Brushology: Badger-Hair Knot-Density Experiment

[ChiefBroom]

I need to look in the yellow pages for cut crystal lab beakers. This needs to be in our wiki.

What's great is that you can get the crystal beaker with a matched, stainless steel camping bowl that has an interior bottom diameter which just about exactly accommodates the beaker.

I assume the cut-glass pattern on the beaker is optimized for grip. If I'd have dropped the thing on the lab, er kitchen, floor, my wife would likely not have had a good sense of humor about it.

 

[ojinsa]

Wow!!! In my former life, I worked for a tech manufacturing company and have seen the analysis that goes into bringing an idea to life, and "the Chief" is as detailed as the work that goes into making mainframes. And, for what it's worth, the work that has gone into the brush is evident as it is freakin' awesome!

 

[ChiefBroom]

Rudy posted here that there are an estimated 1,000 badger hairs per gram. I just PMed to ask what is the average length of hair when weighed and also if he knows the approximate diameter of a typical badger hair.

If those facts were known (or could be approximately assumed with reasonable accuracy), they might be interesting to play around with alongside calculated aggregate hair volumes, estimated knot volumes, and ratios based on the latter two values.

At the end of the day, or rather shave, what matters, of course, is a brush's performance and the quality of the user's experience (whether real or imagined or some of both). As for myself, I find stark reality rather, ... well, stark. Nonetheless, I also think it's a hoot to dig into this stuff.

 

[oldtrout]

My apologies. I posted a question here last night and then deleted the post as I thought it might offend.

My question to Chief was, as far as I can remember, did the data derived from this experiment correlate to Chief's personal experience. What I mean is, assuming Chief is familiar with the performance of the tested brushes, did the brushes he considers to be of similar density produce similar data?

********************


This morning I was thinking about brush density. This is a damn odd thing to think about, thank you Chief!

I would opine that unless the knot diameter and loft where exact in each brush, the math would be too complicated for most of us to grasp. And I've read somewhere that badger hair is wider at the tips than at the base. I don't know that this is true, but if it is, this would complicate things further.

I do find the thought process behind the experiment fascinating, as I find this whole endeavor to be.

Just now, the YMMV brush backbone trait entered my pea brain.

Brush density is different than backbone, although we may tend to lump these traits together. I think backbone is influenced as much or more by the ratio of loft to knot size than how densely the knot is packed. And hair type plays a big role in backbone as well. I don't know why I assume these things, I have no data.
**********************

I did draw a conclusion regarding Ken's experiment. Of my conclusion, at least, I am certain.

I wish he made brushes!

 

[ailevin]

Just a thought. What about going full on Archimedes?

1. Submerge entire brush, knot and handle in container and mark level of water in container
2. Remove brush, squeeze out knot into container and mark level of water in container
3. Submerge only handle in container almost up to knot and mark level of water in container
4. Weigh container when filled with water at each of the three marks.

The difference in weights of water between the three marks should tell you the volume of the handle, the volume of the knot hairs, and of course the volume of the handle + knot hairs. Of course there is still the issue of trapped air and shape of the knot to get true hair density, but normalizing the hair volume by loft *(knot diameter)^2 seems like a good start and if you wanted to get fancy you could use an "adjusted loft" to deal with shape of knot. (YVolumeMV)

Alan

 

[ChiefBroom]

Just a thought. What about going full on Archimedes?

1. Submerge entire brush, knot and handle in container and mark level of water in container
2. Remove brush, squeeze out knot into container and mark level of water in container
3. Submerge only handle in container almost up to knot and mark level of water in container
4. Weigh container when filled with water at each of the three marks.

The difference in weights of water between the three marks should tell you the volume of the handle, the volume of the knot hairs, and of course the volume of the handle + knot hairs. Of course there is still the issue of trapped air and shape of the knot to get true hair density, but normalizing the hair volume by loft *(knot diameter)^2 seems like a good start and if you wanted to get fancy you could use an "adjusted loft" to deal with shape of knot. (YVolumeMV)

Alan

Thanks for reading the OP. And thanks for the thoughtful suggestion!

What would be best, I think, is to find a container with level-lines to indicate mls (I'm not really interested in finer gradations). It would have to be large enough, however, to accommodate manipulation of the brush to remove air.

Thanks again!

 

[ChiefBroom]

Thanks for re-posting, Don.

First, and I will probably be repeating this, I tend to have a contrarian nature; I'm generally allergic to dogma; and I'm absolutely not advocating some new brush orthodoxy based on amateur-science as applied in a half-baked, kitchen experiment.

What I mostly care about with respect to a brush is whether 1) it works well for me and 2) I like it. Although relatively high-density brushes seem to have been trending over the past 6+ months (and I have absolutely no issue with that), based on my own experience I've concluded that highly dense brushes neither work particularly well for me by my standards, nor do I especially like using or looking at them.

Of course, the foregoing statement presupposes that I somehow already knew how to distinguish a highly dense brush from a not-so-highly dense brush before concocting the method of measurement described in the OP. And in fact, I believe I could do just that, at least well enough to adequately serve my practical purposes. So this wasn't about solving some problem I had in figuring out what kind of brush to buy. It was about exploring an interesting question, which involved solving an interesting problem related to quantitation of something that to the best of my knowledge hasn't heretofore been quantitated (i.e., the relative density of a knot in situ).

What you opined (below) with regard to knot complexity is what I started to post (much less succinctly) into this thread referenced in the OP. I completely agree with you. I might even surpass you in agreement. I would add that different hair characteristics (e.g, 3-band vs. 2-band vs. Old Rooney Finest vs. etc.), knot shapes, and even the depth to which a knot is sunk into a handle can also have significant bearing on how a brush behaves, performs, and feels. A focus on density alone, or even a strong emphasis on density, could result in obscuring more than it illuminates.

On the other hand, I've learned that there are sweet spots for me in terms of knot diameter, loft, and shape. I can can identify what those are on the basis of paying attention to what I prefer to use. It bears repeating that as in most things related to shaving, mileage varies, and I have no desire to try to tell anyone else what they ought to like. I do, however, enjoy discussing what I like (and don't like), as well as what others like (and don't like) and the differences between them.

It is very simple to discuss knot diameter, loft, and even shape. It isn't so easy, however, to have a discussion about density. I had particular experience of the challenge involved in meaningful discussion of density last week. That opened my mind a little to the possibility there might actually be some merit in trying to come up with a better way to measure density. I think that's what Dave was trying to get at in his thread referenced in the OP. And so I starting thinking about measuring density, and upon getting an idea about how one might possibly go at it, decided to give it a try. Even though it's highly doubtful a method as complicated as the one I tested would be broadly applied, it still might possibly serve as a means by which to quantitatively benchmark an important variable in brush behavior/performance and user experience.

Now to your question regarding correlation between my measurements and my experience. The answer is yes, maybe, to some extent, kinda. But if you read all of the OP, you caught the part about deriving a useful ratio. I proposed a method, but only roughly calculated a ratio for one brush. It's very important to note that aggregate hair volumes aren't density values. They are half of what is needed to calculate a ratio of aggregate hair volume in relation to knot volume. I think I'm going to try to elaborate on that idea a little more in a separate post. That would best be done with some drawings.

I'll give you an example of what I'm talking about. I have three 2XLs: a Rooney and two M&Fs, which I bought at the same time. The aggregate hair volumes I derived for them are 20 ml for the Rooney (with a 28 mm x 48.5 mm, fan-shaped knot), 21 ml for the M&F faux-briar (with a 27.5 mm x 51.5 mm, bulb-shaped Finest knot), and 15 ml for the M&F faux-jade (with a 27.5 mm x 47.5 mm, bulb-shaped, Finest knot). I was so excited to have scored the pair of M&F 2XLs with Finest knots! They're both terrific brushes. I've used each of them once or twice, no more. The Rooney is also a beauty. I've used it maybe three times. All three brushes are too dense for my taste, and the faux-jade M&F at 47.5 mm loft and the Rooney at 48.5 mm are also too short-lofted for me. On the other hand, I've used my two bulb-shaped M&F Chief prototypes for over 90% of my shaves since April. I came out with aggregate hair volumes for them of 17, 18, and 19, i.e., very closely similar. That doesn't mean I like brushes with hair volumes of 17-19. The faux-jade M&F 2XL has too much hair volume for at 15 ml in combination with its knot diameter and loft. And my Chubby 2 has too much hair volume for me at 18 ml. That, BTW, still doesn't make much sense to me given its knot diameter and loft. But I think That's explained in large part by it knot shape, which is very pointy. That would serve to increase its density ratio.

Sorry to go on so long (again). You're right. It's complicated. But it also makes some sense.

Thanks again for the post.

My apologies. I posted a question here last night and then deleted the post as I thought it might offend.

My question to Chief was, as far as I can remember, did the data derived from this experiment correlate to Chief's personal experience. What I mean is, assuming Chief is familiar with the performance of the tested brushes, did the brushes he considers to be of similar density produce similar data?

********************


This morning I was thinking about brush density. This is a damn odd thing to think about, thank you Chief!

I would opine that unless the knot diameter and loft where exact in each brush, the math would be too complicated for most of us to grasp. And I've read somewhere that badger hair is wider at the tips than at the base. I don't know that this is true, but if it is, this would complicate things further.

I do find the thought process behind the experiment fascinating, as I find this whole endeavor to be.

Just now, the YMMV brush backbone trait entered my pea brain.

Brush density is different than backbone, although we may tend to lump these traits together. I think backbone is influenced as much or more by the ratio of loft to knot size than how densely the knot is packed. And hair type plays a big role in backbone as well. I don't know why I assume these things, I have no data.
**********************

I did draw a conclusion regarding Ken's experiment. Of my conclusion, at least, I am certain.

I wish he made brushes!

 

[ailevin]

Ken,
Thanks for the kind words. The nice part about step one is that you don't really have to be very careful about the vessel or the brush before you mark the first water level, so you could manipulate knot or use a stirring rod or whatever. After that we would have to get as much water off of brush as possible and keep it in the container for steps 2 and 3.

Volumetric glassware is expensive, and I am not familiar with wide-mouth volumetric glassware other than a graduated cylinder. Perhaps you could buy a large (500ml or 1000ml) polypropylene graduate cylinder and "saw it off" so you could more easily manipulate the brush in it. Following that thought a taller narrower vessel like a glass might work better than a bowl.

Another issue is estimating the measurement error you can tolerate. For a very rough ballpark calculation consider a 24mm knot with 50mm loft:

• A cylinder of diameter 24mm and height 50mm has a volume of about 23 ml
• Assume the volume of hair could be in the range of 50%-90% of that or roughly 10ml-20ml
• Weight measurements have .1g-.2g or .1ml-.2ml error
• Marking or reading level errors are .5ml
• Squeezing brush and drip errors are .5ml-1ml

So we are probably looking at errors of 1ml-2ml depending on technique and we are likely trying to measure differences that are <10ml.

Maybe it would be easier to take the knots out of your brushes and weight them dry. I wonder if brush makers work by weight? I would think so since brush prices seem to track amount of badger hair used within a brand and a given hair type.

Good Shaving,
Alan

 

[cwf71]

Just a thought. What about going full on Archimedes?

1. Submerge entire brush, knot and handle in container and mark level of water in container
2. Remove brush, squeeze out knot into container and mark level of water in container
3. Submerge only handle in container almost up to knot and mark level of water in container
4. Weigh container when filled with water at each of the three marks.

The difference in weights of water between the three marks should tell you the volume of the handle, the volume of the knot hairs, and of course the volume of the handle + knot hairs. Of course there is still the issue of trapped air and shape of the knot to get true hair density, but normalizing the hair volume by loft *(knot diameter)^2 seems like a good start and if you wanted to get fancy you could use an "adjusted loft" to deal with shape of knot. (YVolumeMV)

Alan

To increase the precision and accuracy of the endeavor, would it not be best to remove the variable of the handle and the water volume in handle not related to the knot itself? This would likely have the unfortunate outcome of destroying the handles, but would eliminate a number of issues in executing the experiment. If you're gonna make an omelet...

 

[cwf71]

Ken,
...Maybe it would be easier to take the knots out of your brushes and weight them dry.

Hey, that sounds like a good idea... Not only easier, but removes procedural issues and other variables.

 

[ChiefBroom]

Hey, that sounds like a good idea... Not only easier, but removes procedural issues and other variables.

I think it sounds like a great idea too. How about you do it with your brushes and let us know how it goes.

And remember, photos or it didn't happen.

Can't wait to see the results!

 

[cwf71]

Just testing the limits of your scientific curiosity with your own items... perhaps you can apply for a grant from the Wet Shaving Academy. ;-)

 

[ChiefBroom]

Great to have your participation in this discussion.

I'm not a scientist. Never have been. But do I have access to glassware. So I'll have to do some poking around.

Re margins of error, what I guesstimated in mine was about 1.5=+/- grams/ml, although I have a hunch the spread was closer to 2. I got pretty good at it. Less good at recording values. And once I looked down and caught the balance reading -2 grams. I'd probably touched the Tare button when there was a little water either on the balance or on the underside. I don't think it had been like that for very long. I looked down at the display a lot. And I photographed almost every read-out.

Removing the knots in my brushes is not an option.

I'm responding somewhat randomly, but the only squeezing/dripping errors I was exposed to would have been water that came off on my hand, and I was very careful about that. Whether the squeezed out water went back into the glass or missed it and landed on the balance outside the glass didn't really matter. And it also didn't matter whether I squeezed out more or less water. I weighed each brushe dry. Weighed each brush damp. Whether the water went back into the glass to offset negative displacement from squeezing or in the damp-dry adjustment, all removed water went back into the glass. All appreciable error has to to with 1) water soaked into the base of the knot within the inserted portion, 2) water that stuck to my hand during squeezing, and 3) eye-balling while suspending the brush handle above water level and filling the glass to brim level from a pitcher with my other hand. I think I probably didn't have much more than 2 gram variance over all with respect to 3. I don't think 2 was appreciable (although I'd rather eliminate it, which your suggestion would accomplish. 1 is a mystery.

Ken,
Thanks for the kind words. The nice part about step one is that you don't really have to be very careful about the vessel or the brush before you mark the first water level, so you could manipulate knot or use a stirring rod or whatever. After that we would have to get as much water off of brush as possible and keep it in the container for steps 2 and 3.

Volumetric glassware is expensive, and I am not familiar with wide-mouth volumetric glassware other than a graduated cylinder. Perhaps you could buy a large (500ml or 1000ml) polypropylene graduate cylinder and "saw it off" so you could more easily manipulate the brush in it. Following that thought a taller narrower vessel like a glass might work better than a bowl.

Another issue is estimating the measurement error you can tolerate. For a very rough ballpark calculation consider a 24mm knot with 50mm loft:

• A cylinder of diameter 24mm and height 50mm has a volume of about 23 ml
• Assume the volume of hair could be in the range of 50%-90% of that or roughly 10ml-20ml
• Weight measurements have .1g-.2g or .1ml-.2ml error
• Marking or reading level errors are .5ml
• Squeezing brush and drip errors are .5ml-1ml

So we are probably looking at errors of 1ml-2ml depending on technique and we are likely trying to measure differences that are <10ml.

Maybe it would be easier to take the knots out of your brushes and weight them dry. I wonder if brush makers work by weight? I would think so since brush prices seem to track amount of badger hair used within a brand and a given hair type.

Good Shaving,
Alan

 

[mblakele]

Could it be that the brushes you tested are a little too close together in overall quality? What might you learn by trying out similar tests on cheaper brushes?

 

[ChiefBroom]

Could it be that the brushes you tested are a little too close together in overall quality? What might you learn by trying out similar tests on cheaper brushes?

You make an excellent point. I picked a dozen high-end brushes that only included one 3-band. Although there was some range of density (Chubby, three 2XLs at the high end), none was what I'd call sparse.

I'd done the two first passes measuring positive displacement on Saturday, and woke up with the idea to try negative displacement on Sunday morning. I grabbed a dozen brushes and started while my wife was running a bunch of errands with the hope of being done and having everything cleaned up before she got home. Not that it's really a big deal, but there's a look and an attitude that makes me try to keep as much of this underground as possible. Didn't work. She got home when I was about halfway through the dozen and delighted in asking me repeatedly, "now, tell me again, what is the value of doing this?"

Sorry for the digression. I have a pile of brushes, but they are all pretty close together in terms of overall quality. And I wouldn't do this again until satisfied I'd worked out most of the kinks, had the right equipment, knew exactly what I wanted to record, and had some help. Then I'd probably make a long day of it and try to collect data on a broad range of brushes.

 

[mrb7]

The other distinct possibility is that I've finally plunged over the edge.

My wife would vote for that.

welcome, welcome.

 

[Tulsa77]

...."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."

Got to love the short-lived resolutions.

Great post Ken!!!

 

[OpasRazor]

"...............absolute vacuums don&#8217;t exist in nature."

You didn't date enough or you would know this is demonstrably untrue.

 

[coolbus18]

Ken you get huge kudos for this.Many newbies and long timers can go whacko trying to figure out the info about brushes which I have noticed is as varied as there are stars. In watching various videos about brush making(Shavemac's a goodie) the badger hair is weighed out at x grams of hair for the brush ,which now lead to the type of hair of the badger and from what part of the fur it came.Variable city. Then there's boar which to me seems more hydrophilic than badger. I like and use both,although depending on the soap I'll change the brush I use.
Thanks for this great thread and yes we're over the edge, but that's where the good stuff happens! Enjoy your lathering!

 

[ChiefBroom]

Warning: What follows was a waste of time to put together and will be a waste of your time to read.


This past summer I killed most of a weekend designing and carrying out an experiment to see if I might be able to approximate the density of some representative badger hair knots by measuring the volumes of water they each separately displaced under somewhat controlled conditions. My objective was to determine the aggregate volume of individual hairs in a knot rather than its mass density. This was based on my view that “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.”

The results of that experiment were reported in this thread. I noted at the time that what I’d attempted only addressed half of the challenge. As stated in the OP:

A problem involved in trying to design 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.

I went on to say:

The only … 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.

Not long after that, I made a run at re-learning some geometry and developing a method for determining overall knot volume. I hacked my way through it, but didn't have time back then to put it in presentable form and post in follow-up. Having relinquished Stewarding duties, I presently have more time to pursue useless projects. So here’s another installment.

The knot drawing shown in Figure 1 below is based on a photo I took of an M&F L7 that significantly influenced the design of the M&F Chief. I masked and cut out the knot in Photoshop and then placed it into Adobe Illustrator, which allowed me to adjust scaling by .001s of a millimeter. As shown, it represents a knot with a base diameter of 25mm and loft of 50mm.

First, everything about this exercise involved approximation. I started out with an aim to calculate the knot’s overall spatial volume as it would be if the hairs were compressed to form a cylinder up to the height where they terminate around the outer circumference and taper up towards the center to form a dome-shaped top. There really isn't any definite edge, so I picked 32.5mm as close enough. Note that the sides of the knot when it’s in its natural shape also measure 32.5mm in length, but due to their angles the height of the sides in that conformation is only 30mm.

The equation for determining the volume of a cylindrical section is V = pir[SUP]2[/SUP]h. Given a radius of 12.5mm and a height of 32.5mm, the cylindrical section of the knot in a conformation as represented by the red lines is 15.95mm[SUP]3.
[/SUP]
The top was more of a challenge. As stated above, my previous estimate was derived by calculating the volume of a similarly shaped semi-sphere (or dome) with a radius also equal to that of the column. The problem with this approach is that the knot’s dome, as shown, doesn't really closely approximate a semi-sphere when the knot’s base is in cylindrical conformation. Someone who took calculus or advanced geometry could undoubtedly figure out how to predict the shape of a knot’s dome when in a cylindrical conformation based on its geometry in a natural state. I wasn't up to that. So first I trussed my L7’s knot to form a cylinder leaving only the dome exposed, and then eyeballed it to see whether it looked more like a paraboloid or a spheroidal dome. It was a little hard for me to judge, but my intuitive sense was that it should be the latter.

As it turned out, the image of the knot I placed in Illustrator has a top me that closely fits a sphere with a diameter of 51.25mm (represented by the blue circle in Figure 1). What I did as a basis for estimating the volume of a spheroidal cap formed by compressing the knot into cylindrical conformation was 1) make a copy of the circle, 2) place it exactly on top of the original, 3) reduce the copy’s width while maintaining the same height until it formed an ellipse (representing an ellipsoid of revolution) intersecting the top edge of the cylindrical section of the knot. (I think the center of the ellipse should probably be shifted up slightly relative to that of the circle since the sides of the column are higher than the sides of the knot below its dome when the knot is in its natural, i.e., splayed, shape. But I didn't worry about that and think the difference would be insignificant for present purposes.)

The equation for determining the volume of a spheroidal dome (or cap) is V = pia2h2/3c2 x (3c-h), where a = the horizontal axis, c = the vertical axis, and h = the height of the dome. Those values in this example are 13mm, 25mm, and 17.5mm, respectively, yielding a volume of 4.99mm[SUP]3[/SUP].

So the approximate volume of the knot in cylindrical conformation is 15.95mm[SUP]3[/SUP] + 4.99mm[SUP]3[/SUP] or a rounded 26mm[SUP]3[/SUP].

Although I thought calculation of a knot’s volume when in cylindrical conformation might be more useful for the purpose of comparing density ratios on a consistent basis, the space occupied by a knot when it in it natural (i.e., splayed) shape is also of potential interest.

The base section of the knot in this state forms a truncated cone (or conical frustum). The equation for calculating the volume of a conical frustum is V = (pi/3)h(r[SUB]1[/SUB][SUP]2[/SUP] + r[SUB]1[/SUB]r[SUB]2[/SUB] + r[SUB]2[/SUB][SUP]2[/SUP]). Here, h = 30mm, r[SUB]1[/SUB] = 12.5mm and r[SUB]2 [/SUB]= 25mm, yielding a volume of 34.36mm[SUP]3[/SUP].

Determining the volume of the dome in this case was very straightforward and simple. The equation using height of the dome and radius of its base is V = (pi/3)h2(3r- h), which yields a value of 23.84mm[SUP]3[/SUP], where h = 20mm and r = 25mm.

Therefor, the approximate volume of the knot in its natural state is 34.36mm[SUP]3[/SUP] + 23.84mm[SUP]3[/SUP], or about 58mm[SUP]3[/SUP], which is a little over 2.2 times the volume of the knot when forced into a cylindrical conformation.

I could probably be committed for doing stuff like this. But the fact is if anyone ever makes a serious effort to quantify knot density for comparative purposes, it will probably require making calculations along these lines so as to enable estimated hair volume to be expressed in the form of a ratio in relation to a knot’s overall special volume.

Figure 1.