Holiday Math: Secret Santa

Around the holidays, one fun tradition is to set up an arrangement wherein members of a group anonymously trade gifts with one another, a process which is often called "Secret Santa." In this new video by the Center of Math, we mathematically investigate the odds that randomly assigning people around will create a valid exchange: it's a bit more complicated than you might think! Keep reading to watch it here.

Famous Women of Science and Mathematics

Today, the Worldwide Center of Mathematics is going to talk about some of the most famous and accomplished women in science and mathematics. While the number of women in math and other STEM disciplines today is still small compared to the amount of men, it is important to realize that women can achieve just as much as men can, because gender does not determine intelligence, skill, or aptitude. Keep reading to learn about some inspiring ladies!

Throwback Fact: Stokes' Theorem

George Stokes (1819 - 1903)

The University of Cambridge, founded in 1209, has produced many of the greatest scientific minds of all time. It makes sense that Sir George Stokes, famous in both the fields of mathematics and physics, spent the entirety of his career within those halls.

On this day in 1850, Stokes' theorem appeared in writing for the first time. Stokes' theorem is used in multivariable calculus to relate a line integral to a surface integral. To view the notation used for Stokes' theorem that isn't supported in Blogger, click here; Mathworld did excellent work of describing the math behind the theorem. As a sidenote, this theorem appears as a topic in this Center of Math textbook.

A pictoral representation of Stokes' theorem

At the time that his most famous mathematical theorem was first discovered, Stokes was one year into his position as a professor of mathematics at Cambridge. He kept this position until his death 54 years later in 1903. During his career, Stokes made contributions many fields of science. He published several papers relating to fluid dynamics and motion, fluorescence, polarization, and light. His published, and unpublished, works have done much for the development of mathematical physics.

The Science Behind Pixar

See all Pixar's characters at www.pixar.com

Many people simply aren't sure what applied mathematics can accomplish, or, which accomplishments required applying challenging and exciting mathematics. One wonderful example of creators using mathematics to perfect their craft is Pixar Animation Studios. They're the folks responsible for fan-favorites Toy Story (1995), Finding Nemo (2003), Up (2009) and, most recently, the wildly popular Inside Out (2015).

That's why we recommend you check out the newest exhibit at Boston's Museum of Science: The Science Behind Pixar.

Enjoy a unique, first-time look into the Pixar process, and explore the science and technology behind some of the most beloved animated films and their characters with the world premiere of The Science Behind Pixar. This interactive exhibition showcases the science, technology, engineering, and math (STEM) concepts used by the artists and computer scientists who help bring Pixar's award-winning films to the big screen.

You'll definitely have the opportunity to explore and use mathematics to create virtual 3D models.
If you're serious about 3D modeling and mathematics, check out Matrices, Vectors and 3D Math with Matlab® or an Introduction to Groups, Rings and Fields here.

read more and purchase tickets here
read more about the Museum of Science here

What's your favorite Pixar movie? Which movie do you think was hardest for animators to create? Let us know in the comments!

Throwback Fact: Sally Ride in Space

Sally Ride in space.
image: commons.wikimedia.org

On this day in 1983, Sally Ride became the first American woman in space (Valentina Terishkova was the first woman in space, as part of the Soviet program in 1963). In her biography on NASA's site, Dr. Ride's listed experience includes earning a Physics Ph.D. from Stanford University, developing the space shuttle's Canadarm arm, two space missions, and serving on the committee investigating the Challenger disaster.

Ride was not a mathematician, but an astrophysicist. However, I'd like to take the time to chat briefly about mathematics as it relates to space.

Throwback Fact: German Tank Problem

A WWII-era German Tank
image: commons.wikimedia.org

Last Saturday, June 6th, marked the 71st anniversary of D-Day, when the Allies invaded the beaches of Normandy and began to turn the tide of World War II. So, though it is not linked to any particular date in math history, it's fitting that we showcase a WWII era math problem.

The German forces had an advantage that the Allies had to overcome: more, and technologically superior, tanks. Knowing how many tanks the Germans were producing was the first step to figuring out how to take care of this threat. So, the Allies tried conventional methods of gathering intelligence: interrogation, spies, and decoding messages.

Throwback Fact: Emmy Noether's Math Journey

Emmy Noether

The way we name proofs in mathematics, most often after their founder, immortalizes mathematicians who could otherwise slip into history unnoticed. But 96 years ago today, an event occured that helped to keep one particular female mathematician from obscurity.

Emmy Noether (1882 - 1935) almost missed her chance at mathematical fame. Born to a Jewish family in the German town of Erlangen, Noether showed few signs of her mathematical talent until she reached a college age. Noether originally planned to learn to teach English and French, but she attended math courses at the University of Erlangen where her father lectured. There, she earned her doctorate in mathematics in 1907, and worked at the same university for 7 years, but didn't earn a single payment for her research. 

Throwback Fact: The Birthday Problem

image: www.flickr.com

In 1939, mathematician Richard von Mises proposed a problem. The answer was so counter-intuitive to many people, including mathematicians across the globe, that it continues to be one of the most discussed math problems ever. It is very difficult for people to understand, not because the mathematics behind it are hard, but just because the answer is so different from what anyone would expect. That problem is stated as follows:

Suppose you are in a large room that random people generally enter. How many people must be in the room before the probability that some share a birthday becomes at least 50 percent?

Counting Systems: Ancient Babylon

One of the most fascinating parts of math, for me, is the history behind it. Math is one of those things that shows up in every time period, no matter what civilization was in power or what languages were spoken. As a miniseries, I'd like to share some ancient counting systems and facts about math at that time. First up is the math of the Ancient Babylonians.

The Region: Babylonia was name of the cultural region in the fertile crescent, stretching from the Persian Gulf to modern-day Baghdad and west to the Mediterranean Sea. Their history dates back as far as 4000 BCE with the Sumerians, though very little is known about the Sumerian culture.

The Babylonian civilization rose and fell for millenia, and their culture can be classified into several distinct periods. The mathematical records that have survived come from two different periods: the First Babylonian Dynasty period (1800s - 1500s BCE) and the Selucid period (400 - 0 BCE). Very few documents fall between those two periods. The main focus of researchers are documents from the First Babylonian Dynasty, but the second period is interesting because it overlaps with great mathematicians from Ancient Greece.

One particularly famous Babylonian was Hammurabi, the king of Babylon for much of the 18th century BCE. Hammurabi's Code is the oldest known form of constitution for a government, and the stone on which it is written is the longest surviving work from the Old Babylonian period.

The Numerals: The scholars of Babylonia used cuneiform numerals to write their numbers. They pressed the wedge-shaped end of a reed into soft clay, and hardened it into stone tablets. The numbers are written from left to right like modern Latin numerals, but that is where the similarities end.

Throwback Fact: The Origins of Pi

We've mentioned pi quite a few times on the blog (our pi day post in particular). However, we've never discussed the reasoning behind the symbol.

Pi, of course, has existed in approximations since the time of the Babylonians (who got to 28/8 = 3.125) and the Ancient Egyptians (who found (16/9)^2 = 3.1605 on the Rhind Papyrus). Approximations have become better and better over time, but it will never end. Pi is transcendental- it is irrational, and cannot be found as the result of an algebraic function. We can continue to calculate more and more digits, but the number of trailing digits is an uncountable infinity, and therefore we will never know exactly what pi is.

Throwback Fact: Quaternions

A digital representaion of a Julia set quaternion

In 1843, Irish mathematician Sir William Hamilton was walking across Dublin's Brougham Bridge when inspiration struck him. He had been working on finding a method to multiply together points in space (three dimensions) instead of just points on a plane (two dimensions.) On his walk with his wife that morning in october, he realized that while he could not "multiply triples," he could multiply points in four dimensions. He carved this basic rule for multiplication into the bridge:

i² = j² = k² = ijk = -1

Hamilton named a quadruple of this type a quaternion and he devoted the rest of his career to their study. The simplest definition of such a number is a four-dimensional number. Quaternions are still studied today, and present a number of uses in modern mathematics. According to The Math Book by Clifford Pickover, they've been used to describe the dynamics of motion in three dimensions, and have been applied to fields including virtual reality computer graphics, game programming, robots, bioinformatics, and flight software of spacecrafts.

The plaque on Brougham Bridge in Dublin, Ireland

The invention, or discovery, of quaternions represents a moment of great ingenuity in math history. You can actually still visit Brougham (Broom) Bridge today, and see the plaque commemorating Hamilton's discovery. There is even an annual commemorative walk following Hamilton and his wife's path on that fateful morning. Perhaps we should have added this location to our list of math destinations!

One more fun fact about quaternions is that they generated such polarizing responses from the math community. Scottish physicist William Thomson(1824 - 1907) considered them an "unmixed evil to those who have touched them in any way," while mathematician Oliver Heaviside thought of their invention as a feat of human ingenuity. Curiously, one mathematician (who gained his fame from a harshly different circumatance) who studied quaternions was Theodore Kaczynski, the "Unabomber."

Throwback Fact: Perfect Numbers

Tori's handwriting demonstrates a few properties of the first two perfect numbers

We've mentioned the cult of Pythagoras and his followers a few times in the past. And we know that their mathematical findings did not stop at the Pythagorean theorem. One mathematical idea that fascinated the group was perfect numbers.

A perfect number is defined as a positive integer that is equal to the sum of its positive divisors, excluding itself. The definition appeared as early as Euclid's Elements, and only four of these numbers had been discovered before the Renaissance. The discovery process was so slow because these numbers are inredibly rare- the first is 6. It's a relatively small number, and makes the viewer think that the numbers may common. The next perfect number is 28, and then 496. A Greek mathematician (circa 100 CE) named Nicomachus is credited with the discovery of the fourth perfect number: 8128. Then, more than a millenium passed before an unknown mathematician recorded the fifth perfect number, 33,550,336, for the first time.

Cinco de Mayo Mathematician Highlight

After doing a little research, I’ve discovered that Cinco de Mayo is a holiday celebrated possibly more in the United States than in Mexico, its country of origin. Cinco de Mayo began as the celebration of a battle- it commemorates the Mexican army’s victory over France at the Battle of Puebla in 1862. This victory was particularly important in the Franco-Mexican war because it was an impossible feat as the Mexicans won against a much better equipped army.

In the United States, it is often utilized by Mexican Americans as a celebration of Mexican culture, including festivals and parades in some cities. In typical Center of Math fashion, we will use this opportunity to feature a prominent Mexican mathematician, Fray Diego Rodriguez!

Throwback Fact: Gauss's Birthday

Carl Friedrich Gauss

On this day in 1777, Carl Friedrich Gauss was born. The Prince of Mathematics was an important part of our Math Madness competition, and made it to the Final Four round. But in his short bio on our webpage, we missed some facts about Gauss that make his life even more interesting and impressive.

The great mathematician would have been 238 years old today, but there was a time that he did not know the date of his birth. Gauss was born to poor, working-class parents, and his mother was not quite literate. She did not record the date of Gauss's birth. Years later, she could just recall that Gauss was born on a Wednesday, and it was 8 days before Ascension Day, which lies on the calendar 40 days after Easter Sunday.

Throwback Fact: The AMS

An image from our Instagram, when the Center
attended JMM (by the AMS) in January

Friday, April 24th (misaligned slightly from our Throwback Thursday) marks the 118th anniversary of the first sectional meeting of the American Mathematical Society in 1897. Founded nine years previously in 1888, the AMS was the result of Thomas Fiske's vision to create a version of the London Mathematical Society in his home country. Soon after being founded, the AMS began to publish a research journal periodically.

Today, the AMS has nearly 30,000 individual members across the country, with physical locations in Providence, Rhode Island, Ann Arbor, Michigan, and Washington, D.C. And the AMS works today, like in the earliest days of the society, to publish math content- they've upped their postings from one research journal to 8(!), plus 4 translations journals, and two open source journals. The AMS is a wealth of mathematical knowledge.

The AMS works each year with the Mathematical Association of America (MAA) to put on the annual Joint Mathematics Meeting. In fact, the Center of Mathematics regularly attends- we were at JMM San Antonio this past January!

To read more about the American Mathematical Society, click here or visit their webpage.

Throwback Fact: Leonardo Da Vinci

Da Vinci self portrait

Choosing just one topic for a throwback was difficult this week. On this day in 1705, Isaac Newton was knighted by Queen Anne at Trinity College. If you'd like to read a little more on Newton, a past intern did a throwback fact on him here. And Wednesday, April 15th marked two anniversaries- the birth of our Math Madness Champion Leonhard Euler in 1707, and the birth of Leonardo Da Vinci in 1452. 

While he is definitely not a mathematician, Leonardo Da Vinci was a polymath, and contributed many quality works in his lifetime that interest mathematicians today. The original renaissance man, Da Vinci was a master in many areas. His paintings are likely to remain famous for all of human history, and his notebooks fascinate scientists to this day- he worked with codes, he fantasized about flying machines, sketched human anatomy, and much more.

Throwback Fact: Knot Theory

A trefoil knot, the most simple non-trivial knot
image: commons.wikimedia.org

According to The Math Book by Clifford A. Pickover, "The use of knots may predate modern humans (Homo sapiens). This statement is absolutely fascinating, and prompted a research session on a portion of mathematics that the Center's math intern hadn't encountered yet.

A mathematical knot is different from what people normally think of as knots. Instead of a piece of string with two free ends and a tangle in the middle, mathematical knots come from embedding the unit circle into three dimensions, and twisting and disturbing the continuous line from there. Almost always, knot theorists are studying closed loops. From this point, it's important to note that a knot can be trivial (also known as an unknot), and in this case the loop can be unfurled to be a single unit circle loop. For a knot to be non-trivial, it will always have crossings no matter how the "string" is pulled.

Throwback Fact: President Garfield, Mathematician?

President James Garfield
image: commons.wikimedia.org

Several weeks ago, the Throwback Fact of the week focused on the Pythagorean Theorem. But did you know that the United States has a past president with a published work of mathematics? On April 1st, 1876 (yesterday in history!) James A. Garfield, a congressman at the time, published a unique proof of the Pythagorean Theorem in the New England Journal of Education. 

Throwback Fact: The Abel Prize

Many of our blog viewers are coming from one of our social media sites, so you’ll already know that we gave a shout out to the 2015 winners of the Abel Prize- John Nash and Louis Nirenberg. Their work in geometric analysis earned them the prize, regarded by many as the Nobel Prize of Mathematics. Nash is particularly interesting to those not versed in math research because his was already a household name: he was the subject of the 2001 Russell Crowe movie A Beautiful Mind (both of their pictures are above). 

Throwback Fact: The Metric System

image: commons.wikimedia.org

On this day in history, in 1791, the Metric System was proposed for the first time to the Paris Academy of Sciences. The main features of this system were to standardize a set of interrelated base units and prefixes in powers of 10. The system was first developed for commercial use, but the standardized unit size made it particularly suitable for science and engineering.