Khun_Kong Posted January 9, 2005 Report Share Posted January 9, 2005 OK kids- 50 liters = 50,000 cubic centimeters. 40 mm sphere = 4 cm volume of 4 cm sphere = ~33.5 cc (cubic centimeters) volume of 4 cm cube = 64 cc Because spheres will not stack as closely as cubes, you lose that bit of space around the sphere that would be occupied by the cube. In this case, you lose (1-(33.5/64))*100% = ~47.7% 47.7% of 50,000 cc= ~23,828 cc, leaving you with ~26,171 cc 26,171 cc / 33.5 cc per sphere = about 781 balls, with no squishing. Class is over. Those whose heads have exploded are welcome to use the washroom. BTW, ND- partial credit for approaching the correct answer, but zero points for reasoning. Volume of 4 cm cube and 4 cm sphere are nowhere close to each other. Link to comment Share on other sites More sharing options...
sayjann Posted January 9, 2005 Report Share Posted January 9, 2005 i'm just wondering why in your drunk state that you would be wanting to drop black balls into a temperature controlled box without them being squashed?. Miss RB banished you to the spare room again?...... Link to comment Share on other sites More sharing options...
Neo Posted January 9, 2005 Report Share Posted January 9, 2005 You could fit 781 balls *IF* the 50L box shape was such that the floor fit one ball and they were all stacked vertically (4cm x 4cm x 3125cm which is one strange looking box--try getting a 31 meter long box on a plane!). Consider a square floor of 36cm x 36cm that fit them perfectly would be 38.58cm high to be 50L so it would only fit 729 balls since the top 2.58cm could not be used. Well consider this. Say you lay down the first layer of balls, but instead of stacking the second layer, you drop each ball in the gaps. Might that allow more? Something to think about Link to comment Share on other sites More sharing options...
iuytrede Posted January 9, 2005 Report Share Posted January 9, 2005 a bit over 1000 as spheres will be packed more tightly than their surrounding cubes would be packed (factor is about 0,86) Link to comment Share on other sites More sharing options...
Khun_Kong Posted January 9, 2005 Report Share Posted January 9, 2005 Neo said:You could fit 781 balls *IF* the 50L box shape was such that the floor fit one ball and they were all stacked vertically (4cm x 4cm x 3125cm which is one strange looking box--try getting a 31 meter long box on a plane!). Consider a square floor of 36cm x 36cm that fit them perfectly would be 38.58cm high to be 50L so it would only fit 729 balls since the top 2.58cm could not be used. Well consider this. Say you lay down the first layer of balls, but instead of stacking the second layer, you drop each ball in the gaps. Might that allow more? Something to think about Good points! As this board frquently demonstrates: there are no new questions! Sure enough. Googled "packing spheres maximum"... Foundations of Mathematics > Mathematical Problems > Unsolved Problems v Geometry > Computational Geometry > Packing Problems v Geometry > Solid Geometry > Spheres v Number Theory > Constants > Geometric Constants v Math Contributors > Pegg v Sphere Packing Define the packing density of a packing of spheres to be the fraction of a volume filled by the spheres. In three dimensions, there are three periodic packings for identical spheres: cubic lattice, face-centered cubic lattice, and hexagonal lattice. It was hypothesized by Kepler Eric Weisstein's World of Biography in 1611 that close packing (cubic or hexagonal, which have equivalent packing densities) is the densest possible, and this assertion is known as the Kepler conjecture. The problem of finding the densest packing of spheres (not necessarily periodic) is therefore known as the Kepler problem, where <complicated formula> (Sloane's A093825; Steinhaus 1999, p. 202; Wells 1986, p. 29; Wells 1991, p. 237). Gauss Eric Weisstein's World of Biography (1831) managed to prove that the face-centered cubic is the densest lattice packing in three dimensions (Conway and Sloane 1993, p. 9), but the general conjecture remained open for many decades. While the Kepler conjecture is intuitively obvious, the proof remained surprisingly elusive. Rogers (1958), a well-known researcher on the problem, remarked that "many mathematicians believe, and all physicists know" that the actual answer is 74.048% (Conway and Sloane 1993, p. 3). For packings in three dimensions, C. A. Rogers (1958) showed that the maximum possible packing density satisfies <another complicated formula> Packing % useable space Reference loosest possible 0.0555 Gardner (1966) tetrahedral lattice 0.3401 Hilbert (1999, pp. 48-50) cubic lattice 0.5236 hexagonal lattice 0.6046 random 0.6400 Jaeger and Nagel (1992) cubic close packing 0.7405 Steinhaus (1999), hexagonal close packing 0.7405 Steinhaus (1999, p. 202), Link to comment Share on other sites More sharing options...
Nervous_Dog Posted January 9, 2005 Report Share Posted January 9, 2005 Being close together wont squash them, now if they are fragile you need to add individual packing, as Danish said, it depends on the shape of a box. A 50litre container, that s only 30 mm wide will fit none in it! A cube shape is the max volume DOG Link to comment Share on other sites More sharing options...
Nervous_Dog Posted January 9, 2005 Report Share Posted January 9, 2005 If you put a sphere into a box the best shape is a cube. However, as someelse pointed out, speres not indiviually boxed will fit closer together DOG Link to comment Share on other sites More sharing options...
.. Posted January 9, 2005 Report Share Posted January 9, 2005 Khun_Kong said:the maximum possible packing density satisfies <another complicated formula> Packing % useable space Reference loosest possible 0.0555 Gardner (1966) tetrahedral lattice 0.3401 Hilbert (1999, pp. 48-50) cubic lattice 0.5236 hexagonal lattice 0.6046 random 0.6400 Jaeger and Nagel (1992) cubic close packing 0.7405 Steinhaus (1999), hexagonal close packing 0.7405 Steinhaus (1999, p. 202), Hence, even in my hung over state, I was quite close. I came up with the 781 easily, but used a SWAG to come up with a correction factor needed due to shape. I couldn't be arsed to do the research -- I was much more interested in moaning and avoiding puking (yes, the evening went seriously downhill after I saw you and the Mrs). Back on topic, if I understand the above correctly, then 781 * 0.7405 = 578. Where do I collect my prize? Cheers, S "super engineer" D Link to comment Share on other sites More sharing options...
Redbaron Posted January 9, 2005 Author Report Share Posted January 9, 2005 Thanks KK, and ND. I've ordered 700... I'll bring a photo in February. I knew I could rely on the board bar for this one. cheers Link to comment Share on other sites More sharing options...
Redbaron Posted January 9, 2005 Author Report Share Posted January 9, 2005 i'm just wondering why in your drunk state that you would be wanting to drop black balls into a temperature controlled box without them being squashed?. I won't be drunk when I drop them in... I hope. It's for an aquarium filter. Basically I need to fill, or almost fill it with balls BUT if I order them by the litre from the shop, it costs around 300-400THB per litre. I'm ordering them direct from the manufacturer who supply them at 4 or 5 baht each. Bit of a difference. Most people only buy a few litres of them, so it's often not worth ordering them direct (shipping etc). Miss RB banished you to the spare room again?...... Not yet, I'd like to have it all ready when/if she returns in Feb. Link to comment Share on other sites More sharing options...
Recommended Posts
Archived
This topic is now archived and is closed to further replies.