Today is 2/4/16 or 4/2/16, depending on where you live and how you write dates. Either way, it’s a great day because 24=16 and 42=16. There aren’t many days like that (although we are treated to two this year), so it’s worth taking a moment to celebrate.
Converting between units can be hard, as seen before (and before and before). Fortunately, food containers often include both English units and Metric units. Unfortunately, those two don’t always match. Take, for example, Producers Sour Cream. Their 32 oz container says it has 907 grams, which is about what you’d expect. The 16-ounce container has half has many. Not half of 907, but half of that again: in bold defiance of the laws of physics, it sports a mere 226 grams.
This mistake has apparently gone on for years. What’s equally strange is that the various nutrition sites that include information about this product also say 16 oz (226g) without comment. Because, as stated above, units are hard.
Thanks to Philip Bailey for bringing this to our attention! And speaking of Math Mistakes, as I was, several of the mistakes listed in this very blog are published in the PRIMUS article “Math Mistakes that Make the News” by Yours Truly, which can be downloaded for free during the month of March (2015).
Back in the 80s, there was a commercial for Faberge Organic Shampoo. And even if the shampoo doesn’t sound familiar, you might have heard of the ad (“…and they tell two friends…”)
Hey, it’s exponential functions! 1 friend tells 2 friends, those 2 friends tell 4 friends, those 4 friends tell 9 friends, those…wait, 9? Where did that come from? And then those 9(?) friends tell 16 people. So it almost works, except that after the photo of 2 people they decided to switch to perfect squares.
Fortunately, a later ad brings the whole thing to a halt before reaching 9:
Good job Faberge people – you skipped the 9! Of course, this one went straight from 1, 2, 4 to 16 before diving headlong into a grid of 24 people, so I’m not sure it was much of a mathematical improvement.
Threesixty360…your source for commenting on 30 year old math mistakes that have already been well documented.
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!
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.
Decimal points are small, and so easy to lose. And it appears that many of them were lost on FAFSA (Free Application for Federal Student Aid) forms, which is NOT a place that you would want incorrect data. According to an official document from July 18, people filling out the form were supposed to round monetary values to the nearest dollar, rather than using exact dollar-and-cents amounts. But some people put down cents anyway, and the computer didn’t alert them, or tell them there was a problem. No, it slyly accepted the amounts, and then threw all the decimal points in the trash, so [as the official memo said], and income that had been recorded as $5000.19 was suddenly interpreted as $500019, which is one heck of a sweet income and probably enough to disqualify you from most financial aid.
This didn’t happen with just a couple people, either – The Wire says that 200,000 people are likely to be affected. And because it’s more than just a couple, schools have to look at all those applications, every single one, to catch any errors. Those errors might be that people didn’t get aid who should, which is a bummer, but it could also mean that people got too much money. That doesn’t sound as bad initially, but the July 18 memo says, ” If such aid has already been disbursed the institution may need to change awards and return (or have the student return) any overawarded funds.” I can’t imagine that it will go over all that well for a school to tell someone to give back money that was promised, so I suspect this messiness will last a while.
Hat tip to Yousuf for pointing out this article!
I can think of exactly two times when the word urn is used: as a container for someone’s ashes, and as a container for colored balls. Since I’ve never physically seen an urn that has balls in it, it makes me wonder – when did that become such a standard in probability problems? Why are the balls in an urn in the first place?
When I asked that very question, TwoPi mentioned “surmounted” as another example of an English word that seems to be used exclusively in one context: to describe Norman Windows (a window in the shape of a rectangle surmounted by a semicircle). “surmountable” is more common, and “insurmountable” even more so, so I suppose “surmounted” does actually appear in a related context (as in “that difficulty has been surmounted”), but it’s still relatively unusual. I suspect that there are other words, English words as opposed to mathematical terms, that just don’t show up very often outside of the exercises in a text.
Here is a Norman Window, by the way, from Notre-Dame d’Étretat in Étretat, France.
The other day I found myself wondering what proportion of genes cousins would expect to share compared to biological siblings. This took more time to figure out than I would have expected, in part because I knew that siblings who share two parents have 1/2 their genes in common on average, so I thought cousins sharing two grandparents might have a quarter. They don’t, though – it’s half that. In reasoning it out, it turned out to be easiest to think of moving up the biological family tree to a common ancestor, which led to one general formula and a few specific cases:
Given two people A and B, find their closest common ancestor C. If there are n generations from A to C and m generations from B to C, then the expected proportion of shared genes is (½)n+m. If there are two closest common ancestors (for example, both parents) then this number would double.
In the case of a parent and child, for example, there is 1 generation from the child to the parent (the common ancestor) and 0 from the parent to itself, so the proportion of shared genes would be (½)1, or just ½. Cousins would each be 2 generations from common grandparents, leading to (½)4, or 1/16, for cousins with one grandparent in common (sometimes called half cousins) and twice that for cousins with two grandparents in common (sometimes called full cousins). Double cousins — that is, people who are cousins on both sides of the family tree (for example, cousins whose mothers are sisters and whose fathers are brothers) — would still have grandparents as the closest common ancestor, but now it would be up to four common grandparents instead of just one or two: the expected proportion of shared genes between cousins with four common grandparents would be 4·(½)4, or just ¼. Likewise, an aunt and nephew with two parents/grandparents in common would be 1 and 2 generations respectively from this pair of common ancestors, so the expected proportion of shared genes would be 2·(½)3, also ¼.
Special Case 1: great-great-…-great grandparents
In this case the older relative is the common ancestor, so if “g” is the number of “great”s then the proportion of shared genes is (½)g+2. The additional 2 in the exponent is because the number of “great”s counts the generations after grandparents, who are already 2 generations away from their grandchildren. This is the only case where the proportion is exact: in all the others, it’s only an expected proportion because siblings could have anywhere from no overlap of genes to complete overlap of genes from each common parent.
Special Case 2: great-great-…-great aunts and uncles
In this case the older relative’s parent(s) are the common ancestor. With a great-uncle and great-niece, for example, the great-uncle’s parent(s) are the great-grandparent(s) of the great-nephew. This means that there is 1 generation from the great-uncle to his parent(s), but 3 from the great-niece to that common ancestor, with each additional “great” adding another generation. If “g” is the number of “great”s, then the expected proportion of shared genes would be (½)g+3 if there is one parent in common, and (½)g+2 if there are two. (I personally find it interesting that you can expect to share the same proportion of genes with a sibling who shares both parents as you do with either of the individual parents, the same proportion with an aunt or uncle who shares both grandparents as you do with either of the individual grandparents, and the same proportion with a great-great-…-great aunt/uncle who shares both great-great-…-great grandparents as you do with either of those great-great-…-great grandparents themselves.)
One clarification: great-aunt is the term I grew up with, but in looking around I just discovered that “grand-aunt” may be the technically correct term, since that person is in the same generation as a grandparent; likewise, the sister of a great-grandparent would be a great-grand-aunt. This appeals to me aesthetically. If you were to use these terms, then you’d have one fewer “great” in describing the relationship, and you’d need to add 1 to the exponent in the formulas above.
Special Case 3: second cousins once removed (and the like)
Cousins share at least one grandparent, second cousins share at least one great-grandparent, and xth cousins share at least one great(x-1) grandparents. This means that xth cousins are each (x+1) generation removed from the common ancestor(s), and would expect to share (½)2x+2 of their genes if there is one common relative and (½)2x+1 if there are two. Each removal refers to one of the people being one more generation removed from any common ancestors, and so increases the power of ½ by 1. This means that xth cousins who are y-times removed would expect to share (½)2x+y+2 of their genes if there is one common relative and (½)2x+y+1 if there are two. Second cousins once removed would share either (½)7 or (½)6 of their genes, while first cousins twice removed would share (½)6 or (½)5.
For those who like the visual, there is a handy little chart below, which appears to be in the public domain on Wikipedia. It does make some assumptions, however – namely, that siblings, cousins, aunts and nieces, etc. have exactly two closest relatives in common (both parents, two grandparents, etc.).
- The soccer ball that I think of as typical – in other words, the one that I remember from Days of Yore – is an Archimedian solid, made from 20 regular hexagons and 12 regular pentagons. Specifically, it’s a truncated icosahedron because it can be built from lopping the corners off of a regular icosahedron. It’s also a buckminsterfullerene, although that’s only the formal name: friends can call it a buckyball. The buckyball was Red Hot News in 1985, because it was a new way of putting Carbon atoms together. Scientists Harold W. Kroto, Robert F. Curl, and Richard E. Smalley* named it after architect R. Buckminster Fuller*, whose geodesic domes had inspired them to try and create such a carbon cluster. But this soccerball-shaped soccerball doesn’t limit itself to Ancient Greeks and Modern Scientists, oh no. It also dabbles in the arts, as shown in this photo below from Labor Park in Dalien, China.
- Soccer balls aren’t the only thing math-related in soccer: there’s also the number of people on a team. Each team has 23 players, which means that on any team there is a 50% chance that two people on that team share a birthday. With 32 teams playing in the world cup, you’d expect about half of them to have birthday-sharing teammates, and in fact, as the BBC pointed out earlier this week, exactly 16 of the 32 teams do. For example, tomorrow (June 20) six people have birthdays, including two (Asmir Begovic and Sead Kolasinac) on the team from Bosnia and Herzegovina. Now oddly enough, even though you’d expect half the teams to have teammates sharing a birthday, the fact that it’s exactly half is actually rather strange: with a 50% chance of two teammates getting to share cake, the probability that exactly 16 of the 32 teams satisfy that is only 14% – it’s just that at that point it’s equally likely to be more or fewer days. Ironically, it’s rather unexpected to actually hit the expected value.
- One final math fact about the World Cup: one of the referees for yesterday’s match between Chile and Spain is actually a former high school math teacher! Not all that former, either: Mark Geiger taught in New Jersey alongside his brother, winning the Presidential Award for Excellence in Math and Science Teaching, but eighteen months ago he left teaching in order to referee full-time, hoping for a shot at the World Cup. Not a bad gig, and he always has those math skills to fall back on if he finds he misses teaching.
*Whenever I type “[Occupation] [Person’s Name]” I get the urge to add “renowned” and then go read Da Vinci Code again.
The photo of the sculpture is by Uwe Aranas, Creative Commons License. And if you didn’t follow the link to the BBC article, “The Birthday Paradox ath the World Cup” by James Fletcher, it’s worth a read – it has a lot more detail about the birthday paradox and sports.
Godzilla is celebrating Star Wars Day with a Severed Wampa Arm Cake (recipe from justJenn):
Ewwww — that looks kind of gross. Godzilla is such a monster. Here’s a less bloody angle:
May the Fourth be with you!
Our 9-year old recently introduced us to Kid Snippets, a series of short videos feature adults acting out children’s conversations. (The children are given a scenario and “act” it out vocally; the adults then do the actual acting to the children’s dialogue.) These are now sweeping through the department, since watching funny videos is way more fun than studying for math tests. It’s also possible that I’ve watched more than a few of these for a break myself, even when no kids were around.
The news is no stranger to third derivatives, although it doesn’t sneak in very often – we’ve mentioned before the October 1996 issue of the Notices of the AMS in which Hugo Rossi wrote, “In the fall of 1972 President Nixon announced that the rate of increase of inflation was decreasing. This was the first time a sitting president used the third derivative to advance his case for reelection.”
Well, just like a blog that doesn’t post for so long that you figure it’s basically dead, and then all of a sudden out of nowhere WHOA there’s a new post (Hi everybody!), the third derivative made a new appearance recently. On the appropriately palindromic- (in the US) date of March 10, 2013, Paul Krugman wrote in the Opinion pages of the New York Times:
People still talk as if the deficit were exploding, as if the United States budget were on an unsustainable path; in fact, the deficit is falling more rapidly than it has for generations, it is already down to sustainable levels, and it is too small given the state of the economy.
Did you catch that? The line about the deficit falling more rapidly than it has been? Let’s take a closer look:
Assume that the National Debt at year t is the original function: D(t). This is positive, since we have debt.
Then the Deficit is the derivative, D'(t). It’s also positive, because the National Debt is increasing.
If the Deficit is falling, that means that the Deficit is a decreasing function, so the derivative of D'(t) – that is, D”(t) – is negative. That would mean the Debt is increasing, but concave down.
But the quote said the Deficit is falling more rapidly than is has been: the derivative of the Deficit is getting more negative, so to speak. In other words, the Deficit itself is decreasing and concave down, which means that D”'(t) is negative.
And so we have a third derivative! Welcome back old friend!
A few days ago while in a yarn shop I ran into one of our alumni from a few years ago. Seeing her was fantastic, but the icing on the cake (Sorry. Sort of.) was that she had gotten married last summer to a biochem major, and they designed their own wedding cake. She sent me a couple photos the next day of what is perhaps the coolest wedding cake I’ve ever seen:
Hey, I can answer that question! I’m not as sure about the next one, though.
Congratulations (and Happy Anniversary!) to Emily and Glen — I can think of no better way to start a marriage.
Props to the photographer, Hilary Argentieri, for taking such clear picture!
Oddly, this isn’t even our first math wedding post (see this mathy proposal) although it’s the very first one involving cake.
Edited 9/19 to add: Apparently this came from xkcd — I think that makes me like it all the more.