Rectangular vs. Hexagonal Arrangements

Traditionally, when one places similar objects together in an orderly manner, the items get placed in distinct columns and rows. This is called a 'rectangular' arrangement.

	×	×	×	×
	×	×	×	×
	×	×	×	×

    An alternate approach would be to use a 'hexagonal' arrangement.

	×		×		×
		×		×		×
	×		×		×
		×		×		×

    Besides being less familiar, a hexagonal arrangement seems to be disadvantageous. It has either columns or rows, but not both. When items are arranged in rectangular fashion, multiplying the number of columns and rows can readily attain their quantity, but that's not so easily done with items arranged hexagonally. Also, there's the obvious indents, or 'holes', that would appear to be wasteful of space.

    So, are there any advantages to a hexagonal arrangement over rectangular? It would seem that nature thinks so; bees and wasps arrange their hive units hexagonally.

    One possible advantage of using a hexagonal arrangement, rather than a rectangular one, is in the distance between adjacent objects. In a rectangular arrangement, the items must be further away on the diagonal than horizontally or vertically. With a hexagonal arrangement, a collection of identical objects can easily be placed with all adjacent items equidistant from each other, not only horizontally or vertically but on the diagonals as well. Arranging them at the corners of equilateral triangles accomplishes this.

    It also appears that, by placing objects hexagonally, their arrangement might also have a rectangular boundary that is closer to being a square, as evidenced by the above two designs, as well as the following:



Comparing one thru ten objects arranged rectangularly and hexagonally:

			RECTANGULAR		HEXAGONAL
	1		×		×				Trivial!
	2		× ×		× ×				No difference so far.
	3		× × ×		× × ×				Now they're different.
	4		× × × ×		× × × ×				Okay, the only real difference here is a rotation, although the hexagonal arrangement does use a larger box.
	5		× × × × ×		× × × × ×				This hexagonal arrangement is known as a 'quincunx'.
	6		× × × × × ×		× × × × × ×				rectangle vs. triangle
	7		× × × × × × ×		× × × × × × ×				single row vs. hexagon
	8		× × × × × × × ×		× × × × × × × ×				Hexagonal is more suggestive of a square than rectangular.
	9		× × × × × × × × ×		× × × × × × × × ×				This rectangular arrangement actually forms a square.
	10		× × × × × × × × × ×		× × × × × × × × × ×	or	× × × × × × × × × ×		The triangle-shape makes for a nice alternative.

    It seems that, with a few exceptions, a hexagonal arrangement occupies a setting that more closely suggests a square than a rectangular arrangement does. However, when the quantity is greater than 2, the hexagonal placement doesn't actually fill the enclosing box.




    Many of us Americans regularly see a particular example of a hexagonal arrangement, probably without even paying much attention to it:


    Okay, the 'stars' aren't laid out on the corners of equilateral triangles and the hexagons are somewhat flattened, but the hexagonal principle is being used here. Imagine how this would look if they were arranged rectangularly: the closest we could get to square would be 5 ×10; that would look quite different.

Notice that the number of columns between the two rectangular grids differ by  no more than one. The  same is true for the rows   as well. This condition is important in the design of   a hexagonal arrangement. In the first example, they both increase. The number of columns and/or rows could also remain the same, or one could increase while the other decreases.

    A hexagonal arrangement can be viewed as consisting of two overlapping rectangular structures. In the above illustration, we have 50 'stars' in a blue field. 50 is the sum of 20 and 30. 20 is the product of 4 and 5; 30 is the product of 5 and 6. Thus, the design can be broken down into two rectangular arrangements, 4×5 and 5×6.

4×5 → 20 (four rows, five columns)
5×6 → 30 (five rows, six columns)
            50



    Suppose we were to add two more states to the union - oh, how about Guam and Puerto Rico. We would need two more 'stars' to make a total of 52. We proceed as follows:

1. Factor 52: 52=2(2)(13)
2. Noting that '13' is the largest factor, we split it into two numbers with a difference of 1, namely 6 and 7.
3. The other two factors are within 1 of each other - they're the same - so, we're good to go.
4. So far, we have 2(2)(6+7). Using the distributive property, we obtain 2(2)(6)+2(2)(7), which equals 4(6)+4(7).
5. We need to superimpose a 4×6 rectangular arrangement within a 4×7 rectangular arrangement. This gives us alternating rows of six and seven 'stars' each.
   

    Incidentally, this approach does not always generate a design to use. But as it has in this case, we are now able to arrange the 'stars'.




    So, this is not all that close to being a square. But, it's much closer than if we had gone with a straight arrangement of thirteen columns and four rows.





    What about the density? That is, is one of these arrangement schemes more efficient at maximizing the number items per unit of area? Actually, the hexagonal pattern is more space-efficient.
    As an example, suppose someone wants to arrange bottles or cans in a carton. Let's quickly examine the size requirements for 25 items arranged rectangularly, and then consider arranging the same 25 items hexagonally. Our scale will be the diameter of one bottle or can as the unit of length.

    The rectangular arrangement of 25 items can be accomplished by constructing an arrangement of five columns and five rows. It requires 25 square units of space. (Actually, slightly more, for practical reasons, but we can ignore that here.) So, that's 1 square unit per item.

    For the 25 items arranged hexagonally, we use a 3×3 rectangular arrangement superimposed into a 4×4.


    From this layout, we easily see that the shorter dimension is 4 units.
    The length of the longer dimension, however, is more difficult to determine. The number of units in this direction is not a whole number, so we it makes sense to go to the diagonal, which has a length of 7 units. Problem is, the diagonals don't go all the way into the corners. For this longer side, a somewhat novel approach is at hand - we'll measure distances between the midpoints of the items.
    The length from the centers of the items, in opposite corners, is 6 units. If we look at the 6 as the hypotenuse of a triangle, we find a convenient right triangle with another side having a length of 3 units, measuring from the centers of the items. From that, we can use the Pythagorean Theorem to find the 'unknown' side, which is 33. This still leaves us ½ unit away from the actual edge on both ends, so the full length is 33+1, or approximately 6.196 units.
    We haven't achieved a whole lot of space savings, but we have made a more efficient use of space, nonetheless.

    Now, suppose we have a manufacturer wanting to fill cartons with a product. They presumably would want to stack several of these cartons on a pallet. This 4×6.196 ratio is rather awkward to deal with; a simpler ratio is preferable. The ratio of 4×6.196 is the same as 2×3.098, so we'll seek 2×3.
    To change the height and width ratio, we'll need to adjust the angle of the diagonals. The math gets somewhat involved, but the new angle is

Arcsin(

2+3467 78

)

or approximately 58.96°.

    This gives the construct the dimensions of approximately 4.094 by 6.141 units. So, we have achieved the 2×3 ratio.
    Unfortunately, the amount of space requirement increases to approximately 1.0056 square units per item, so now we're less efficient on space. Evidently, it's best to stick with rectangular after all.