Wednesday, December 26, 2012

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


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.

am 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."

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.

Wednesday, December 19, 2012

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.

Wednesday, December 12, 2012

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.

Wednesday, December 5, 2012

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.

[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. 

Saturday, December 1, 2012

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.

Relative pull
Side closest to Moon
374,000 km
Center of Earth
380,000 km
Side farthest from Moon
386,000 km

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.

Relative pull
Pull relative
to the Earth
Side closest to Moon
374,000 km
3.00 X 10-6
Center of Earth
380,000 km
2.91 X 10-6
Side farthest from Moon
386,000 km
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.

[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."

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.