360

In spherical geometry, the shortest-length curve between two points on the surface of the sphere turns out to be part of a Great Circle – an equator-line circle that cuts the sphere in half.  So lines are circles, which is fun to share with philosophers.  (Note – taxicab geometry provides that same amusement, where circles are squares.)

So a natural question, where “natural” means I never actually thought of it but wish I had, is What is the longest line along the surface of the earth that goes entirely through water?  This would be the longest possible straight-line sailing distance, if you ignored all the physical aspects of sailing like wind and water currents.  Fortunately, before I even thought of the question, someone had answered it.  Behold!

Longest straight-line distance through the water on earth.

This gif appears to be from a youtube video by Patrick Anderson of 2012 (here) which has the advantage of being a little slower.

So that raises the question of the longest straight-line distance through land.  And here’s a guess at it:  http://i.imgur.com/nbNfl.jpg and then another one  https://sites.google.com/site/guybruneau/fun-stuff/longest-distance-on-land, although that second one it doesn’t quite look like part of a Great Circle so possibly the projection imposed a different geometry. Or possibly I have trouble visualizing projections of Great Circles, which is also possible because they are weird.  (The cool kind of weird, of course.)

Thanks CJ for sending me that gif, although now that I’m finding myself asking questions like “What line passes through the most countries?” I can tell that it’s going to keep me from my grading for longer than it should.

Remember that post on August 1 about the math mistake?

The mistake here is that while individual dimensions were correctly converted to centimeters (by multiplying by 2.54), the Capacity was incorrectly converted, since the 4720 cubic inches were multiplied by 2.54 (and not the cube of 2.54) to get 11988.8 cubic centimeters.

I found this error a month or so before posting it so I know it was around for a while, but I just discovered that the error has been fixed.  Behold, the new stats!

In this case, the somehow-determined capacity of 4720 cubic inches is multiplied by (2.54)³, or approximately 16.387 cubic cm per cubic inch, to get 77,346.9 cubic centimeters.  But one liter is, conveniently [although not exactly coincidentally], exactly 1000 cubic centimeters, so this translates to 77.3469 liters, which does round to 77.3 liters.  The stats for other pieces of luggage were similarly updated.

I’m so happy!  I’d like to think that the People In Charge actually read this blog, or at least the email I sent them about it, but alas I have no evidence of this.  Still, it’s nice to see the mistake corrected.

As far as fractions go, halves and quarters get a lot of the glory — it’s fairly easy to cut things in half for one, and one could argue that more integers are divisible by two than by any larger integer (although one could also argue the opposite, so that’s probably moot).  Lately I’ve been thinking that thirds are really the sleeper fraction.

Take the calendar year:  we divide it into four seasons, but they’re not quite equal, at least in upstate NY.  Winter spans three of them most years, which can feel a little depressing at times.  So about a year ago I started wondering if there was a way to redistrict the year in order to contain winter more thoroughly, and the best configuration I could come up with was:

Fall:  August-November
Winter: December-March
Spring-Summer:  April-July

It’s not quite perfect, but actually fits our academic schedule pretty well — school starts at the end of August so most of August is spent getting ready for school, and not so much in summer mode.  November is probably more like winter, but at least the first half of it is usually pretty fall-like.  Then December through March are clearly winter, no two ways about it, but April…ah, April.  I love April.  It might be my favorite month, except it’s always so busy, but that’s when the daffodils come up and the grass thinks about growing, and we can open the windows more often.   Like I said, April tends to be really busy, but it also has that happy Summer’s Coming feel, and the seniors are thinking about graduation, and it’s really a jump into summer.   And May is finals and graduation, which lead to June and July.  So for me at least, these thirds are a little more natural that the quarterly seasons.  [And for schools that start in September, a one-month shift might work, depending on whether school or the weather is the dominant feeling of winter.]

I suspect that thirds might work well for a clock, too, where the AT hours are for work/school, the BT hours are for homework and play, and the CT hours are for sleep, although when I try to put numbers to this it doesn’t work quite as well, since our work day itself tends to be split into three parts (early morning at home, day at work, and then evening at home again, stealing time from the other parts).  But still, in theory it makes sense, at least as much as 12-hour divisions.

Goooooo…thirds!

If you haven’t been there yet, head over to Google [their image links to a search for Pi Day).  If you’ve missed it, it might show up in their archive page here.  Happy Pi Day everyone!

There was an interesting article in ScienceNOW last week about how a study has shown that people walk in circles if they are blindfolded and set loose in a field (unless it’s sunny out; then they stay in a straight line).  The article referred to various novels and movies, indicating that as a plot devise walking in circles isn’t unrealistic.

As I read it, I kept thinking, “Circles?  Really?  But wouldn’t maintaining constant curvature be about as remarkable as walking in a straight line?”  Yet no mention was made of this, and when I looked at the accompanying picture it turns out that circle was used in a more literary sense:  the circles were really just closed loops occurring within the path.  And although that too is interesting because I’d be freaked out by crossing my own path again, my mathematician self was sorry that they weren’t actually circles.

This squiggle picture actually has a lofty pedigree:  it appeared on page 1 of La Peau de Chagrin by Honore de Balzac in 1901.

We adopted kittens this week, within 24 hours of arriving home.  This was the culmination of several months of deciding whether or not it was a good idea given that we already have two adult cats who weren’t actually asking for younger siblings.  So far, the kittens are happy, one adult cat is curious and I think will be fine, and the other adult cat has at least progressed to the point where she’ll eat snacks right by the doorway to the room the kittens are primarily staying in.

Anyway, the place we got them from advertises that one adult female can produce 420,000 offspring over 7 years.  That seemed a little high, so I decided to check the math.

The first thing to wonder about was how to count offspring.  Clearly this was more than just kittens:  it must be counting all future generations.  But predicting how many offspring males can sire was difficult, because it really only takes a few males to father a LOT of kittens.  So I decided to initially look only at the female lines.

In looking around, it seems that cat pregnancies last about two months, and a female can get pregnant again about a month after that.  In theory this would mean 4 litters a year, but The Internet implies that most cats have 2-3 litters per year.  I decided to go with 3, since I was trying to see if that 420,000 was even resonable (as opposed to typical).

There’s also the question of how old cats have to be until they are able to mate.  It turns out to be around 6 months, though for convenience (since I was going by 4 month intervals) I decided to go with 8.  Or, looking at it another way, I figured that kittens would have their own litter on their 1st birthday (and every 4 months after that), so that’s assuming mating at about 10 months old.

Finally, there’s the size of the little.  I decided to go with 4 kittens per litter, two of which were female.

Then I made a table.  It started off like this:

0 months:  1 adult female
4 months:  1 adult, 2 (female) newborn kittens
8 months:  1 adult, 2 newborns, 2 kid-kittens
12 months: 3 adults, 2 newborns, 2 kids
16 months:  5 adults, 6 newborns, 2 kids
20 months:  7 adults, 10 newborns, 6 kids

By the end of 7 years, there were 35951 adults, 42410 newborns, and 25006 kid-kittens, leading to a grand total of 103,367 cats, including the original one, all along female-descendent lines.  If you double it in order to count the males in each litter (but no separate offspring of those males), you get around 206, 732 offspring.

Initially it sounds like the 420,000 is an overestimate, but the wording only is that cats can have that that many youngsters, not that they normally do.   If you assume that there are 3 females in each litter (all of whom procreate, etc, since this is just a ballpark estimate) you actually get 898798 offspring along the all-female lines, or over 1.5 million offspring in 7 years.   So while I doubt that having half a million offspring within 7 years is the norm, it certainly seems possible in extreme cases.

Or, rather, Math Teachers at Play #9 is up at Homeschool Bytes (who have hosted MTaP before as well!)  The theme is Game Time, and the post has several games and magic tricks.

Speaking of games, did you know that they have a bunch of math games at coolmath?  You can play Mancala (although it seems like there are a million variations to that), the Tower of Hanoi, and lots of other stuff.  Watch out for the sound effects, because that can get a little overwhelming at times.   (Fifteen minutes later:  I just finished a game of Sudoku, helped out by the 8-9 year olds in the living room.  Why didn’t I play with this site before?  I think Sudoku is one of the best ways to teach Proof by Contradiction, since once you get past the infinitude of the natural numbers, the infinitude of the primes, and the square root of ___ is irrational it’s harder to come up with good examples.

(Incidentally, in courses where I teach that the square root of 2 is irrational I usually make it homework to show that the square root of 3 is irrational.  Once they’re comfortable with that, I have them prove that the square root of 4 is irrational, and to find the flaw in the proof.  This proves to be really challenging to the sophomore math majors.)

I think I’ll end my stream of consciousness here.

This is hardly worthy of a blog post (but, really, if I waited for stuff that was there would be like 3 posts a month), but it still really amused me.

WordPress lets you look at stats for individual posts, which is fun because you can see which posts were the most popular over time (Scoring March Madness by a landslide, thanks to Basketball Guy).  There’s also a column for the % change each week, so if 2 let’s say 200 people look at that particular post one week, and 3 300 people look at it the next week, the percentage increase is +50%, from (300-200)/200.  But what happens when you first post?  The previous week clearly no one looked at the post, so finding the percentage increase causes problems.  But not for WordPress!

The percentage increase (in this case of the views of Godzilla makes a hexaflexagon) is  ∞.  That little attention to detail makes me happy.

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.

Yesterday I posted about the biggest bills in the US, and that was interesting to me in a historical sense.  However, the part of me that likes big numbers really liked seeing what other countries had to offer.

This proved to be difficult.  There are a lot of countries, and even individual countries change their currencies from time to time, especially if there is high inflation [which would be tied to really big numbers on the bills].  Here is but a small sample.

There’s a 100,000 rubles bill (2000) from Belarus, worth about US$36.

There used to be a 2,000,000 złotych (1992) in Poland but in 1995 the złoty was formed by dropping 4 of the zeroes, leaving the largest current note as 200 złoty.  It’s worth about US$61 and has Ignacy Jan Paderewski on the front, who was a Prime Minster of Poland and also a world-class pianist.

From 1990-2005 there was a 20,000,000 note in Turkish Lira.  Isn’t it colorful?  It’s got Mustafa Kemal Atatürk on it.  He was the first president of Turkey, and in 1927 gave a speech that lasted 36 hours.

But the numbers don’t stop in the millions, no siree.  About 15 years ago Yugoslavia, right before it split up, had a  500 billion dinar note featuring Serbian poet Jovan Jovanović Zmaj:

From the National Bank of Serbia (NBS Copyright)

And that’s not even the biggest.  Zimbabwe is still undergoing hyperinflation, and has a 100 billion dollar bill (2008):

Even with all those zeroes it’s not worth much, less than US$1, and by the time you read this the value will have decreased even more.   Less than four months ago newspapers reported that Zimbabwe would start printing notes in the trillions, up to and including a 100,000,000,000,000 dollar note, worth about US$30 at the time.

Last week I was all excited about an article showing a connection between schools with playgrounds and higher math scores.  As it happens, at the very same time there was another news report about higher test scores.  This one linked them to gum.

The first article I read, from the LA Times, explained:

The study was conducted by researchers at Baylor College of Medicine and was sponsored by the Wrigley Science Institute. The study included 108 students, ages 13 to 16, who were assigned either to chew sugar-free gum during math class, while doing math homework and during math tests, or to refrain from gum-chewing. After 14 weeks, the students took a math test and their grades were assessed.

The result was increased test scores for the gummy kids on the Texas Assessment of Knowledge and Skills achievement test.  The increase was small — only 3% — but the National Post says it was statistically significant.  Several articles drew the connection that chewing gum can lower stress, and that may have helped with the testing.

The National Post article turned out to be one of the more detailed ones that I ran across (explaining, for example, that the gum-chewers chewed gum 86% of the time in school and 36% of the time while doing homework) and it shared a fact that I didn’t see in many other articles:  there were TWO standardized tests given, not just one.  The other standardized test was the Woodcock Johnson III Tests of Achievement, and on this test there was no statistically significant difference in the scores of those who chewed gum and those who didn’t.

Furthermore, when I looked further, I found  this nutrition site that revealed that the course grades for both groups actually decreased over time, although the gum chewers had less of a decrease.    So all in all, it’s kind of interesting but perhaps not worth the hype of headlines like “Chewing Gum Improves Math Scores… No Lie” and “Teenage Brain Power Boosted by Chewing Gum:  Wrigley Study”.

Isn’t that a gross photo at the top?  That’s Bubblegum Alley in my hometown [photo taken by Samir and posted on wiki under Creative Commons License].  When I was about 10 a friend and I wrote “Sit on It” in gum on the wall of Bubblegum Alley.  We felt very sneaky.