Upcoming Carnival at The Math Less Traveled

The next Carnival of Mathematics #53 will by hosted by Brent Yorgey  at The Math Less Traveled.  He’s taking submissions through Friday at 8am (US Eastern Daylight Time) — there’s more information about how to submit here.

(Pausing to figure out if it’s Daylight time or regular time always reminds me of one of the more memorable times that  TwoPi and I drove from California to Tuscon visiting various relatives.  Initially we were on Highway 8, starting from Yuma.  But then at Gila Bend we saw that there was an alternate route to Tuscon.  And it looked like it was just about the same distance, if you forget to take into account speed limits and the fact that the southern route had lots of twists and turns.  Which we did.  Benoît Mandelbrot would have had a field day.

It was scenic, but it soon became clear that we’d be arriving in Tuscon late.  And that still seemed okay until we realized that even though Arizona is usually in the same time zone as California, that’s a consequence of Mountain Time Arizona NOT going on Daylight Savings, and since this occurred around Christmas neither state was on DST, and they were an hour apart.  Are you confused?  So were we, but the end result was that we were an hour later than even we realized, and that was the point at which we pulled over and called TwoPi’s brother.  In good news, he was nice about it because by then he was used to our meandering ways and knew we tended to go off track, as anyone still following this story can attest.)

Math Teachers at Play #8 is up!

Hooray!  It’s another Math Teachers at Play, up at Let’s Play Math! There are like 7000 posts okay, maybe less than that, but it’s still a lot of mighty fine reading.  As a bonus, it’s spiced up with math jokes.  Everyone likes math jokes.

Denise also threw in a comment about the Carnival of Mathematics, wondering if anyone knows it’s next location.  Anyone?  Anyone?  Bueller?

Longitude, Part I

So yesterday we talked about how to find your latitude if you’re in the Northern Hemisphere (and I’m pretty sure there are ways to find it in the southern hemisphere with an astrolabe, but I don’t think they’re as straightforward).  Longitude, however, turns out to be tough no matter where you are.

For one thing, although the Earth spins so there are two natural poles, and therefore you can measure Latitude in relation to those poles, there isn’t any natural Longitude:  it’s all in relation to an arbitrary point.  According to this official site at Greenwich, Hipparchos was the first guy to use longitude formally, about 150 BCE,  and he used Rhodes in southern Greece as the key spot.  About 250 or 300 years later Ptolemy joined the fun, but he used the Canary Islands in Spain as the starting point.    Mecca was used as a Prime Meridian for many mathematicians.  France had one, as did the US and Canada.  Altogether there were lots and lots of 0°, which could get a little confusing, and so about 130 years ago a bunch of countries got together and decided that Greenwich could be #1 Number 0, and everyone was happy.  Except France.  France didn’t want to give up its own Prime Meridian through Paris, so they refused to vote for Greenwich and kept using their own meridian until…until… actually, I don’t know if they ever formally adopted the Greenwich one.  Here’s a picture of the Paris one:

FredA took this picture (published under GNU-FDL).

So the first problem is deciding on a reference point, and clearly that was a big one.  The next problem is figuring out how far away from that reference point you are, and that’s hard too because the earth keeps spinning.  The simplest way is actually to use time: the Earth makes one complete rotation in 24 hours 23 hours and 56 minutes, so the Earth turns 15° in just under an hour.  Which means that if your buddy calls you and says, “Dude, did you see that amazing eclipse that happened at 1:23?” and you say, “Yes I did, but here it happened at 2:15!” then it means that you are 52 minutes ahead, so that’s (52 minutes)*(1 hour/60 minutes)*(15°/1 hour)= 13° longitude further east than your friend (assuming the AM/PM match up).    You could also set a clock to noon, and then if your clock keeps working  you could compare it to local noon (when the sun is at the highest point in the sky) as you travel away from your starting point.

You don’t HAVE to use time to determine longitude, and in fact some of the great minds did look into ways to use the stars to find a way, but it did turn out that clocks were the key (as we’ll see in part II).

Latitude

Suppose you’re on the open seas, and your GPS falls overboard.  Or maybe it’s 400 years ago.  How can you tell what your latitude and longitude are?

We’ll start with latitude, because it’s easier.  A LOT easier.  If it’s a nice night, and you’re somewhere in the Northern Hemisphere, the easiest way is to take a gander at the North Star using an astrolabe.  Some astrolabes are simple, others are complicated, but they can all be used to find the North Star.

Hey, how about if Godzilla shows us how to use a homemade one!

This model was originally designed by…someone.  I don’t actually know who came up with it.  But all you need is a protractor, a straw, tape, string, a weight, and someone to drill the hole in the protractor so you can attach the string [or you can use a paper protractor].

In theory, the bottom of the straw should be held up to the eye.  Godzilla’s short arms aren’t really conducive to holding things up (he’s had to survive using his brains as much as his brawn), so here’s a drawing of what it would look like:

When you look through the straw at an object like the top of a tree or the North Star (Hmm, why didn’t I draw a star?  Because Godzilla was actually facing tree in our yard at the time, so I got distracted).  Anyway, when you look through the straw, the weight causes the string to fall right at the angle that the object is above the horizontal!

Here’s why:

So sailors just grab their handy dandy astrolabe, measure the distance of the North Star above the horizon, and that’s the latitude!  [I always have to think that one through.  On the equator, the North Star would pretty much be on the horizon, so that's 0° above the horizon, and at the North Pole the North Star would be directly overhead, which is 90° above the horizon.]

Next up:  Longitude (Part I and Part II).

Memorial Day 2009

Last year, we posted a brief discussion of the history of Memorial Day, a US holiday of remembrance of Americans who died in military service for their country.

Following up on last year’s post:   Frank Buckles is now 108, and [naturally] is still the last known surviving American veteran of World War One.  Last year’s NPR interview with him is still available on-line.

Photo taken by TwoPi in January 2008 at Fort Rosencrans National Cemetery, on Point Loma, San Diego,  CA.

Dieting the MATH Way

There are many diets in the world, but most of them fail for the simple reason that they use the wrong kind of mathematics.  Simple logic suggests that food choices based only the arithmetic of

Pounds Lost = (Calories Used – Calories Eaten)/3200

are insufficient for good health.  The math is just too elementary!  If you want an Advanced Diet, you need to use Advanced Mathematics!  Now with the power of Theorems.

The Harmonic Diet
If you’re a person who starts off strong but has trouble sticking to a plan, this is the diet for you!  On the first day, lose 1 pound.  Tough, but possible with that initial surge of motivation.  On the second day, lose 1/2 pound.  On the third day, lose 1/3 pound.  On the fourth day, lose 1/4 pound, etc.   As your motivation wanes, so will your weight loss; however, thanks to the power of Calculus you are guaranteed to eventually reach your goal weight.  One of the amazing features of this diet is that it will still work even if you don’t start with the 1-pound loss:   you can just as easily jump in on Day 3 or whichever day suits your fancy.

The Zeno Diet
Endorsed by Ancient Greek Philosophers everywhere, this plan is perfect for the daytime snacker!  Start by choosing a total number of calories that you’d like to consume in a day:  say, 1000 calories.  Your first meal should be 500 calories; after all, breakfast is the most important meal of the day, and skimping is unwise.  Your second meal can be whatever you want, whenever you want:  just be sure to limit it to 250 calories.  Want another snack?  No problem!  Treat yourself to any 125 calorie food.   Your next tidbit can be 62.5 calories, the next 31.2 [no rounding up!] and so on.  With this diet plan you can snack as many times as you want, and you’ll never exceed (or even reach!) your total allotted calories.

The Banach-Tarski Diet
This diet uses the power of Set Theoretic Geometry to help those people who want larger portion sizes.   Start with small amounts of your favorite foods.  Roll each item into a ball and grab a sharp knife.  Then cut each ball into 5 pieces and reassemble into two balls OF THE SAME SIZE!  Repeat as desired.

The Fibonacci Diet
Is your focus more on healthy eating than actual calories?  The Fibonacci Sequence 1, 1, 2, 3, 5, 8, 13, … is more than just a fancy way to convert between miles and kilometers:  you can arrange your entire plan around these special numbers.  Break your eating into 3 meals with 2 snacks.  Make sure each meal is made up of 1/2 carbohydrates,1/3 protein, and1/5 fat.

It’s OK that the amounts add up to more than 1, because THIS diet was based on a Bestselling Novel and is featured both in Women’s World Magazine in 2006 and in the book  The Diet Code:  Revolutionary Weight Loss Secrets from Da Vinci and the Golden Ratio by Stephen Lanzalotta, which you can buy for as little as 13¢.  Now that’s value.

In a fortunate coincidence, Walking Randomly makes all of this even easier by showing show to use Wolfram Alpha to compute calories.

The Wolfram Alpha Bandwagon

I knew that Wolfram Alpha was coming, but couldn’t quite figure out what it was so I didn’t keep too close an eye out for it.  Then I read on Teaching College Mathematics and the Number Warrior that it was up, and I was pretty impressed with the screen shots.    Since then, I’ve been playing around with it, and I’m impressed.

It solves problems:

(See how you can switch from exact forms to decimal approximations?  With series, you can even tell it to use more terms.)

It gives you data:

(My favorite part is the info at the bottom, about population density and population growth.  I can see those as being useful for writing problems in stats or calculus.  I wasn’t able to get it to predict the population in a given year or predict when it would be a certain population, except once accidentally when it said the US population would be something like 4 quadrillion some year in the distant future.)

It also converts units.  I know that you can do this easily on Google, but this gives you a whole selection and you can pick the one that you like best.

And you get to look up cool stuff, like cities, movies, colleges, and names:

So all in all, it seems like it’s a combination of many of the things I like about Wolfram’s MathWorld, Wikipedia, and Google.  It doesn’t supplant any of them, but it’s quite user-friendly and I’m looking forward to seeing what else it does.

Math Teachers at Play #7 plus a video

Math Teachers at Play #7 is living this week at Homeschool Bytes, with lots of neat posts (like this one about Math and Language Reversals).

Incidentally, if you haven’t seen it yet, on April 30 The Daily Show had a great piece [not quite safe for work in places] about the Large Hadron Collider (currently due to start up again in the fall).   Pay careful attention to the calculation that the world had a 50% chance of being destroyed; Walter probably checks this site regularly.  (There’s a lot more discussion at Good Math, Bad Math.)

[Hmmm, the embedding isn't working.  Bummer.  You'll have to go here.]

Sunscreen confusion

An article in the New York Times describes consumer confusion over the ever-rising SPF numbers (used to rate the efficacy of sunscreen lotions), and their interpretation.

Unfortunately, the NYT adds to the confusion with the following:

The difference in UVB protection between an SPF 100 and SPF 50 is marginal. Far from offering double the blockage, SPF 100 blocks 99 percent of UVB rays, while SPF 50 blocks 98 percent. (SPF 30, that old-timer, holds its own, deflecting 96.7 percent).

Technically they’re right:  doubling the blockage is not the same as halving your radiation exposure.  But in terms of safety, the issue isn’t how much UV exposure you’ve avoided, but rather how much UV actually gets to your skin cells (which would then be a 2% versus 1% comparison).

According to the article, SPF measures how much longer a person wearing sunscreen can be exposed to sunlight before getting a burn, when compared to someone wearing no sunscreen.  Someone wearing SPF 50 can remain in the sun 50 times longer than someone with no sunscreen, and so SPF 100 sunscreen provides the wearer with twice the protection (in terms of time) as SPF 50 sunscreen.

It turns out there is a sense in which SPF100 is not twice as effective as SPF50 in protecting your skin, but it has nothing to do with the 99%/98% comparison.

According to the NY Times, “a multiyear randomized study of about 1,600 residents of Queensland, Australia” found that most users applied at most half of the recommended amount of sunscreen.

“If people are putting on about half, they are receiving half the protection,” said Yohini Appa, the senior director of scientific affairs at Johnson & Johnson, of which Neutrogena is a subsidiary.

But in fact they are receiving far less than half the protection:   a 2007  British Journal of Dermatology study noted that cutting the amount of sunscreen in half did not reduce the effective SPF in half, but rather reduced it geometrically to its square root.

If a person uses half of the recommended amount of an SPF50 sunscreen, they’ll get the protection of an SPF7 (since 7.1 is roughly √50), while similarly underapplying SPF100 sunscreen gets the protection of SPF10.

Apparently, if you’re looking for the protection of an SPF30 product, but like most people tend to under-apply sunscreen, you should be shopping for sunscreen rated as SPF 900.   No word yet on when such products will hit the marketplace.

(One wonders: does this work the other way ’round?  If I apply TWICE the recommended amount of a cheaper SPF8 sunscreen, do I end up with the protection of SPF64 sunscreen?)

Godzilla makes a hexaflexagon

When Godzilla isn’t trampling buildings, flipping pancakes, or making cookies, he likes to engage in the fiber arts.

So he decided to crochet a hexaflexagon.  This is a hexagon that seems flat, but can be twisted to show hidden sides.

Here’s the hexaflexagon that Godzilla made using a pattern from Woolly Thoughts (link updated 1/1/10). It initially looks like this:

but he can twist the inside…

and there is purple in the middle instead of blue sparkles!

Then he can twist it again…

And it’s orange in the middle!

And those aren’t the only colors.  If you look at the other side, there is this

and this

and, finally, this!

(Crocheting it can go pretty quickly, depending on how many meetings or TV shows are on your schedule; you can also make paper versions using patterns from here or here or, of course, here.)

Counting Trouble

There have been posts here before about how bees and elephants can count small numbers.  Sometimes it’s a little difficult for us humans, however.

see more pwn and owned pictures

DragonFable Math

The 8½ year old in our household recently announced, “I have an idea for a blog post!  DragonFable Math!”   DragonFable is a role-playing game in which you “walk around, go on quests, and slaughter stuff.”  Young E insisted that there was math in it, however, explaining:

If you don’t look at the melee (aka damage) [the damage points that you've scored against your opponent] and just look at the hit points, you can figure out how much damage was sent.

A character’s hit points measure their ability to endure damage, and they decrease by the number of damage points in each fight.  When your hit points reach zero, you’re toast!

So what Young E was describing is the missing addend model for subtraction.  And actually, it turns out that there’s more than just that.  By Googling “DragonFable Math” painstaking research, I found several  DragonFable Game Formulae.  For example, there’s :

EXP To Next Level= (Your Level)*(Your Level)*(100)-(Current Experience)

and

Total Stats Possible At Any Given Level= 3*(Your Level-1)

complete with examples. Many of these are at a good level for upper elementary school, and sort of real-life (virtual life?).  Moreover, the formulas were derived by players,which to my mind suggests opportunities for even more interesting problems  of the what’s-the-pattern variety.

So credit to E, for showing me that there’s more to DragonFable than just dragons.

Carnival of Mathematics #52

YES!  It’s Carnival of Mathematics #52, hosted by The Number Warrior.   He starts with a great math problem that had me wondering, “Huh?  Is this the Carnival?” until I got to the end and then I loved the punchline.    There are a ton of posts there, and I’m running into my regular carnival problem that I check out the links and get completely distracted by the fact that I’m in the middle of writing a blog post.

Thanks for another fantastic Carnival!

Unit Comparison

Here’s the price of Flax Oil at our local Wegmans:

Is that too hard to read?  The bottle on the far left is 16 fluid ounces for $15.49, and the bottle next to it is 8 fluid ounces for $8.49.  As it happens, even though the bottles are different sizes it’s pretty easy to compare the prices for this particular example:  the large bottle is just under $1/ounce and the small bottle just over, so the prices are similar but the large bottle is the better deal.

In general, though, it can be difficult to compare different sizes of the same product so the store provides a unit comparison to the left of the price.  In theory this gives the price per ounce/pound/quart or whatever makes sense.     The most important part of this is that you use the same unit for the different brands/sizes — that’s what makes for the comparison.  Yet if you look at this picture [which is a little grainy, because Wegmans doesn't like people taking photos so this was done on the sly with a cell phone] you can see that the small bottle is $33.96 per quart, while the larger bottle is only $15.49 per… pint.    Every other bottle on that shelf has the price per quart except for that bottle of Flax Oil, which happens to be the Store’s Name Brand.

What makes it worse is that when TwoPi first noticed this a couple weeks earlier, Wegmans Flax Oil was more expensive and so the large bottle was not the better deal.  That make the sneaky “per pint” price even worse, because it hid the truth.

Wegmans, we love your store but in this particular regard, you blew it.

Math in the Movies

A recently ran across a great resource page of  “Math in the Movies” from MathBits.  This page lists clips of movies (with links if they’re available online) and worksheets that can be used in the classroom.

Two of the examples relate to Star Trek, which as you all know (right?) will be in theaters starting this weekend.  In Episode 20 (“Court Martial”) of The Original Series, James T. Kirk is accused of murdering Benjamin Finney, the Records Officer.  At one point late into the episode, Kirk uses the computer to hear the heartbeats of everyone on The Enterprise.  As he explains:

Gentlemen, this computer has an auditory sensor.  It can, in effect, hear sounds.  By installing a booster we can increase that capability on the order of one to the fourth power.  The computer should be able to bring us every sound occurring on the ship.

One to the fourth power?  Not so impressive (and was I the only one thinking that you’d hear a lot more than heartbeats if every sound was magnified?  Wouldn’t breathing and moving be really LOUD?  But I’d better be careful with my critiques, because I’m a big fan of The Next Generation and a recent viewing revealed that it isn’t immune to problems either.)

The MathBits worksheet to accompany the scene is here.  They don’t link to a video clip, but you can see it on YouTube.  Actually, you can see the entire episode on YouTube:  the mistake starts at 38:10.

(Edit 5/9:  the embedding no longer works, but you can see the episode at this link.)