(Dis)proof by Construction

[unixronin]

In the New Scientist article "It's a wrap" (New Scientist, December 20-27, 2008, pp66-67), it is asserted that mathematicians have proven that in two dimensions, given six or less circular objects and an elastic wrapper, a "sausage" of the circles arranged in a straight line encloses the minimum possible area within the wrapper.

They may have "proven" it mathematically, but the assertion can be disproven by construction in a matter of minutes, without even having to do any calculations.

Consider, please, the following.

In Figure 1, we arrange six identical circles in a straight line.  We number the circles 1 through 6, left to right.  The minimum elastic perimeter that can be drawn around them can be generated by drawing tangents across the top and bottom of the row from circle 1 to circle 6.  The area enclosed in this perimeter is the sum of the area of the six circles, plus the sum of the ten trianguloid areas A through J enclosed between the two tangents and the circles 1 through 6.

(Scaled here to 50%; you'll probably want to open it in a separate tab or window, as it's a QCad line drawing and doesn't scale down well.)

Now, in Figure 2, we detach circles 4 through 6 and move them up and above circles 1 through 3, nesting them together so that the circles are in contact, allowing our elastic perimeter to reshape itself as we go.

The circles are, of course unchanged, as are the regions A, B, D, E, F, G, I and J.  Region C rotates around circle 4 and drops down to lie against circle 1, but does not change in area.  Likewise, region H rotates up around circle 3 to contact circle 6, without changing in area.

However, region A - which we have just noted has not changed - now overlaps circle 4 and regions C and I.  Region B similarly overlaps circle 5 and regions I and J.  Region I overlaps circle 2 and regions A and B.  Region J overlaps circle 3 and regions B and H.

In figure 3, the yellow region is that region in Figure 2 in which circular or trianguloid regions from Figure 1 overlap.  It is possible to see by examination that in each of the twelve sub-regions of the yellow-shaded region, exactly two regions from Figure 1 overlap.  There are no multiple overlaps to complicate the issue, and thus the total overlapping area is exactly the visible area of the yellow-shaded region.

Since none of the circles 1 through 6 nor the regions A through J have changed in size or area between Figure 1 and Figure 2, the total area enclosed within the elastic perimeter of figures 2 and 3 must necessarily be less than the total area enclosed within the minimum elastic perimeter of Figure 1 by the area of the shaded overlap region.

Referring to Figure 4, we can show by similar construction that a triangular packing of the six circles is also more efficient than the sausage packing.  In fact, it has exactly the same total overlap area - and thus is exactly as efficient - as the staggered-row packing of figures 2 and 3.  If we were to cut out the two shaded areas from Figure 4, we could assemble them to exactly form the shaded area of Figure 3.

It is immediately obvious, since we have just shown by construction that the area enclosed in Figures 2, 3 and 4 is less than the enclosed area in Figure 1, that any mathematical proof which says that Figure 1 represents the smallest possible enclosed area must necessarily be false.

Further, we could use the same method to show that if we arrange the six circles in neat vertically-aligned rows on a rectangular grid, the area thus enclosed is exactly equal to that enclosed in Figure 1.  So even if we do NOT nest the circles for the most efficient packing, the sausage arrangement is still no better than the rectangular-grid arrangement.

(Calculation of the exact amount, in grid units, of the yellow-shaded area is a fairly simple geometrical construction, and is left as an exercise for the reader.)

What is more, by further examining Figure 4 we can also show that for five circles, a staggered-row arrangement of three and two is more space-efficient than a sausage of five objects.  We can go on to demonstrate that a rhombus formation (staggered rows of two and two) is more space-efficient than a sausage for four circles or cylinders, and that a triangle of three is more space-efficient than a sausage of three (visible by examining Figure 4 again and considering only circles 4, 5 and 6 and regions D, E, H and I).

We can continue this process to show that the ONLY case for packing circles in two dimensions in which a "sausage" is equal to the most efficient packing is the two-circle case.  This is true only because a "sausage" of two circles is equivalent to the degenerate two-object case of hexagonal close packing.

In fact, for circular or cylindrical objects in a plane, a correctly chosen hexagonal close packing will ALWAYS (except in the degenerate 2-circle case) be more space-efficient than a "sausage" packing.  Further, for any N>2 circular objects factorisable into integer n and m, not only will a sausage never be the most efficient packing, but a sausage will never be more space-efficient than a n-by-m rectangular grid.

Comments

[dafydd.livejournal.com]

Did New Scientist share the proof itself, or give you means to contact the author of the original paper?

[unixronin.livejournal.com]

Nope.

[lebo-superman.livejournal.com]

Very nicely done. You should submit your challenge to the authors of the "proof".

[ilcylic.livejournal.com]

Man, I'm dying to hear the mathematician's explanation for this. I mean, what 5 year old doesn't know that hexagonal packing is the most efficient way to "stack" circles?

[ithildae.livejournal.com]

An algebraist or Calcist will not understand a geometric proof. The study of mathematics has become so focused and specialized that the researchers of the sine and cosine can no longer understand each other's work.