Tuesday, August 4, 2015

Big rocks, cannon balls, and permeable paving

There always is more than one way to tell a story. Hear are three that are related, but emphasize somewhat different things. The first is a time management story that appeared in the 1995 book First Things First by Stephen R. Covey, Roger Merrill, and Rebecca R. Merrill on pages 88 and 89. They didn’t say who it really came from though. 


“One of our associates shared this experience.

‘I attended a seminar once where the instructor was lecturing on time. At one point, he said, ‘Okay, it’s time for a quiz.’ He reached under the table and pulled out a wide-mouth gallon jar. He set it on the table next to a platter with some fist-sized rocks on it. ‘How many of these rocks do you think we can get in the jar?’ he asked.

After we made our guess, he said, ‘Okay. Let’s find out.’ He set one rock in the jar...then another...then another. I don’t remember how many he got in, but he got the jar full. Then he asked, ‘Is the jar full?”

Everybody looked at the the rocks and said, ‘Yes.’

Then he said, ‘Ahhh.’ He reached under the table and pulled out a bucket of gravel. Then he dumped some gravel in and shook the jar and the gravel went in all the little spaces left by the big rocks. Then he grinned and said once more, ‘Is the jar full?’

By this time we were on to him. ‘Probably not,’ we said.

‘Good!’ he replied. And he reached under the table and brought out a bucket of sand. He started dumping the sand in and it went in all the little spaces left by the rocks and the gravel. Once more he looked at us and said, ‘Is the jar full?’

‘No!’ we all roared.

He said, ‘Good!’ and he grabbed a pitcher of water and began to pour it in. He got something like a quart of water in that jar. Then he said, ‘Well, what’s the point?’

Somebody said, ‘Well, there are gaps, and if you really work at it, you can always fit more into your life.’

‘No,” he said, ‘that’s not the point. The point is this: If you hadn’t put those big rocks in first, would you ever have gotten any of them in?’ ” 



The practical situation which started this idealized math problem is the stacking of cannon balls, as shown above at the famous civil war Fort Sumpter. A quarter century before First Things First was published I took a college course on An Introduction to Materials Science. That course included a discussion of the crystal structures for metals. Metallic bonding involves free electrons, and the crystal structures for many elements can be described by considering the close packing of equal sized spheres. How much free space is there left?

There are two common close- packed crystal structures - face centered cubic (fcc) and hexagonal close packed (hcp). For fcc the layers alternate in an A, B, C pattern. For hcp they alternate in a simpler A, B pattern.  Aluminum, copper, gold, nickel, platinum, and silver are face centered cubic. Cadmium, magnesium, and zinc are hexagonal close packed. For both structures the spheres fill about 74% of space (roughly 3/4), and the other 1/4 is empty. (Some rare earth elements instead have a double hexagonal close packed [dhcp] structure which is A, B, A, C, A). When you instead just randomly close pack spheres, they only fill 65% of space (roughly 2/3) and the other 1/3 is empty.

If you try to tell the BIG ROCKS story to engineers (chemical, mechanical, or materials) they may well sneer So What? and quote you those percentages.


Paving large areas like parking lots creates problems when dealing with runoff of rain water from storms. One solution is porous or permeable paving, which lets the water trickle through to the underlying soil and be absorbed normally. You can make pervious concrete by bonding together coarse gravel (rather than rocks, gravel, and sand), with Portland cement.   

An image of cannon balls came from the Library of Congress, and the permeable paver demo  image came from J. J. Harrison at Wikimedia Commons.

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