Turning the crank

Years ago - I was perhaps 14 - my Dad went on a business trip to check out an inventor who was trying to sell some of his ideas. He took me along. The main gizmo that my Dad was dispatched to look at was something that was similar to a Wankel engine. I won't bother with the details of the design here, since everyone already has a working knowledge of how this engine works [1].

Turning the crank on the Ramelli water pump

Perpetual motion machines

The talk of the engine went over my head, but I was intrigued by the inventor's perpetual motion machine.

Perpetual motion machines I understood. At the tender young age of 8, I had responded to a "sell us your inventions" ad in the back of a comic book with a perpetual motion machine of my own. I had the brilliant idea to connect a generator up to a wheel that was running on a motor that was run by the generator. If only I had had the foresight to apply for a provisional patent before I sent off my idea! How was I to know that that those unscrupulous folks would sell my idea to Toyota. The intellectual property that I so naively gave away was to become the cornerstone for their regenerative braking system. [2]

"This invention is a pendulum structure that operates as near as possible 

as a perpetual motion machine to provide activity and entertainment to one using same."

US Patent 4,422,530

So, let me describe the perpetual motion machine that I saw at this inventor's shop. It was a large brass disk, about 18 inches in diameter and a few inches thick. It was mounted on a shaft so it was free to rotate. Magnets were embedded  along the circumference of this disk.

On the outside of the disk, there were magnets mounted on a mechanism of gears, springs, and levers. By virtue of this beautiful collection of parts, the magnets were brought almost into contact with the magnets in the brass disk, north pole to north pole, as the disk spun. Each near contact would give the disk a little push and momentum would carry the disk on to the next magnetic propulsion station.

The ingenious inventor gave the mechanism a little spin and it whirred away for ten seconds before coming to a stop. "I just need to make a little adjustment here." I wanted to stay and what him tweak the incredible machine into perpetual life [3]. I was not very happy with my Dad when he whisked us away. "The guy's a crank. You can't build a perpetual motion machine." "Are you going to buy his other invention?" "Absolutely not. He has proven himself to be a fool."

"But Dad, the momentum will always carry it up to the point where the magnets can push it again. How can it ever stop?"

Dad was so dismissive.

A tale of another crank

Just recently, I had an interchange with a crank [4]. I had posted information that he disagreed with. Fair enough. I make mistakes. Maybe I made an incorrect assumption or faulty argument. I provided him with a detailed proof and asked for him to help me find the error. He replied with a lengthy post that I couldn't comprehend. I asked him once again to find the error in my proof. I copied the proof again, this time with line numbers, making it completely clear that I wanted to know exactly which line was in error, and what the error was. 

John the Math Guy, puzzling over his proof

This was a long an frustrating discussion. Over the course of it, I requested no less than five times for him to provide me with the location of the step in my proof that was wrong. I never did get an answer.

I started asking myself... is there something important I am missing? Something I don't understand? Or is this a hopeless task? Is my proof just complete and utter gibberish? Do I have any bananas in the cupboard?  

The fellow had a rather uncommon name, so it wasn't hard for Google to find him. I found eight forum discussions that answered all my questions. Except for the question about bananas.

This gentleman has been referred to by the following epithets: persistent cretin, stupidity, complete nutcase,  trolls and/or insane, ignoramus, monumental ignoramus, delusions of grandeur, crackpot (twice), pseudoscientific, crank,  "are you still in high school?", and "stupid or dishonest". His posts have been refered to as patented cretinisms, masterpieces of idiocy, tortuous, horribly wrong, and utter stupidity.

Wow. I thought I had played to some tough audiences.

My own experience with him - him avoiding direct questions with inane rambling - was common. Here are some quotes from various people who were expressing frustration at this fellow in various discussions:

"I would love to follow what you are saying but for me you are not precise enough in your description. You use too many words."

"I have given up trying to follow [name elided] and your huge posts."

"I hope that there is some relevance to your tortuous conversations."

"I can never understand the point that you are trying to make. You make 10 half points and I don't understand any of them."

"In the Space that you spent going around the question without actually bothering to answer it , you could instead have answered it." 

"I am afraid that I cannot read your dialogue. You are not succinct enough."

"I've read through your response a few times and I have no real idea what you are saying."

"No matter how long-winded your explanations get, please bear in mind that you cannot defeat the fact that ..."

"I realize that English may not be your native language. But I can't respond to something that is not only incoherent..."

"You have simply written a very complicated paper (which appears to have errors) to state the obvious and draw an incorrect conclusion from it."

Two people had interchanges similar to mine. They tried to get the crank to answer a direct question. Here are the repeated requests from one frustrated person. The crank responded either off topic or incoherently in between each request. 

"Longer posts aren't going to do it for you [name elided]. Respond to what I said."

"Your refusal to respond to the questions in my post shows that you are avoiding the issue."

"I asked questions you're unable or unwilling to answer. Why is that?"

"I agree that further "discussion" is probably fruitless because you're not discussing the issue. I asked a couple of simple questions here to try to have you look at my point of view. You have refused to do that. I can only conclude that you see that you are wrong and are simply avoiding the issue."

Resistance is futile

Here are the frustrated requests from another person.

"Just post the scalar function for which you think the curl of its gradient in not zero. You don't need an article just to write down a function."

"Just post the function. If it exists and you know what it is, then nothing prevents you from posting it, especially given the fact that you would be quite happy to rub my nose in being wrong..."

"Just post the function. I'm not going to read nine pages of a convoluted argument about the curl of a gradient not being zero when you can post the function for which you claim that relation doesn't hold in a single line."

"I conclude that despite a lot of motivation to post such a function, you haven't done so because you know that if I take the curl of the gradient of that function."

"Either post the function or get lost…  If you had a real result, you wouldn't hesitate to post 1 function rather than post a lot of crap to cover for not having a function."

"Post it and I'll see it."

When the crank said that he had already posted the function, this was the frustrated reply:

"Where? You have spent this entire thread making excuses for not providing it."

Yet another crank case

As if this wasn't enough fun, another crank turned up on LinkedIn during the same week. This one claimed to be able to trisect an arbitrary angle [5], and claimed to have a "correct" value of pi (3.1547...). Naturally  there was a firestorm of explanations and proofs that he was mistaken. 

Pi to the World,

twenty two over (eight minus one),

Irrational though it be.

Let every round thing

Now fit the expounding

With each and every degree,

With each and every degree,

With each, with each, and every degree.

(from John the Math Guy's Math Carols, 1974 [6])

Here are some ridiculous (as in, worth of ridicule) links to websites, each with their own true value for pi:

      pi = (6 + 2 sqrt (3)) /3 = 3.1547005...

      pi = 201 / 64 = 3.140625

      pi = 3.146264   (Just send $14 for their 12 page pamphlet!)

      pi = 4   (Of course, this is only true for kinematic situations.)

      pi = (14 - sqrt (2)) / 4 = 3.14645...

      pi = 3.125

My own favorite true value for pi is based on the golden ratio, phi. You will no doubt recall that phi = (sqrt (5) + 1) / 2. If you square this number (or add 1 to it, same thing!), and then multiply by 6 / 5, you get 3.14164...   [7] I'm gonna start my own campaign to get this number into textbooks as the correct value.

The wonderful thing about pi is that there are so many values to choose from!

And more...

Here is a quote from a recent post on LinkedIn:  "Reality and Fact are that we are NOW in the transition period of the Magnetic Pole shift."  The poles are going to shift - north becomes south and vice-versa.

Conclusion

I feel the need to add a conclusion to this. I wish I had one. I don't understand what is going on in these crank's heads. I understand the excitement of finding something that no one else has found. The potential glory of going down in history as the person who corrected longstanding misconceptions... I don't understand how people who are clearly intelligent can delude themselves into not seeing the glaring mistakes in their proofs, especially when others are kind enough to point them out.

-------------------

[1] I really honestly don't want to bore anyone with the inner workings of the Wankel engine. Really, seriously. My lack of explanation has nothing to do with me having no knowledge whatsoever about what a Wankel engine is. Abject ignorance has never been an impediment for my pontification. 

[2] After this and a few unfortunate incidents with xray specs and Charles Atlas abs, I learned not to trust these comic book ads.

[3] The motion of the machine was almost perpetual! As an aside, I once used the phrase "almost infinite" in a group of math-savvy people and got the little titter that it deserved. Then I started noticing the phrase being used seriously. Sigh. As a further aside, here is how a mathematician responds to a funny joke. "Oh yes. That is really quite humorous. I will have to share that one with my wife." This is to be spoken without the barest hint of inflection.

[4] Wikipedia defines crank as follows: "Crank" is a pejorative term used for a person who holds an unshakable belief that most of his or her contemporaries consider to be false. A crank belief is so wildly at variance with those commonly held as to be ludicrous. Cranks characteristically dismiss all evidence or arguments which contradict their own unconventional beliefs, making rational debate a futile task, and rendering them impervious to facts, evidence, and rational inference.

[5] Trisection means to cut the angle into three equal pieces, using only a compass and straightedge. Everyone knows that this can't be done. My dog Truffle was eight weeks old when he came up with his first proof of this.

[6] Also included in the Math Carol songbook are such favorites as "Arcs the mighty compasses bring", "We wish you a semi-circle", and "I'm proving the Pythagorean Theorem", and "Oh Sum, all ye Mathful". Classics, every one of them. I wrote these in high school, and honest to god, my math class walked through the halls singing these one day before Christmas.

[7] A simpler form of this computation, which does not use phi as a stepping stone, is (3 / 5) (3 + sqrt (5)). This is the answer to the riddle: "3  5  3  5" - add plus, minus, multiply, divide, and square root signs to this series of digits to make an approximation to pi that is accurate to four decimal places. I asked this riddle of Mike, and he cleverly came up with "3 + (5 / 35)", which is only accurate to two places.

Logarithms and the end of the world

Hardly a day goes by when I am not asked "Hey Math Guy!  What do logarithms have to do with the end of the world?"  Now that the end of the world has come and gone, I will share an interesting coincidence from the annals of math history.

When the natives are restless, they use log rhythms to communicate over long distances

Prosthaphaeresis

I am not talking about prosthetic devices, like artificial limbs and fake noses. At least not yet. I predict that I will, however, be talking about fake noses soon. And I am definitely not talking about electrophoresis or paresthesia[1]. Prosthapheresi Prosthespaeoius Prosthaphaeresis is a technique used to simplify multiplication.

Here is the simple approach. When you have two numbers to multiply together, you first compute arccos and arcsin of both. Then you add and subtract to get some other numbers. Then you compute the sine or maybe cosine of these and add them together. Or maybe subtract. Then divide by two.

Easy enough, right? Except for the silly fact that all these trig functions will probably involve multiplying, this simple method turns multiplying into the easy process of addition and subtraction. The trig functions may be seen as an impediment, but if you happen to have a trig table handy (or a scientific calculator app for your smart phone) then you can just look them up. And lemme tell ya, 16th century astronomers always had their trig tables handy.

Required reading for all 16th century astronomers

So, when some unknown person came up with the idea that a trig identity could be used to make the multiplication process faster, it got a lot of airplay. The phrase "(sin x + sin y)/2 = sin ((x+y)/2) cos ((x-y)/2)" was on everyone's lips.

I have a similar formula that turns multiplication into addition, subtraction and a quick look up in a table of squares: ((x+y)/2)² - ((x-y)/2)² = x y. Let's say we wish to multiply 9 X 17. The sum and difference are 26 and 8, which become 13 and 4 when you divide by 2. I happen to know that 13 squared is 169, and also that 4 squared is 16. The difference (169 - 16) is 153. I'm so confident in this method, that I am not even gonna check that with my calculator.

This formula has become known as John the Math Guy's Prosthaphaerisoid. I have not found any evidence that this formula was known back in the 16th century. I have looked in the index of all my math history books (there are 127 of them on my shelf), and have not found a single mention of the formula under that name. It's too bad. If they had known the formula earlier, Pluto would have been discovered so much earlier and the whole "is Pluto a planet" thing would have been settled already.

Invention of logarithms

Enter John Napier (1550 - 1617), who is remembered today as the inventor of logarithms. In 1614, he published a book of logarithms called Mirifici Logarithmorum Canonis Descriptio. I am sure most of my readers have a well worn copy of this book sitting on their night stand.

The father of modern-day ciphering

Napier took inspiration from several sources for this boon to cipherophiles [2]. Primarily, he drew from the idea of Prosthaphaeresis which was indirectly introduced to him by Tycho Brahe, an eminent astronomer of his time [3]. Probably at a dinner party of 16th century math geeks. Napier also drew on the work of Michael Stifel (1487 - 1567). While Stifel most certainly did not invent the formula that makes logarithms possible, he was likely the first one to state that logarithms turn multiplication into addition, and division into subtraction.

a^m aⁿ = a^(m+n)

The formula that made it all possible

Doomsday prophecies 

It would perhaps be to Napier's dismay that he has not been remembered for his book A Plain Discovery of the Whole Revelation of Saint John (1593). This book was a bestseller during his day, and even after his death, was printed in twenty-one editions, and was translated into Dutch, French and German. Both the words and the numbers were translated, by the way [4].

Napier was a vehement Protestant. In his book he viciously attacked the Catholic Church, claiming that the pope was the Antichrist. Napier also offered his prediction that the Day of Judgment would come between 1688 and 1700.

Michael Stifel entered an Augustinian monastery and was ordained in 1511. After much dissatisfaction with the Catholic Church, he sought the help of no less a personage than Martin Luther in setting him up in Protestant pulpits. Unfortunately, Stifel's career as a minister was tumultuous, having to move from several parishes due to anti-Lutheran sentiment, and leaving another due to war. Stifel predicted that the world would end on October 3, 1533, and many believed him. As a result he was arrested and forced to leave yet another post. There is not much profit in being a prophet.

Actual unretouched photo from Oct 3, 1533

It would be to Stifle's dismay that he has not become known for his own bestseller, A Book of Arithmetic about the AntiChrist. A Revelation in the Revelation. [5] 

Not only do Stifel and Napier share the discovery of logarithms and prediction of the end of the world, but both men also accused the pope of being the Antichrist.

Much as he would have liked, Isaac Newton himself could not lay claim to the invention of logarithms, but he  had something to do with logarithms. He came up with an estimate of the sum of the harmonic series that involved logarithms. And, much like Stifel and Napier, Newton would have been dismayed that he has not been remembered for his endless religious writings. And, needless to say, he had a few things to say about the end of the world. He declared that is could not end until after 2060, when his pants would be done at the cleaners.

"This I mention not to assert when the time of the end shall be, but to put a stop to the rash conjectures of fanciful men who are frequently predicting the time of the end, and by doing so bring the sacred prophesies into discredit as often as their predictions fail."

http://www.isaac-newton.org/update.html

Now there's a clever guy. If his prediction were to fail (the world were to end before 2060) who would be left to call him on it? And if 2060 comes and goes without incident, then it's well past his lifetime. Why worry? 

One last mathematician's prophecy

Some of you are no doubt familiar with the greatest living mathematicomedian, Tom Lehrer. Well, maybe he is one of the greatest. There is at least one other vying for that title. Lehrer also had his predictions about the end of the world.

Oh we will all fry together when we fry.

We'll be french fried potatoes by and by.
There will be no more misery
When the world is our rotisserie,
Yes, we will all fry together when we fry.

--------------------------------

[1] Just to be clear, electrophoresis is not a way to remove unsightly, embarrassing hair. Paresthesia is not waking up one morning thinking that you are Paris Hilton.

[2] Cipherophile - One who loves cipherin'. I coined this word just for this blog. Just look for it to be used on the Jerry Springer show in the next week or two. My own prediction.

[3] Tycho Brahe was an interesting fellow. His chief contribution to science lay in providing his collaborator and rival, Johannes Kepler with data on the positions of the planets. Kepler used this data to do a little curve fitting that showed that the planets run their courses in elliptical paths. Newton in turn drew from this to establish the inverse square law for gravity.

Aside from having an odd sounding name that I don't know how to pronounce, Brahe also had a prosthetic nose made of gold and silver. His original nose was of the standard design with flesh and cartilage, but he lost part of it in a sword fight with a cousin about a mathematical formula. Ah! to be back in the days when scientists were real men, rather than sarcastic little blog-writing cipherophiles! Brahe owned a pet elk, which met it's demise when it was fed too much beer at a party. It toppled to it's death on a set of stairs.

[4] I know it is unlike me, but this is a silly statement. Oddly enough, the symbols for numbers were standardized across Europe at this time. This was before ISO.

[5] I personally am a bit dismayed that this title has already been taken. I had been planning to use that very title for my memoirs.

Math carols

In this week's blogpost, I share the intersection between John the Math Guy, music parodies, Christmas, and silly stuff from high school. By popular demand, and just in time for Christmas math caroling of your own, I present the complete collection of John the Math Guy's Math Carols. At least as complete as I can remember from 1974.

Pi to the World (sung to the tune of  "Joy to the World")

Pi to the world,

twenty two over (eight minus one),

Irrational though it be.

Let every round thing

now fit the expounding

with each and every degree,

with each and every degree,

with each, with each, and every degree.

Arcs the Mighty Compasses Bring (sung to the tune of  "Hark the Herald Angels Sing")

Arcs the mighty compasses bring,

constructed with a steady swing.

Geese are worth not twenty Yen,

Shucks! I've torn my diagram of Venn.

A triangle within a semicircle lies.

Incongruent triangles are not the same size.

But, congruent ones are the same.

Geometry drives one insane.

But, congruent ones are the same.

Geometry drives one insane.

We Wish You a Semicircle (sung to the tune of  "We Wish You a Merry Christmas")

We wish you a semicircle,

we wish you a semicircle,

we wish you a semicircle,

and a parallelogram.

We wish you an acute trapezoid,

we wish you an acute trapezoid,

we wish you an acute trapezoid,

and a doggie named Sam.

Constructions we make of geometric figures,

constructions of figures,

and a doggie named Sam.

There's No Place Like Rhombus the Holidays (sung to the tune "There's No Place Like Home for the Holidays")

Oh, there's no place like Rhom.... bus us there today,

cuz division by zero's undefined.

If you set out to graph this maladay,

You will find it's mis-inclined.

Oh Sum, All Ye Mathful (sung to the tune of  "Oh Come, All Ye Faithful")

Oh sum, all ye mathful,

corresponding pupil.

Oh sum ye, oh sum ye to find the total.

Desist your secant,

Archimedes' streak can't

Move the Earth,

Eureka, move the Earth,

A streakah move the Earth,

No can't move disjoint.

Go work Ye Merry Protractors (sung to the tune of "God Rest Ye Merry Gentlemen")

Go work ye merry protractors

make straight lines here today.

Remember every angles not made of just one ray.

Euler was a greaser,

but that's not what textbooks say.

Oh, tangents of circles and spheres,

circles and spheres.

Oh, tangents of circles and spheres.

Follow up on colorblindness testing

I posted a blog on colorblindness testing (and apps for tablets, and gamuts of displays, and my dog Truffle) that has led to some interesting discussions in various LinkedIn groups. The comments are great, but unfortunately, they wind up in a variety of different groups. People from one group don't necessarily see what people in another group have had to say.

In this blog post, I am coalescing the various discussions.

How can you test for colorblindness?

The one true standard test for colorblindness is the Ishahara test. The picture below shows slide 4 from this test. People with normal color vision will be able to clearly see a picture of Eddie Albert [1] in this picture. This was developed back in 1917 by an ophthalmologist whose name was (oddly enough) Dr. Ishahara. The test is quick and easy to perform, and you can find 867,324 versions of this test online, although I did notice that there are a few that are not quite as good as the other 867,319.

A picture of my Academy Award winning spaghetti

The other one true standard for colorblindness is the Farnsworth Munsell 100 hue test, which, according to the XRite website is "one of the most widely used tests in industries where color decisions are critical." You can by the boxed set from various places including "the Munsell store". I am not sure how they came up with that name. Just another fluke, I guess.

The Farnsworth Munsell 100 Hue test with 84 hues

In the test, the contestant is given one rack of tiles at a time, and is asked to put them in order according to hue. This is definitely a tough test - the difference in hue between adjacent tiles is very subtle. For that reason, there is some substantiation for the claim of the Munsell Store that this is an important test for people who are making judgments on color matches where the color is critical.

David A. (one smart guy I know) pointed out the difference between the two true standards for colorblindness: "As the Ishahara test has fewer testing points than the FM test, the FM test is also a great test to test for subtle discrimination..." David is making a distinction here between colorblindness and something more like "color acuity". I have seen myself that there are (loosely speaking) three types of people when it comes to the FM test: 1. those who fail miserably and are truly colorblind, 2. those who do very well on the test, putting almost all of the colors in the proper order, and 3. those who get a bunch wrong. I honestly don't know what distinguishes the second and third categories.

Paul raised this point in a way that sounds like he is pretty smart, too: "to find out who in the staff that has perfect colour discrimination capacity (roughly only 10% of the male and female part of the staff), you need to use the FM100 test (or similar)." [2]

Two other people (separate LinkedIn discussions) Shoshana and Michio provided links to the online version of the Farnsworth Munsell test. This is certainly cheaper than the physical version, and it is part of the webiste from XRite, so you know it's not from some fly-by-night guy who claims to be a color scientist and maybe has a silly blog where he pretends to know what he is talking about.

Screenshot for the XRite FM test, which has nothing to do with radio

But this might not be a good test. Here is a comment from that European (and therefor presumably smart) guy, Paul: "One of the teachers at Malmoe University, Graphic Arts Department, tested the online version of the X-Rite (simplified) FM-100 test, and failed miserably. I pointed out to him that he probably needed a better quality monitor, and a calibrated one. I suggested he try my monitor, a LaCie at the time, which was fairly recently calibrated. Now he suddenly managed very well in the same test."

I tried out the online test myself. My score was not all that spectacular. I appreciate the fact that Paul has offered me a way to save face. My monitor is not calibrated.

This brings up the next discussion topic...

Are the tablets up to the task of accurate color testing?

Here is the seed that got me started on this blog post. Bob C. apprised the group "ISO TC 130, WG 13, Task force Harmonizing" of the EnChroma app. Bob was wondering the basic question as well: "I'm also interested to know if EnChroma test correlates with the Ishihara's test for color blindness."

Here is my opinion, for what it's worth. I think the EnChroma app is likely to work well to screen for colorblindness.

David A. asked a similar question in a different thread: "I wonder if the three tablets tested would show any significant differences with an FM type of test."

It would appear that we have two pieces of anecdotal evidence that suggest that color management is important, at least for the FM test on the display of a "real" computer. This makes sense. Colors change when I move a window from my laptop to the monitor that it drives. It is not that far of a stretch to say that the difference between two colors can change as well. The display on my laptop compresses some areas of color space. If the compression happens in areas that the FM test is looking for, then the test will just get harder.

This in turn, brings out the next question...

What about color management on tablets?

Reem pointed out that the Datacolor SpyderGALLERY is an app for iPad and iPhone that provide some color management solutions for photographers. This gadget will measure the colors on your iPad and make adjustments so that it will provide accurate color rendition of images.

The Sypder4 monitor calibration device

She asked if there were any similar apps available for Android devices. I am not aware of any. Sounds like an opportunity for some company... Maybe Datacolor or XRite have some interest in this?  Or maybe the Android operating system does not facilitate calibration?

Does anyone have an answer to this?

One last thing

For those unfortunate readers who are not from the Midwest, I may need to clarify a subtle joke in the previous blog. I called the folks who took the test "Ollie", and "Lena". I should have called the fellow "Ole". Steve T corrected my spelling on this.

The names came from one of the favorite pastimes of Midwesterners: telling Ole and Lena jokes. Ole and Lena were an old Norwegian couple, and they were always getting a bit confused. I wrote these two joke sespecially for this occasion.

Ole: "De doc says dat I am color blind to vun color."
Lena: "Vut color is dat?"
Ole: "I don't know. I couldn't see it."

Sven noticed his old friend, Ole eating a bag of rabbit turds.
"Vat are you eating der, my friend?"
"Lena gave me a bag of M&Ms for my lunch. Wasn't dat sveet of her?"
"Ole, didn't you notice anything funny about de color?"
"Ach, yes! Lena always takes out all de red ones cuz knows I'm color blind. Such a dear voman she iss"

-------------------------------

[1] Eddie Albert played the itinerant peddler Ali Hakim in the movie version of Oklahoma, and was nominated for Academy Awards in two other films. After sunc accomplishments, how would you like to be remembered for Green Acres? Life is not supposed to be fair.

[2] Another clue that Paul is smart is that he puts the "u" in "colour". That means that he is either European or pretentious. Since I know that he is European, and all Europeans are smart, then... I don't think that David A. is from Europe, although I would guess that he has been there a few times.

Does my dog appreciate my KindleFire display?

I read another interesting blog post from Jeff Yurek [1]. In his post, he compares the displays for the iPad Mini, the Nexus 7, and the Kindle Fire HD. Unlike the scores of other reviewers, he doesn't expound on the size or number of pixels. Jeff looks at the color. Specifically, he has checked out the size of the gamut - the total range of colors that can be displayed.

His conclusion is that the Nexus and Kindle have bigger gamuts than the iPad. This is important, because what self-respecting guy doesn't want a tablet that has a big gamut? After all, women realize that it's not about the number of pixels you have. It's what colors you can get with them.

Here is an image that I have remorselessly stolen from his post. You can see that the iPad is somewhat lacking in the saturation of red and green, but where it really falls down is in the blue. The iPad just does not have as saturated of a blue. I was quite pleased with this, since I own a Kindle Fire, and secretly have a bit of iPad envy going on. My wife has an iPad.

Comparison of the gamuts of three tablets

This all got me thinking. Naturally, it got me thinking about whether my dog, Truffle, can appreciate the extra gamut of the Kindle Fire.

Truffle the Teddy Bear Guy

Colorblindness and dogs

One of our dogs, Scrabble, has been featured previously in my blog. He played the Shih-Tzu in the Mathematical Misnomers post, and he played the dog in the painting in the Card Shuffling post. It's about time that Truffle gets mentioned.

Everyone has told me that dogs are colorblind, but I wanted to find out myself. I just heard about an app for color blindness testing from EnChroma. (Thank you, Dr. Bob!) So I thought I would give it a try. This app is a version of the Ishahara colorblindness test. I downloaded it for my Kindle and administered the test to Truflle to check to see if he really is colorblind. He licked the screen. I'm not sure I know what that means.

Screenshot of the colorblindness test app showing an orange cross sign

The competition

My wife and I both have very good color discrimination. I took the Farnsworth Munsell 100 Hue Test [2], and had a nearly perfect score. Now, my wife (who is not competitive at all) took my performance as a challenge. She took the test and proceeded to get a perfect score. But, she's not competitive. Honest. Just ask her. [3]

My wife has an iPad, and I have a Kindle Fire. The EnChroma app is free for both of these. And as I have said, my wife is not competitive. Can you see where this is going?

We played the game color test, a total of 16 times. I took it four times on the iPad, and four times on the Kindle. She did the same. Before running the test, I made sure that none of the usual experimental safeguards were in place. I did not normalize the whitepoint of the two displays. I did not set us each down in a neutral room with low lighting. I did not test either contestant experimental subject for any psychotropic drugs that might effect the outcome of the experiment. In short, I wanted to make sure that I had an out if I didn't do well.

The table below shows the results. Note that the subjects were randomly assigned names to protect the identities and marital harmony. Note that EnChroma gives you 5 points for every incorrect score and there are 60 opportunities to screw up. A score of 0 is perfect. A total score of 300 is perfectly awful.

Person	Device	Protan	Deutan	Tritan	Total
Ollie	iPad	5	10	5	20
Ollie	iPad	0	0	0	0
Ollie	iPad	5	0	0	5
Ollie	iPad	0	5	0	5
Ollie	Kindle	5	10	0	15
Ollie	Kindle	0	0	0	0
Ollie	Kindle	0	5	0	5
Ollie	Kindle	5	5	10	20
Lena	iPad	10	5	5	20
Lena	iPad	5	10	0	15
Lena	iPad	5	5	0	10
Lena	iPad	0	5	5	10
Lena	Kindle	10	10	10	30
Lena	Kindle	0	0	0	0
Lena	Kindle	15	5	0	20
Lena	Kindle	0	5	10	15

Whoever is named "Ollie" attained the astoundingly remarkable feat of an absolutely perfect score an incredible 20% of the time. Ollie's wife only did that once throughout the entire experiment. To be fair, Ollie's wife also complained that the Kindle Fire touch screen is not as responsive as the iPad. Twice, this caused an error. She wanted to make sure that she had an out as well. [4]

When comparing the Kindle Fire and the iPad, we have average total scores of 13.13 for the Kindle and 10.63 for the iPad. By my reckoning, this is just on the edge of being statistically significant. But, allowing for the two reported mis-keys in the test, the difference becomes insignificant.

My summary: Based on this awesomely scientific experiment, I conclude that the difference in the color gamuts of the displays does not cause color acuity problems.  

One more acknowledgement

I have acknowledged two people who inspired this blog post, Jeff, and Bob. I should mention one other.

I just had a nice long chat with an old friend of mine by the name of Steve the Tall Guy [5]. (I should make it clear that this Steve is different from "Steve the British Guy", who provided me with inspiration for my blog post on the Monty Hall problem.)

Anyway, I was talking with Steve the Tall Guy and he pointed me to a very interesting podcast about color from RadioLab. The podcast is about the subjectivity of color. It makes the point that color is not strictly some inherent property of an object, but rather, it is partly something that is manufactured somewhere after a photon enters our eye.

Illustration from RadioLab website showing 

some weird kinda Pink Floyd thing that I wish I could do with my eye

They gave color blindness as one simple example of this subjectivity. Someone who is colorblind may perceive two objects as being the same color, whereas I may see those two objects as being a different color, in much the same way that there are some rare individuals who don't think my wife is beautiful. I don't understand them, but something happens to the photons after they go in their eyes. As far as I can tell, my two dogs think my wife is beautiful even through the poor sods are colorblind. 

The podcast has given me grist for all kinds of future blog posts. One subject I wish to bring up is why we have three color sensors in our eyes, and why the mantis shrimp has 17. I just happen to be reading a book that proposes an answer. I'll finish the book and get back to you.

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[1] I mentioned a blog post of his before in my blog post "Red is the color of..." I enjoy his posts.

[2] For some odd reason the Farnsworth Munsell 100 Hue Test has 85 tiles. What do I know? I do math. I can't count.

[3] The Scrabble app for my Kindle Fire is currently broken, so I am not able to get routinely trounced by my very-noncompetitive wife. She is not at all competitive about Scrabble. Did I mention our first dog's name is Scrabble? I hope they take their time making sure that the app is thoroughly debugged before they release an update. I mean... a few years of testing would be good.

[4] This is a legitimate complaint about the Kindle Fire. It often appears to not be listening. I get that a lot.

[5] I'm not making this up. This guy actually calls himself "the Tall Guy".  Someone has to be pretty full of himself to select a moniker like that! John the Math Guy would never do that. 

Quantification of my post on tides

I was asked a quite reasonable question on my post about tides. Azmat Hussain posted the question to my blog and on LinkedIn. Here is what he had to say on LinkedIn:

"I liked your writing style, and but it could have used some mathematics and some quantification. Like how much is the force on the body of water on this side vs the other side, and is the difference significant?"

I like it when people try to keep me honest!  

I looked on Wikipedia to get some rough numbers. The perigee and apogee of the Moon (closest and farthest points from the Earth) are 362,570 km and 405,410 km. Let's just say that the Moon is 380,000 km away from the Earth when we happen to be looking at the tide. The Earth has a radius of 6,371 km. I'm going to round that to 6,000 km just for ease of reading.

So, here are the numbers that I am going to play with: 374,000 km from the Moon to the closest point on Earth, 380,000 km to the center of the Earth, and 386,000 km to the far side of the Earth. [1]

Next, we throw in out good friend the inverse square law. Someone wrote a very entertaining blog that had something to do with that. It says that the pull of gravity falls off as the square of the distance between two objects. Based on that, I have assembled this table that shows the relative strength of the Moon's pull.

Location	Distance	Relative pull
Side closest to Moon	374,000 km	103.23%
Center of Earth	380,000 km	100%
Side farthest from Moon	386,000 km	96.92%

So... we have a total swing in gravitational pull of 6%. Wow. I was actually expecting a smaller number! I think this proves this is a huge effect, right? [2] And it also explains the whole monthly weight gain problem that women complain about. It's not bloating, ladies. It's the gravitational effect of the Moon!

But, then again, this isn't the whole story. I am talking about the gravitational pull of the Moon. Isn't that tiny? Who cares if it is varies, if it is too small to measure?

Hmmm... If I am standing on the Earth, how does the gravitational pull of the Moon compare to the gravitational pull of the Earth? The gravitational pull goes as the square of the distance. I am 6,000 km from the center of the Earth and 380,000 km from the Moon, so we are talking (6000 / 380,000) squared, which is about 0.025%.

Wait!! That's only a part of the answer, since gravitational pull also goes as the product of the masses, that is, the product of my mass with that of the Earth, and the product of my mass with that of the Moon. My mass is the same, so relatively speaking, we can just look at the relative masses of the Moon and Earth.

The Moon is about 7 X 10^22 kg, and the Earth is about 6 X 10^24 kg. The ratio here is about 1.2%. 

So, I conclude that the gravitational pull of the Moon is on the order of 0.025% X 1.2% of the pull of the Earth. If I am using my calculator correctly, I cipher this out to about 3 parts in a million.

I have gone through the whole calculation below.

Location	Distance	Relative pull	Pull relative to the Earth
Side closest to Moon	374,000 km	103.23%	3.00 X 10-6
Center of Earth	380,000 km	100%	2.91 X 10-6
Side farthest from Moon	386,000 km	96.92%	2.82 X 10-6

Wow. I was actually expecting a larger number! I think this proves that the Moon has a negligible effect, right? 

Then again, this effect is tiny, but we are talking about tides of a few meters... just a tiny distance compared with 380,000 km, about 5 parts per billion.

Azmat has asked a very reasonable question. At this point, I am afraid I must shrug my shoulders and say "flies walk on the ceiling". I say this whenever I am confronted with a problem that is orders of magnitude outside of where I normally live.When I make this comment about flies, I mean that the glue on the foot of a fly that holds it to the ceiling is tiny in my experience, and big for a fly. Alternately, gravity is a pretty big deal for me, but flies seem to be able to take it or leave it.  

My next inclination would be to put together an experiment to test the hypothesis. I would measure some tides with the Moon in place where it is, and then remove the Moon and measure the tides again. Simple experiment, really. Maybe I would repeat it a few times. I have applied for a grant to the NSF. Right now, it's hung up on the "romantic impact assessment". Some silly folks are concerned that the removal of the Moon might have a negative impact on the mating habits of homo sapiens. Darn tree huggers! I'll let you know when I hear back from the NSF.

Seriously, my next approach would be a computer simulation in which I modeled the Earth as a thousand little balls. Each ball would be given an initial position and direction vector. They would move through space under the effects of momentum and gravity, and subject to the constraint of non-compenetrability [they can't occupy the same space). Simply enough to write the code, right? (John rolls his eyes.)

In the mean time, I think that Azmat has provided a significant question about insignificance.

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[1] A thought occurred to me when I was figgering those numbers. I have made the assumption that perigee and apogee mean the distance between the centers of the Earth and whatever satellite  we are talking about. I wasn't sure about that, so I went to look it up on Wikipedia:

"An apsis ... is the point of greatest or least distance of a body from one of the foci of its elliptical orbit. In modern celestial mechanics this focus is also the center of attraction, which is usually the center of mass of the system. Historically, in geocentric systems, apsides were measured from the center of the Earth."

http://en.wikipedia.org/wiki/Apogee

Uh-oh. Things are getting complicated. I'm gonna stick with the geocentric model.

[2] Rule number 1 about reading John the Math Guy blogs. When I end a statement with "right?" it means that it's not right.

[3] This footnote is not referred to in the main text, but I thought I should put it in for completeness.