Monday Morning Math: Hippocrates of Chios

No, not that one!

Hippocrates of Chios was a Greek mathematician who followed Pythagoras and lived around 470-410 BCE. He was a contemporary of the more famous Hippocrates, the physician Hippocrates of Kos (c460-370 BCE). He is perhaps best known for his work on the classical geometry problems of squaring the circle and doubling the cube.

Astute readers will note that both of the geometry problems above are impossible to solve in general, but Hippocrates’ work led him to discover how to compute the areas of certain lunes, regions bounded by two circular arcs. (See this video for a pleasing application of Hippocrates’ discovery.)

The “Lune of Hippocrates”

The Greek philosopher Proclus credits Hippocrates with writing the first version of Elements of Geometry, much of which Euclid would later incorporate into his own Elements. In this text, Hippocrates introduced the use of letters to represent points, as well as naming objects using the points that defined them (e.g., “triangle ABC”). It is also believed that he had developed a method of “proof by exhaustion” (approximating circles by polygons with an increasing number of sides), later used by Eudoxus and Euclid. Unfortunately, Hippocrates’ text has been lost to history.

While respected as a mathematician, Hippocrates was viewed as “stupid and incompetent in the business of ordinary life” by Aristotle. He lost a large sum of money due to fraud, and had to teach geometry in Athens to make up for it.

Sources:

https://en.wikipedia.org/wiki/Lune_of_Hippocrates
https://mathshistory.st-andrews.ac.uk/Biographies/Hippocrates/

How many before the end?

Let’s start with ultimate (yes, that seems a bit backwards – stay with me.), as in “last”.  The next-to-last, then, is penultimate, easily one of my favorite words.  But there’s more!  The next-to-next-to-last is the antepenultimate.  Need another?  The -last is the preantepenultimate, a completely real word that Chrome’s built-in dictionary has never seen (it suggests “prearrangement”).

Why am I telling you all this?  I mean, besides the sheer ridiculousness/awesomeness of a word for the fourth-to-last item in a list?  I needed an excuse to post this video:

Granola Fail

From a recent Nature Valley ad in the London Metro newspaper:

Perhaps the second bar is twice as delicious as the first.

Via Language Log.  Photo from Spiderham.

Alpha’s Curious Filter

For no reason that I can think of, I decided to see how much Wolfram Alpha knew about probability, so I typed “probability of a full house” into the search box and got the following:

I thought that was pretty cool, especially since it includes the derivations, so I asked a few more questions, such as “probability of at least 2 red cards in a 5 card hand“:

Odd that it will count the numerator but not the (easier) denominator .  At this point, I thought I’d try a standard probability question (balls in an urn) that might be harder to parse because of the additional statements: “probability of drawing a blue ball from an urn contaiing 5 blue balls and 7 red balls“.  However, I missed the ‘n’ key when typing “containing” and got the following:

So, yeah, OK, Wolfram Alpha doesn’t provide “adult” content (why the quotes?), and I’m pretty sure I know what it’s reading as “adult”, but c’mon.  Note that fixing the typo doesn’t alleviate the problem, but it does cause Alpha to hiccup and request more computing time.  With variations on the wording, I’ve also had it return a picture of a blue ball along with the HTML code to generate it.  Nice.

The Carnival Lives!

The Carnival of Mathematics is still going strong.  This round – #76 – is hosted over at Walking Randomly and has, as usual, something for everyone, including a post from one my favorites: Language Log.  (Yes, they use math there.  Fairly often, in fact.)  Go check it out, and while you’re at it, contact Mike if you’d like to host one.

The Difference

Friday’s Saturday Morning Breakfast Cereal.  (Click to view the original along with the bonus content.)

So which are you?

Fall Newsletter Is Up!

Actually, it’s been up for almost a month, but I’m just now getting around to telling anyone.  See it here, and as usual, it’s probably more interesting for our current students and alumni than others.  But also as usual, we have a few fun problems to ponder:

Problem 5.1.1: (2000 AIME I) Let be relatively prime positive integers and suppose that the coefficients of  and are equal in the expansion of .  What is ?

Problem 5.1.2: An envelope contains 12 bills: 3 ones, 3 fives, 3 tens, and 3 twenties.  Two bills are drawn at random without replacement.  What is the probability that their sum is at least $20?

Problem 5.1.3: (From a Martin Gardner collection) An absentminded teller switched the dollars and cents when cashing a check for Mr. Brown. After buying a 5-cent newspaper [this is an old problem], Mr. Brown found that he had exactly twice as much as his original check. What was the amount of the check?

Rounding Up – Way Up

Ever heard of Dudeney numbers?  Neither had I, until yesterday, when I discovered them completely by accident while reading (Wikipedia, what else?) about narcissistic numbers.  A Dudeney number (named after famous English mathematician and puzzle author Henry Dudeney) is a number that is the cube of the sum of its digits.  For example,

There are only six Dudeney numbers.  Neat numbers, but I was a little disappointed by that.  What to do next?

Generalize, of course!  Generalized Dudeney numbers (discussed here, but the link appears to be dead, so I used Google’s cached version) are numbers that are some power of the sum of their digits:

The largest number on the above site is , which has 147253 digits.  The site links to Wolfram Alpha to confirm this.  Here’s where it gets weird:

How many digits is that?  About ?  About a million?  What kind of rounding is that?  It gets worse.  Try a number with just 100,002 digits (despite what Alpha says).  I think Alpha is a great tool, and I’ve had (far too much) fun playing with it, but I’m a tad disappointed (that’s twice in one post).  So, hey, get on that, Wolfram.

A 100-letter word…and a song

Maybe that should read “in a song”.  In response to What’s a seven letter word for “seven letter word”?, Kurt gives us the following centiliteral:

I hope…I mean, I know my students aren’t doing anything like this during class.  Right?

Pi Day Sudoku Is Back!

You laughed in 2008.

You cried in 2009.

This March, from the producers of Naked Sudoku*, comes…

PI DAY SUDOKU 2010!

As in past years, there is also a contest** associated with the puzzle, but who needs a contest when you’ve got a puzzle with the (conjectured) minimum number of clues (18 – in this puzzle, the first 18 digits of π) for a unique solution of a rotationally symmetric puzzle?  Here’s a printable PDF so you can take the puzzle with you to your next meeting lunch break.

Happy solving!

*Brainfreeze Puzzles, in case you were wondering.
**Also as in past years, we will not allow a solution to be posted until after the June 1 deadline.  Thank you for your cooperation.

A Couple Food-Related Fails

I’m a sucker for a gimmick, particularly food gimmicks.  I’ve tried just about every flavor of Mountain Dew, as well as every variety of Reese’s peanut butter whatevers.  So when Starbucks released their new VIA instant coffee, I lined up with everyone else to take the taste test.  (It’s not bad, if you like Starbucks coffee.)

As a reward for tasting the coffee, I got a coupon for $1.00 off any VIA purchase.  I then heard the barista explaining to another customer that with the coupon, the 3-pack was only $2 (actually $1.95), but the 12-pack was a better deal.  So I looked at the price of the 12-pack: $9.95, or $8.95 with the coupon.  And then I did a quick calculation in my head, and discovered that no, the 12-pack isn’t a better deal after all.  With the coupon.  Of course, it is a better deal without the coupon, and I’m sure that’s what she meant, so maybe this isn’t a FAIL, but it’s still a fail.

Next up, however, is definitely a FAIL.  The following was spotted at my local Target store, where Halloween candy is on sale right next to the Christmas decorations:

Can anyone figure out where this number came from?  The box weighs something like 38 ounces (so it’s about $4.21 per pound).  If it really were $159.84 per pound, a Butterfinger (2 oz) would cost $20.  I think I’d frame it instead of eating it.

Information Gain in “Manager-Speak”

There’s a neat post over at Language Log on determining whether or not someone is a manager if they say, “at the end of the day.”  It is the latest in a recent thread (parts 1, 2, 3) about irritating phrases being associated (often incorrectly) with irritating people (see here for an earlier discussion).

DIY Beanbags, or Tiling a Sphere

As an avid juggler, I have a rather large supply of props to juggle, most of which are balls or beanbags.  (Also on the shelves: clubs, rings, scarves, devil sticks, and a diabolo.)  As anyone who’s ever bought juggling equipment can tell you, this stuff isn’t cheap: decent beanbags can run $10-15 apiece, and rings and clubs are much more expensive.  So when I discovered these instructions to make your own beanbags, I was understandably excited.  (Of course, I’ll have to ask Batwoman to sew the pieces together for me.  Needles have a tendency to end up stuck in me instead of the fabric.)

Then I started clicking around the IJDb, and I found these.  Marylis Ramos has clearly spent a lot of time thinking about tiling a sphere.  Certainly any Platonic or Archimedean solid can be adapted to a sewing pattern to approximate a sphere, but a great deal of experimentation among jugglers and sewers has led to only a few becoming popular: the tetrahedron, cube, dodecahedron, icosahedron (one of the best, but really hard to sew), truncated tetrahedron, cuboctahedron, and the “lemon” (with 3, 4, 5, 6, or 8 panels).

The pictures are, of course, idealized beanbags with perfect 1-dimensional seams that cannot be achieved by terrestrial sewing machines (at least not the Singer in my basement).  Or maybe they’re just from Wikipedia’s spherical polyhedron page.

I’ll be making some 4-panel lemons, and I’ll post a follow-up to discuss their sphericity (or lack thereof).

CSI: Calculus

Via xkcd:

Eat your heart out, David Caruso.

“Guesstimating”

Given that there are no Sonic restaurants withing a 150-mile radius of my home, I spend a surprising amount of time talking about them, or at least their commercials (which they show on local TV stations for…no reason?).  I recently showed a friend the Food Math commerical, and he responded by showing me something several orders of magnitude better (you’ll have to watch it on  YouTube):

So.  Good.