Mixing my ink with my beer

I have had a lot to say in the John the Math Guy blog about beer. There was a recent post about ruminations on beer, but the key post on beer, my seminal post on beer, is the post where I cleverly used beer to illustrate Beer's law. I keep going back to that one because I just can't get over how brilliant the idea was.

I have referred to this Beer's law post in heaps and gobs of other posts:

     Where are my CIELAB knobs? [1]

     Why does my cyan have the blues  [2]

     Why does my cyan have the blues? addendum

     The color of a bunch of dots, Part 4

     Density is ink film thickness

Green ink being shamelessly added to beer

Several people have asked questions on this Beer's law thing, and how it connects with ink. Way back in January (2013), I got an email from a PhD student in the UK:

When I scoured the internet for the derivation of this equation I only found the original equation based on absorption coefficient, path length and concentration.

I would like to understand where the alternative equation is coming from. Could you point me in the direction of a useful paper or similar? Any help would very much be appreciated. Thanks a lot in advance.

Kind regards,

Anja

I just recently got a similar question from Michael, who is not a PhD student in the UK, but is nonetheless a smart guy. He just aced the Science and Technology Quiz that was put together by Smithsonian magazine and the Per Research Center. This is quite an accomplishment. I'm proud to be a "virtual" friend of yours, Michael.

I thought Beers law was related to transmission of light through something, not reflected - but, well, same difference ?

Michael's question came to me on through that website that everyone uses for scientific collaboration.: FaceBook. If you haven't heard of it yet, I suggest you check it out. That's where I do most of my serious research.

The plethora of questions (there were two...) show that I have clearly messed up big time in my desire to educate the world about ink and beer. I left one little step out, the mixing of ink and beer. Just how is it that Beer's law applies to ink?

Beer (on the left) and ink (on the right)

Ink is soooo not like paint

One of those wonderful things that we can count on in this world is that paint is not like ink. Oh... they may seem the same to the untrained and unscientific eye. You put them on something and it changes the color. But there is one key difference, as illustrated in the image below.

This time, paint is on the left and ink (on the right)

This image was created by smearing ink and paint on a sheet of paper [3]. Before smearing, a large black area was printed on the sheet. Note that on the left, the paint completely obliterates the black underneath. Paint has a great hiding power, at least when you pay more than $8 a gallon for it. The ink, however, does a perfectly lousy job of hiding the black. You really can't tell that the yellow ink is overneath the black.

What gives? Is ink just really, really cheap paint? Oh contraire! Let me assure you, ink does a pretty decent job of doing exactly what it was trained to do. And paint also does a pretty decent job at what it was trained to do. That is, if you aren't cheap like me, buying the ultra-cheap paint at $8 a gallon from Fast Eddie's Paint Emporium and Car Wash.

The actual photomicrograph below illustrates what an ideal cyan ink is trained to do. Red, green, and blue light hit the surface of the ink. [4] As can be seen, the blue and green light go right through the cyan filter ink. The ink is transparent to green and blue light. These two flavors of light hit the paper (or other white print substrate) and reflect back. Why? Cuz the substrate is white, and that's what white things are trained to do. 

Cyan ink, sitting contentedly on paper while being bombarded with red, green, and blue light

The red light suffers a completely different fate. For anyone who has visited a red light district, this should be no surprise that one's fate may change. The red light is absorbed by the cyan ink. Few of the poor hapless red photons ever even get a chance to reach the paper, and even fewer make it through the ink in the hazardous journey back through.

I may have shattered some illusions about ink here. I apologize, but it's time you learned the facts about the birds and the bees and the inks. It is customary to think of light just reflecting off the ink. Sorry. It's more complicated than that. The only reflecting that's done is done by the paper.

Ink is a filter, a filter laid atop the paper.

Why is ink that way?

This bizarre behavior is not just some side effect of some bizarre organic chemistry that is only understood by some bizarre color scientist locked in the lab at Sun Chemical. This bizarre behavior is a property that is specifically engineered by some some bizarre color scientist locked in the lab at Sun Chemical. 

To see why this would be a good thing to engineer in, consider what happens when magenta ink is placed overneath cyan ink. Magenta ink works a lot like cyan ink, except that it absorbs green light and passes red and blue. The excitement starts when you put on ink on the other. The cyan ink absorbs red light and the magenta ink absorbs the green. What's left? Just blue light.

Magenta ink, sitting contentedly on cyan ink

The exciting part of this is that new colors are created. We start with cyan, magenta, and yellow inks. By putting one ink overneath another, the additional colors red, green, and blue are created. Try doing that with paint! It ain't gonna work. The paint on top defines the color, hiding whatever is below. 

This feature of ink is what allows us to have a much wider gamut. With three inks (cyan, magenta, and yellow) we can theoretically get eight different colors: white (no ink), cyan, magenta, yellow, red, green, blue, and black (all three inks).

Getting a bit more quantitative

I need to put some numbers on this if I'm going to get Beer's law involved. I painted a rather black and white picture of cyan. Well, ok, I should say that I inked a black and white picture rather than painted it. And black and white aren't quite the correct colors. But, the point is, inks are not perfect. Cyan does not capture all the red photons. Nor does it pass all the green and blue photons.

 A typical cyan ink might allow 20% of the red photons to pass through on their way to the substrate. That is, 80% of the red light gets absorbed and the other 20% makes it down to the paper. Let's just assume that all of those photons reflect from the paper. (I am telling a little white lie here, but it's for a good purpose.) 

Ok, so if we start out with 100 red photons heading downward into the ink, 20 of them will reach the paper. Of these 20, 80% of them (I think that would be 16) will get absorbed on the way up. That leaves just 4 red photons, out of the original 100, that make it back out. For those of you who are all into the density thing, this would mean about 1.40D. If you understood that, then you know Beer's law, and can apply it to ink on paper. Who said that ink and beer don't mix?

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[1] You may have seen this blog post in Flexo Global Magazine. So, we are talking popular here.

[2] I am pleased to say that this blog post was picked up just recently by the Australia New Zealand Flexographic Technical Association magazine (August 2013). Look for it in your mailbox. This blog post was also picked up by Flexo Global magazine. So... we are talking really popular here.

[3] Truth in advertising here... this is not an actual photo, but a digital simulation. It is very nearly photorealistic due to my vast artistic ability, and it simulates what really happens, but I repeat, this is not an actual photo.

[4] Someday I will get around to writing a blog about how light comes in three flavors: red, green, and blue. That's all. All other colors are a combination of those three. Based on this simplification, you can explain why inks are CMY and computer monitors are RGB. It will be a totally cool blog. I will send you a text when I finally publish it.

The color of a bunch of dots, Part 4

If you have been wondering for years about TVI bananagrams, you have come to the right blog post. This blog post is the definitive blog post on TVI bananagrams. But if (for some crazy reason) you have not been wondering all your life about TVI bananagrams, then this could still be a watershed moment in your understanding of dot gain and its fuzzy sister, tone value increase.

Cyana bananagram

Dot squish

Pat Noffke and John Seymour presented a paper at TAGA in 2012 entitled A Universal Model for Halftone Reflectance. [1] In the paper, they developed an equation for what they called Dot Area Increase. This is a measure of how much a dot squishes out when it hits the paper. This is similar to what Murray and Davies called dot gain, but there is one novel difference. In both models, dots get bigger when they hit the paper, but in the Noffke-Seymour model, they also get thinner because of the whole law of conservation of ink thing. Because the dots are thinner, the color of the dot is less rich than the solid.

A halftone dot, before and after the steamroller

If there isn't any dot squish for a particular hypothetical printing, then a 30% dot would cover 30% of the area, and the thickness of the dot would be same same as the thickness of the solid ink. This hard dot has no dot area increase. The equation for reflectance in this case is the Murray-Davies equation.

Murray-Davies equation for hard dots

At the other extreme, the dots squish out completely so there is no longer any semblance of the dot structure (think gravure). Perfectly soft dots. Contone - continuous tone. At this extreme, one can use Beer's law to estimate the reflectance of a halftone. You will no doubt recall the equation from my blog on Beer's law, 

Beer's law equation for continuous tone

I should add here, that the two equations here should be applied on a wavelength by wavelength basis.

I should also 'splain a bit about the "Ain" that is being used as an exponent. Normally in Beer's law, this is where you would put something to do with ink film thickness. So, why did I stick the dot area in that spot? Imagine that you start out with perfect, hard halftone dots that are, I dunno, 30% dot area. Since they are perfect, let's just assume that the ink thickness of the dots is the same as the thickness of the solid.

Now, let's say that these dots get stepped on by an elephant. They are squished out so as to cover the whole area uniformly. How thick is the ink now? If the original dot area is 30%, then the new ink thickness is 30% of that of the solid. So, the exponent of Ain represents the thickness of the fully squished out dots.

Poor defenseless halftone dot, about to become a continuous tone

The dot squish equation

The pure genius of the Noffke-Seymour paper is that they considered what happens in between. In the figure below, the left side shows the starting condition. The halftone dot covers 25% of the area, and has the full thickness of the solid. The right side shows what happens after squishing [3]. The dot now covers 39% of the area, and as a result, is thinner by whatever ratio is necessary to preserve ink volume. I dunno? Maybe the ratio is 25% / 39%? I guess that's about 0.64. [4]

A tale of two halftone dots

Using the halftone dot at the right to illustrate the Noffke-Seymour formula, Beer's law is used to estimate the reflectance of the light blue area covered by ink. A thickness of 0.64 (as compared with the thickness of the solid) is used. Since a wandering photon has a 39% chance of hitting the area covered by ink, this reflectance is multiplied by 39%. This accounts for the light that reflects from the ink. The remaining photons will hit the paper, so (in true Murray-Davies fashion) the reflectance of the paper is multiplied by 61%, and this is added to the first number.

The equation below tells the whole story. "Ain" is the dot area going in. In the example above, this would be 25%.. "Aout" is the final area of coverage after squishing, 39% in the example.

The beautiful Noffke-Seymour equation

Note that at the extreme of no squish, Ain = Aout, the equation simplifies to Murray-Davies. At the other extreme, then Aout = 1, and this equation simplifies to Beer's law. 

In the first big finding, the authors of this paper looked at spectra from a big pile of tone curves, and came to the conclusion that pretty much every printing modality (web offset, stochastic web offset, gravure, newspaper, flexo, and ink jet [5]) all fit conveniently between these two extremes. This is huge. (But that's just my opinion.) Tone value increase for any type of printing can be described in terms of how broadly the dots squish. That's all you need to know.

Finally, the bananagram

I have saved the best for last. The other huge finding is shown below, the invention of the bananagram [6]. The bananagram below is an a*b* plot of all possible tone curves for a given cyan ink [7]. The left edge of the banana is the tone curves generated by assuming that the halftone dots are perfectly hard. The right hand side is a similar curve made with the assumption that the dots are perfectly flattened out.

Cyana bananagram

Now, lemme tell you about the rainbow colored lines. The yellow line, as an example, is all possible a*b* values that a 40% cyan halftone dot could take. Starting with a perfectly hard 40% cyan halftone, as you gradually squish it out, you will see it trace out the curve from one side of the banana to the other.

Wow. The position along that line tells you how hard the dots are. If you know the dot hardness (along with the spectra of the solid and the paper, and the original tone value), you can figger out the color of the halftone.

Foreshadowing the next blog post

I need to eventually tie up at least one loose end. I have been throwing dot gain kind of equations around willy-nilly, or perhaps yuley-niely. We have (so far) the following three equations to explain the color of a halftone: Murray-Davies, Yule-Nielsen [2], and Noffke-Seymour.

Murray-Davies (we all know) is a lump of over-cooked turnips when it comes to accurately predicting or measuring color. Yule-Nielsen, seems to be all the rage. Then these young upstarts come along with yet another formula that is gonna save the world! How can this all be reconciled?

Stay tuned for the thrilling conclusion!

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[1]  I know these guys. One of them seems to be around whenever I stop at the bar for a beer. I guess he must hang out there a lot. Anyway, these two guys do bunches of seriously good stuff. And they're modest, too. Well, at least one of them is.

[2]  If you have been paying careful attention, you will notice that I have reverted back to the more common spelling the the latter gentleman's name. In a previous blog, I cited the original paper from the TAGA Proceedings, along with a scan of the heading for this paper. In the original published paper, the name is spelled "Neilsen", which is contrary to virtually every citation of the name. I ranted and raved about how 11,400 patent citations use the misspelling "Nielsen". As such, I would imagine that these 11,400 patents are potentially invalid. Gary Field did some excellent detective work, and has convinced me that the TAGA paper is a transcription error. The gentleman's name is Waldo J. Nielsen. Assignees of these patents can breathe easier.

[3] I keep talking about squishing, but this might not always be the case. In a web offset press, where there is a lot of pressure between the plate and the blanket and the paper, then squishing is probably a valid term. But in the case of gravure, where the ink has a very low viscosity, maybe it's not so much squishing as it is just spreading out. In newspaper, where the paper does not have a coating, maybe the significant effect has more to do with the ink being wicked into the paper. All of these I have put under the umbrella term "squishing." Whatever you squish under your umbrella is your own business.

[4] How should I know what the ratio is?  Am I called John the Arithmetic Guy??!?!?

[5] No data was harmed in the filming of this experiment.

[6] I expect to see bananagram T-Shirts available on the internet. There will be bananagram support groups for people who have family members sucked into the cult. I expect this will be a topic in the next state of the union address, with plenty of polarized commentary on Fox News and MSNBC.

[7] In three dimensions, this is a surface, sort of like a fly's wing or a sail. In other words, the possible range of colors of a halftone of a given ink can be described by a three dimensional figure that looks like a fly's wing.

Why does my cyan have the blues? (addendum)

I was asked a question about my previous blog about why the hue of ink sometimes changes when the ink film is increased.

I had a lovely plot (see below) that showed that showed that Beer's law doesn't do all that bad of a job at predicting the ink trajectory (and the hook) of a magenta ink. The plot shows how close the match is in a*b*. Erik pointed out that I didn't show what is going on with L*. It could be that Beer's law works well in a*b*, but really messes up when it comes to L*.

The magenta hook, real and estimated

So, I had a look at this same data from a few other perspectives. Here is what the data looks like in the L*a* plane.

And here it is on the L*b* plane.

My conclusion is that it doesn't do so bad. Thanks Erik, for keeping me honest. Naturally, if it hadn't worked out I would have suppressed the results.

Why does my cyan have the blues?

When I started in the print industry as an apprentice to Gutenberg, I noticed that the folks in the press room called the inks red, yellow, and blue. This confused me. Everything I had read in color theory books said that cyan, magenta, and yellow were the subtractive primaries. These were the primaries that you use to make a wide range of colors with pigments and filters. Pigments and filters work by subtracting certain wavelengths of light. On the other hand, red, green, and blue were the additive primaries, and these were used to make all the colors when you are mixing light, as in a TV or computer monitor.

Polaroid snapshot of me working at my first job

Why were those silly printers using some of the additive and some of the subtractive primaries? Didn’t they realize that this reduced their gamut? That was the theory, anyway[1].

Just a naming issue?

Anyone who knows me, or who loves me[2] can attest to the fact that I am a firm believer that ignorance is the main explanation for every cultural and scientific phenomenon. In this case, my previous blog about counting colors provides a clue as to the sort of ignorance that might explain why magenta is so curiously called red.

The eleven people who read my previous blog learned that there are only eleven basic one-word color names in our active vocabulary. Neither cyan nor magenta made that list[3]. Clearly the folks on press were calling the inks “red” and “blue” because they have no other words to describe the colors.

Cyan ink is blue, and magenta is red

In my normal incisive way, it took me a few years to realize that the pressmen were not quite as ignorant as I thought they were. I guess I spent too much time running for buckets of halftone dots to actually put my head in a bucket of ink. When I finally did put my head in a bucket of ink (as part of a hazing[4] experiment) I could see that cyan ink is blue, and that magenta ink is red when you look at them in a bucket.

Cyan and magenta inks are blue and red in the can

Cyan and magenta inks are cyan and magenta on paper

So, this confusion is obviously beyond my original explanation. Just like when a fellow accidently calls his wife by the name of a former girlfriend, you can bet there is something deeper going on.

Beer’s law revisited

In yet another very popular[5] blog of mine, I provided a charming explanation of Beer’s law. This blog post is a prerequisite for the following exciting discussion.

Let’s just say that we have a perfect magenta ink. A perfect magenta ink will reflect all the red light and all the blue light that hits it. As for the green light, a light shade of magenta might reflect about 10% of the green. A rich shade of magenta will reflect about 1%.

Now we bring in Beer’s law. Let’s say we start with that light magenta and add another layer of the same ink. Beer’s law would predict that the reflectance would multiply. Since perfect magenta reflects 100% of red and blue light, Beer’s law predicts that the double layer of magenta will reflect 100% of the red and blue light. Beer’s law would further predict that the green light would reflect at only 10% X 10%, which is 1%. A double layer of light magenta becomes a rich magenta.

Key point here: for this perfect magenta ink, the hue is still that of magenta. It still reflects most of the red and blue light, and absorbs most of the green light.

Let’s just say that we now switch over to a magenta that is less pure. Let’s just say that for some inexplicable reason, the publishers of Schlock magazine are unwilling to spend $100,000 per gallon for their ink. The bargain ink they decide to use does not reflect quite as much blue light as we would hope; maybe it only reflects 40% of the blue light when we put a thin film down, and maybe 10% of the green light. Let’s say that the red light is still reflected at 100%.[6]

What happens when we double the amount of ink on the paper?  Beer’s law takes over, and we see that blue light is reflected at 40% X 40% = 16%.  Green light? The reflectance goes from 10% down to 1%. Red light stays at 100%. The table below summarizes the Beer’s law estimation.

Blue Green Red Thin layer 40% 10% 100% Thick layer 16% 1% 100%

From this table, it would seem that the thick layer of magenta is a lot closer to red. The plot below shows the actual spectra of two magenta patches, one at a larger ink film thickness than the other. The plot leads one to the same impression – that a thick layer of magenta is closer to red in hue than a thin layer.

Spectrum of a magenta ink, normal thickness and thick

The tentative conclusion is that magenta turns red when it is thick because it is impure, or more accurately, because there are several different reflectance levels in the spectrum. When Beer’s law kicks in, the areas of the spectrum where the reflectance is “mid-level” (i.e. 40% reflectance) are grossly effected by the ink film thickness.

The plots below are the spectra of cyan and yellow inks. If the previous rule applies, then we would expect that cyan ink will have an appreciable change in hue as it gets thicker. From the plot of cyan ink, we see that the reflectance values between 500 nm and 600 nm are “intermediate”, somewhere between the highest value and the darkest value. This is the green range. As cyan ink gets thicker, we would expect the amount of green light reflected to drop.

Thus, based on Seymour’s rule of ink hue shift, a quick look at the plot below would suggest that thick cyan ink will be blue, just like thick magenta ink will be red. Yellow ink has very little in the way of intermediate values. It basically has either 75% reflectance or 3%. From that, you would guess that yellow ink will not change in hue. Note that a bucket of yellow ink does indeed look yellow.

Plots of cyan and yellow ink

But spectra can be a bit misleading when trying to discern color. I don’t know many people who can look at a spectrum and tell what the color is. So, I offer a little computational experiment to further validate Seymour’s rule of ink hue shift.

First, I will show the results. Then I will explain how I got them. The chart below shows the a*b* values of a set of ten magenta patches with increasing ink film thickness. These values are the ten blue diamonds in the plot. There is clearly a strong hook. The first five are pretty much along a line without much hue shift. The sixth one goes around the bend, and the last four are changing a lot more in hue than they are in chroma.

The magenta hook, real and estimated For the other views of this data, I have published an addendum to this blog post.

The magenta colored line in the plot is a prediction of what I call the “ink trajectory”. This is the set of all L*a*b* values that an ink will go though as you change the ink film thickness. To compute this estimated trajectory, I started with the spectrum of the sixth patch and that of the paper. (You will note that the magenta line goes right through that point.) I loaded these spectra into a spreadsheet, and used Beer’s law to estimate the spectrum over a range of ink film thickness. You will note that the estimated trajectory comes reasonable close to predicting actual measured values, and definitely predicts the hook.

For those who want more detail, I have a little more description below. This is excerpted from a paper I presented at TAGA in 2008.

Excerpt from Building a Bridge from Dense City to Colorimetropolis

This pretty well settles it in my mind. Magenta ink on paper is magenta. Magenta ink in a bucket is red. I have explained this with some simple ciphering with Beer’s law. This led me to define Seymour’s rule of ink hue shift, which allows you to tell (just by looking at a spectrum), whether an ink will have an appreciable hook.

I then showed some really, really impressive results that show that, armed with just the spectrum of your paper and that of your ink on that paper, you can determine the magenta hook. This is clearly a triumph of modern science.

I have come a long way since I was ransacking the printing plant to find those elusive halftone dots!

Caveats

This is where I admit to some of the lies in the previous section.

First off, Beer’s law is only an approximation. It makes the simplistic assumption that a photon will either pass right through the ink, or get absorbed. It does not make allowances for photons that reflect directly from the surface, or for photons that bounce around a bit in the ink and maybe come out of the ink without ever having visited the paper.

Despite those simplifications is does fairly well. For the standard process inks. I do not have data to see whether it works for Pantone inks. If anyone has a cup of data to spare…

One limitation that I glossed over is that it does not do well at predicting the reflectance of a double layer of ink. Us folks in the know like to say that ink is “sub-additive”, which means that Beer’s law does not do well at predicting the reflectance of a double layer of ink. It will, however, give you a spectrum that is attainable, however. Just not at that particular ink film thickness.

Well, that was kind of a lie as well. There are limitations, especially when you get up to the very high densities. You will note that my hook graph fits the data pretty decently, but it would not be nearly so good if I tried to predict the lightest density from the darkest, or the other way around.

There is one more lie, or one more pair of lies actually, but they are subtle. I demonstrated two ways of deciding whether the spectra of magenta showed a hue change. The first way was kind of hand-wavy. “Look at the spectra and see that it looks a lot like red. Ignore the little bump behind the curtain at 450 nm.”

Well, this argument may fly for someone who has not spent thousands of hours looking at spectra. But, if you have devoted a lifetime to deciphering spectra, you would know that sometimes the stuff happening down at the dark end is important. That little bump at 450 nm might just have a big effect on the color.

In this case, it didn’t. Converting to CIELAB demonstrated that the magenta is definitely turning red.

Or did it turn red? This is where the lie gets very subtle. We are trained from childhood to believe that colors with the same CIELAB hue angle are actually the same hue. But I have stubbornly disagreed with this all along. My first grade teacher almost flunked me over this point. I was glad to come upon a paper by Nathan Moroney where he made an off-hand comment that agreed with me.

The issue has to do with the fact that the CIELAB formula performs a nonlinear function on the XYZ values, which are a linear combination of the actual sensors in the eye, but which probably don’t actually exist in the eye or the brain. But that is grist for another blog.

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[1] Yogi Berra said “In theory there is no difference between theory and practice. In practice there is.”

[2] I am still baffled as to why there are so few people who both know me and love me. Why is there no intersection between these two sets?

[3] Both words came into our language relatively late. Magenta became a word shortly are 1859, and cyan became a word in 1879. You wouldn’t expect them to become common words that quickly, would you? After all, look how long it took “internet”, “email”, and “perifarbe”  to become common words.

[4] “Hazing” of course is some sort of print defect for gravure printing. Nothing to do at all with the old guys picking on the newbie.

[5] Popular? So far, seven people have read the Beer’s law blog post. Well, I should clarify. Seven people stumbled upon the blog post. It is perhaps optimistic of me to expect that all seven of them took the time to actually read the blog rather than just look at the really cool pictures.

[6] Standard process magenta ink is not all that perfect, and there is a ”magenta” ink that is a bit closer to perfect: I am exaggerating just a tiny bit about the price of the alternative. I have not checked the price of Pantone Rhodamine just lately, but I think I can hook you up with a guy who can get you a gallon for something less than $80K a gallon. Unless of course, you are looking for ink jet ink.

One Beer's law too many

Some people may think that Beer’s law has to do with underage drinking, and that August Beer is what comes before OktoberFest. Beer’s law is, however, one of the coolest laws of photometry, and August Beer is the guy who it is named after. (For a complete discussion of how it got that name, skip to the end of this blog post.

This blog post is a re-enactment of a seminal experiment that a preeminent researcher reported on back in 1995. This phenomenal scientist has had such a profound influence on the worlds of printing and colorimetry, that I am tirelessly committed to the promulgation of his work. I am speaking, of course, about myself.

Experimental setup

The picture below details the equipment to be used in this experiment. At left is a constant current power supply, which provides power for the blue Luxeon LED. This LED shines into the optical assembly, which is supported by one of the biggest books I have on color science. At the far right is the sensor for an expensive light sensor, with the control unit show on the expensive black carpet. The observant reader will no doubt be impressed by the huge expense that I must have gone through to dig this pile of junk out of my basement.

Expensive equipment used for this experiment

The lights were turned down and the system calibrated so that the light meter read 100.0 banana units when there was nothing between the light source and the detector, as shown below.

Expensive optical stuff, bored, with nothing to read

Now the party begins. I cracked open a cold one and set it in the beam. Note that the reading has dropped to 90.0 banana units, indicating that 10.0 banana units of light got caught by the amber fluid and never quite made it home. I can definitely identify with these photons.

Same set up, but with one sample cell

As they say, you can’t milk a camel while standing on one leg, so let’s order another one. But before it gets set down on the bar, let’s take a guess at what the light meter will read. Hmmm…. The first sample dropped it by 10.0, so it would make sense that the second one would so the same. My guess at the results: 80.0.

Same set up, only this time with two samples

For those of you who agreed with my guess, it was commendable, but wrong. There was indeed a pattern established, but not the one you were thinking of. Why did it go down to 81.0, instead of 80.0? For every 100 photons that entered the first sample, 10 of them were absorbed, and 90 were transmitted on to the second sample. Upon reaching the second sample, the same probabilities apply. Of the 90 photons that made it to the second sample, 90% of those made it out, so that there were 81.

You now know Beer’s law.

But just to make sure the concepts are all down, let’s take this one step further. How about three samples? 90% X 90% X 90% = 72.9%, as verified by the highly sensitive experimental set up below.

Results for three samples

One last thing… Can you guess what kind of beer was used?

Miller Lite – the official beer of color scientists everywhere

Disclaimers – Do not try this experiment at home. I am a trained professional. The mixing of beer and scientific equipment is not recommended. No beer was wasted in the photoshoot for this blog. I cannot say the same for the scientist who performed the experiment.

Who invented Beer’s law, anyway?

Some folks may have just assumed that Beer’s law was named after William Gosset, who was a pioneer in statistics, and worked for Guinness. That would be a good guess, since he was a smart guy. It would have been just like him to have a really cool law of physics named after him, since he invented the t test, which was named after Student, which was actually his pen name. But that’s another interesting story.

The guess is unfortunately wrong, since Beer’s law was named after August Beer. This is yet another in my series of mathematical misnomers.

This law of physics was first discovered by the father of photometry Pierre Bouguer in 1729. August beer didn’t discover this law until over a century later in 1852. Beer worked with  Johann Heinrich Lambert on a book (“Introduction to the Higher Optical”) that was published in 1860. So naturally, the law has become known as “Beer’s law”, “Beer-Lambert law”, “Beer Lambert-Bouguer law”, “Lambert-Bouger law”, “Lambert’s law”, and “Bob”.

Why is it known in the printing industry as “Beer’s law”? There are two key influences that led to this egregious misnomer. The first was a landmark 1967 book by J.A.C. Yule, “Principles of Color Reproduction”. Any book with the word “reproduction” in the title is apt to move quickly. I just checked with Amazon. They only have two copies left.

The second thing that probably had an even greater effect was the frequent use of the eponym by the eminent applied mathematician, color scientist, mathematics historian, and all around nice looking guy, John “the Math Guy” Seymour. He has made no bones about why he decide on this name among all the potential candidates. I quote here from his paper delivered at the 2007 Technical Association of the Graphic Arts:

Since there seems to be little agreement about who is responsible for which law, I have chosen to refer to the statement that optical densities of filters add as Beer’s law. My decision is not based on historical evidence, but on the gedanken I introduced in a paper given at IS&T (Seymour, 1995). In this, I demonstrated the law by using a varying number of mugs filled with beer. My hope is that my further corruption of already corrupt historical fact will help remember the law!

Brilliant words by a brilliant man, indeed.