Barry Birkey has been editing a series of articles on the question of honeycomb cell size for his site at http://www.beesource.com.
Ever since I first received some of the articles by email, I've been looking through them and had great trouble understanding them, largely because the writers and scientists wander back and forth between several measurement schemes. Is this a scandal or merely a misunderstanding? It's hard to tell at this point.
This page has three purposes:
The table shown as a GIF below is an attempt to bring all the measurement schemes together on one reference sheet.
If you are going to read and try to understand the articles, I recommend that you download the spreadsheet or the GIF and print it, maybe in several copies, because things get pretty confusing and you'll want to highlight and mark up the sheets.
This chart lists the various methods of scientifically measuring cells and gives a conversion to each other scheme by simply scanning across the table. It should prove invaluable for trying to understand Dee Lusby's articles, or the Grout article. I have also written an article about measuring which follows below.
The following spreadsheet underwent minor changes Nov 11/03 to correct minor computational errors.
I hope everyone will read this article, because I have a hunch that a lot of discussion in the next weeks will depend on understanding what is under scrutiny here. Some of these points are not at all obvious and I may even have some of them wrong, so please read carefully, get a sample of comb or foundation and if you find a flaw in what I am saying, please correct me.
For background on the discussion that revolves around natural cell size, its implications regarding bee health and comb measurement, go to http://www.beesource.com/pov/lusby/
The thing that has tripped me up in trying to understand the whole cell size question and the idea that North American comb foundation forces bees to make cells that are too large for the bees to function properly is this: How does one best measure the cells that honey bees construct? As I read the literature, a suspicion has been growing growing that not everyone is using the same assumptions. I can see reasons for misunderstandings that could explain what is starting to look like a either blunder on a huge scale or a widespread misunderstanding -- or both.
I think everyone who wishes to discuss the cell size question needs to understand the basic measurement methods and the reasons that several intelligent, educated people might look at the same sample of comb and come up with different measurements. As we will see, this may have happened historically, and may have resulted in damage to beekeeping in developed countries. How serious this damage, if any, may have been remains to be proven. The whole matter revolves around how to accurately measure cells, calculate density per measured area, and communicate the conclusions. Unless everyone is using the same assumptions, methods, and terminology, there are bound to be errors in any conclusions reached.
The ramifications -- both positive and negative -- of using artificially sized cells is something that will be thrashed out over the next months, but no one can participate intelligently without understanding the underlying measuring methods, and they can be very confusing. I have spent several full days on the matter and am still working on it.
There are a number of methods of measuring cells. We are not concerned with depth at this point -- merely the dimensions and the areas of each hexagonal cell opening that one sees when looking perpendicularly at the surface of a comb -- disregarding any coping that may be present.
Looking into an open cell, one can see that placing a ruler across a cell presents a quandary: at what angle should one measure? There are three ways that result in the smallest possible measure, and there are three angles that would result in a maximum number, and there are many other angles that would get results in between.
Although honey bee cells are an approximation to round and are sometimes considered circular, each one is, in fact, a close approximation to an ideal hexagon: the end projection of each cell has six more or less equal and straight sides joined at equal angles and is symmetrical around six possible axes. Cells on a comb face only line up coherently for easy counting in straight rows in three of those six directions every 120 degrees. Since these directions are not orthogonal, the cells do not lend themselves to the simple square measurement which everyone can do easily, and which -- by definition -- requires square corners. The various methods of measurement and calculations of area are simple and obvious to a mathematician or an engineer, but not to most laypersons. Hexagon references: Hall of Hexagons Hexagon Facts The Area of a Hexagon
The smallness of the cells, the fact that they are not always exactly the same, and the thickness of the cell walls complicate measurement. As with all things relating to bees, we find that we are so large in comparison to them and their constructions that many of us must don glasses or a loupe to see well enough to proceed very far.
As long as we measure and calculate area for individual cells, we are not faced with the tricky problems that face us as soon as we try to figure out more than one cell at a time or calculate surface area and cells per area. Calculating the area of a single cell, once the dimensions are established, is simple high school math, since the cell can be reduced to six identical triangles. Unfortunately the discussion seldom deals with single cells.
Since we are interested in the average measurement, we must measure many cells, and divide to get the characteristics of a typical cell on that comb. We must keep in mind, though that cells naturally vary, from a little to a lot, over the face of any comb, although in many cases the individual differences may be almost imperceptible to us. For the above reasons, we normally measure across ten cells, then divide by ten to get the individual cell average size. Usually metric units are used for comb measurement.
Linear measurement: If we are going to measure, the obvious and easy and unmistakable way is measure along a row of contiguous cells, either in any of the three angles that give the minimum measure or any of the three angles that give the maximum number. Measuring the long way on the cells may be possible, but if you try it, you will see it is quite confusing because the cells stagger right and left of the line of measurement..
Measuring and comparing cells per inch or per centimeter, or stating the width of a cell as described above, has seldom caused any misunderstanding. If we tell ten people to take a ruler and measure across ten cells the shortest possible way and then give the answer, all ten are going to get it right most of the time.
Area measurement: BUT, if we ask ten people to tell how many cells there are per square inch or centimeter or square decimeter (the preferred measure) on the same comb, three will get it right, three will get it wrong, and four will tell us they haven't got much of a clue how to do it. It is not a simple problem for most people. 'Most people' includes some non-mathematical scientists, I suspect. This is where the confusion sets in.
Why measure area and cells per square decimeter or square inch, anyhow?
Such calculations are *not* necessary for simple comparison of comb from different bees and different areas of the world. A simple linear measurement made by counting ten cells in a row and measuring the distance carefully, then dividing by ten results in a useful, easily understood number, which for A.m.m. should be in the 5.2 mm range. Repeating and averaging the results from different angles (120 degrees apart) and positions on the comb will give a very meaningful figure. When the variation of size from average is also stated, a complete picture can be communicated.
A need for calculation of area and cell density comes up when one is trying to determine how many cells are in a specific area, such as on a frame of comb. Most of us when faced with this problem, realize that it is a non-trivial mathematical problem for us and just measure off the area, then count. Assuming we choose a large area, like the entire surface of a frame, what we decide to do with the part cells along the edges won't have a huge material effect on our count -- especially if these cells are going to occur along a top, bottom or side bar. The number we get is pretty accurate and satisfactory for our purposes. And we are done PDQ.
For scientists who want to discuss things abstractly, and who are considering brood density on a frame, this is not good enough. They wish to be able to talk about cells per square decimeter without measuring off a square 10 cm by 10 cm and counting the cells. They don't want to count because that is not very elegant, and because when one tries to do a count, he/she is faced with the same problem of how to estimate the areas of the cells that are on the boundary and not completely inside -- or completely outside -- the area under consideration. Scientists don't like to estimate if they can measure or extrapolate or use a table or formula.
A rhombus, which is like a square in that it has four equal sides but different from a square in that it has non-square corners, can be used, but the formula for a rhombus (B x H) must be used where B is the length of one side, and H is the orthogonal distance between that side and the opposite side. In the case of a square, B=H. However, on comb, merely multiplying two adjacent sides together as in a square leads to an error, since each corner is either 60 or 120 degrees, not 90 degrees as in a square.
Since we know the base angle of a rhombus drawn on honey bee comb to be 60 degrees, we can just use the sine of 60 degrees which is known to be about 0.86603, and multiply that factor times the product of two adjacent sides of the rhombus, to determine area if we like. What we must remember is that if we just multiply the two sides together as in a square -- without also multiplying by 0.866 -- we will overstate the area by about 13 -1/3 percent and make small but serious errors in any conclusions we reach using the data.
Using the 60 degree sine factor, 0.866, we can now just put pins into the centres of 4 cells on the corners of the rhombus and measure the sides, which all should be almost exactly the same. If we choose to put a pin in a cell, then count 10 cells in a straight line, then insert another pin, we should have 9 complete cells between, and -- since the pins are in the middle -- half a cell on each end for a total of ten cells between the pins.
If then we go at 60 degrees to the line between them and place pins at the other two corners, we should have enclosed exactly 100 cells. An average of the measurements in centimeters between the four pins along the sides of the rhombus should give us the number for B and, when multiplied by 0.866, for H. This gives us the square centimeters per 100 cells. If the discussion includes the cells on both sides of the comb, then the number must be doubled.
In conclusion: Linear measure is a simple and reliable and non-confusing way to measure and compare comb, however the literature written by scientists and technicians uses other, area-based considerations and can be very confusing to laypersons.
Nonetheless, laypeople can follow the literature using the above chart to convert the arcane area measurements into easily understood linear measure.
-- April 1, 2000 (no foolin')
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