Jr. ihuxr!lew
kwmc@hou5d.UUCP (08/16/83)
The following 'proof' appeared in net.math recently ....
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It is often said that a sphere has the smallest ratio of surface area to
volume among possible shapes. (See net.sf-lovers)
... Well, here's proof to the contrary!
For the sphere: ratio = ( 4*pi*r^2 ) / ( 4/3*pi*r^3 ) = 3/r
For the cylinder circumscribed on the sphere:
ratio = (height * circumference + 2 * base) / ( height * base ) =
( 2*r * 2*pi*r + 2 * pi*r^2 ) / (2*r * pi*r^2) = 3/r
The same as for the sphere!!!
Lew Mammel, Jr. ihuxr!lew
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well it seems very tempting to believe it at first UNTIL you realise
that the ratio is dependant on 'r'. The shape of the cylinder could
be molded into a sphere ( of the same volume ) but in which the value
of 'r' is greater, and thus the ratio 3/r IS smaller as it should be.
Remember, for the same volume the 'r's are different, also 3/r is NOT
a constant it varies with the value of 'r'.
Ken Cochran hou5d!kwmcjames@umcp-cs.UUCP (08/16/83)
Okay, the ratios are the same, but they are both functions of r! Isn't the original point 'How much can x surface units enclose?' or 'How many surface units are necessary to enclose y volume units?' I mean, the examples you used have unequal surface areas and unequal volumes...how about a comparison of equal surfaced figures and/or equal volumed figures?
ecn-ec:ecn-pc:ecn-ed:vu@pur-ee.UUCP (08/16/83)
NO ! The ratio is *not* the same for sphere and cylinder. Saying that the ratio is largest for the sphere means that given volume V, then the surface area will be largest for the sphere **among objects of volume V**. Now the cylinder with base radius r (same as sphere's radius) and height 2r is way BIGGER than the sphere (i.e. not the same V) and thus not comparable. Remark that for a fixed r (e.g. r=1) the ratio for a sphere volume (4/3)pi is 3, and a cylinder will have ratio = 3 only if it has volume 2 pi = (6/3) pi . Hao-Nhien Vu (pur-ee!vu)
john@hp-pcd.UUCP (John Eaton) (08/18/83)
#R:ihuxr:-55700:hp-pcd:6100001:000:568 hp-pcd!john Aug 17 09:13:00 1983 Well You Managed To Show That A Sphere With Radius R Has The Same Surface Area To Volume Ratio As A Cylinder Of Radius R, But That Wasn'T The Problem. The Problem Is How To Maximize The Enclosed Volume Of A Shape Given That The Surface Area Is Constant (Or How To Minimize The Surface Area Given That The Volume Is Constant). Assume That You only Have 36 * Pi Of Material To Make A Shape: If You Build A Shere Then R = 3 And That Surface Area To Vol Ratio Is 1 If You Build A Cylinder Then R = 2.45 And Its Ratio Is 1.22 John Eaton hplabs!hp-pcd!john
darrelj@sdcrdcf.UUCP (Darrel VanBuer) (08/18/83)
Lew Mammel Jr claims that a cylinder inscribed on a sphere has the same ratio of surface area to volume, namely 3/R, but there is one flaw in the analysis, for the same r, the cylinder is 50% larger in both area and volume; if you scale the cylinder down to equal volume, it has an area which is the cube root of 1.5 times larger (1.146 by my old slide rule). The sphere wins by a nose!
bill@utastro.UUCP (10/03/83)
med to say) until I found that you get the same ratio for a cube: Let a cube be circumscribed about the same sphere, so that its sides have length 2*r. Then the ratio of surface area to volume is ratio = (6* area of 1 side)/volume = (6 * 4 * r^2)/(8*r^3) = 3/r!!! When I calculated this I saw the fallacy in Lew's argument. The quantity he calculates is not dimensionless, so it is not a legitimate ratio for comparison. What we ought to do is to compare the surface areas of a sphere and a cylinder *whose volumes are the same*. Then we can tell which one can be manufactured with the least amount of "skin". Thus: Let the radius of the sphere be r, and the radius of the sphere inscribed within a cylinder of the same volume be s. Then Volume of sphere = volume of cylinder (4/3)*pi*r^3 = 2*pi*s^3 s = (2/3)^(1/3)*r Now compare the surface areas of the two solids: Area of cylinder/area of sphere = 6*pi*s^2/(4*pi*r^2) = (3/2)^(1/3); So the surface area of the cylinder is about 1.14 times that of a sphere of the same volume. I think that Lew's "ratio" is very cute, and a little curious. It's a lovely swindle. Where did you get it, Lew? Bill Jefferys 8-% Astronomy Dept, University of Texas, Austin TX 78712 (Snail) ihnp4!kpno!utastro!bill (uucp) utastro!bill@utexas-11 (ARPA)