Updates

New Virtual-Reality Technology Literally Allows You to Play Ping-Pong with Yourself:

Playing a game of ping-pong by yourself in the past typically involved hitting a ball against a wall or playing against a computer-generated opponent in a video game. Now, thanks to the Mathematical Computing Laboratory at the University of Illinois at Chicago, you can play against the perfect opponent: yourself.

Students and professors working in the lab created a virtual reality (VR) game in which you hit a ping pong ball with a paddle in a three-torus. This space is created by “gluing” opposite sides of a cube together: top to bottom, left to right, front to back. So, if you move through one face of the cube, you will reappear through the opposite side. Because of this, if you hit the ball forward through the front face, it will come back at you from behind with the same speed and trajectory at which the ball was hit.

To help understand what a three-torus is, let us first look at the two-torus. Instead of gluing opposite ends of a cube together, a two-torus is created by gluing opposite sides of a square together, creating a 3-dimensional object. A common representation of this is a donut. A three-torus, like a two-torus, exists one dimension up from its base shape — a cube — in a four-dimensional space. Thus, providing an accurate visualization of one is physically impossible. However, we can still express a three-torus in three dimensions, which looks something like the image found below. If the space is small enough, you will look forward and see the back of your body. Likewise, if you look to the side or up or down you will see yourself through the opposite face of the cube. This will create seemingly infinite versions of yourself, but it is important to note that the three-torus is a finite space. Regardless, playing ping-pong in a three-torus can be quite distracting, as you may struggle to figure out which ball to go for.

The original article did not mention if or when this new game would be marketed to the public for use. Regardless, this cool new innovation in VR technology allows us to play games like ping-pong by ourselves in a way not previously possible. In this new form of ping-pong, the only person stopping you from victory is yourself. The only question is: would you be up to the challenge?

 

Top: A three-dimensional visualization of a two-torus. Bottom: A three-dimensional visualization of a three-torus.

An article by Joe Gyorda

View the original article and a video of the ping-pong VR program in action:

https://blogs.scientificamerican.com/roots-of-unity/ping-pong-for-introverts/

 

More information about the three-torus and source for the image:

https://blogs.scientificamerican.com/roots-of-unity/a-few-of-my-favorite-spaces-the-three-torus/

New Breakthrough in the 82-Year-Old Riddle known as “The Collatz Conjecture”

This past September, mathematician Terence Tao of UCLA released a paper summarizing his near solution to a problem that has stifled the mathematical community for over three quarters of a century, “The Collatz Conjecture”. Given the function f(n) (shown below), plugging in an even value results in it being divided by 2, and plugging in an odd value results in it being multiplied by 3 and added to 1. The conjecture proposed by German mathematician Lothar Collatz in 1937 is that for any natural number n, the end result will always be 1. The example provided by Linkletter (2019) goes as follows:

“For example, let’s use 10. It’s even, so the rule says to divide by 2, taking us to 5. Now that’s odd, so we multiply 5 by 3 and then add 1, landing us on 16. Now 16 is even, so we cut it in half to get 8. Even again, so halving gets us 4. Now 4 is even, so we take half, getting 2, which is even, and cuts in half to 1.”

 

The issue facing mathematicians hoping to prove this conjecture is finding a way to show that it is true for all natural numbers n. So far, it has been proven for numbers with less than 19 digits, and supercomputers have the ability to check for numbers with hundreds or thousands of digits. However, this still does not prove the conjecture for all natural numbers.

Tao’s latest findings were published in a post titled Almost All Collatz Orbits Attain Almost Bounded Values. Collatz orbits are the sequences of numbers obtained when plugging a natural number into the function above. The Collatz Orbit for 10 is (10, 5, 16, 8, 4, 2, 1, 4, 2, 1, …). The orbit is “bounded” by 4, 2, and 1, because 2 is half of 4, 1 is half of 2, and 4 is 3 multiplied by 1 plus 1. So, the orbit will cycle through 4, 2, and 1 infinitely. What Tao demonstrates in his paper is that nearly all natural numbers have such orbits and thus satisfy the Collatz Conjecture. Although he was unable to prove the conjecture, his results suggest that the likelihood of a counterexample to the conjecture decreases logarithmically as you go further down the number line. In other words, Tao’s results suggest that the occurrence of potential counterexamples to the conjecture is rarer than previously proven.

 

A simple math riddle with a difficult solution has puzzled mathematicians for years, but Terence Tao has taken a huge stride towards discovering its solution. However, Tao believes that an entirely new approach than his may be needed to bring about the full solution to this problem. So, it may be some time before The Collatz Conjecture is put to rest.

An article by Joe Gyorda

View the original article:

https://www.popularmechanics.com/science/math/a29033918/math-riddle-collatz-conjecture/

 

View Tao’s Results:

https://terrytao.wordpress.com/2019/09/10/almost-all-collatz-orbits-attain-almost-bounded-values/

Penn State Wind Probability Modeling

Researchers at the Pennsylvania State University have recently designed a new model to determine where wind farms should be built. In the past, considerations have generally only been focused around the terrain and the wind speed/consistency in the area, but this can be hard to determine. This level of unpredictability has been a drawback in promoting this renewable energy source. With sources like fossil fuels and nuclear energy, there are consistent and reliable methods to determine how much energy you will get, creating an advantage over wind farms. These researchers have designed a new method that is more accurate and efficient to track wind predictability, allowing them to more accurately determine the amount of energy that can be produced.

The researchers studied historical data on wind speeds for various locations spanning several months. They were able to use this data to create a probability model which allowed them to forecast available wind for power production. Their model showed that more consistent wind speeds were generally related to sites with lower average wind speeds. Their modeling allowed them to draw the conclusion that rather than building wind farms at sites with the highest average wind speeds, as if often done now, those building the sites should instead focus on areas with lower average wind speeds that will have more reliability.

An article by Victoria Hoffner

https://www.nsf.gov/discoveries/disc_summ.jsp?cntn_id=299302&org=NSF&from=news

 

 

New Big Data Analyses with Topology

New Big Data Analyses with Topology

The term ‘big data’ has become increasingly prevalent in modern dialogue. Professionals throughout a variety of fields have referred to the importance of  ‘big data’ in their progress and many others’.

According to an article titled “Big data is being reshaped thanks to 100-year-old ideas about geometry,” scientific developments in Switzerland recently amounted in the creation of a 3-D brain cell atlas using the brain of a mouse. Although the production of such a tool was a feat, scientists have begun undergoing the laborious process of trying to understand this atlas.

In the past, mostly mathematicians and statisticians analyzed data sets, specifically through developments in the Internet and other technologies. Currently, a transformation is occurring in the data analysis field in which experts will be able to better understand certain types of big data sets.

The mouse brain map problem constitutes much of the mathematical subject of topology, the study of characteristics of certain shapes. During the 1900s, topology was studied most theoretically. Since then, however, applications, such as the mouse brain, have emerged, deeming the subject increasingly pertinent to our understanding of different complexities.

As topology focuses on characteristics, it is well-versed to maintain pattern analyses despite interference or outliers. Because of this, topology is seen as a medium to create newer drugs, particularly through its shapely applications to ‘molecule space’ and mapping.

Specifically, the field is called topological data analysis, or TDA. Essentially, TDA influences and categorizes data sets depending on how large they are and what the sets look like. It has become a preferred method of showing data.

While the field of topology and its newfound purpose in big data continue to develop, humanity is getting just a little bit closer to understanding the world around, inside, and outside of us.

By Archita Harathi

https://phys.org/news/2019-02-big-reshaped-year-old-ideas-geometry.htmlhttps://www.ayasdi.com/blog/bigdata/why-topological-data-analysis-works/

Math and Urban Burglaries

In the past, mathematical modeling has focused on burglaries in residential, suburban communities. However, the highest burglary rates occur in urban areas where it is easier for thieves to disappear and sell their goods after committing the crime. Many suburban burglary models have a focus on repeat victimization. This trend shows that homes that have been burglarized before or that are surrounding homes with similar architecture are more likely to be burglarized in the future. Researchers in a new study decided to take a different modeling  approach to examine urban burglary trends. This past fall, Joan Saldaña, Maria Aguareles, Albert Avinyó, Marta Pellicer, and Jordi Ripoll published their work on a new, non-linear model that studies urban burglarization with a focus on the deterrent effect of police presence. The study looked at data from the Catalonia region in Spain.

The study also focuses on the times at which these robberies occur instead of their physical location. Researchers used the “age” of both the burglarized houses and the burglars. By “age of a house” and “age of a burglar”, they refer to the amount of time elapsed since a house was last burglarized and since a burglar last committed a robbery. Some limitations of the study include that researchers assumed a constant number of houses and burglars (no turnover) and that burglars would always repeat the act again.

They believe that their model is both simpler and more flexible than previous burglary models used for suburban areas, and hope that these traits allow for further study of results and model adjustment when needed. Ultimately, they hope that their model can be useful in helping police departments optimize their resources, as police presence was shown to be a burglar deterrent in their research. They have also concluded that in urban areas houses with a higher age are more likely to be robbed; however, this conclusion does not hold in the long run. Overall, as stated by Pellicer, they hope that their work can, “…highlight the increase over the last few years in the association of mathematics with criminology to produce models that ultimately help with crime prevention.”

 

https://www.sciencedaily.com/releases/2018/12/181204102532.htm

 

By Victoria L. Hoffner

 

Math, Infectious Disease, and Human Behavior

The Department of Applied Mathematics at the University of Waterloo recently did a study on how the spread of infectious diseases is related to human behavior. The study, performed by Joe Pharaon and Chris Bauch, investigated whether or not human behavior plays a role in both the spread and evolution of infectious diseases. To test this claim, Pharaon and Bauch used computer modelling to add social interactions to pre-existing models for the outbreak of disease. Their models supported the idea that human behavior affects the spread and evolution of infectious diseases. Although the model studied by Pharaon and Bauch was a general model for pathogens, their research has opened the door for more testing to be done in order to better understand the relationship between human behavior and the spread of specific disease. In the future, this social modelling could have an impact on public health policies at times of an outbreak and could improve public health responses to epidemics.

https://www.sciencedaily.com/releases/2018/08/180816081446.htm

 

Dartmouth Team competes in Putnam

Last year, the Dartmouth Team competed in the Putnam Math Competition and placed 53rd place in North America. The Math Society will be hosting weekly Putnam training sessions in the fall 2016, and sending teams to compete in the competition in December.

Math in Music: Tristan Perich

Musician and mathematician Tristan Perich is visiting Dartmouth this week! The Dartmouth Math Department, Dartmouth Math Society, and AWM are co-hosting a dinner discussion event on Monday, May 16, to welcome Tristan and have him give a talk on how he has used math to compose music. Refreshments will be provided outside of Kemeny 007 at 6PM, and the talk will start at 7PM in Kemeny 007.

Dartmouth Team places at Trading Competitions

The Dartmouth Math Society sent multiple teams to the MIT Trading Competition, the Rotman International Trading Competition, and the UChicago Algorithmic Trading Competition in the past year, and achieved excellent results. Harry Qi and Ethan Yu placed 3rd at MIT Trading Competition, and the A team consisting of Harry Qi, Ethan Yu, and Daniel Kang placed 3rd at UChicago Algorithmic Trading Competition. The Dartmouth Math Society hopes to send and train more teams to trading competitions next year.

Meet a math artist! 3/5 @ 5pm

Interested in the ways in which math, art, and music intersect? Come meet artist and composer Tristan Perich to hear him talk about his work both past and present!

Thursday, March 5 @ 5pm
Kemeny 108

About TRISTAN PERICH
Artist and composer Tristan Perich works with code and electronics to create visual art and music that explores the intersection between the physical world and the abstract world of computation. Influenced by the foundations of math and physics, his work is inspired by ideas from the studies of logic, randomness, information theory, computability and the limits of mathematics and computation, such as Gödel’s undecidability theorem and Turing’s halting problem. The artist will give a short presentation of his work and its connections to these topics.

More information: http://www.tristanperich.com