Tuesday, August 27, 2019

Where did my indigo (part 2)

I answered a question on Quora recently. What is the true color of indigo? There are many answers to this question. In my last post, I gave one of them: indigo is a dye. This blog post expounds on Isaac Newton's answer.

We all know the seven colors of the rainbow: ROYGBIV. For those who missed that day in kindergarten, this is an acronym for red, orange, yellow, green, blue, indigo, violet.

But, what slice from the rainbow does indigo get? I consulted my high school optics book, Hardy and Perrin. It told me that indigo is the slice from 446 to 464 nm.

Question answered! Indigo is a very specific slice from the rainbow, which has been scienterrifically defined. 👍

But there is more to the story
But, as you might expect from my blog posts, there is more to the story. I didn't mention that this reference book, Hardy and Perrin, was from 1932. I also didn't mention that I had to dig deep to find this definition in any physics textbook. Perhaps I am being just a tiny bit disingenuous by implying that the definition "446 to 464 nm" is today's scientific consensus, when practically every science textbook I could find neglects to define indigo?

I dug my handy monochromator out of the closet and dialed in 455 nm. This is the location smack dab in the middle of the range that Hardy and Perrin gave for indigo, so that should give me the truest indication of what indigo is.

I looked at it and said, "Oh. It's blue." I asked my wife (who claims to be the most color-literate person who I know) what the color was. She said "blue", and then modified it to "cobalt blue", and then pointed at a lovely flower vase of hers. "Did you notice that there aren't any flowers in my cobalt blue vase?" She smiled and batted her eyes. I'm not sure what she meant by that.

I took a picture of the monochromator output with my cellphone camera. There are lots of caveats here, like RGB cameras don't do a good job at measuring color, and computer monitors don't always produce reliable color, but it kinda looks like my camera is identifying 455 nm light as blue.

Hmmmm... Who had the crazy idea of naming that part of the rainbow indigo? Why not just call it blue?

Newton's rainbow
To answer my rhetorical question, Isaac Newton was the person who had the crazy idea of naming part of the rainbow indigo. In 1665, Isaac Newton took leave from his schooling in Cambridge in order to escape the Great Plague. His work over the next two years proved to be one of the most productive in the history of science. Beyond the whole bit about the inverse square law of gravity, and inventing calculus to do that, Newton made some fundamental observations about light during his sojourn.

He passed sunlight through a prism and demonstrated that, among other things,

       white light is comprised of a whole lot of different flavors of light,

        a prism doesn't impart color to the white light that passes through (as was thought at the time), but rather bends light by different amounts depending on the flavor, and

        these individual flavors could be recombined to make white, or to make a host of other colors if some of the flavors are left out.

This remarkable time period is when he labelled the parts of the rainbow, or rather, the parts of a spectrum projectected on a wall through a prism. The results of his experiments were published by the Royal Society in 1672. These results challenged some long-held notions about light and color... but that's a good topic for another blog post.

Blue vs. blue
This may sound like a change of topic, but bear with me here. In English, there are eleven basic color terms (BCT): white, black, gray, red, orange, yellow, green, blue, pink, brown, and purple. I have written about these before when I tried to identify unambiguous color names. These basic color terms have been the subject of much research since the work of Kay and Berlin.

I offer a quote from a brilliant scholar, one who I respect immensely. I just can't say enough about the guy. Here is the quote from the esteemed John Seymour in his blog post How well do we remember color?

"In some languages, such as Russian, Japanese, and Italian, there is a separate word for light blue which stands on its own as a distinct color."

I have another quote, this one from a guy who is actually a real chromolinguist. Again, I respect him immensely. This is from Dimitris Mylonas

"[Two studies]  found that Russian and Greek languages both have 12 BCTs, differentiating ‘light blue’ from ‘dark blue’."

In Russian, we have two monolexic (single word) names for blue: синий and голубой, which mean "blue" and "sky-blue", respectively. In Japanese, the same concepts are in the words kon and mizu.In Italian, there is blu and azzuro. In Greek, kyaneos refers to dark blue, but it could also mean dark green, violet, black or brown. The ancient Greek word for a light blue is glaukos.

Consider the plight of Isaac Newton when he was trying to assign names to the colors of the rainbow. He looked at this wide expanse of colors which slowly pass from violet to green. If he had been conditioned by being a native speaker of Russian, Japanese, Italian, or Greek, then it may have been obvious to him to call the colors violet, dark blue, light blue, green, and so on.

But Newton spoke English. He did not have basic color terms for the two different types of blue. He had three choices.

1. He could use the terms dark blue and light blue. He probably felt this was kinda dorky. Or at least awkward. Or maybe he just wanted all the color names to be monolexic.

2. He could have chosen blue for the dark blue and found another name for light blue. I don't know what color names were in vogue at the time, but today, he might have used: aqua, aquamarine, azure, baby blue, cerulean, cyan, robin's egg blue, sky blue, teal, or turquoise. But these all strike me as being somewhat ambiguous.

3. He could have instead chosen blue for the light blue, and then found another color name for dark blue.

Newton went with option #3, and chose indigo as the name of dark blue. Here is a quote from Newton:

"So there are two sorts of colours: original and simple colours and colours made by compounding these. The original or primary colours are red, yellow, green, blue, and a violet-purple, together with orange, indigo, and an indefinite variety of intermediate shades."

Why did he pick the word indigo? I have a suggestion that I am dyeing to share. If I may be so bold as to quote the blogger who wrote the first post in this series: "Dutch ships carted nearly 170 tons of the [indigo] dye from India to Europe in 1631." Newton did his work with prisms in 1665. Indigo dye was quite popular in Europe at this time. So (my contention) is that the word indigo was at the time associated with a dark blue dye and that it was a common word at the time.

So, indigo is just another name for blue, only bluer than blue can be.

But, why didn't Newton just call them both blue? Why not just six colors for the rainbow? That is the thrilling question I will answer in my next blog post on this subject!

An earlier version of this post omitted green from the list of basic color terms. I apologize to any verdiphiles who were offended. Robin Myers is to be thanked for his ever-vigilant corrections to my blog posts.

Brent Berlin and Paul Kay, Basic Color Terms: Their Universality and Evolution, CSLI Publications, Stanford, California (1999)

Arthur C. Hardy, Arthur C.  and Fred H. Perrin, The Principles of Optics, McGraw-Hill Book Co., Inc., New York. 1932, p. 16

Dimitris Mylonas and Lindsay MacDonald, Augmenting Basic Colour Terms in English, Color Research and Application, Volume 41, Issue 1, February 2016

Isaac Newton, A New Theory of Light and Colours, Transactions of the Royal Society, 1672

John Seymour, How well do we remember color?, John the Math Guy blog, May 30, 2018

Thursday, August 22, 2019

Creating a circle that goes through three points on your lawn

I get some off-the-wall questions once in a while. Today, I got an off-the-porch question and decided the answer might make a good blog post. I'm probably wrong about that, but here goes.

My friend Dark Laser (I have changed his name to protect his identity) is putting a flower garden in his backyard. He wants the edge of the flower bed to be a circular arc that goes through three points . Those three points have been defined by a higher power. Maybe the higher power is his wife, or maybe it's just cuz of where the house is. I dunno. Below is the drawing that he sent me.  I hope you appreciate his obvious artistic skills.

Before he sent the email, Dark did a little googling and YouTubing. He came up with one answer in a YouTube video, which I show below.

I'm not sure how to interpret an email that asks me a question, and also sends me the answer. I'm not sure what he meant by this juxtaposition of Q & A, but due to my insecurity, I took the email as a challenge. "See if you are as smart as this guy, John!!" I am certainly not going to allow any dufus on YouTube to out-answer me on any dumb old math question!

The MathTuber who answered this question used analytical geometry, which lies between geometry (which is all lemmas and compasses) and algebra (which is all factor this polynomial and take the square root of both sides). But mostly the video is algebra.

Here's my opportunity to show off my brilliance. The abracadabra algebra video is all well and good, but it doesn't completely answer the original Dark question. I mean, how does Dark use this equation? Does he need to go out and buy a huge piece of graph paper to lay on his lawn?!?!?

I chose to forego the algebraic approach and go for a solution that is more along the lines of Euclid and his book Elements. A lot of this ancient text has to do with making constructions of various figures with a pencil, a compass, and a straight edge. But I don't think Dark has these tools, especially not in the size required to lay this out on his lawn. So, I improvised with more appropriate tools: rope and stakes.

(While I am at it, let me take a moment for more self-congratulations. Pure mathematicians are perfectly content with theoretical answers. But I am an applied mathematician, which means that my math isn't happy until it answers a real world problem. You can't get any more practical than building an aesthetically pleasing flower bed!)

Step 1

Put stakes at Points 1 and at Point 2. Make two equal lengths of rope, and tie them together at one end. You don't need the ropes to be red and blue, as in my diagram, but it may help.

Tie the other ends of the ropes to the stakes at Point 1 and to Point 2. Grab the ropes where they join and pull them taut. Place a third stake at that point. This third stake is shown as a blue circle in the drawing below.
To avoid any confusion, I use the word stake in a general sense. If any of the stake positions should happen to be on lawn then a tent stake could meet the purpose. If the point is on a wood porch, then a hefty nail or a bolt could work. If the point is on a cement slab of a porch, then maybe a can of spray paint could be used to mark the point. Or a bathroom plunger? 

Step 2

Shorten the two ropes and repeat the process to locate a position for a fourth stake, as shown below as a second blue circle.

Now the aha! part, which I mention in order to create a sense of making progress. All circles which go through both Point 1 and Point 2 will have a center someplace on the blue dotted line! I hope you are as excited as I am.

A practical comment -- If the lengths of the ropes in Steps 1 and 2 are very close to the same, then the two stakes will be pretty darn close together. This is not such a good thing. This will lead to uncertainty in the angle of the blue dotted line, which will lead to inaccuracy in the position of the final circle.

An even more practical comment -- As I wrote that last comment (the so-called  practical comment), it occured to me that in Step 2, you could have kept the rope the same length, and merely pulled it to a position above Points 1 and 2. Too bad you already went through the process of shortening the ropes.  

Step 3

If you want to get technical on me, Step 3, is really two steps. It's a repeat of Steps 1 and Steps 2, only with different points. Repeat Steps 1 and 2 with Points 2 and 3. You probably will need to stretch the ropes out if you already cut them.

The green dotted line in the drawing below is analogous to the blue dotted line. All circles which go through both Point 2 and Point 3 will have a center someplace on the green dotted line! 

I know some of you may have jumped right ahead to the big climax, but I will state it here for anyone who might not have quote caught the significance: All circles which go through Point 1, Point 2 and Point 3 will have a center at the intersection between the blue dotted line and the green dotted line. Assuming the two dotted lines intersect, and assuming the two dotted lines are not along the same line, we have uniquely defined the center of the circle.

Step 4

A pure mathematician would stop at Step 3, since the point has been theoretically defined. But an applied mathematician, being of a superior breed, would realize that we still need a way to mark that physical intersection point on the porch.

Here is my suggestion. Place a skinny pole at a possible location for the circle center. Move the skinny pole around until it lines up with both of the blue stakes. Then turn your head and line the skinny pole up with the two green stakes. Then go back to the blue stakes to make sure they still line up. This may take several iterations. (My own experience suggests that copious quantities of beer can reduce the number of iterations necessary, not because beer increases your skill level, but because it gives you a more realistic view of just how important the position of the center of the circle is in the grand scheme of things.) Put a nail or a stake or a plunger at the point of intersection. 

If neither nail nor stake nor plunger will work on the porch, then buy another 12 pack and set it next to the intersection point. When a neighbor stops by to see what you're doing, hand him a beer and ask him to sit at the intersection point. The extra beers will keep him from moving.

A comment for those who remembered taking geometry... The task of finding the intersection would have been done with a straight edge. If Dark happens to have a 2 X 4 that is long enough, he could certainly use that to mark the dotted lines. A can of spray paint could serve to make that line indelibly, so the next owners of the house can appreciate the mathematics that went into constructing the flower bed. I haven't looked in Dark's garage lately, but I am guessing that finding a 2 X 4 that is long enough is a tall order. Or a long order. So, we must resort to an iterative procedure which would have been scorned by Euclid. But being scorned by Euclid is not a big deal. We are using non-Euclidean geometry.

Step 5

We're now ready to finish the project. Attach a rope to the nail/stake/plunger/neighbor at the center, and stretch the rope out until it reaches Point 1, Point 2, or Point 3.

Tie a spike to the  rope at that point. Holding the rope taut, move the spike from Point 1 to Point 2 and then on to Point 3, scratching the lawn to indicate the edge of the circle. If you are not skilled in the art of lawn scratching, feel free to tie a can of spray paint to the rope. I suggest a color of paint which is different from that of the grass. Although I show purple in the drawing below, green paint would provide a real good contrast to the color of my lawn.

Step 6

Pull up a lawn chair and finish the beer, content in having accomplished a good day's work.

Sunday, August 18, 2019

The allure of tweeny colors

I recently saw an interesting question on Quora about colors that are positioned between the basic colors. Here is the question, and the answer I gave.

Question: “Why are those color hues so intriguing that linger right between two known colors: between blue and gray, between pink and purple, etc.? They keep me captivated with the visual ‘tease’.

Interesting question! I have seen this before, and my own observations are that these tweeny colors are interesting and either beautiful or ugly because of this. I don’t know if I have the answer to why, but I have one plausible explanation.

Basic fact #1: There is much information compression and loss of information as an image makes its way from the retina to the upper parts of the brain. If I look out on the room before me, and then close my eyes to attempt to remember what is there, I come up sadly short. I won’t remember anywhere near all the objects, or remember much in the way of details about them. I certainly wouldn’t be able to paint a picture from memory (even if I could paint).

Basic fact #2: There is evidence that colors work the same way. When the lower brain tells the upper brain that a car is red, it doesn’t report the exact color coordinates of the color either an an RGB value or a number from a color matching book. According to the theory, it will generally put the color of the car into one of a small number of buckets. There might be eleven buckets, or maybe there are just a few more.

I suspect you may be inclined to disagree with such a small number. Surely there are hundreds or maybe thousands of nameable colors? The small number in the last paragraph comes from an experiment where people were shown a color and moments later asked to pick that color out of a lineup. This experiment showed that our remembrance of a color skews toward a quintessential version of that color family, and there are not hundreds or thousands of color families. A somewhat desaturated blue is remembered as blue. A slightly orange version of yellow is remembered as yellow.

Here is my blog post on the experiment: How well do we remember color?

This explanation is consistent with the way we perceive the world. I look at a car and say that it is red, completely ignoring the fact that one part of the hood is lighter because of the position of the Sun, and the lower door panel is darker because it is partially shaded. In some areas of the car, there is a strong delineation in color, and I can easily choose to be conscious of that. In other places, the change in shade is gradual enough that it is difficult to be consciously aware of it.

Let’s apply this knowledge to your question. Consider looking at a car that is somewhere between blue and gray. I may glance at it once, and my lower brain will decide that the color is in the gray family. I look again, and my lower brain may change its mind and put the color in the blue family when it reports to the higher brain.

If this happens, we have a cognitive dissonance - the upper brain has to deal with two conflicting thoughts: “the car was gray” and “the car is now blue”. This conflict draws our attention to the color, hence it is interesting.

Monday, August 5, 2019

Where did my indigo? (part 1)

I answered a question on Quora recently. I am so proud of my astute answer that I decided to expound on it to make it into a blog post.

The question: What is the true color of indigo?

In today's blog post, I provide the first answer to that question.

Dyeing is a pigment of my imagination

The words dye and pigment are often used interchangeably. Just don't try that in the presence of any colorist. At best, they will roll their eyes and/or laugh at you.

Cool pic of dyes from Alliance Industries

The general term is colorants, something that is used to impart color. Dyes and pigments are two types of colorants.

Dyes are molecules (often organic) that are incorporated into fibers. As a rule of thumb, they are soluble, and we work with them typically in solution, so they are individual molecules. Also generally they require a binder molecule to attach them to whatever we are dyeing. Dyes are most commonly used for dyeing fabric. Dyes need to penetrate into the substrate (e.g. the cloth fibers) in order to become permanent.

Pigments are made from grinding stuff up -- usually non-organic compounds. Since they are ground up, they come in the form of solid particles with lots of molecules clumped together. when used, they are generally suspended in a vehicle that is evaporated when the paint or ink dries. The vehicle might be water (as in latex paints and inkjet inks) or oil (oil-based paints and lithographic ink) or a solvent such as alcohol or toluene (sometimes used in gravure printing ink). Pigments generally coat the substrate rather than becoming incorporated into the substrate.

I expect everyone to use these terms correctly from now on. This will be on the final exam.

Mordant or less?

When I was young, I remember my sister showing me a neat trick. She showed me how to crush the flower buds of a certain flower between my fingers. The flower excreted a gorgeous rich blue fluid. My sister told me that this was a dye used by Native Americans.

Naturally, I was scolded. Maybe my mother was angry because the fluid stained my hands. More likely, she was angry that I stained my clothes. Then again, maybe she scolded me because I was destroying her irises. But, my memory is hazy. Maybe it was a wildflower. And maybe the Native American dye that my sister told me about was beets. I dunno which version is true, but one fact is perfectly clear. It was Nancy's fault. She forced me to get into trouble. This last part will be on the final exam.

This all left me with the impression that dyeing was pretty easy. You find some natural color in the woods or in a field. You crush the plant, dissolve it in water, and then soak your cloth in it.

Indigo dye is almost that easy to use. You just soak the cloth in a solution of the dye, rinse it, and then dry the fabric. Well, maybe that's a simplification, but I think I have the basics of it.

But making the dye is a bit more involved. The indigo plant has pretty pink flowers, but that's not where the dye comes from. Oddly enough, the dye is made from the leaves. These leaves are mashed and then fermented. The gunk left over is then dried and beaten to aerate it, since the process needs oxygen. Then it is left to dry into cakes.

When I first heard this, I was surprised. Who'da ever thunk to ferment leaves? Who'da ever thunk to ferment something and then not drink it? That part boggled my mind. (I'll get back to that in just a bit.)

Getting back to the dye, indigo is odd in the world of dyes. Indigo is called a non-mordant dye, because it does not require a mordant.

Ok, so what's a mordant?

Most dyes do not have a natural affinity for cloth. That is, they don't stick. Cloth is treated with mordants (like alum or tannic acid) so that the dye molecules will stick. The mordant molecules have an affinity for both the cloth and the dye, so they bind dye molecule to cloth by linking the left arm to one and the right arm to the other.

Here is the lesson that will appear on the final exam: Indigo was one of the earliest dyes because of two properties. First, the dye is created naturally in a way that humans could readily see and imitate. Second, the dye did not require pretreatment of the cloth to fix the dye.

Who invented indigo?

Indigo comes from the the indigo plant, specifically from the species of the indigo plants named indigofera tinctoria and indigofera suffruticosa. I called up the ancient Roman historian, Vitruvius, to help answer the question of who invented indigo dye. He told me that "indigo comes from India... where it attaches itself as mud to the foam of the reeds." Hence the name, indigo, which is derived from the Latin word indicum, meaning substance from India.

Now I can understand how someone would have come up with the idea of fermenting indigo leaves. I surmise that someone noticed that the muddy foam on the indigo plant stained fingers and cloth. They then sought to duplicate the natural process, and likely found that they needed a bit of this sticky foam as a starter. Whatever animicula was in the foam -- maybe it was yeast? -- would greedily ingest something in the leaves and go through some sort of chemical reaction that liberates the indigo dye molecules.

A bit of historical pedanticness -- just which Roman historian I spoke with is a matter of disagreement among historians. The quote above is from Ball, p. 201. Another account (Phipps) refers to Pliny, without a fancy quote. Another author, DeBonnet, attributes a similar quote to Dioscorides in 23 BC. I am not really sure who I talked to on the phone. I had spent the whole night sampling my fermented indigo, and ... well...

Marco Polo brought this dye to Europe from his travels in India. Indigo dye became a much desired item for import from India to Europe, first along the Great Silk Road, and then around the Cape of Good Hope. For example, Dutch ships carted nearly 170 tons of the dye from India to Europe in 1631. This factoid will not appear on the final exam.

Cool picture of the indigofera plant, from Phipps, 1832

I pause here to admit to a gap in my personal recollection of world history. I'm a bit surprised that the ancient Greeks had trade routes going with India. Here I thought Marco Polo was the guy who first connected European commerce with Indian commerce. Did I sleep through that day in history class?

So far, the story goes like this: The recipe for making indigo dye from the indigofera plant was devised in India, which became the source of indigo dye in Europe. There are reports of indigo dye in Harappan Civilization in the Indus valley somewhere around the time 2,000 BCE, so this all makes sense.

But that's only part of the story. A clay cuneiform tablet, dating back to 600 or 500 BCE, was found in southern Iraq with the recipe for creating indigo dye.

This is a bit of a conundrum having to do with the history of indigo. Suppose you were transporting indigo from India to Babylonia. Would you load your camels or rafts or what-have-you with bundles of leaves, or would you load up with cakes of the processed dye? I'm thinking I would go with the more compressed form. Keep up with me now... so if we have an ancient Babylonian tablet, written in ancient Babylonian cuneiform, that was found in the area that was ancient Babylon... and that tablet gives a recipe for turning indigo leaves into indigo dye...

Do you see where I'm going here? Why would a Babylonian go to the trouble of describing he process to manufacture indigo dye to a Babylonian when Babylonians are only receiving shipments of the processed dye???!?

I think that the indigo plant may have been cultivated in ancient Babylon. I think that indigo seeds were transported from India to Mesopotamia at some point before 500 BCE. Then again, maybe the seeds went the other direction at some time before 2,000 BCE?

Here's another indigo sighting from about that same time period. Herodotus (a Greek historian from about 450 BCE) wrote that in Caucasus "They have trees whose leaves possess a most singular property: they beat them to a powder, and then steep them in water; this forms a dye which they paint figures of animals on their garments." Herodotus doesn't actually use the word indigo, but it sure sounds like he was talking about how to make indigo dye. He also doesn't mention India. Strange.

So, now I'm confused about whether the process to manufacture indigo dye actually came from India.

Let's muddy the waters a bit more. In ancient Egypt, mummies were wrapped in linen. The strips of linen were dyed with indigo. This puts the invention of indigo back to around 2,400 BCE, and perhaps earlier. Did the very ancient Egyptians get their indigo from India? Or did the Indians get their indigo from Egypt? Were there even trade routes between these civilizations at that time? Tell me, just where did the indigo?!!??

There are also early reports of indigo dyes being used in the Xinjiang province of China around 1,000 BCE. Were there trade routes between China and India? Here my meager world history completely falls on its face. I done got edicated in 'murica. We don't need no stinking urapean history there.

Earliest occurrences of indigo dye in various regions of the Old World

I look at the map above, and it seems likely to me that indigo dye was developed independently in Egypt and India, and possibly also in China.

But I omitted one last piece of the puzzle. Indigo dyed fabric was found in scraps of cloth in Huaca Prieta, Peru that date back as far as 5800 BCE! I hope you're as excited about that as I am.

My conclusion: a process to extract indigo dye from the indigofera plant was developed independently in multiple places around the globe. That will be on the final exam.

Stay tuned for part 2 of this series of blog posts, where I investigate Isaac Newton and the indigo that he put in our rainbow!

Want to know more about pigments and dyes?
Have a look at a blog post about mauve, Tyrian purple, and magenta.
Or, check out the blog post about the invention of Klein blue.
Or better yet, have a quick read about vermilion and cinnabar.


Ball, Philip, The Bright Earth, Art and the Invention of Color, University of Chicago Press, 2001

Beloe, William, Herodotus, Translated from the Greek, 1814, p. 254

DeBonnet, Maurice, Origin of Paint Pigments, Varnishes, Vehicles, National Painters Magazine, Vol 48, Jan. 1921, page 26.

Finlay, Victoria, The Brilliant History of Color in Art, Getty Publications, 2014

Mattson, Anne, History of Indigo in the Early Modern World,

Nassau, Kurt, The Physics and Chemistry of Color, The Fifteen Causes of Color, John Wiley and Sons, 1983, p. 285

Phipps, John, A series of treatises on the principal products of Bengal, 1832

Splitstoser, Jeffery C., Tom D. Dillehay, Jan Wouters, and Ana Claro, Early pre-Hispanic use of indigo blue in Peru, Science Advances  14 Sep 2016: Vol. 2, no. 9,

St Clair, Kassia, The Secret Lives of Colour, pps. 189 - 192

Wikipedia, Indigohttps://en.wikipedia.org/wiki/Indigo

Wild Color, History of Indigo & Indigo Dyeing,