Sofia Kovalevskya

Good morning!  In honor of Women’s History Month we are featuring Sonia Kovalevskya, the first woman to receive a (modern) doctorate in mathematics.

Sofia Vasilyevna Korvin-Krukovskaya was born on January 3, 1850 in Moscow, Russia, although her birthday is sometimes listed as January 15 (the equivalent date in the Gregorian Calendar, which Russia adopted in 1918).  Her name, too, is written many ways: Sophie, Sofia, Sofya, Sonia.  She grew up literally surrounded by mathematics: the walls of her room were covered in her dad’s Calculus notes from when he was a student. She wrote later:

The meaning of these concepts I naturally could not yet grasp, but they acted on my imagination, instilling in me a reverence for mathematics as an exalted and mysterious science which opens up to its initiates a new world of wonders, inaccessible to ordinary mortals.

Her parents were well off and she and her siblings had private tutors, but Sofia liked math so much that she ignored her other subjects and her dad put stop to the math studying.  Or tried to, at least – she studied on the sly after her parents had gone to bed.

Sofie wasn’t able to go to university in Russia, what with being female and all, so she married Vladimir Kowalevski (aka Kovalevskij or Kovalevsky) and they moved first to Austria and then to Germany where, because of her continuing femaleness, she still couldn’t take classes. She was, however, able to take private lessons from the mathematician Karl Weierstrass. She wrote several papers and with Weierstrass’s support [and influence at the University of Göttingen] was granted a well-earned doctorate. 

Sophie had a daughter Sofia (who was called Fufa), moved multiple times, and after the death of her husband, became a professor at the University of Stockholm in Sweden. She wrote mathematical papers, non-mathematical works, and was recognized for her contributions even in her lifetime.  Short as it was – she was only 41 years old when she passed away from pneumonia, a complication of the flu.  Many schools have hosted Sonia Kovalevsky Days in her honor, bringing her love of mathematics to new generations.

Sources:

• The Potential to Inspire by Laura P. Schaposnick (written in verse in both English and Spanish)
• Wikipedia
• MacTutor, which itself referenced her autobiography: A Russian childhood: Sofya Kovalevskaya
• SK Days at the Association for Women in Mathematics

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/

Another Math Mistake:

This mistake was printed almost a year ago, but it’s still relevant, and math mistakes never go out of style.  This was posted by Richard Fuhr, who I believe is the original author.

The author was looking at an article about the Gobi desert in China, which read in part: “Temperatures may vary up to 95°F (35°C) in one day in the Gobi.”  It also indicated that the average temperature in winter was -40°F (-40°C) and in the summer could be 122°F (50°C)

The -40°F being equal to -40°C is correct – it’s the only place the two temps have equal numerical designation, and I am a little sad that I’ve never gotten to experience it except in windchill form.  The 122°F being equal to 50°C is also correct, and something I have exactly no desire to experience, although it’s still lower than the 129.2°F (54°C) recorded in Kuwait last month.  Both of those conversations can be found by using one of the formulas

• Temp in °C = (5/9) (Temp in °F – 32)
• Temp in °F = (9/5) (Temp in °C) + 32.

The issue is that these are temperature readings, not changes in temperature.  For a change in temperature, the 32 in either formula will disappear, leaving

• Δ°C = (5/9) (Δ°F )
• Δ°F = (9/5) (Δ°C)

This means that a variation of temperature of 95°F would actually correspond to a change of about 52.8°C, not 35°C.  And a variation of 35°C would be a change of “only” 63°F, not 95°F.    It’s not possible to tell mathematically whether the correct variation was  95°F (53°C) or 63°F (35°C), but looking through The Internet at temperature variations, it appears to me that although either one is possible, the printed variation was likely intended to be 35°C, not  95°F.

The photo above is by Doron, with a Creative Commons license.  Thanks to YG for bringing the original article to my attention!

 

Long lines on Earth

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.

Luggage Math Mistake No More!

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.

Counting with 0

We’ve just painted our living room, for the first time since moving in ten years ago, and decided to celebrate the new jazzed up look by getting a new lightswitch for the porch door to replace the old, painted over one.  (Question:  Why did the previous painters think it was a good idea to paint over every light switch and every electrical outlet?)  I eventually found a place that had some pretty neat ones, but I noticed that the two pages of lightswitches [11 items, listed 6 to a page] were ordered in an unusual way:

YES — they started counting at 0.  Can you imagine how happy that made me?  Although the fact that our living room was newly painted, with 10 years of crayons and scratch marks obliterated in just  2 4 6 days of work might have also had something to do with it.

Thirds

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!

Google loves Pi Day!

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!

I think it’s time to revoke our grading licenses

Here’s what happens when you didn’t get any all of your grading done this weekend (but the Saints won!!!!!!! Yay New Orleans!), and then instead of catching up before class your department gathers informally and starts to talk about Friday’s wacky trillion point grading scheme, and if there’s a class that lends itself to that, and someone mentions that Physics would really be the ideal course for this, because they have to deal with measures of scale so often.

Well, there’s only one place for this conversation to go:  the other extreme.  What if you had a class that, instead of having 500 points for the semester, or even 1,000,000,000,000, had only 1 point.  Total.  Exams could be worth 2/10 of a point, each homework assignment might only be worth 1/250 of a point, etc.  If someone had an unexcused absence, you’d knock 1/500 off a grade.   It might make it easier to defend points deducted, too, because you could say, “Look, I only took off 5 thousandths of a point for that mistake, so it’s hardly worth arguing about.”

Plus, this could be a learning experience because you could use Official Prefixes:  “This exam is worth 2 decipoints (not to be confused with decapoints)”.   The trouble is, most science students are pretty familiar with deci, centi, and milli so if you really want to make it…memorable….having 1 point for the whole semester is still too much.

The problem is, in looking online, we don’t really seem to have enough prefixes.  The prefixes start off simply enough (deci, centi, and milli being 1/10, 1/100, and 1/1000 respectively) but then decrease by factors of 1000:  the next smallest amount would be a micropoint, which is actually a thousandth of a millipoint, or 1/1,000,000 of an actual point.     Then there are nanopoints (1/1000 of a micropoint) and picopoints (1/1000 of a nanopoint) and while those words are fun to say, we’d need something closer in scale to a micropoint to be able to distinguish amounts, or else it becomes essentially a 1000 point grading scheme with a twist:  “This exam is worth 200 yoctopoints out of a total of 1 zeptopoint for the semester.”

I guess maybe this isn’t so practical after all.  And that’s truly a shame, because who wouldn’t want to write a grading scheme that used yoctopoints?

What has 8 legs and no room for anything else?  A yoctopus!

So what’s interesting about 2010?

It seems (perhaps only to me) like it ought to factor nicely, because 20 is twice 10.  But once you factor out that 10 and that 3 you’re just left with a prime, since 2010 is just 2·3·5·67. (Speaking of four primes, did you know that you can get four prime New York Steaks for $132.95 on Amazon?  I was relieved to see that they were not available for Super Saver Shipping.)

Wolfram Alpha points out that 2010 itself is a factor of 29⁶-1.  And  Number Gossip adds that it’s untouchable, which means that there aren’t any numbers whose proper divisors add up to 2010.

It can be written as 133122 in Base 4, which is kind of cool, and as 6, 3, 12 in Base 18; my favorite, however, is that it is 5, 10, 15 in Base 19.

Finally, it’s equal to:
669+670+671
400+401+402+403+404
127+128+…+141 and several others
[Hmmm…I can find a string for each of the 7 odd factors, but I’m not sure that exhausts all of the possibilities.]

While getting ready to post this, I noticed that MathNotations has a similar post from yesterday.   Whoops!

And by circles we mean….

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.

Kitten math

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.

Math Teachers at Play #9 is up at HB

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.

WordPress does the math right

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.

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.