Throughout much of the 1600s through the 1800s, Japan was
largely isolated from the West. During this time, the Japanese samurai thrived,
along with feudalism and the Buddhist religion. A tradition known as Sangaku (roughly
translated: Japanese temple geometry) was born in this era. Japanese people
from all social and educational classes would work to solve difficult geometry
problems and inscribe the solutions on tablets, then hung the tablets on the
roofs of Shinto and Buddhist temples as votive. Many of these problems are
heavily focused on circles, an uncommon method in the West in the same time
period.

Happy Monday from the still-frozen tundra of New England. Like each Monday, I have shared the above problem to our Twitter, Google +, and Facebook pages. Today's problem, more logic than straight mathematics, was adapted from this puzzle site. I had actually encountered this problem once before, but I thought it merited a detailed explanation.

Of course we all know of Plato, Aristotle, Pythagorus… but
what about female mathematicians in the ancient world? There weren’t many. In
fact, the first woman mathematician about whom we have reasonably secure
knowledge is Hypatia (circa 370 – 415 CE).

It is likely that the reason Hypatia became a skilled
mathematician, philosopher, and astronomer is because she was exposed to
science at an early age. Her father, Theon of Alexandria, was a distinguished
professor of mathematics. Hypatia went on to teach mathematics at the same
university as her father, and was always cited as a charismatic tutor by her
students.

A depiction of Hypatia just before her death, by Charles William Mitchell image: en.wikipedia.org

Hypatia is
unfortunately not remembered for any significant contributions to
mathematics. Her works were all lost, though it is known that she created a
commentary on Diophantus’s Arithmetica.
The legends surrounding Hypatia stem from curiosity about her death. According
to the limited sources available, Hypatia (a self-proclaimed pagan) drove a
wedge between two political figures of Alexandria. She was blamed for
increasing tensions and even riots between Christians and non-Christians as the
church and state fought for power. Legends tell that Hypatia was brutally
flayed and burned alive by Christian zealots in a church.

Hypatia’s legacy is so popular that it was adapted for
screenplay in the 2009 film Agora. While very little facts remain
about her, it is important to remember Hypatia, the first woman in mathematics. Sources: The Math Book by Clifford A. Pickover

The closing years of Greek mathematics by Jesse Osborn

Another Monday, another Problem of the Week. This problem was found on this website and while it's not mathematically complex, it involves a new layer of decoding that we haven't featured in a while. I thought it was particularly fitting because of the Alan Turing movie that is up for a few Oscars! So our problem of the week lies below:

Click to enlargen!

Once again, I've written up my solution to the problem. The image came out blue- I'm not quite sure why. Before you scroll down to see my answer, here's a hint: "Crispin" Bacon (or, you know, Francis Bacon) was particularly interested in cipher systems.

I’m a college student. And I love learning- I really do. But
students today have to jump through more hoops
than ever before in order to complete their education. There are more
competitive applications and admissions exams, more scholarship applications,
more loans, more hidden costs, and, most importantly, there is more student
debt. I did well in high school AP classes, I aced the SAT and ACT, and I
applied early to colleges. While I’m extremely happy with where I ended up,
there is one aspect of my two years at university (so far) that I would change
in a heartbeat: the textbook bubble.

Students across the nation recognize that the rising price
of academic textbooks is a problem. I’m very lucky- I have a great academic
scholarship, so my parents are able to financially support me in the textbook
department (thanks so much again, Mom and Dad!), but that doesn’t stop me from
feeling sticker-shock and price-induced guilt.

“Wow, Mom, I’m so sorry,” I said over the phone while
standing in the aisle of the university bookstore in the Fall of 2014. “My
physics book is $280.” Of course I had to buy the book new- it came with a
package for the online software we used for homework in the course. And that
was just physics. I was also enrolled in a French class (a different book from
my previous semester of French), an upper-level math course (two smaller books,
think about the size of Harry Potter and the Sorcerer’s Stone for each), and an
honors seminar on modern art (special ordered for the university bookstore
book). My total price tag at the bookstore: $450.

And that’s just one semester of my story. College students
around the country have different circumstances than mine: some struggle to pay
the campus bookstore prices, some search online for discounts and end up buying
sometimes outdated editions, and others -- a staggering 65 percent of students
according to this article [9] -- are
willing to forgo buying an assigned book altogether because of the price. If
anything is more ridiculous than the prices students are forced to pay, it’s
this: Students are going to great lengths to save money at the expense of a
coherent learning experience. Professors choose a specific book to accompany
their course because the material is relevant. If 65% of students who can’t
afford full-price textbooks simply don’t buy the book, they’re missing out on
all of the material not covered in lectures.

David Massey, a full professor of mathematics at Northeastern University and the founder of the Worldwide
Center of Mathematics, sat down with our math intern Tori for an interview this
week. He is the first in hopefully a string of mathematicians who will be
interviewed for this new blog series.

David B. Massey, Ph.D

Are you currently
involved in any research?

Yes - I study abstract notions of space in any number of
dimensions. I prove things about what those spaces look like at places where
they’re not smooth [singularities].

How do you picture
those?

You don’t really. You picture what happens in low dimensions
and hope that it gives you intuition for the higher dimensions. You can’t
picture the higher dimensions, so you prove theorems that describe them. I’m
interested in singularities because they’re a great blend of topology,
commutative algebra, algebraic geometry… a lot of stuff.

In honor of Valentine’s Day, we wanted our throwback fact of
the week to be heart themed. What in math looks like a heart? A cardioid, of
course!

The cardioid was named in 1741 by Italian mathematician
Giovanni Salvemini de Castillon, though it was already a studied shape. It is a
degenerate case (with no internal pedals) of the limaÃ§on, a figure formed when
a circle rolls around the outside of a circle
of equal radius.

How does one generate the cardioid? This Wolfram Mathworldpage gives us the equations.

What makes a cardioid relevant today? Aside from its
importance in geometry, cardioids are a shape seen in acoustics. The cardioid
microphone is nam

ed so not because the microphone is shaped like a heart, but
because the sound pick-up pattern is heart shaped. The shape allows a
microphone to be used in a loud setting, because the microphone will pick up
only sounds very close to it within the cardioid spread. For more detail on
this topic, click here.

Possibly the most important application of the cardioid for
this week, however, is that they produce beautiful looking, heart-shaped
curves. You can see a few members of the Center’s attempts to draw hearts
below. Although perhaps we should leave the heart-shaped curves to geometrically precise cardioids.

In the image above, Tori drew out a few sample heart curves with some imprecise geometrical backing. Then Zach, Adam, Ruairi and Tori all drew heart curves their own way.

We hope you have a good weekend, whether you celebrate Valentine's day or not! Have you had a chance to watch our new video on the Monty Hall problem? See it here:

It's another snowy, snowy Monday in New England, but the Center of Math team is hard at work to continue bringing you math content. Like our Problem of the Week! This week's problem (found on mathschallenge.net) is short, but tricky. It doesn't take long to solve as long as you can make the creative first two steps.

First, I'll tell you to click here to visit the Kickstarter's main page.

Second, I dare you to tell me those dresses aren't cute.

Third, I'll explain why this is important to me.

A young lady in the "Pi" print play dress

Gender biasing in academia is slowly, slowly dissolving. I'm lucky to be going through college at a time where women are encouraged to go into Science, Technology, Engineering and Math (STEM) fields. This infographic from 2013 describes that women in undergraduate studies earn 50% of all bachelor's degrees in Science and Engineering. There are many disparities within STEM, for example, men earn 82% of bachelors degrees in engineering, while women are more likely to major in psychology or biology. But even with the disparities, there is no denying that women are (finally) becoming a true force in the STEM fields.
Even so, when I tell aquaintances that I'm majoring in mathematics, I can't count how many times I've heard something like "Just math? No business?" or "Wow, you must be so smart," or "That's so brave!" There's always a tone of surprise.

The Pythagorean Theorem image: commons.wikimedia.org

Let’s talk about something everybody knows: the Pythagorean
Theorem. It’s been drilled into students’ heads from the age of around 13. A
squared plus B squared equals C squared. It’s necessary for 9^{th}
grade level geometry. This theorem seems second nature to almost everyone. We’ve
used it for years, we know it works, and it’s so easy that even, for example, a
novelist who never works with math would still know it by heart. Who came up with it? What makes it so simple, so clean and important?

Here at the Worldwide Center of Mathematics,
Tori, an intern working as the Assistant Director of Mathematical Content, has
been hard at work creating new math resources and posts. She’s recently starred
in several short videos that were made to help students study for the math
section of the ACT (the next exam is February 7^{th}). Watch the playlist here! Tori is currently a math student at Northeastern University in Boston, and
truly enjoys breaking down math problems into simpler, understandable pieces.

Here's one of Tori's first ACT videos

From a big-picture perspective, we want
you, the students, parents, or teachers who watch these videos, to think of the
Center of Mathematics as your main resource for math help. Our founder, David
B. Massey, has given a multitude of lectures on his books and upper-level
mathematics that are all available on our Youtube channel. We are not
just a textbook publisher; we are a network of teachers and resources, and we
are here to help students learn the intricacies of their mathematics courses.

A video from the second ACT shoot

We know that standardized,
college-admission tests may seem daunting. They put a lot of pressure on the
student, both in terms of time and emotion. They require quick recall of
subjects learned all throughout high school. Our videos are meant to help train
you to approach ACT or SAT math problems as simply as possible. Often you’ll
see Tori use the same methods in several videos- that’s because she knows the
techniques to simplify the problems. Math isn’t everyone’s favorite subject,
but with Tori’s help (and the help of past and future interns), we can
hopefully ease the burden and relax the stigma that is associated with
mathematics on standardized tests.

If you have any questions or
suggestions for future videos, let us know in the comments!

In recent weeks, The College Board has made national headlines over their decision to revamp the SAT, the score that is perhaps the most important number on a college application besides the hallowed Grade Point Average. The College Board has lowered the top possible score to 1600, down from 2400, heavily revising what the old three sections contained...

Yes, this PoW post is a day late; we apologize. We were unexpectedly out of the office yesterday due to Winter Storm Linus (our second huge snowstorm within a week). To make it up to you, we have an interesting problem that intern Zach found here on reddit, which we wrote out to post on our Facebook, Twitter, and Google+ pages.

Click to enlargen!

See Tori's solution by clicking through the jump...