Interactive Explorations of Hilbert Curves

One of the most famous and elegant constructions in mathematics is Hilbert’s space-filling curve. A nice description of Hilbert curves can be seen in Grant Sanderson’s (@3Blue1Brown) video “Hilbert’s Curves: Is Infinite Math Useful?” These curves have an impressive number of applications such as tilings, paper-folding, number systems, and art just to name a few. They can even be used to find locations on a game map such as in “Zelda: Breath of the Wild” as showcased in Axel Wagner‘s blog post, in spatial indexing for data applications that rely on locations (e.g. Zenly as described by Alex Sitton), and to visualize the similarities between a human’s and a chimp’s genome (as described by Martin Krzywinski).

Hilbert Curves is a unique app authored and illustrated by Doug McKenna in the form of a book that shows, explains, and lets you explore and play with, you guessed it, Hilbert curves. It is designed for high-school or college students, math professionals, and any math-curious person interested in two-dimensional design patterns and space-filling curves and/or fractals. The app presents you with over 130 illustrations in the form of a 160-page electronic document entitled “Outside-In and Inside-Gone”.

The idea of an interactive and highly visual book is super exciting to me. So, I decided to chat with the author to talk about his experiences and inspirations in the realm of interactive visualizations. McKenna’s made his first discoveries of space-filling curves constructions in 1978 and worked as Benoit Mandelbrot’s two full-time research programmer/illustrators, working on pictures for his influential treatise “The Fractal Geometry of Nature” in 1980. In the decades since, he has built both research and commercial software, and has explored various visual mathematical ideas related to recursive geometries, tilings, and fractals.

VRQ: In a few sentences, how would you describe yourself?

Doug McKenna: “I have always been interested in visual math, algorithms, software, user interfaces, computer languages, simplicity, self-similarity, fractals, music, creativity, discovery, and generally How Things Work or How Can They Work Be. I find it super-satisfying to discover or construct simple, never-before-seen mathematical patterns, such as the new tendril-based, half-domino curves I present in this eBook/app. So I think of myself as a software developer and mathemæsthetic explorer who relies upon my computer programming skills to find and/or illustrate and/or play with interesting, infinitely detailed, geometric forms uniquely suited to being drawn only with automated tools.”

The book’s mathematical content was inspired by his collaboration with Dr. Erez Lieberman’s group who had previously used space-filling curves in 3 dimensions to model DNA and was interested in space-filling curves with fuzzy, fractal borders as a better model. As a consultant for the group, he used his previous experience of tuning and/or maximizing the fractal dimension of a space-filling curve’s border. The group’s eventual report studied the distribution and mechanism of near-loops in DNA, and only his Meta-Hilbert construction (called the “Inside-Out Curve” in the paper) was included. His motivation for this book/app stems from multiple reasons, one of them being sharing his beautiful results with a wider audience.

VRQ: What inspired you to write this book? Why  did you choose this particular format?

Doug McKenna: “When one has performed a comprehensive study of the ways one can generalize a highly visual (and famous) mathematical idea, you want others to see it and learn from it. I wanted to publish an account of some new, very cool, and both mathematically and aesthetically beautiful results that I have recently devised/discovered for a wider mathematical audience than just academic journal readers. After my collaboration with the DNA project,  I was left with my notes having over 100 highly detailed illustrations (hand-programmed in PostScript) that documented my findings and journey to the constructions of maximally “fuzzy” space-filling curves in the plane. Space-filling curves created interesting technical problems under the usual forms of publishing.  I set out to create a custom electronic book with dynamic content and excellent mathematical typesetting that I had been imagining for a while.  And I expect and hope it can be made useful to other authors as well as myself. Eventually, I hope to port this publishing system to other platforms.”

VRQ: What is personally your favorite aspect of the book?

Doug McKenna: “Ouch!  This is a little like asking which of one’s children one likes the best!  Some highlights that I’m proud of are getting to discover, report, and give animated life to a visually rich world of half-domino space-filling curves, whose boundaries are self-similar, self-negative, infinitely detailed, and sometimes beautiful and eye-catching forms that live between the linear and the fractal worlds.

Fig. 1: Examples of the author’s favorite pairs of order-12 half-domino curves at stage2. a) Navajo rug pattern. b) Anasazi pottery pattern. c) Ancient greek pattern.

Also, some of the self-negative half-domino motifs  are reminiscent of indigenous craft designs like you might find in a Navajo rug or Anasazi pottery  or an ancient Greek vase (see Fig.1).  The human eye has been fascinated by self-negative forms for millennia. Finally, the reader/user of my eBook/app can see and explore one of these mandala patterns (see Fig.2) as a second approximation to its space-filling curve.  That approximate fractal mandala is a filled polygon built from $92^4$ (over 71 million) tiny, piecewise-connected, self-avoiding line segments, all accurately drawn in front of the reader’s eyes, all of it magnifiable to view any part.  It might be the most detailed geometric illustration ever to be made in any math book. Rather than asking, is it art or is it math, the answer is really both. They are beautiful either way.”

Fig. 2. Illustration of the mandala-like patterns as a second approximation to its space-filling curve.

As Jeffrey Ventrella mentions in his blog post,

“McKenna’s newer curves extend the basic concept of the Hilbert curve, making it just one instance of a larger class of curves. Even within the square, there is an infinite variety of plane-filling sweeps. But some of these curves bust out of their square homes and push the fractal dimension of their boundaries to the point of them becoming their own space-filling curves. That’s meta! ” – Jeffrey Ventrella

This book/app showcases the intersection of math, art, and technology in a very innovative way. The number and quality of the illustrations are astonishing. What I’ve enjoyed most about the book/app is that in the interactive figures, you can do an animated infinite zoom into its construction, or explore the mapping at different levels of precision. If you are a space-filling curve enthusiast this book/app is a great way to explore their beauty and the math behind them.

Do you have suggestions of topics you would like us to consider covering in upcoming posts? Reach out to us in the comments below or let us know on Twitter (@MissVRiveraQ)

Posted in Applied Math, Biomath, Book/App, Interactive, Math Communication, Mathematics and the Arts, Publishing in Math, Visualizations | Leave a comment

Posts to Ponder

I have recently read some posts that don’t necessarily have a common theme uniting them, except that they all grabbed my attention. Without further ado, here’s a little bit about a few of them.

What we mean when we say ‘Anyone can do math’”

The post on the Math for Love blog makes suggestions about what the phrase “Anyone can do math” might actually mean (including “Everyone is capable of mathematical literacy…Everyone deserves to see some beautiful ideas of mathematics” and “A great mathematician can come from anywhere”).

“This is what we have to mean when we insist that anyone can do math. For it to be more than an empty platitude, or a blatant falsehood, we have to be precise,” the post notes.

“For People Of Color, Succeeding In Academia Is A Political Statement”

This post on the e-Mentoring Network blog for the AMS was written by Melissa Gutierrez Gonzalez, a junior mathematics and philosophy student at Occidental College in Los Angeles, who is concurrently enrolled at the California Institute of Technology in Pasadena.

She wrote about what her mother told her before she left for college:

“Vas a la escuela para demostrar que los mexicanos no solo están aquí para limpiar casas o servir como mano de obra, y también para demostrar que las mujeres no solo sirven para casarse y tener hijos.
[Translation: You go to school to show that Mexicans are not only here to clean houses or serve as labor, and also to show that women not only serve to marry and have children].”

She then wrote about pressure she has faced “to speak in cogent and intelligent dialogue whenever I opened my mouth in my discussion-based seminars (not because I wanted to seem like an intelligent person, but an intelligent Mexican). I couldn’t make a mistake, because if I did, what would others think of Mexicans?” She also wrote about the minority tax and the responses she has received when she has tried to talk with her peers about her experiences.

“It’s Not a Competition…But We’re Still Ahead”

On the PhD + epsilon blog for the AMS, Katherine Thompson draws on her experience writing for Who Wants to be a Mathematician and MATHCOUNTS, as well as her personal experience with competition (in classical piano) to explore some of the impacts math competitions can have on young competitors.

While Thompson noted the potential for particiants to become discouraged by low competition scores, especially for students who “are putting in the time to prepare, and think they’re really good at math, and they’re being TOLD they’re really good at math, and realistically they probably legitimately are good at math” but then receive a low score, she also discusses the importance of these competitions for students who are hungry for a challenge.

She summarizes what one of her friends told her about why he writes for math competitions:

“Smart kids first need to be challenged, among other reasons so they don’t get bored and move on to a subject other than math. Realistically, they are in part flocking to competitions because their curiosities aren’t piqued from their standard curriculum and we as a mathematical community don’t want to suffer that loss of talent. Prepping for these competitions, which start with pedestrian topics but take them in remarkably creative directions, addresses that. But just as crucially, these smart kids are with VERY few exceptions sincerely and severely lacking in humility…competitions can show them they still have something left to learn.”

While you’re on the PhD + epsilon blog, consider reading Thompson’s recent post about extra credit in math courses, including the complications surrounding giving extra credit and the different roles extra credit can play in math departments.

In other news for AMS blogs, there are a couple of new things I encourage you to check out, if you haven’t already. Did you read Brian Katz’s post about changes that will be happening to the inclusion/exclusion blog? He was recently named as Editor-in-Chief of that blog, a position that was formerly held by Adriana Salerno. (Salerno was the blog’s founding EIC.) Also, have you visited the new Living Proof blog yet? I’m excited to read upcoming posts on that blog!

Posted in Issues in Higher Education, K-12 Mathematics, Math Communication, Math Education, people in math | Tagged , , , , | Leave a comment

Diving into the DeepMind podcast

Deepmind, famously known for creating the computer programs AlphaGo and Alpha Zero, features a blog that showcases their current research efforts in artificial intelligence (AI). Their more recent posts include: How evolutionary selection can train more capable self-driving cars” by Yu-hsin Chen, Using AI to give doctors a 48-hour head start on life-threatening illness” by Mustafa Suleyman and Dominic King, and Using machine learning to accelerate ecological research” by Stig Petersen, Meredith Palmer, Ulrich Paquet, and Pushmeet Kohli. While the breadth of the blogs cover a lot of topics, I was extremely excited to see the launch of the DeepMind podcast hosted by Hannah Fry (@fryrsquared) author of a number of books including “Hello World: How to be human in the age of the machine”, “The Indisputable Existence of Santa Claus”, and “The Mathematics of Love”.

Why make a podcast? As mentioned on their website,

“Put simply, we love the convenience and format. We thought podcasts were a great option for a series about AI because they allow nuanced discussion and lets listeners hear directly from the people doing the work.”

This podcast is aimed at people who are curious about AI but may not have the technical background. In this eight-part series, listeners get an inside look from researchers themselves to the challenges the field of AI is tackling today. Curious? See the trailer below.

What I enjoyed most about the podcasts were the many analogies used to explain how AI connects to human experiences and other fields. Also, each 30 minute episode includes notes/resources to learn more about the topics covered. Here, I summarize my top three episodes.

 “AI and Neuroscience: The Virtuous Circle”

How do we define intelligence? Jess Hamrick mentions that the debate centers on two camps: should we create AI that is smarter than humans or as intelligent as humans? Matt Botvinick describes how neuroactivity suggests that human brains learn by replaying memories and a very similar idea has a place in AI research. For example, an AI can beat Atari games such as Space Invaders, mainly by learning from the previous games played and maximizing its rewards. Also, human abilities such as liking memories to each other, using mental simulations, and adapting to new situations, give AI a better capacity for solving problems. By studying AI and neuroscience together we can create a virtuous circle where knowledge in the fields flows between one another.

Interviewees: Deepmind CEO and co-founder, Demis Hassabis; Matt Botvinick, Director of Neuroscience Research; research scientists Jess Hamrick and Greg Wayne; and Director of Research Koray Kavukcuoglu.

“Out of the Lab”

Can we use AI to solve real-world problems outside the lab? Pearse Keane discusses how as the number of patients increases there is a growing challenge in diagnosing urgent and common conditions such as age-related macular degeneration (AMD), which can lead to blindness, accurately and quickly. Using AI could promote the early detection and treatment of diseases. Sandy Nelson explores what can AI tell us about proteins, which play a role in many neurodegenerative diseases like Alzheimer’s. Proteins fold in on themselves (in about $10^{300}$ ways!) and their shapes are of great interest to scientists. AI can find clues to reduce the number of shapes being considered for a particular problem. Finally, Sims Witherspoon describes how our use of technology, which relies on data centers, has a great energy demand. For example, data centers consume 3% of the world’s energy. We can ask AI to tell us how to adjust dials in data centers to reduce energy use.

Interviewees: Pearse Keane, consultant ophthalmologist at Moorfields Eye Hospital; Sandy Nelson, Product Manager for DeepMind’s Science Program; and DeepMind Program Manager Sims Witherspoon.

“AI for Everyone”

How can we ethically implement, develop, and use AI? One of the concerns is that Verity Harding mentions is that AI could be used in different ways than intended. If AI is transformative in a good way, it can also be transformative in a negative way. Lila Ibrahim makes the point that there is a lot of responsibility when building technology, especially now that it is available to more people. For example, when using AI in the criminal justice system to reduce inconsistencies among rulings, one must tread lightly and account for racial prejudice and bias in both data and algorithmic implementation. William Isaac highlights that algorithms are not necessarily more objective than humans thus we still have to grapple with ethical questions. Along with Silvia Chiappa, both point out how difficult it is to define and technically measure fairness. Thus, interrogating data and involving more voices in the conversation is and will be crucial to making sure we build a world that belongs to all.

Interviewees: Verity Harding, Co-Lead of DeepMind Ethics and Society; DeepMind’s COO Lila Ibrahim, and research scientists William Isaac and Silvia Chiappa.

Other podcasts include: “Go to Zero”, “Life is like a game”, “AI, Robot”, “Towards the future”, and Demis Hassabis – The interview”. Overall, it was a great way to make AI research more accessible!

Do you have suggestions of topics you would like us to consider covering in upcoming posts? Reach out to us in the comments below or let us know on Twitter (@MissVRiveraQ).

Posted in Applied Math, Artificial Intelligence, Current Events, Math Communication, Mathematics and Computing | Leave a comment

Math Class and Comics Blogs

It’s that time of year again: For those in school, the fall semester is in full-swing and approaching the stretch where the winter holiday season doesn’t seem quite on the horizon. If you’re anything like me (averse to cold weather but living in an area that usually gets hit hard with snow and frigid temperatures), the quickly shortening days make winter weather seem like a looming possibility soon to be realized.

Since Vanessa wrote about an important and serious topic in her last post —  the climate strikes and sustainable mathematics — I’m in the mood to write about something a little more whimsical: comic books in math class.

For The Classroom Blog from Pop Culture Classroom, Jim McClain, the creator of the Solution Squad, a graphic novel about math-themed superheroes wrote “Using Comics to Teach Math!”

“I didn’t set out originally to create a graphic novel, or even a comic book,” McClain wrote. “I created Solution Squad to be heroes that I could use to replace the Marvel and DC characters I was using on my dozens of classroom activities. I had access to a wide variety of the worksheet sort of stuff early on in my career, and they were so mind-numbingly boring that I tried to jazz them up a little with the superheroes kids were watching on TV and in the movies at the time,” he added.

He describes some of the ideas behind his Solution Squad, such as using “characters themselves to embody math concepts.”

“The heroes are teenagers with math-themed powers. Absolutia controls temperature, which requires effort whether she raises or lowers it…Radical, whose symbol is a square root sign, can create invisible electromagnetic prisms whose bases are right triangles. He can move things, including himself, along the hypotenuse face of such a prism. Every construct he makes is itself an application of the Pythagorean Theorem,” he wrote.

Besides describing ways to use his book, McClain also mentions other resources, including The Manga Guide to Calculus and Prime Baby.

McClain also has a blog for the Solution Squad where he recently announced plans to complete Solution Squad Volume 2.

Math Teachers Take a Page From English/Language Arts: Comic Books!”

Education Week recently published this piece by Catherine Gewertz. While there aren’t currently many graphic novels about math themes, math-focused ones are “just starting to get traction” and “it’s a real opportunity for teachers to shape the market,” John Shableski, the director of education development at Udon Entertainment, said to Gewertz.

The piece quotes teachers who describe using math-themed comic books along with other curricula. The piece ends with a discussion of misconceptions about comics in the classroom.

Cindy S. Ticknor, author of The Mysterious I. D. Vide & Newton’s Nemesis math comic book series, also writes Half a Blog. She has written a few “Math & The Characters” posts explaining why she gave the characters some of the traits that she did. For instance, in Math & The Characters: Theo’s Mother,” Ticknor wrote about creating a character to help “the underlying narrative to reinforce growth mindset, or more specifically, mathematical mindset” as described in Jo Boaler’s Mathematical Mindset.

Posted in K-12 Mathematics, Math Communication, Math Education | Tagged , , | Leave a comment

Sustainable Mathematics

Hands cupped together holding soil and a plant On September 20, 2019, a series of strikes around the world demanding action against climate change began as part of Global Week for the Future. It inspired me to look into ways mathematics contributes to the growing challenge of sustainability. Sustainability is the ability of a system to meet the needs of the present generation without compromising the ability of future generations to meet their own needs. In her blog post “Planet Math”, Anna Haensch shared some of the resources in the math blogosphere to tackle climate change. As she mentions,

“… when someone suggests that mathematicians have a lot, they can contribute to our fight to save the planet, but for many people, it can be hard to imagine just why or how (other than gathering lots of data and making graphs that are totally scary).”

I agree! To add to the discussion, I wanted to highlight ways we can practice and incorporate sustainability in mathematics.

In our classrooms

One way we can use math for sustainability is to add it to our syllabus. If you are looking for ideas, the blog “Sustainability Math” by Dr. Thomas Pfaff (Ithaca College) has a collection of projects in statistics and calculus related to sustainability. Thomas Pfaff and Jason Hamilton expand on these ideas in the book “Mathematics for Social Justice: Resources for the College Classroom” by considering social justice and sustainability as the same system. They do this by discussing three sustainability issues: climate change, income inequality, and lead exposure and crime.

If you are a fan of differential equations like me, you can take a look the recent special issue of the CODEE Journal “Linking Differential Equations to Social Justice and Environmental Concerns”. It includes 11 ideas on how students can use applied mathematics to make a difference. Using mathematics to solve real-world problems in their differential equations classroom can shape student identities as Karen Kleen, Next CODEE Editor-in-Chief, shares,

“[The students] talked with me about working together on problems of population growth, and climate change, and other questions that involve finding answers to society’s and Earth’s issues. These students, who were all from underrepresented groups in STEM, showed how they are developing mathematical identities that includes being mathematical problem solvers to make a difference.” – Karen Allen Kleen

You can also share with your students the Society of Industrial and Applied Mathematics (SIAM) has a videos such as “Math Behind Sea Ice & Our Changing Planet” and “Renewable Electricity: Where are we now and where are we headed?” which explain in a few minutes how mathematicians contribute to global sustainability problems.

At conferences and for community building

A big concern in academic social media is the environmental impact of travel. In the opinion piece “Reducing the Carbon Footprint of Academic Travel”, 12 scholars offer concrete ideas on how lower the environmental impact of academic travel and suggest some questions for consideration when planning an event, such as

  • Calculate the carbon impact of your event. What steps can you take to minimize that impact while achieving your goals?
  • How might decarbonization improve your event — whether in terms of intellectual exchange, equity and inclusion, or some other factor?
  • What goals for your event could be achieved by means other than an in-person meeting? How can your event make the most of its setting and the physical travel that it involves?

Some of the suggestions include minimizing the carbon impact of an event by serving vegetarian food, encouraging digital and not paper programs, minimizing swag and working with your venue to serve food and drink in reusable or compostable containers. Also, holding fewer conferences or minimizing overseas travel, organizing conferences at regional hubs near airports or as part of bigger conferences, and building carbon offsets into the conference budget.

These ideas extend to other aspects of academic life. For example, the use of virtual platforms (i.e. Skype, Zoom, Webex, Connect and GoToMeeting) to hold conferences, give talks remotely, and build communities through online discussion groups. There are several online communities such as Quantitative Undergraduate Biology Education and Synthesis (QUBES) Hub,  SIMIODE, and SUBgroups which encourage collaboration, resource sharing, and building community.

The intersection of research and sustainability

In his 2013 opinion piece, “The Mathematics of Sustainability, Simon Levin, identifies three mathematical challenges towards achieving sustainability: developing the statistical mechanics of ecological communities, socio-economic systems, and the biosphere, modeling the emergence of an ecological pattern, and determining indicators of impending critical transitions between states. He also points out the great challenge of achieving cooperation with problems at a global scale, especially in the case of common resources, and the mathematical theory needed to tackle it.

“The greatest challenge facing us is to achieve cooperation in dealing with problems of the Global Commons, especially as regards public goods and common pool resources. This brings to the fore a different set of mathematical tools—control theory, game theory, voting theory, and mechanism opinion design — for identifying under what conditions cooperation is possible and how best to achieve it.” – Simon Levin

Also, SIAM hosts the Conferences on Mathematics of Planet Earth (MPE) which looks to offer a forum for mathematicians and computational scientists to discuss planet earth as a physical system, which supports life, organized by humans, and is at risk.  It will be held again in June 2020 and is co-located with the SIAM Conference on the Life Sciences. This conference is part of a larger initiative described in Evelyn Lamb’s post, “Strength in Numbers: Mathematicians Unite to Tackle Climate Change and Other Planetary Problems” whose goal is to “convince more mathematicians that climate change and other planetary problems are not only important but also interesting”. Besides, talk of “Green AI” has surged in the artificial intelligence community due to a recent paper by Roy Schwartz, Jesse Dodge, Noah A. Smith, and Oren Etzioni from the Allen Institute for AI. The ideas of the paper are summarized very nicely in Juan de Dios Santos’ blog post “Sorry, but your cat or dog AI is damaging the world”,

“The authors propose using efficiency, alongside accuracy and overall performance, as an evaluation metric for future AI implementations. Furthermore, the writers suggest that Green AI could turn the field into a more inclusive one. By doing so, not only researchers but students and others that don’t have access to state-of-the-art machines would have the opportunity to contribute to the field.”- Juan de Dios Santos

Big challenges need out-of-the-box thinking and a lot of collaboration. Be it in your classroom, research, or at your next conference, make sustainability part of the discussion.

Have suggestions of topics you would like us to consider covering in upcoming posts? Reach out to us in the comments below or let us know on Twitter! You can find me @MissVRiveraQ.

Posted in Applied Math, Artificial Intelligence, Biomath, Current Events, Data Science, Math Education, Mathematics and Computing, Sustainability | Leave a comment

Wolfram Blogging

“A Very Brief History of Mathematics” is a recorded lecture recently posted to Stephen Wolfram’s podcast. A Boing Boing post describes it as “a fascinating lecture” but also “a spoken-word illustration of the problems with his incredible (and incredibly difficult) book A New Kind of Science which was…both brilliant and rambling and unstructured.” Want to listen for yourself? Here’s a link to Wolfram’s podcast episode.

In case you missed it, Wolfram made A New Kind of Science freely available online (with the option of downloading it) two years ago in celebration of the book’s 15th anniversary. He announced its free release while reflecting on that momentous occasion in a post to his blog.

In the last few years, Wolfram has written about diverse topics on his blog. Here are some of his recent posts.

“A Book from Alan Turing… and a Mysterious Piece of Paper”

In this post, Wolfram wrote about receiving a copy of Dirac’s Die Prinzipien der Quantenmechanik that once belonged to Alan Turing. George Rutter, one of Wolfram’s former high school teachers, gifted him the book after Norman Routledge, Wolfram’s former high school math teacher and friend of Turing, gave it to Rutter.

Wolfram describes Routledge as someone who “was charmingly over the top in many ways, and told endless stories about math and other things. He’d also been responsible for the school getting a computer (programmed with paper tape, and the size of a desk)—that was the very first computer I ever used.”

The rest of the post discusses a four-page note from Routledge to Rutter that was tucked into the book and Wolfram’s last interaction with Routledge before he died. It also covers more about the book itself and mysteries surrounding who else may have owned it.

“Fifty Years of Mentoring”

Wolfram has been mentoring others in some form for “a shockingly long time: my first memories of it date from before I was 10 years old, 50 years ago,” he wrote. “Somehow I always ended up being the one giving lots of advice—first to kids my own age, then also to ones somewhat younger, or older, and later to all sorts of people,” he added.

He describes mentoring as different from teaching in that it’s “about answering the specific ‘What should I do about X?’ questions, and the general ‘What should I do given who I am?’ questions.” In the remainder of the piece, he discusses the commonalities between mentoring two populations which seem quite different but are actually “at some level, surprisingly similar”:  kids and CEOs.

“Testifying at the Senate about A.I.‑Selected Content on the Internet”

Wolfram was invited to testify at a hearing of the US Senate Commerce Committee’s Subcommittee on Communications, Technology, Innovation and the Internet. In this post, he explains the issues behind the hearing (determining whether Congress should consider pursuing options for algorithmic transparency or algorithmic explanation policies) and why peering inside the AI to see what’s happening isn’t a solution.

“If we want to seriously use the power of computation—and AI—then inevitably there won’t be a ‘human-explainable’ story about what’s happening inside,” he wrote.

The piece includes the full-text version of his testimony.

In addition to his blog, Wolfram also writes for the Wolfram Blog (which has multiple authors). Some recent posts to that blog include “Authorship Forensics from Twitter Data” by Daniel Lichtblau, “Embracing Uncertainty: Better Model Selection with Bayesian Linear Regression” by Sjoerd Smit and “Spherical Aberration Optics Problem Finally Solved Using the Wolfram Language” by Swede White.

Have ideas for topics or blogs you would like us to consider covering in upcoming posts? Reach out in the comments or on Twitter (@writesRCrowell).

Posted in Mathematics and Computing, people in math | Tagged , , , , , , , , | Leave a comment

A tribute to Hispanic Heritage Month

It’s almost that time of the year again!

Hispanic Heritage Month (September 15 – October 15) is a national holiday in the United States that began as a way to promote the history, contributions, and culture of Hispanic-Americans. Its observation started with a week-long celebration in 1968 and was enacted into law to cover 30 days on August 17, 1988. As a Latinx mathematician myself, this month embodies a time of reflection of all we have achieved and the challenges still ahead of us. Most of all, it allows us to know the stories of fellow Hispanic and Latinx mathematicians that contribute to mathematics through their research, teaching, mentoring, and/or service.

During this month,  features the profile of a Latinx/Hispanic mathematician daily, providing a biography and information on their research, teaching and service contributions. It was founded by Alexander Díaz-López, Pamela E. Harris, Alicia Prieto-Langarica, and Gabriel Sosa on 2016 and continues to grow each year (read more about its origin here). Last year’s calendar featured early-career mathematicians including graduate students and post-docs. You can visualize the locations,  institutions, and research areas represented by the 2018 honorees’ in this map. (Note: The data displayed is based on the honorees profiles and might have changed since the calendar launch.)

This year’s theme is Mathematics Education  and you can already see a sneak-peak to this year’s calendar on the AMS September Notices communication, “2019 Lathisms: Latinxs and Hispanics in the Mathematical Sciences”, which features the profile of four of the honorees: Dr. Hortensia Soto (University of Northern Colorado), Dr. Enrique Treviño (Lake Forest College), Dr. Vilma Mesa (University of Michigan), and Dr. James A. Mendoza Álvarez (University of Texas at Arlington).

It will also continue its podcast series featuring interviews with previous honorees led by Evelyn Lamb. What excites me the most about getting to know this year’s honorees is summarized in Evelyn Lamb’s post “The Dangers of a Single Story in Mathematics”,

“There really is no one kind of person who becomes a mathematician.”

What a wonderful thing that is!  A theme of many of the past honorees’ profiles and interviews has been the importance of mentorship, early access to opportunities, and the big sense of responsibility to they feel towards making mathematics and inclusive environment for all.

“Hispanic Heritage Month means someone recognizes that we contribute to this country. It means that the sacrifice that my parents made for us was not in vain. It means that the work that my elementary teachers did for me is recognized. It is a mechanism to serve as a role model for others—regardless of gender, race, sexual orientation, religion, socio-economic status, level of education, etc. And, I get to do this in the same manner that it was bestowed upon me: through compassion. It means that as a Hispanic mathematics educator, I am valued.” —Dr. Hortensia Soto

In her blog post “For People Of Color, Succeeding In Academia Is A Political Statement” Melissa Gutierrez Gonzalez, a junior mathematics and philosophy student at Occidental College in Los Angeles, highlights some of impacts women and minority students face in academia.

“I couldn’t make a mistake, because if I did, what would others think of Mexicans? As someone who has always been a part of predominantly white institutional environments, ignoring the fact that I am acting as a representative for people of my ethnic background is incredibly difficult, because for many, I am the first Mexican they meet. For most, I am the first Mexican or Latino/a they meet in the mathematics field.”

These impacts are sometimes aggravated by the current political status of being an undocumented student. In “Requiem for a Dream”, Dr. Adriana Salerno shared the story of an undocumented mathematician and provides resources to support students affected by the changes in Deferred Action on Childhood Arrivals (DACA) program, which shields young undocumented immigrants from deportation. Representation and acknowledging the full identities of our students, teachers, professors, and colleagues matters. The importance of having faculty that can serve as role models for our students was highlighted in the following student quote from “Cluster Hiring Is Working for Us: Two Early Career Latinas in Math” by Dr. Selenne Bañuelos and Dr. Cynthia Flores,

“The fact that they were both Latina gave me the opportunity to communicate with them about my DACA status, and their support was invaluable to me.”

In the past years, there has been a growing recognition of how mathematics is political. For example, in her 2013 commentary, Dr. Rochelle Gutierrez proposes that,

“By virtue of mathematics being political, all mathematics teaching is political. All mathematics teachers are identity workers, regardless of whether they consider themselves as such or not. They contribute to the identities students construct as well as constantly reproduce what mathematics is and how people might relate to it (or not).”

When we are in the mathematics classroom, whether it is at the K-12 or college level, we are promoting more than mathematical knowledge, we are shaping the identity of  who is and can be successful in mathematics. This extends to other mathematical spaces such as workshops, conferences, and departments. As Dr. Pamela Harris shares in her recent blog post sometimes “It’s the little things” that make you question belonging in the mathematical community.

“It reminds me of little things that I have experienced and which have affected my self identity as a mathematician and made me question my place within the mathematical community. Itʼs the small actions or words or micro aggressions (regardless of intent) that take me out of doing mathematics and bring only selective parts of my identity front and center.”

What it means to be a Hispanic/Latinx mathematician is a multi-layered experience and we will not have a way to condense it all into a single narrative. Conferences such as SACNAS, Blackwell-Tapia Conference, Latinx in the Mathematical Sciences Conference (LatMath) play an important role in bringing together members of the Hispanic/Latinx community. Read some of the reflections on LatMath 2018 by previous participants Emily Alvarez, Dr. Adriana Salerno, and myself. These conferences hope to achieve true diversity in STEM, encourage and showcase the research being conducted by Hispanic and Latina/os at the forefronts of their fields, and to build a community around shared academic interests.

This Hispanic Heritage Month, let’s celebrate how far we’ve come, recognize the unique journey of the members of our community while continuing our work to make mathematics a place where all can belong.

Do you have suggestions of topics you would like us to consider covering in upcoming posts? Reach out to us in the comments below or let us know on Twitter! You can find me @MissVRiveraQ.

Posted in Current Events, Hispanic Heritage Month, Math Education | Leave a comment

Uncovering ‘What if?’ and ‘Why?’ in the A.I. era

Artificial intelligence, which has been extensively developed in the last few decades, cares about the power of a machine to copy intelligent human behavior.  As humans, we make decisions every day that rely on the cause and effects of our actions. For example, we know if we work out at the gym it will cause the number of calories we burn to go up. However, the implications this may have on our overall health is more difficult to address. This boils down to the difference between two statistical concepts: correlation and causation.

  • Correlation: measures the relationship between two things.
  • Causation: means that one thing will cause the other to happen.

The distinctions between the two can have important implications. In the website, “Spurious Correlations” by Tyler Vigen, you can explore a wide variety of correlations that are due to chance.  One of my favorites can be seen in Figure 1, which illustrates the correlation between math doctorates awarded and the Uranium stored at United States nuclear power plants. While these two variables have a correlation of 95.23%, it is highly unrealistic to think my degree caused an increase in the amount of Uranium stored in the United States.

Chart that illustrates how math doctorates awarded correlates with Uranium stored at US nuclear power plants over time.

Figure 1: Example of a spurious correlation by Tyler Vigen.

When we think of causality we want to prove that there is a direct relationship between two variables. This can be harder than expected since, as the famous phrase goes, “correlation doesn’t imply causation”. One of the first examples I encountered  as a student was based on the question: Do storks deliver babies? Many parents may wish the answer was yes to avoid explaining where babies come from to their kids. While the number of storks and human births exhibit a positive correlation (see “Stork Deliver Babies”  by Robert Matthews), again this is not true.

I like this simple example that Adam Kelleher uses in his article “If Correlation Doesn’t Imply Causation, Then What Does?”. Think of your daily commute, if your alarm doesn’t go off or there is traffic, you will be late for work. There are many events on your morning routine that could also make you late for work (i.e. traffic is fine but you spilled coffee on your way or your alarm goes off and you oversleep). We think of all this as noise and as the author mentions, “it takes care of the host of “what-if” questions that come up from all of the unlikely exceptions we haven’t taken into account”.

In the new era of big data, how do we discover the underlying relationships between big datasets and which relationships can we trust? “The Book of Why” by Judea Pearl and Dana Mackenzie, which was recently reviewed in the Notices of the AMS by Dr. Lisa R. Goldberg, tackles the question of how we can use the theory of causality to model and interpret data. In the review, the concept of “The Ladder of Causality” is summarized nicely:

“The bottom rung is for model-free statistical methods that rely strictly on association or correlation. The middle rung is for interventions that allow for the measurement of cause and effect. The top rung is for counterfactual analysis, the exploration of alternative realities.”

Figure 2: Illustration of the ladder of causality in “The Book of Why” by Judea Pearl and Dana Mackenzie.

To achieve intelligence, Pearl proposes a machine’s reasoning should move through the ladder illustrated in Figure 2.  Machines should move from seeing associations in data to doing and planning interventions to obtain a desired outcome, towards  becoming counterfactual learners  that can imagine what does not exist yet and infer from observed data. As mentioned by Pearl in an interview by Kevin Harnett from Quanta Magazine,

“If we want machines to reason about interventions (“What if we ban cigarettes?”) and introspection (“What if I had finished high school?”), we must invoke causal models. Associations are not enough — and this is a mathematical fact, not opinion.”

Reaching the top of the ladder of causality may still be out of our grasp. As our understanding progresses, I would love to see how we integrate the qualitative and quantitative aspects of building a world around data. As Andrew Gelman points out in his review of the book,

“If you think you’re working with a purely qualitative model, it turns out that, no, you’re actually making lots of data-based quantitative decisions about which effects and interactions you decide are real and which ones you decide are not there. And if you think you’re working with a purely quantitative model, no, you’re really making lots of assumptions (causal or otherwise) about how your data connect to reality.”

In my perspective, as humans, we are only able to imagine when we consider both. For machines to answer “what-if?” and “why?” they must do so as well.

Posted in Artificial Intelligence, Data Science, Statistics | Leave a comment

Mathematical oncology blog posts

In July, The Mathematical Oncology Blog was launched. This community blog, which focuses on mathematical and computational oncology, is looking for contributors. Presently, the blog has several posts and seems to be off to an great start.

I find it convenient that links to manuscripts are included at the tops of some of the posts — such as “Quantifying Tumor Heterogeneity” by Elana Fertig and “Space Accelerates Evolution” by Jeffrey West. I am also excited to see that so far, the contributors on the blog have chosen to cover varied topics.

For instance, in “The Colors of Cancer,” Thomas Hillen uses a paint analogy to argue against the “popular myth that the pharma companies have a Golden-Bullet-drug against cancer in their drawers and refuse to use it as to maximize profits” (emphasis is Hillen’s). He then takes his argument further by explaining why “even more so, it is not even theoretically possible to have such a Golden Bullet drug against cancer.”

Taking a totally different approach and critiquing the modeling in a Nature Communications paper published in 2017, Fred Adler’s “All pithy maxims about modeling are wrong, but some are useful” on the blog begins with this:

“In science, criticism is perhaps the highest compliment, showing that we take work so seriously that we are inspired to spend time thinking about and formulating how work can be improved. It is in this spirit that I lay out some of the many issues that I have with the model of Zhang et al (2017) developed by my friends and colleagues at the Moffitt Cancer Center. This model has been widely cited as providing support for the appealing idea of adaptive therapy in prostate and other cancers, the approach of adjusting therapy continuously in response to tumor status in order to maintain long-term control by delaying or avoiding resistance. However, I have been arguing for some time this model is severely flawed in its assumptions, the way it implements those assumptions, and in the presentation and interpretation of results. I take this opportunity to present a brief outline of these flaws. How much they weaken the justification for adaptive therapy is a broader question, as is the question of whether I have wasted my career trying to write models that avoid flaws of this sort.”

Artem Kaznatcheev, who is part of the group that launched the blog, has also written posts about cancer on the Theory, Evolution and Games Group Blog at the University of Oxford. That blog is in its ninth year. Just a few of the posts Kaznatcheev has written about cancer for TheEGG include “Hamiltonian systems and closed orbits in replicator dynamics of cancer,” “Symmetry breaking and non-cell-autonomous growth rates in cancer” and “Abstracting evolutionary games in cancer.”

On the SpringerOpen blog, Jorge Gómez Tejeda Zañudo wrote “Using physics, math and models to fight cancer drug resistance.”

Know about particular blogs or topics you would like us to consider covering in upcoming posts? Reach out to us in the comments below or let us know on Twitter! You can find me @writesRCrowell.

Posted in Applied Math, Game Theory | Tagged , , , | 1 Comment

Let’s Talk About Viral Equations

Recently, there was a viral post about solving the equation below:

Many mathematicians and social media powerhouses have weighed in on what the answer should be. But, why has this equation led to a lot of debate? This is not the first time that an equation or math problem has become an internet sensation and I suspect it won’t be the last (see “10 viral math equations that stumped the internet”). Depending on the order of operations you apply to this equation you will end up with an answer of 16 or 1.

Every time a new viral equation storms the internet, I casually scroll through all the responses.  What draws my attention the most is the way people interact with these types of posts. Many seem to jump at the opportunity to flex their math muscles while others reach for their calculators. The perception seems to be that this basic math question should have solely one answer and however disagrees is wrong. The answer to this viral equation, as Steven Strogatz explains in his recent article, depends on what conventions you are using.

What jumps at me is that, when presented with solving an equation, it’s is rare to discuss how we write mathematics and why we interpret it in a particular way. Mathematical grammar matters and conveying the “right” statement of a problem relies on using the symbols that have become our language. As told in this 2013 article “What Is the Answer to That Stupid Math Problem on Facebook?”  by Tara Haelle, ‘Consider how often people debate grammar. Math has syntax just as language does—with the same potential for ambiguities. And just as word-based riddles exploit the ambiguities of language, so do these math problems’. As Hanna Fry mentions, ‘This is like a maths version of the sentence “He fed her cat food”. Does it mean the man gave some food to a cat? Or – slightly darker- fed some cat food to a woman. It’s impossible to tell from the information we’ve been given’. We have allowed the perception that math is unambiguous to reign. But different from a language, being right or wrong in math, tends to be associated with how smart a person is. This mindset ultimately is what leads to these heated debates and causes a divide. As expressed in this essay  by Kenneth Chang, “It implies that the point of mathematics is to trip up other people with stupid rules.”

Don’t get me wrong, mathematicians spend a lot of time trying to be as unambiguous and precise as possible. However, math can be written in ways that make the reader interpret its meaning in unintended ways. Last year, I taught a course that was aimed at future elementary school teachers. This class was a great way to see how we can build our interpretation of math when most of the tools we use are suddenly are unavailable. Students struggled with playing with assumptions, definitions, and ultimately engaging with mathematics in a playful way. We explored the number systems that were used in different cultures, build proofs using physical objects, and played with what makes and breaks definitions. Yet, the idea that there is only one way to interpret a problem stayed throughout the course. The more time I’ve spent teaching (and learning) mathematics the more dangerous this perception feels. It hides away the fact mathematics was built, written, and interpreted by humans, and sometimes this leads to fun internet debates.

Mathematics has a rich history of how we’ve built the conventions that we use every day. In his blog “What is 00, and who decides, and why does it matter? Definitions in mathematics”, Art Duval shares why definitions (I would also extend this notion to conventions) are useful and highlights how we can still have choices to make even for precise definitions.  For example, Terry Moore’s short TED video, “Why is ‘x’ the symbol for an unknown?”, illustrates that even some conventions have origin stories outside of mathematics. But often, students learn acronyms like PEMDAS (parenthesis, exponents, multiplication, division, addition, and subtraction) without knowing the history of these conventions. We may not have the time to unpack the history of all the conventions that we use but we can highlight how they help us communicate math. Next time a viral equation floods the internet, don’t fret! This can be the perfect opportunity to share a bit of the history of our conventions and the importance of how we write (and interpret) mathematics.

Posted in History of Mathematics, Math Communication, Recreational Mathematics, Uncategorized | Leave a comment