1. Background
About a week ago I tried to post into another thread on the topic of knot density, but Chrome froze and I lost what I had written. I decided that was probably for the best and resolved to make more productive use of my time. It was a short-lived resolution.
A day or two later I joined a discussion in which the description of a particular knot’s density was at issue. Part of the problem – probably the greatest part – had to do with attaching different meanings to the term “density” as well as to adjectives (e.g., “medium”
used in relation to it. I had previously described a brush as being of “medium density”. Another member thought it was “not dense at all.” Although he made that statement with what I took to be complete sincerity, it struck me as literally devoid of meaning. A knot cannot be entirely lacking in density: absolute vacuums don’t exist in nature. Of course, he didn’t mean to claim the knot was a vacuum. What he meant was that he regarded the knot as being relatively sparse. But that begged the questions: based on what frame of reference and to what degree? And thinking about that brought me back to the thread linked above. After reading through it, I made a run at coming up with a method by which to approximately measure knot density. What resulted is flawed in multiple respects. But I think it’s sufficiently interesting to share. I decided to start a new thread because this is going to be long, and I think it would serve best have a thread dedicated to this approach and what it yielded.2. Terminology: what is knot density?
First, it’s important to clarify that in discussing knot density we generally don’t use the term with reference to mass density, i.e., mass per unit volume. I would also argue that numeric density, or per unit volume, isn’t the best (or at least not an independently sufficient) measure of knot density. If two knots possess exactly the same base diameter, height, shape, and number of hairs, but Knot A comprises individual hairs with an average thickness 50% greater than Knot B, it stands to reason that Knot A will likely be experienced as having greater density as between the two. So I think it’s the aggregate volume of individual hairs in a particular knot in relation to that knot’s overall volume (i.e., the space it occupies as a whole) that constitutes what we generally think of as knot density. Besides, counting badger hairs is a pain in the ***.
3. Theoretical principles
In short, I tried to apply Archimedes' observation that for any submerged object, the volume of the submerged portion equals the volume of fluid it displaces.
I reasoned that if a dry knot were fully immersed in a water-filled container, a volume of water would be displaced from the container which should equal a) the aggregate volume of all the individual immersed hairs composing the knot plus b) the volume of any air that remained trapped in the knot. My idea was to measure how much water each of several representative knots separately immersed in water would displace and evaluate whether such measurements might somehow be meaningfully compared.
4. Initial trials
Immersion in tap water and measurement of overflow
I first tried placing a glass full of tap water inside a flat-bottomed bowl that was resting on a lab balance. Then I individually submerged the knots, which displaced water from the glass into the bowl. After that I removed the glass along with the brush and calculated the weight of the displaced water by subtracting the weight of the bowl. (It is convenient that 1 gram of water equals 1 milliliter, which makes it easy to convert water-weight measurements to volume measurements.)
The range of values that resulted from measuring displaced water remaining in the bowl after removal of the glass was 18-21 grams. Amazingly, I obtained measurements of 20 grams each of the three times I immersed my faux-bone-handled, M&F group-buy prototype with a fan-shaped knot. Despite the apparent consistency, however, a problem I predictably ran into from the start related to water surface-tension. As I immersed knots, water level in the glass rose above the rim almost immediately and didn’t begin to spill over the top until it had reached what I estimated to be about 3 mm higher than the rim (I didn’t measure it). Consequently, although I did the best I could to compensate for the effect, I wasn't able to get satisfactorily accurate results.
Immersion in soap-water solution and measurement of overflow
I next tried dissolving a small amount of dish soap in the water. Soap typically reduces the surface tension of water, which I thought might reduce or eliminate the problem of displaced water rising above the rim rather than immediately spilling over the side of the glass and into the bowl.
The result of that modification was surprising to me. There was somewhat less apparent surface tension such that displaced water didn't rise as high above the rim of the glass before spilling over and into the bowl. However, the soapy water didn’t rise as fast initially and it yielded less displaced volume. The amounts that overflowed into the bowl were closer to 10 grams, which was somewhat mystifying. My assumption is that soapy water was absorbed into the knots faster and to a greater extent than pure tap water.
5. Issues/concerns
The initial trials described above left me dissatisfied with simple immersion into a full container and capture of overflow as a method for measuring displacement.
First, there was the problem of water surface-tension, for which using soapy water didn’t appear to be a good solution.
Second, I suspect a significant volume of air might remain trapped in submerged knots, such that what is displaced into the bowl substantially exceeds the actual volume of the hair in the knot alone (i.e., it includes space taken up by the trapped air).
Third, some volume of water remains in a knot after its immersion notwithstanding whatever amount is squeezed out and back into the glass.
6. Modified approach
After some further thought, I formulated a different approach to try. Instead immersing a knot and measuring positive displacement based on spilled overflow, I would measure negative displacement produced by removal of an immersed knot from a container filled to a known volume. There’s undoubtedly a better way to go about this than the one I came up with, but I made do with the equipment I had available to use. The steps I followed are summarized below. I won’t elaborate the crude techniques I employed (e.g., removing air bubbles, adding water while eyeballing its level in relation to knot base, etc.).
1) Note the weight of a brush when completely dry.
2) Place a container on scales, note its weight, then fill it with water exactly level to the rim and note its filled weight. Subtract the container’s empty weight from its filled weight to determine its volume capacity.
3) Using the same container, fully immerse a knot in water without worrying about spillage, and then gently work out any air trapped in the knot.
4) Once no more bubbles appear, position the brush such that only the knot is lower than the rim of the container.
5) Slowly and carefully top-up the container level to the rim with the knot fully immersed to the base of the handle.
6) Raise the knot out of the water and hold it directly above the container.
7) Hand-squeeze water remaining in the knot back into the container.
8) Note the weight of the container with the water it holds after completing Step 7 above.
9) Remove the container from the scales and weigh the damp brush.
10) Determine the volume of the container’s remaining water content by subtracting the empty weight of the container from the weight noted in Step 8.
11) Determine the volume of water still held in the brush after completion of Step 7 by subtracting the weight noted in Step 1 from that noted in Step 9.
12) Subtract the value obtained in Step 11 from that obtained in Step 10.
The aggregate volume of immersed hairs should be equal to the volume capacity of the container less the sum of i) water remaining in the container after the immersed knot is removed and ii) any water removed from the container along with the previously immersed brush, i.e., the result of Step 12.
Summary results, along with some related brush measurements, are set out in the tables below. I strongly suspect some of the measurements as shown were incorrectly taken and/or noted. In particular, I would expect the Chubby 2, the Thater, and the Simpson Gary Young LE to produce higher Derived Aggregate Hair Volume values. At one point well into the project I noticed the balance needed to be reset. I also caught two errors I made in recording values. There could easily have been more, although I have photographs that confirm most of the balance readings. Carrying this project out with only two hands was a challenge. It would have been very helpful to have at least one other person available to assist, verify both observations and recorded results, take photographs, and fetch beer.
7. Discussion
Deriving a useful ratio
As stated above, I think it’s the aggregate volume of individual hairs in a particular knot in relation to that knot’s overall volume (i.e., the space it occupies as a whole) that constitutes what we generally think of as knot density. Even if water displacement principles can be applied to approximately measure aggregate hair volume, one would still need to settle on a consistently reproducible method by which to measure (or at least closely estimate) a particular knot’s overall volume so that a density ratio could be determined based on aggregate hair volume in relation to knot volume.
A problem involved in trying to conceive a method for consistently and reproducibly measuring the space a knot occupies as a whole is that the space a particular knot occupies can vary. The knot in a new brush will likely display a different geometry when it is first removed from its box than after it has fully bloomed subsequent to initial lathering and use.
The only possibly decent idea I’ve come up with in this regard is to estimate the volume of a knot when its conformation most closely approximates that of a cylindrical column up to the height where the knot begins to taper. This columnar segment of the knot would have a radius equal to one half the knot’s base diameter. The knot’s additional volume above that height could be estimated by calculating the volume of a similarly shaped semi-sphere (or dome) with a radius also equal to that of the column.
I calculated knot volume in this manner for my bone-handled Chief prototype with a fan-shaped knot. It was most convenient to use by reason of the fact that although it is fan-shaped when fully splayed, when compressed into a cylindrical column as described, its tapered segment roughly forms a hemisphere with a radius of 12.5 mm, the volume of which was easy enough to calculate. (Note that I rounded the measured 24.75 mm knot diameter of this brush to 25 mm for ease in serving this limited purpose purpose.)
The equation for determining the volumes of a column and hemisphere, respectively, are given below.
Pi × radius2 × height
.5 x (4/3) × pi × radius3
Based on column dimensions of height = 37.55 mm and radius = 12.5 mm, and a hemisphere dimension of radius = 12.5 mm, I came up with values of 18,408 mm3 (or 18.4 ml) and 4,091 mm3 (or 4.1 ml), respectively, for a total knot volume of 22.5 ml. Using the derived aggregate hair volume value from the table above, this approach yields a ratio of 17/22.5 or .76. I have no idea what that might mean, but it’s a rationally derived number.
Water absorption
I’m pretty sure the hairs in a knot absorb some amount of water that doesn’t affect what we think of or experience as density in terms of altering hair volume (i.e., the space a hair occupies). I haven’t given much thought to how that either should or might be taken into account.
Refined methodology
This project was mostly curiosity driven with an objective to test an approach and see what might come from it. A lot could obviously be done to improve on the crude methods I used. Also, some assistance would have made a big difference.
Other measurements
If something along these lines were to be done again with more rigor, it would be good to record additional data, e.g., drying times based on brush weight. I checked some, but didn’t bother to write them down. Another interesting measurement would be average hair width. I did check a few with an old micrometer I have. I’ll note here that the average hair thickness for the M&F knots I measured was about .003” (which converts to .076 mm), whereas that for the Rooney Finest knot was closer to .002” (or .05 mm).
8. Conclusions
I’m not at all sure that what is presented here has any useful meaning. Most likely the project will turn out to have been a waste of time. But it was fun, interesting, and, at least in some small ways, informative to me.
Previously, I hadn’t been all that intrigued by the subject of knot density per se, and I’m still not convinced there would be much practical utility in coming up with 1) a standardized method for measuring knot density, and 2) a corresponding unit or ratio by which to express such a measurement. Brushes are complex. It has seemed to me that there are too many other variable factors (e.g., hair characteristics, knot diameter, knot loft, knot shape) for a measurement of density by itself to serve meaningful comparison. On the other hand, we discuss other brush characteristics with respect to which much the same can be said. In any case, the more consistent we are at establishing and maintaining common usage of terms to serve in our discussions, the more meaningful and useful our discussions will be. Whether or not what I tried to do here is deemed to be worth repeating, I think it’s at least worth sharing for what it offers in terms of different ways to think about density.
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The good news is the night wind can always carry one more scream.
When my stunned body unfreezes, there will be applause. 
