Wednesday, May 29, 2013

What? Are you tone deaf???

I would guess that all of us who are musically inclined have had the unpleasant experience of singing (or trying to sing) next to someone who just could not get the notes. I'm not talking about the people who can't quite reach up to an E or an F and wind up just a bit flat. I'm talking about the folks who just can't sing the right notes even when the pitch is well within their range. These people are tone deaf.

Florence Foster Jenkins was one such lady with a tin ear. She was a lady of society in the early 1900's who aspired to be an opera star. And she had the funds to back her own performances. A bad combination. She appeared once at Carnegie Hall, and sold out the venue with people wishing to participate in the ridicule.

Jenkins was apparently completely unaware of her total inability to sing on pitch. She thought that people who ridiculed were just jealous. In the play Souvenir, there is a scene where her lifetime pianist is mortified that she was going to listen to a recording that had been made of her. He relented, and she heard her voice for the first time. She was thrilled.

What causes this? How can people not know?

Where hearing takes place

Hearing takes place deep in the ear. People talk about the ear drum, and the hammer and anvil and stirrup and I dunno, the spurs and the staple gun. All those things are great to have in one's inner ear, but the actual sensing of sound comes in that snail-like thing called the cochlea.

Something inside your ear, and something that would be gross to have in your ear

The cochlea is a long tube, filled with hairs. It just happens to be twisted around into the shape of a snail shell, but that's not a functional feature. It just conserves space. If you were to stretch the cochlea out and slice it open, it would look exactly like the diagram below. Absolutely identical. Sound travels from the left to the right. The left side of the picture is near where the sound enters, and the right side is where the tube ends.
Actual photomicrograph of the inside of the cochlea
Ok. I lied. This is not a perfect representation. It is a conceptual drawing. 

But anyway. Each of the hairs has its own resonant frequency due to its size and stiffness. When a specific frequency of sound enters the cochlea, it will set certain of the hairs to vibrating like a bunch of little tuning forks. The vibration is converted to nerve impulses and transmitted to the brain. The louder the sound at that frequency, the more the corresponding hair vibrates, and the stronger the signal passed to the brain. Thus, the cochlea is performing frequency analysis on the incoming sound wave.

This next image is another conceptual drawing, illustrating the mapping between frequency and position along the tube. The opening of the cochlea is sensitive to sound around 16 kHz, which is the upper limit of our hearing. The hairs that are one quarter of a trip around the spiral are sensitive to sound one octave below this, that is 8 kHz. Each subsequent quarter trip around the spiral represents another octave drop, or halving of the frequency.
Where the frequencies hit the hairs in the cochlea
This continues over the entire span of human hearing, or about ten octaves - a factor of 1,024 in frequency. Each octave turn around the spiral has at least 12 discrete steps, since we are able to readily distinguish 12 half steps in a musical. Our frequency resolution is not likely to be more than a factor of five more than this, however, since a half-step feels fairly small.

I should point out that an octave is not necessarily one quarter spin around the spiral. That's just my guess. The salient point is that the individual frequency receptors are spaced out logarithmically.

Maybe the hairs are not working?

If someone is tone deaf, then perhaps the cochlea is not functioning according to spec? This is conceivable... but I'm going to argue that it's not likely.

Here's my argument: if a person can tell the difference between "oooooo" and "ahhhhhh", then the basic hardware must be working. 

I sang an oooooo into the microphone on my laptop and then did a little frequency analysis. The graph below shows the amount of energy at each of the frequencies from 0 Hz to 1.4 kHz. For the "oooooo", there are two spikes: one at the fundamental frequency of about 140 Hz, and the other at one octave above that, at about 280 Hz. The second frequency is called the first harmonic. Almost everything (vocal chords included) vibrates in a complected way that includes multiple frequencies that are multiples of a fundamental frequency [1]. 

Oooooooooo
The diagram below shows roughly which hairs in the cochlea will be stimulated by the "oooooo" sound.
Hairs that  respond to an "oooooo"
I also recorded and analyzed an "ahhhhh" sound. The frequency breakdown is shown in the next graph. The difference is startling. The ahhhhh has the fundamental frequency, and the first harmonic, but it also has an appreciable amount of energy in the third through ninth harmonics [2].
Ahhhhhhhhh
Here is my argument again. If a human ear is capable of differentiating between the oooooo and ahhhhh at a fireworks display, as well as eeeeee, and oh, and also mmmmmm, and nnnnnn, and ellllll... then it is likely that a lot of the mechanism in the cochlea is intact. The fault must lie somewhere in the brain. The part of the brain that decodes sound for the purpose of speech must be getting everything it needs, but the part of the brain that interprets sound as music must be broken somehow.

What do the real scientists say?

A recent article in the Journal of Neuroscience remarkably provides support for my back of the napkin analysis. The folks who wrote this paper had a look at the "arcuate fasciculus" portion of the brain. The AF is a pathway that connects the part of the brain that controls our perceiving of sound with the part of the brain that controls the production of sound. There are two such pathways. People who are tone deaf lack one of these connections. The other connection, presumably, is responsible for distinguishing between ooooos and ahhhhhs.

This answers the question of what causes tone deafness. It's not a lack of training, or a problem in the ear itself, but an anomaly in the brain.

Thanks to Rachel for getting me thinking about this question!

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[1] Softly blowing across a pop bottle gives a fairly clean tone that has just the fundamental frequency. If you  were to simultaneously blow across two bottles, tuned one octave apart, the result would sound similar to an oooooo.

[2] This is equivalent to blowing pop bottles tuned to C, the C that is the next octave up, the G above that, the C that is two octaves above that, the E, G and B flat of that octave, the C that is three octaves up, and the D above that.





Wednesday, May 22, 2013

My favorite "photon walked into a bar" jokes

It wasn't that long ago that I posted about the fate of four photons: one reflected from the surface of the beer, one passed right through the beer, one was absorbed into the beer, and the last one got scattered. Naturally, that got me thinking about whether I could come up with four golly whumpus knee slappers about the quirky behavior of photons in a bar. I got more than four. A few of these tuned up in Google searches. The rest are original.

An infrared photon walked into a bar and said, "is it hot in here, or is it just me?"

An xray photon walked into a bar. The bartender says, "can I get you something to drink?" The xray photon said, "no, I'm just having a look inside!"

A green photon walked into a bar. The bartender said "you look fluorescent!" The photon turned red, and left.

A pink photon walked into a bar. The bartender knew that pink photons don't exist, so he said, "say... you know... we don't get many pink photons around here..." The pink photon shot back "Not at these prices you won't!"

The bartender was fed up with the revelry of the photons, so he turned out the lights and they were gone.

A photon stopped at the bar and asked if there was a room to rent. The bartender said "Sure thing. Can I take your bag up to your room?" The photon said "no, I am traveling light."

A 450 nanometer photon walks into a bar. The bartender says, "why so blue?"

A photon walks into a bar. The bartender says, "what'll you have?" The photon says "light beer".

Wednesday, May 15, 2013

Is your green the same as my green?


Here is a common question when I give a class on color theory: "Do we all see color the same way?"

Often the question starts with the words "My wife...", or "My husband..."  There is, at the very least, a perception that men and women see color differently. But what about me and some other guy? Do we see color the same?

There are lots of answers to this question: How much variability is there is color perception? That makes it fun to blog about!

High level evidence of gender differences

There is a well known anecdote that if you ask a guy about his car, he will give a lengthy discourse about the engine. A woman will tell you the color. If you can get a guy to mention the color of his car at all, he will use basic color names like "green" or "blue". A woman will use a nuanced color name like cerulean or mauve to describe her car.

My wife and I, discussing automotive colors

I am not aware of any experimental evidence of gender bias in car automotive descriptive preference, but I know of a brilliant study from a brilliant color researcher [1]. He provided some evidence that somewhere between a photon entering the eye and a neuron recalling a word, there is some difference in the way men and women perceive color. This incredibly smart and good-looking color scientist asked 50 people to write down all the single word color names that they could think of in two minutes. Here is his conclusion:

"There was a statistically significant difference between the number of colors that the men could recall versus the number that the women could recall. Men averaged 15, and women averaged 18."

(A bit more on this groundbreaking experiment can be found in the section "Eleven" in the blog post "How many colors are in your rainbow?")

Where does this happen? Is this in the physiology of the eyes? Or maybe it's a brain thing -- woman are, on average, better at verbal skills than men? Or is it a cultural bias because women in general are encouraged to be interested in fashion?

Let's consider the differences in physiology...

Color blindness

Color blindness is one explanation I have heard for why there is a gender difference in color perception. Men are more likely to be colorblind than women. Dogs are colorblind. Therefore men are more likely to be dogs than women.
Comparison of how I see apples, and how my colorblind dog sees apples

It's true that women are less likely to be colorblind. Color blindness is a sex-linked trait [2]. The weakness is something expressed on the X chromosome. Basically, you need to have one copy of the color vision genes on either chromosome to have full color vision. Since women have two X chromosomes, they have two chances to get all the pieces for color vision. Men, who only have one X chromosome, only have one shot. As a result, color blindness is more prevalent in men than in women. About one man in a dozen is colorblind. For women, it's about one in 200. [3]

As I said, this is true, but it's not an explanation for why there is a gender difference in the color naming experiment. Even with the elimination of any possible colorblind men from the experiment, the results were still statistically significant. And what about the 11 out of 12 men with normal color vision who still prefer to talk horsepower and miles per gallon?

Tetrachromacy

As I described in Organizing your crayons, color is three-dimensional. The reason for that is simple. We have three types of cones (light receptors) in our eye. Each cone responds to a different range of wavelengths. These three cones are called "L" (which stands for red), "M" (which stands for green), and "S" (which stands for blue). [4]

There is some controversy on this, but there are reportedly a number of people who have four different types of cones. To them, color is four-dimensional. 

Just to give a feel for what that means, it is a rough approximation that each cone allows us to distinguish about a hundred levels. For a combination of two cones, it is possible to distinguish 100 X 100 different colors. (For those who don't have a calculator handy, that number is 10,000, or ten thousand.) This is a rough idea of the number of colors that a dichromat (someone who is color blind with two types of cones) can see. 

For most of the population, there are three types of cones, so there are 100 X 100 X 100 different combinations that could be distinguished; one million perceptible shades of color.

A tetrachromat has four different types of cones. As a result, there are one hundred times as many different shades of color that this person could distinguish. One hundred million colors. Such a person could sense subtle differences in hue that I could only see with the help of LSD. Imagine how long it would take such a person to match their socks in the morning! [5]

Some folk's eyes are just more colorful than others

This is all pretty new research, but by one estimate, only a few percent of the female population (and none of the male population) has this super-human ability. This estimate came from a researcher who used to be right here in Milwaukee by the name of Jay Neitz.

Always wanting to make significant contributions to science, I did a little research on my own on the prevalence of this anomaly. I stopped a hundred woman in the mall and asked them if they were tetrachromats. Unfortunately, I had to curtail my data gathering when the police showed up.

As I have said, the existence of people like this is still under debate. As near as I can tell, the argument against goes something like this: "People who claim to be tetrachromats are just people looking for attention. If the whole tetrachromat thing didn't come up, they would claim to have been abducted by aliens, or would be singing Justin Beiber songs at a karaoke bar." Scary stuff indeed.

But... this still doesn't answer the question of why there is an apparent difference in the way men and two women see color. A few per cent of the female population is not enough to allow all women to be able to point to a color and tell whether it is ecru or oyster.

Physical variation in the eyeball

I just received a copy of the PhD thesis from a fellow by the name of Abhijit Sarkar [6]. The first comment I want to make--and this applies to everyone who is now working on a PhD thesis--is this: Why do you gotta make those things so darn long? And why do you gotta use such big words? I'm just saying... if you want your thesis to be tweeted about, you need to shrink it down to like two or three pages tops. And cut out all those darn equations, ok?

Dr. Sarkar describes several mechanisms within the eye that could cause two people to see color differently. The simplest one to describe is the yellowing of the lens in our eye. This happens gradually as we age. This yellowing has the effect of subduing the effect of light at the blue end.

I see the equivalent of this effect every day. My wife has our bedroom painted a pale lavender color. On a sunny day when the shades are open, the ceiling is a delicate pastel lavender. At nighttime, though, when we view the ceiling under incandescent light, the color is more of a gray [7].
The effect of a yellow filter on purple flowers

The yellowing of the lens in our eye has a similar effect. This effect is most striking for colors in the purple family. Sadly, as we grow older, we start to lose some of the subtle shades of purple. When I am old, I will wear purple, but I will need to have a younger woman around to appreciate it. [8].

Dr. Sarkar talks about a number of other mechanisms in the eye that act like filters to change the tint of what we see. His thesis is littered with phrases like "macular pigment optical density function". This of course, limits my ability to understand what he is talking about. But the big idea is that these tinted sunglasses that are built right into our eyes vary from one person to the next.

Luckily, this gentleman's thesis had a very clear and succinct abstract that even I could understand. There are differences in the make up of people's eyes [9] that mean we all see color just a bit differently. This effect, called observer metamerism, is a bummer. In industry, it makes it really darn hard to match colors. One person with "normal color vision" might say two colors match, and another may say they don't.

The whole crux of his thesis is that the problem is potentially solvable. His research says that people can be grouped into eight categories for their color vision, and he has developed a relatively simple test to categorize them. [10]

The practical application of Dr. Sarkar's work is when you are trying to match a color on a computer monitor with the color of a physical object held next to that monitor. Doing this is common in the printing industry, where people display what is called a proof on a computer screen. Customers sign off on a color based on this "soft proof", and press operators use that soft proof as a reference for when they are printing.

If you knew which category an observer fell into, it would be possible to adjust the computer monitor accordingly. There are some holes in implementation of this. In particular, I am not sure what happens when you have two people from different categories looking at the same screen.

But none of this sheds any light on gender differences. There is nothing in the thesis that is gender specific.

Conclusion

I don't have a final pronouncement on this, but I have yet to find a physiological explanation for why men and women seem to think about color differently. Until I hear differently, I am going to assume that if such a difference really exists, then periwinkle and wisteria and orchid and lavender originate somewhere in the murky canescent matter of my wife's brain, and not in the gray matter of my brain.


As for the simpler question, is there variability in the color response to different eyes, the answer is most certainly "yes".

----------------------------
[1] When I say something like "brilliant study from a brilliant color researcher", it is generally a euphemism for "I did this work". I prefer the euphemism because I am far too modest to take credit for it.

[2] I am not aware of any research that goes into how often color blind people get lucky, so I would prefer to call color blindness a "gender-linked trait" rather than "sex-linked".  On the other hand, there is a certain blue pill that is taken by some (often older) men that may cause a temporary color deficiency, and use of this blue pill is strongly correlated with getting lucky. I have heard of this pill, but have no personal experience to verify the effect. 

[3] That whole X/Y chromosome thing is a rip off, if you ask me. Men are also more likely to have hemophilia and male pattern baldness. I would give my manhood for another X chromosome!

[4] Ok... LMS stands for long, medium, and short, referring to the range of wavelengths that each accepts. It  is a simplification to call them red, green, and blue, since they are not exactly in the red, green, and blue region of the spectrum. The S cone is close to what we would call blue, but the M cone is broader than just the green part of the spectrum. And that darn L cone overlaps just a whole lot with the M cone. It would not be such an awful faux pas to call this the yellow-green cone. 

[5] If sock matching is an onerous burden for your lifestyle, I recommend checking out this iPhone app for sock matching.

[6] "Identification and Assignment of Colorimetric Observer Categories and Their Applications in Color and Vision Sciences", 2011 from Ecole polytechnique de l’UniversitĂ© de Nantes.

[7] Of course my wife keeps insisting that the color is neither lavender nor gray, but rather a grayish lavender.

[8] My wife objected to this statement. But I won't tell you which one of us is older.

[9] Just to be clear, he is not talking about makeup, as in eye shadow and mascara. He is talking about how the eye is built.

[10] In my own experience, there are two categories of people: those who like to be put in categories, and those who don't.








Wednesday, May 8, 2013

Calculating odds at the racetrack

My wife and I watched the Kentucky Derby this last weekend. She asked me a simple question that got me thinking. "How do they calculate the odds on the horses?" I told her, "Gosh, that sounds like the sort of question that John the Math Guy should be able to answer!"



So I pondered a while, scratched my head, and then scratched my beard. I almost got around to scratching my...but I had it basically figgered out, so I went online to see if my suppositions made any sense. Sure enough, I was right.

Before explaining how it is done, I'll give you a multiple choice test.

Q. How does a racetrack compute the odds for a horse race?

a) They determine the probability density functions of race time for each horse, based on previous performances, and corrected for the jockey and the condition of the track. These functions are then corrected for the horse's predilection to either go faster or slower when racing against other horses. Monte Carlo analysis is then used to simulate the probabilities of each horse winning.

b) They adjust the odds so that no matter which horse wins, the racetrack always get a certain percentage.

Disclaimer

Before I go on, I feel that I should share a bit about me, just so I don't give anyone the wrong impression. I am not a gambler, and I have never played one on television. I was in Vegas, once, for a business convention. I saw some good shows there, but the biggest bet I made was when I put three quarters in a machine, pushed the button, and got a can of Coke [1]. I figger I came out ahead. I was thirsty.

I did enjoy watching the race this last weekend, as I always do. The horses are beautiful, and the race is exciting. 

Now for the "Truth or Dare" part -- my embarrassing public confession. The thing I like about the Kentucky Derby? You really wanna know? It's the hats. I love seeing women with huge hats. The bigger and more ostentatious the better. I just go weak in the knees.

Any caption under this picture would be funny

Back to the topic

Now that I have that off my chest, I will resolve the suspenseful multiple choice test. The answer is b). The game is rigged so that the racetrack can't lose. Ever. Even if there is a huge payoff on a long shot. Should you feel sorry for the track cuz they lost a lot of money? Nope. They make pretty much the same percentage regardless of the outcome.

So, just how do they figger?

Let's take an example. Let's say that the track brings in one thousand bets on a given race, each a bet of one dollar. [2] That means they have $1,000 coming in. Each track has it's own "take", but let's just say that this particular track has a take of 15%. They will automatically pocket $150 before doing any ciphering. This leaves $850 for the payoffs.

Now let's just say that The Lovely Mare in the Big Hat received a total of $200 in one dollar bets. The track has to consider, in the unlikely case that this nag actually makes it to the finish line without having to go back to get her hat, what can we afford to pay? The pot is $850 dollars, and it has to get split 200 ways. That means that everyone who bet on The Lovely Mare in the Big Hat should get $4.25 for each $1 bet they made. That's how they set the payoffs. It's that simple.
The fillies in the lavender hats are always my favorites

Let's take another example... Let's say that I Can Barely Get Outa The Stable in the Morning has a grand total of four bets. Hmmmm.... an $850 pot split evenly between four tickets? I would say that each ticket should  be paid $212.50.

Aside from a bit of rounding (which is always done in the favor of the track) this is how the payoffs are calculated. No fancy math, no determining of probabilities, or looking at the weight to stride ratio of the forelocks and fetlocks. No Bayesian analysis or conditional probabilities or Markov chains.

But I left one little confusing thing out, the special lingo that is used to explain what the odds are..

What does "6 to 4 odds" mean?

Here is something I learned today. Odds are communicated in a funny way. Go figger. When someone says that the odds on a given horse are 6 to 4, it means that if the horse wins, I will have a profit of 6 dollars for every 4 dollars that I invested. I find that confusing. Of course nine out of every seven adults have trouble with fractions, but odds are no stated as a simple fraction.

I'll try to 'splain it again. Odds of 6 to 4 means that if I put 4 dollars in, I will get 10 dollars if I win. 4 dollars of that 10 will be returning what I put in for bet, and 6 dollars will be the payoff.

In the case of the $850 pot, split 200 ways... I said that a $1 bet should get a payout of $4.25. In other words, a $1 bet gets a profit of $3.25. Thus, the odds would be 3.25 to 1. Maybe this would get reported as 13 to 4? Or maybe it would be rounded (in favor of the racetrack) to 3 to 1. I dunno.

Another thing that makes me go "hmmmmm"

I wonder...

I have made the point that the racetrack doesn't use any historical information on horses and jockeys to figure odds. They don't go through any fancy math to determine the probabilities that each horse will win.

Now that I said that, I'm going to disagree with myself.

Consider this. Let's say that the odds for Old Stewball [3] are hanging at 50 to 1. That means, of course, that a $1 bet will return $51 if you win. Now, the folks at the track will have been doing their analysis. Maybe some will look up Stewball's track record. Some will have a look at the jockey to see if he has fire in his eyes. Some will just say "What a cute name! My parents had that song on an album!" Based on their individual analyses, the folks at the track will put money on Old Stewball, or not. Essentially, they will use their dollar bills to vote on whether the odds are too low or too high.
Old Stewball might be just a bit long in the tooth [4]

The final odds are a crowd-sourced estimate of the probability that Old Stewball will win the race. The racetrack uses a sophisticated crowd-sourcing algorithm to figure the odds.

I find that pretty cool. Of course, I think the whole idea of doing crowd-sourcing is cool. One may think that crowd-sourcing is a new thing that came about only after the internet, but as I recall, there were racetracks around long before TCP met IP.

It makes me wonder... how accurate is this crowd-sourcing?  I will take a wild guess that any race that brings in a lot of professional gamblers will probably wind up with fairly decent estimates. Sounds like an interesting question to throw some data at.

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[1] A can of Coke from a vending machine for 75 cents?  Yes, this was a long time ago.

[2] Yeah, I know there's no such thing as a $1 bet.  I was just trying to make the math more simple for all of you who aren't John the Math Guy.

[3] Bonus points to anyone who recognized this horse name from the song by Peter, Paul, and Mary or Woody Guthrie. If you caught this, send me an email (john@johnthemathguy.com) and I will email you back with one bonus point.

[4] The phrase "long in the tooth" actually came from horses. Unlike people, equine teeth grow their entire lives. If an old horse is not getting enough heavy chewing in, the vet will have to go in with a rasp and file down the back teeth. Honest. I've been the guy holding the tongue.

Wednesday, May 1, 2013

What color is water?

Four wise people were asked a simple question: What color is water? Their answers help explain the physics that imparts color to ink.
The four colors of water

The four colors of water

I asked my friend "Dennis the PhD Chemistry Guy" what color water is. "Well John the Math Guy, pure water is clear." Now, Dennis is a smart guy. I would even go so far as to say that Dennis is a pretty darn smart guy. But I don't remember seeing a crayon labeled "clear" in my disorganized box of crayons. Maybe if I had a PhD in chemistry I could afford the primo box of crayons.
Dr. Dennis calmly explaining to an agitated John the Math Guy that water is clear

So I asked my friend "Bruce the I've-Got-A-Sailboat Guy" what color water is. He told me about the bluest lagoons in the world in the Cayman Islands, and snorkeling in the blue depths in Bermuda, and the bar in Aruba that makes the best blue Hawaiians. "Dude! Water is bluuuueeee. Doncha see it, man? Blllllooooooooooooo!"  I started to get flashbacks to this one Jimmy Buffet concert that I don't remember, and hope to never remember. I quickly left to seek out another sage.

Sea Turtle, pic taken just before he ate my friend Mark [1]

Lo and behold I ran into another friend of mine, "Frosty the Snow Guy". Such a cool-headed guy should be able to tell me what color water is. "Snow is water, and snow is white. Clouds are water, and clouds are white. So, my warm-blooded friend, water is white." Now there's a crayon I can hold onto.

Having run out of friends to ask, I sought the advice of one of the most smartest people I know, "Sammy the Neighbor Kid". Without a second's deliberation he told me. "Water is the color of whatever KoolAid mommy puts in." So, I sat down to have a cold one with Sammy and ponder the fates of the photons. He had a tall lime KoolAid, and I had the same with just a splash of Jose Cuervo. 

The last thing I remember, the bartender was waking me up for last call and I asked him about the color of water. He looked at me quizzically. "Water? Never hoida da stuff." 

The fates of the photons

For the uninitiated, a photon is a tiny little itty-bitty piece of light. The smallest little particle of light possible. How small is a photon? A photon is smaller than the weekly paycheck I get for writing these blogs!

As the joke starts out, four photons walked into a bar...  A lighthearted joke. Very illuminating. I wish I could remember the rest of the joke.

Let's say that the four photons of the apocalypse are shining down ink on paper [2]. The first photon just glides through the ink, bounces off the paper, and glides back through again [3]. For him, ink is clear. In general, not many photons get off this lucky. It might be that only 1% of  them do. But, for certain inks and colors of photons, a lot of them see the ink as transparent. For example, a red or green photon hitting yellow ink is unlikely to even notice that yellow ink was in its path.

The second photon enters the ink, and something or other distracts it. My wife tells me that she can identify with this photon. It may have been that very cute molecule in the very tight shirt? Or the photon might change direction so as to avoid the much more stern Lord Rayleigh [4]. In a typical ink at a typical thickness on a typical substrate on a typical day, very few photons will do much in the way of changing direction (scattering). For this photon, the ink might be clear, but it depends on where she goes after changing direction.

The third photon sadly never makes it into the ink. He bounces right off the top. For him, the ink is white. We call this photon by the name of gloss. Roughly 3% to 5% of the photons hitting the ink suffer this fate. Now, you might think that the ink on a glossy cover of a fashion magazine might have a lot more specular, (surface-reflected) photons than the ink on a dull, matte stock like a dull newspaper. But you might be wrong. The big difference between a glossy and a matte stock is not the amount of surface reflection, but the direction that the photons head after reflecting from the surface. See my blog post on flat paint for more explanation. 

The fourth photon is the most glamorous photon of all, since she is the one who imparts color to the ink. This photon encounters "wavelength selective absorption". This means that sometimes the photon will get gobbled up by a molecule of pigment, and sometimes not. The really cool thing is that the likelihood of  being gobbled up depends on the wavelength of the photon. For example, if the photon resides at the red end of the spectrum, she is likely to get gobbled up by cyan ink, but is not so likely to get gobbled up by magenta ink.

Thus, there are four potential fates for a photon when it comes upon some media, like glass or water or ink or bubble gum. It may reflect from the surface, scatter within the media, get absorbed in the media, or transmit through the media.

What you see in the magazine

When we look at ink on paper, we are viewing primarily the effects of the first and fourth photon paths: transmittance and absorption.

Paint is designed to cover whatever colors are beneath it. To make this happen, they add lots of little particles of stuff like titanium dioxide to scatter light. On the other hand, for ink, you want exactly the opposite. When you put yellow ink on top of cyan, you want to be able to see the combination of the two inks - green - and not just yellow. In this magic way, we can see a wide gamut of colors when we print one ink on top of another. Thus, in ink, there is little effect of the second path. Photons in ink are not easily distracted like my wife.

The specular light, that is, the light following path three, is the color of the incident light, so it imparts no information about where ink is on the paper, and what color it might be. Thus, we generally tilt a glossy magazine so as to avoid seeing the specular light. Ink on a non-glossy stock, on the other hand, scatters the specular light in all directions so that we cannot tilt a newspaper so as to avoid this contamination. This puts an upper limit on the richness of the color of matte objects.

Those of you who are regular readers of my blog might be feeling a bit thirsty right now. You might have been subliminally reminded of a certain related blog. If you happened to be thinking about beer and Beer's law, then you had correctly come to the conclusion that the first and fourth photon paths are described very well by Beer's law.

When light enters the ink, the probability of being absorbed depends upon how long it spends in the ink. The longer the trip - that is, the thicker the ink - the more likely it is that the photon is absorbed. The thicker the ink, the darker the color. If we ignore the effect of surface reflection and scatter, then Beer's law describes it perfectly. I have, of course, previously alluded to the fact that other factors limit the accuracy of Beer's law on ink.

I'm thirsty. It's time for me to experiment with wavelength selective absorption.
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[1] Mark is married to Teal, who technically is more of a shade of green than blue. Mark took this gorgeous picture while snorkeling off some island that I never heard of. Both Mark and Teal are alive and well, and so is the turtle as far as I can tell.

[2] Naturally, the bar is a color bar. For those not in the print industry, a color bar is a stripe of color patches that resides between pages of a printed product. These are used for quality control.

[3] "Some photons' lives roll easy..." http://www.youtube.com/watch?v=kbagtXi4HBs  Love that song.

[4] Rayleigh was the lead singer for the group The Sky Blue Scatterers. Their popularity peaked about noon, and then trailed off into the sunset.