Copyright ©2008 by Paul Niquette. All rights reserved.
Science has a modern definition -- "reverse engineering nature."
-- Paul Niquette Sophisticated: The Magazine 1996
asked in the puzzle, for
various sizes of cylindrical
containers, "Is a
height-to-diameter ratio of 1.5
especially economical?" The implication is that,
to production of living things by natural selection,
creations by intelligent
design are not necessarily constrained by the Square-Cube
In other words, size does not influence shape. To
answer the question,
solvers of the Tin-Can Mystery must
out an analysis that somehow separates size from
shape. Let's get
A cylindrical container with diameter D inches and height H inches requires a surface area...
D H + 2 (/4)
2 square inches of
material to accommodate a
To facilitate the analysis, let us define a dimensionless ratio...
 C =
H / D, so
that substitution of H = C D into
Substituting  into , produces the following relationship:
2/3 (C +
-2/3 in which we
identify two factors...
Observe that the size factor FSIZE
imposes the Square-Cube Law as follows:
Meanwhile, the shape factor FSHAPE
plays an independent role...
ur economic objective calls for the minimum value of A; however, FSIZE is fixed by the assumed value of V in . Thus, for the solution to Tin-Can Mystery puzzle, we must ascertain the value of C that produces the minimum value of FSHAPE and compare it to 1.5. Sophisticated solvers will remember their elementary Differential Calculus and proceed as follows:
+ 1/2) C
-2/3] / dC = (1/3)(C
-2/3 - C
-5/3) = 0,
which simplifies to...
That will not surprise sophisticated solvers who know that for any volume, a sphere has the smallest surface area. A tin-can with C = 1.0 mandates that H = D, which is about as close to a spherical shape as a cylinder can get. For maximizing volumetric efficiency, designers of internal combustion engines exploit H = D (stroke = bore) in what they called the "square cylinder," by which swept area and therefore heat-loss is minimized for a given displacement.
Substituting  into , we obtain the minimum surface area for a cylindrical container required to accommodate a given volume...
 AMINIMUM =1/3(4V) 2/3 (1.0 + 1/2) 1.0 -2/3 = 1/3 (4V) 2/3 (11/2).
Any value of C > 1.0 or C < 1.0 will require more than the minimum amount of material...
 AEXCESS = 1/3(4V) 2/3 [(C + 1/2) C -2/3 - 11/2], as shown as a percentage of AMINIMUM in the graph below...
hether the product of natural selection or intelligent design, size is not the only determinant of shape. In the Tin-Can Mystery puzzle, the oil barrel accommodates more than 500 times the volume of a soup can but has the same shape (C = 1.5). According to the graph, both will take about 2% excess material over the minimum for an enclosed cylinder.
Cupboards and refrigerators, boxes and box-cars, trailers and trucks, cartons and crates, warehouses and shipping containers -- all are rectangular in both elevation and planform, none are themselves cylindrical. Cylinders are indeed round pegs in square holes. It is easy to show that every can-in-a-box takes up 21.46% more volume than it holds (1- /4). Not much can be done about that. Intelligent design of the box or carton may be another story.
Let a box enclose cylindrical containers (cans) having dimensions D and C D. Assume the cans are arranged in m by n rows. The internal surface area of the box to accommodate the cans would be given by...
 ACANS = 2 D2 [C (m + n) + m n].
Of course, the box will require more material than that for structure and strength.
ou might enjoy reverse engineering the following real-life case: A take-home cardboard carton is designed to hold 12 aluminum softdrink cans in two rows of 6. Each takes up a volume inside the carton V = C D3. Unfold the carton and you will find a narrow flap for gluing along one of the long edges (m D), plus 100% glued overlaps on both ends (2 n D H).
Try substituting these numbers into . You will find the value of C that requires the least amount of cardboard is 1.9. Coincidence?
The following explanation was received in July 2008 from Myles Buckley:
The 4% excess material is a function of burst strength vs crush load resistance, according to Duncan Hosford in "The Aluminum Beverage Can" in the September 1994 edition of Scientific American Magazine. Hosford described the efforts of Coca Cola and Pepsi Co. to minimize the need for aluminium, which imposes by far the single highest cost of production for their canned soft drinks.
The pop-can is very thin on its side-wall. More aluminum by mass is found in the top ridges (not to cut the drinker's lips) and more again at the base. The can has three "pieces." A slug of aluminum is extruded to form the base and sides; a disk forms the top (including the "button" that holds the pull tab); the pull-tab itself is the third piece.