The holidays are a perfect time to unwind, reflect, and spend time with loved ones. For me, it is also a great time to browse the internet for fun activities to do. In this post, I highlight some of the mathematics inspired holidays internet treats you can explore and share with others!
Icosahedron Ornament Balls from the “Math = Love” blog.
If you are looking for Christmas and Hanukkah puzzles take a look Almost Surely Math’s blog post “5 Holiday Math Puzzles”. One of my favorites is the 2n-Menora puzzle.
The 2n-Menorah: In an alternate universe, instead of the flask in the temple lasting for just 8 days, it lasted for 2n days for a positive integer n. Thus, Hanukkah in that universe is celebrated for 2n days, and the menorah has 2n+1 candles (including the shammash):
A 2n-Menorah illustration from Almost Surely Math.
On the first day, 1 regular candle is lit and also the shammash. On the second, 2 regular candles and the shammash. And so on until the 2n-th day where 2n regular candles are lit alongside the shammash. How many candles do you need for the whole of Hanukkah in that universe? (Note that the shammash is just the name for the candle put in the center, which is traditionally used to light the other ones. No candle is reused.)
Best Math Moments of 2019
If you are a fan of lists like me, you’ll enjoy reading about the best math “treats”/moments of 2019. In the “Biggest Math Breakthroughs of 2019″, David Linkletter highlights that 2019 was a great year for getting closer to answering old questions and developing new techniques.
“In 2019, math seemed to have many mainstream moments—and that’s not including the viral problemsthat made us want to rip our hair out. This year saw a steady stream of answers (or at least partial answers) to tough questions that had puzzled mathematicians for decades, as well as new techniques that captured our attention in a big way. Here are the numbers—and the minds behind them—that mattered most this year.” – David Linkletter
“For mathematicians and computer scientists, this was often a year of double takes and closer looks. Some reexamined foundational principles, while others found shockingly simple proofs, new techniques or unexpected insights in long-standing problems. Some of these advances have broad applications in physics and other scientific disciplines. Others are purely for the sake of gaining new knowledge (or just having fun), with little to no known practical use at this time. Other fun insights into the world of numbers this year include finally discovering a way to express 33 as the sum of three cubes, proving a long-standing conjecture about when you can approximate irrational numbers like pi and deepening the connections between the sums and products of a set of numbers.” – Bill Andrews
Book lovers rejoice! If you were already preparing your 2020 “to-be-read” list, take a look at Evelyn Lamb’s “Math Reading Challenge” in which she has provided 12 prompts along with suggestions to help you find mathematics-related books for the coming year. My favorite three prompts are,
A math book that helps you make something Crafting Conundrums: Puzzles and Patterns for the Bead Crochet Artist by Ellie Baker and Susan Goldstine Making Mathematics with Needlework, edited by sarah-marie belcastro and Carolyn Yackel Crocheting Adventures with Hyperbolic Planes by Daina Taimina
A nonfiction math book written by a woman Beyond Infinity: An Expedition to the Outer Limits of Mathematics by Eugenia Cheng Mathematics in India by Kim Plofker Power in Numbers: The Rebel Women of Mathematics by Talithia Williams
A math-related book published the year you were born
For me, that’s 1983. Your mileage may vary. Discrete mathematics: A Computational Approach Using BASIC by Marvin Marcus Invitation to Geometry by Z. A. Melzak
For the last prompt, if you are born in 1990 like me, you might try reading “Reaching for Infinity” by Stan Gibilisco or “Journey through Genius: The Great Theorems of Mathematics by William Dunham. You can find inspiration on the Goodreads group or if you want to try getting “Free Springer Science Books” you can use the R code provided in the Learning Machines blog to update your library.
As we close out the year, I wish you all happy holidays and look forward to the many new math treats to come. Do you have suggestions of topics or blogs 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)
Decorating for Christmas but running behind? The Chalkdust Magazine blog has a post about tangled strings of lights, formulas for “perfectly” decorating trees, and more. Image credit: Yuvarani N (Wikimedia CC)
Chalkdust Magazine (“for the mathematically curious”) and the associated blog are a treat. Anna wrote a post last year in which she described the magazine as “filled with as much mathematical goodness as a fresh unopened box of Hagoromo “Fulltouch” chalk.” Now I’m here to share links to a few of my favorite pieces on the blog.
Earlier this year when we purchased our artificial holiday tree, my partner and I had a great debate about the relative merits of pre-lit versus unlit trees. I’m more old-fashioned — I would rather hang the lights myself and have the option of rearranging them in different configurations from year-to-year.
This blog post written by Hugo Castillo Sánchez in 2018 discusses connections between tangled strings of lights, knot theory and the second law of thermodynamics. As a last minute tree decorator and the self-appointed “official light string detangler” of our house, I appreciate the explanations in this post about how tangles form and why it’s often true that if one tangle is present, there are more tangles to be found.
Speaking of last minute decorations, if you’re like me (planning to decorate a tree but running behind) and want mathematical insights for decorating your tree, this post also includes “treegonometry” formulas for “perfectly” decorating a tree. Those formulas were developed in 2015 by members of the Sheffield University Maths Society.
The post shows each step needed to create a tiling art project mimicking the style of cover artist Samira Mian, whose cover image was inspired by the work of Ali Reza Sarvdalir, a Persian geometer and architect.
In this 2018 post, John Pougué Biyong, who was in graduate school studying mathematical modeling and scientific computing at the time of its writing, wrote about his experiences. He describes studying Ebola virus propagation during the 2013 outbreak in West Africa and researching connections between droughts and food insecurity in Mauritania households.
His post also discusses a few disconnects between mathematics and people. “Looking back at my French middle and high-school years, I cannot remember a lecture during where a teacher put a mathematical concept into a more realistic context,” he wrote, adding. “As the years went by, the more complex the tools, the fewer the illustrations… when it should actually work the other way around. Consequently, most pupils would lose track of mathematics because it doesn’t speak to them anymore. And maybe it has never done.” He also noted, “As a maths student, I know as a fact that most people do not know what mathematics deals with and what it is for.”
He rounds out the post by describing the importance of representation and communication, ending with this inspiring line:
“At the end of the day, [mathematics] is all about passion and love so, until the sun sets, let us keep nurturing ourselves all together.”
In November, I ran across a very interesting article in QuantaMagazine “Neutrinos Lead to Unexpected Discovery in Basic Math“ by Natalie Wolchover. She described the discovery that three physicists —Stephen Parke (Fermi National Accelerator Laboratory),Xining Zhang (University of Chicago) and Peter Denton(Brookhaven National Laboratory) had made about eigenvalues and eigenvectors while studying neutrinos.
“They’d noticed that hard-to-compute terms called “eigenvectors,” describing, in this case, the ways that neutrinos propagate through matter, were equal to combinations of terms called “eigenvalues,” which are far easier to compute. Moreover, they realized that the relationship between eigenvectors and eigenvalues — ubiquitous objects in math, physics and engineering that have been studied since the 18th century — seemed to hold more generally.” – Natalie Wolchover
Neutrinos, which are sub-particles that interact with matter weakly, have oscillations between different types that can be described by eigenvalues and eigenvectors. In particular,
“The mathematical description of how neutrinos interact with matter involves square arrays of numbers called matrices. Every matrix has a set of characteristic numbers called eigenvalues; and along with each eigenvalue goes a direction in space called an eigenvector.” – Peter Lynch, “Particle physics gives maths potentially powerful new tool”
With the help of Terrence Tao and Van Vu, the Eigenvector-eigenvalue identity (shown in Figure 1) was proven! Formulas to convert from eigenvalues to eigenvectors exist but they can be hard to compute. This “new” identity states that if you have a square Hermitian matrix (such as the matrix associated with neutrino scattering), one can relate its eigenvalues to its eigenvectors through the eigenvalues of its minors (i.e. a sub-matrix of the original matrix with some rows and columns removed).
“Within a few weeks we became informed (through private communication, online discussion, and exploration of the citation tree around the references we were alerted to) of over three dozen places where the identity, or some other closely related identity, had previously appeared in the literature, in such areas as numerical linear algebra, various aspects of graph theory (graph reconstruction, chemical graph theory, and walks on graphs), inverse eigenvalue problems, random matrix theory, and neutrino physics.” – Terrence Tao
For example, What I love about this article is how it portrays two ideas that resonate with my experience in mathematics:
(1) Mathematics is full of discovery (and rediscovery) even when studying well-known objects.
(2) Nature is indeed written in the language of mathematics and often mathematics is also written in the language of nature.
“I think this new paper is an incredible lucky break for anyone teaching linear algebra now or in the future. It really isn’t that often that a new math paper has a result that is accessible to young students.” – Mike Lawler
An advantage of being a mathematician is that the more that you immerse yourself in math the more you see it all around you. At least for me, even the most personal self-reflections have a math flare to them. As a new decade approaches, I’ve seen a lot of Twitter posts reflecting on what they have accomplished and how they’ve changed over the last decade. This made me ask myself, “how has my experience of mathematics changed during the last 10 years?”
Back in 2009, I was a sophomore barely starting to learn the vastness of math. I would go to the library and promise myself I would come back one day and be able to read any math book in the stacks. I used to think mathematics as a sorta abstract and above all else. When I started to meet other mathematicians and learned of their experiences, how they played with concepts, and their own passion for the subject, mathematics became much more than ideas, it became about the people and the world around me too. Fast forward to now, I’ve realized how naive I was and what a beautiful dream that was. In the video “The Map of Mathematics”, Dominic Walliman captures the richness and evolution of math and as he shares,
“Now the thing I have loved most about learning maths is the feeling you get where something that seems confusing finally clicks in your brain and everything makes sense. Like an epiphany, kind of like seeing to the matrix” – Dominic Walliman
And it clicked, I started thinking about matrices, and in particular, eigenvalues and eigenvectors. Historically, the prefix eigen, has been a cause of debate. As described in “Math Origins: Eigenvalues and Eigenvectors“ by Erik K. Tou,
“By the middle of the 20th century, there were at least five different adjectives that could be used to refer to the solutions in our particular type of matrix equation: secular, characteristic, latent, eigen, and proper. In general, though, two naming conventions dominated: eigen- (from Hilbert’s German writings) and characteristic/proper (from Cauchy’s French writings and von Neumann’s translation of eigen-). In the United States, the peculiar prefix eigen– won the debate.” – Erik K. Tou
Even in many languages, the idea behind this concept remains the same: eigenvalues (and eigenvectors) tell us something about ourselves and definitely as this decade ends I will be reflecting on my own eigenvalues, maybe I’ll also rediscover new identities.
Do you have suggestions of topics or blogs 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)
“When members of the mathematics community are made to feel unwelcome in our profession, the success of mathematics as a whole is put into jeopardy,” an announcement about the workshop notes, adding “This workshop is focused on understanding and creating welcoming environments (providing actionable information and process change plans to mathematics department interested in driving cultural change at their respective institutions) so as to invite more people to enter and persist in STEM disciplines.”
The workshop, which is funded by the NSF and supported in part by the AMS, is co-organized by Maeve McCarthy, Elizabeth Donovan, Vrushali Bokil, Ami Radunskaya and Karoline Pershell. In a brief interview conducted over email, McCarthy and Bokil answered some additional questions about the event. (The following interview has been lightly edited for length and clarity.)
Rachel Crowell: Is this the first workshop of its kind?
Maeve McCarthy: Yes, this is the first workshop like this that AWM has held.
RC: What inspired the idea for the workshop?
MM: Mary Anne Holmes from the University of Nebraska–Lincoln gave a talk at Murray State about ethics in the professional societies based on her work with the American Geophysical Union. It was part of Murray State’s ADVANCE program. Two mathematicians, Maeve McCarthy and Elizabeth Donovan, heard the talk and got excited about doing a similar workshop on harassment for the mathematics community.
RC: I read on the posting about the workshop that there will be sessions related to developing an action plan for departments. Can you explain a little more about what those action plans entail and how those sessions will help participants develop those?
Vrushali Bokil: We have adopted the idea of using action plans to provide an avenue for each participant of our workshop to participate in social transformation in the mathematical sciences. The idea of using action plans is based on the work of the ADVANCE program at Oregon State University (https://advance.oregonstate.edu/advance-seminar-action-plans).
As part of the Moving Towards Action Workshop, all participants are requested to apply the knowledge of issues around sexual harassment in the mathematical sciences that they acquire in our workshop by proposing actions to create change within their spheres of influence. An action plan template will be provided to all participants to help them think of the actions they could take to further the work of improving the climate at their institutions or within the mathematical community.
Oregon State ADVANCE co-PI Tuba Özkan-Haller will give a talk on how action plans have been created and implemented at Oregon State University within its ADVANCE program activities and the impact that they have had at the personal, symbolic and institutional levels. Professor Özkan-Haller will also lead participants in drafting their own action plans using the provided template.
RC: What are some of the key things you hope participants will take away from the workshop?
MM: We hope to equip participants with tools for making cultural shifts in their departments and in the mathematics community.
RC: If you had to choose just three pieces of advice about how departments can be more welcoming to all, what would you pick?
MM: 1. Harassment must be stopped. 2. An atmosphere of inclusion is imperative. 3. Changes can have a bigger impact than we think.
RC: Is there anything else you would like to share about the workshop or related to the topic of improving the culture and climate in math?
MM: We’re particularly excited to have the University of New Hampshire’s PowerPlay group do a Bystander Intervention program led by Stephanie Goodwin from Wright State. It’s a powerful approach and will really get the audience thinking.
Editor’s note: Unable to attend the workshop but interested in learning more on about the topics it will cover? See the list of resources and pre-readings available here.
A new AMS blog — Living Proof: Stories of Resilience Along the Mathematical Journey — was recently launched. It follows the publication of the book by the same name (jointly published by the AMS and MAA). The book, which was edited by Allison K. Henrich, Emille D. Lawrence, Matthew A. Pons, and David G. Taylor, tells the stories of 41 mathematicians. It’s available as a free e-book download here.
Henrich is the editor of the new blog and Pons is the co-editor. (They are also joined by associate editors Jen Bowen, Susan Crook, Chawne Kimber and Anisah Nu’Man.) In an interview conducted over email, I asked Henrich and Pons some additional questions about the book and their plans for the new blog. (The following has been lightly edited for length and clarity.)
Rachel Crowell: For folks who might be unfamiliar with the Living Proof project, can you please share in a few sentences what you would like for them to know about it?
Allison Henrichand Matthew Pons: Living Proof is about sharing our stories—the good, the bad, and everything in between. We hope that people will find comfort in reading about others who have had experiences similar to their own. Our aim is for Living Proof to help our community to become more open and accepting by recognizing, on the one hand, struggles that are common across the community and, on the other hand, long-standing biases the community has had that have disadvantaged mathematicians with different backgrounds. We also hope the stories in the blog point the math community to places where we need to focus energy to eliminate those biases.
RC: Since Living Proof: Stories of Resilience Along the Mathematical Journey was published earlier this year, what kind of feedback have you received about it?
AH and MP: We have received incredibly positive feedback. Many folks in the community have voiced their support for us, the editors, and those authors who contributed to the book. Several people have wondered why it took so long for a project like this to come about. Students have told us in person and via email how much the stories have encouraged them or opened their eyes to situations they wouldn’t have imagined.
RC: How did you decide whose stories to include in the book?
AH and MP: We were pretty open here. We reached out to about 100 people and accepted most of the stories that were submitted to us. For most of the submissions, our editorial team worked with authors to fine-tune their writing in order to clarify what happened and what message they want to pass on to others. After submitting a partial draft to the MAA and AMS, we discovered the need for a bit more diversity in several areas. So, we sent out a second round of invitations to a smaller, targeted group of people. In general, we tried not to exclude any piece unless it really didn’t fit with the goals of the project.
RC: Can you tell me a little bit about your vision for the blog?
AH and MP: The main goal of the blog is to continue sharing stories. The book contains 41 stories, which cannot possibly represent the variety of experiences out there. For instance, we just had our first anonymous post by someone who struggled with alcoholism and severe depression on the tenure track. The story is quite different from anything in the book, but a valuable one for people to read—especially those in our community who might be silently struggling with similar issues of addiction, substance abuse, or depression that involves suicidal ideation.
Another goal behind sharing more stories is that more themes will begin to emerge. A single story about imposter syndrome has limited impact, while 20 stories that all involve people struggling with self-doubt can help others realize just how common the phenomenon is.
We have also talked about having posts that synthesize themes in several stories. In addition, we are excited to have posts about how people are using Living Proof in their classes or with advisees.
RC: What are some of the things you hope readers will be able to take away from the blog?
AH and MP: The takeaways are essentially the same as mentioned above for the book. We want the conversations about how to address struggles and build resilience to continue. We want to shine a light on the places where folks don’t want light shone. We want to acknowledge and appreciate the path that each of us has to walk and work to make it a little smoother for the next few generations.
RC: If you could share three pieces of advice with math students (especially those from underrepresented groups) who are overwhelmed and feeling uncertain about whether there is a place for them in math, what three things would you pick?
AH and MP:
Whatever you are feeling or struggling with, it is highly likely that someone else who has had a very successful career has been there. And even though you might not be able to talk to them or read their story (although, our hope is that you can!), at the very least you can find some comfort in the fact that you aren’t the first one experiencing this particular struggle.
Find mentors. Find professors, colleagues, and friends who can listen to you when you need to be heard, who can advocate for you when you need someone to speak to others on your behalf, and can give you advice when you’re not sure about how to navigate a situation. Tell them what you are struggling with. Ask them questions.
Your self-worth is not defined by how well you did on a math test or what you got on the math GRE. It is not defined by what others tell you about yourself. If you are struggling with learning math, that does not mean that you are stupid or a failure. If you got an A in your math class, that does not mean that you are a genius. The more we can decouple our self-worth from our performance and accolades, the more resilient we can be to challenges and the less likely we’ll be to develop egos that will push others away from us. (For more on these ideas, see Francis Su’s story in the book and the story by Anonymous in the blog from November 2019.)
RC: Are you open to folks contacting you if they’re interested in contributing their stories for the blog? If so, what is the best way for them to get in touch? Are there any guidelines they should keep in mind about the types of stories you’re looking for, beyond the general theme of resilience along the mathematical journey?
AH and MP: We are absolutely interested in folks contacting us. We are pretty open about what types of stories we are looking for. People are welcome to contact us to discuss ideas before they start writing if that works best, or they can write something, send it to us, and we can get back to them with general feedback before an official acceptance/editorial process begins. Folks can feel free to reach out by email to Allison Henrich (henricha@seattleu.edu), Matthew Pons (mapons@noctrl.edu), or any of the other members of our editorial team (Jen Bowen, Susan Crook, Chawne Kimber, and Anisah Nu’Man).
I am a huge fan of Tai-Danae Bradley’s blogMath3ma. Why? In her blog, Tai-Danae explains concepts related to Category Theory and many other fields of math with beautiful illustrations in an accessible way. Math3ma was launched in 2015 when Tai-Danae was in her second semester of graduate school and has been going strong ever since. Tai-Danae, was also a co-host in “PBS Infinite Series”, has written a collection of introductory, expository notes on applied category theory, “What is Applied Category Theory?”, and was featured in the “Mathematically Gifted & Black” website. A while back on Twitter, I read a post asking for advice on how to start a blog as a graduate student and was inspired to revisit Math3ma and chat with Tai-Danae about her blog. In this tour, I hope to give you a glimpse of the blog’s style, content, and insights from Tai-Danae herself!
As she describes in her post “What is Category Theory Anyway?”, Category Theory provides a bridge between different areas of mathematics and its objects (see Figure 1). Informally, “each has objects in it (set theory has sets, group theory has groups, topology has topological spaces,…) that can relate to each other (sets relate via functions, groups relate via homomorphisms, topological spaces relate via continuous functions,…) in sensible ways (composition and associativity).”
Figure 1: Example of categories showcased in Math3ma’s “What is Category Theory Anyway?”
Similarly, Tai-Danae’s blog and its great content is abridgebetween many wonderful areas of math and it’s readers.
VRQ: Can you tell our readers a bit about yourself and your blog?
Tai-Danae Bradley: “I’m a 6th—and final! —year PhD candidate inmathematics at the CUNY Graduate Center. My research interests lie in the (non-empty) intersection of quantum physics, category theory, and machine intelligence. I also have deep admiration for ideas within algebra, topology, and homotopy theory. Some quick facts about my blog: Math3ma is pronounced “mathema” as in mathematics. Mathema is a Greek word that means “a lesson.” The domain mathema.com was owned by an Italian tech company. I didn’t want to give up the name, though. That’s why the “e” became a “3.” The blog’s official logo is an “M” surrounded by little electrons. I double-majored in math and physics as an undergrad, so the logo is a nod to my interest in both subjects.”
Figure 2: Math3ma’s Logo.
VRQ: What is the inspiration behind your blog?
Tai-Danae Bradley: “Math3ma began as a tool to help me adapt to graduate school. I knew that the transition from an undergraduate program to a PhD program would be a challenging one for me. The blog’s About page opens with this idea: The learning process often involved writing. That is, I learn best by putting my thoughts on paper. That’s because the process of writing helps me to slow down and think carefully about the mathematics. This is when the ideas have a chance to slowly simmer and marinate to eventually become “aha moments,” which are hard (impossible?) for me to get otherwise, without putting in this concentrated time and effort. I developed writing as a study skill pretty early on—in middle school, I think—so it naturally carried over into high school and college and later to graduate school. In particular, after my first semester I’d already amassed lots of mini expositions I’d written for myself. So I had all these little essays laying around and eventually decided to share them online in case other students could find them helpful, too. “Math3ma was originally created as a tool to help me transition from undergraduate to graduate level mathematics. Quite often, I’d find that the ideas of math are hidden behind a dense fog of formalities and technical jargon. Much of my transition process was (and still is!) learning how to fight through this fog in order to clearly see the ideas, concepts, and notions which lie beneath.”
Figure 3: From “Learning How to Learn Math”, Tai-Danae shares: “In the third line, I’ve written, “…ask yourself…what is it that they all have in common?” I suspect this may be the origin of my affinity for category theory, a unifying language in mathematics.”
VRQ: Has blogging changed how you view math? If so, how?
Tai-Danae Bradley: “Blogging hasn’t changed how I view math, but my view of math affects how I blog. This goes back to the idea mentioned earlier: “Quite often, I’d find that the ideas of math are hidden behind a dense fog of formalities and technical jargon.” The articles on Math3ma are my personal attempts at fighting through the fog. I believe that much of advanced mathematics can be made accessible to a wide range of people, even though it may feel inaccessible because of the sophisticated language. So I spend a lot of time distilling ideas for myself. As I mentioned above, this takes a lot of work that often involves putting my thoughts on paper. Occasionally, I’ll polish up some of these writings for a blog post. Blogging, then, ultimately began as a tool to help me gain access to the simple ideas that lie beyond the misty fog. This is one of the main reasons I wanted to go to graduate school—to learn how to see mathematics clearly. Math3ma is a public record of this personal endeavor.”
VRQ: Any advice for other mathematicians interested in creating their own blog?
Tai-Danae Bradley: “The blog’s most frequently asked question is how I make the illustrations. It’s super easy! I draw them by hand with pen and paper, snap a photo with my phone, Airdrop to my computer, and edit (add color, etc.) with Photoshop. In the early posts, I used a pen tablet but that didn’t last long. I like the personal touch of a real pen. So, I’m not sure if this counts as advice, but if you want illustrations then old-fashioned pen and paper is an affordable place to start! Another thought I often hear is, “I really want to start a math blog, but I just don’t have time to post regularly.” I don’t have advice on time management, per se, but I will share something that I’ve personally found helpful: don’t worry about frequency! Or rather, I don’t worry about frequency. I blog only when I have time, and I think this has worked out well so far. My blog articles are really written for me, so there is no internal (or external) pressure to publish on a certain schedule. If you enjoy mathematics, and you enjoy sharing it, then I think others will be just as delighted to share in your joy, whether it’s once a week, once a month, or once a year.”
Some of her favorite blog posts include:
1.”A First Look at Quantum Probability (Part 1 & Part 2)”, where she shares some simple mathematics that’s a combination of linear algebra and elementary probability theory (see Figure 4).
Figure 4: Comparison of classical and quantum probability from “A First Look at Quantum Probability”.
2. “The Yoneda Perspective/Embedding/Lemma“ (a three-part series), which is one of her favorite theorems, and says, informally, that a mathematical object (a set, a group, a vector space,…) is completely determined by the set of all mappings into or out of that object (see Figure 5).
Figure 5. Illustration of the relationships showcased in “The Yoneda Perspective/Embedding/Lemma”.
3.“The Tensor Product Demystified”, where she illustrates through examples and pictures, how you can make new vectors from old using the direct sum and the tensor product (see Figure 6)!
Figure 6: Illustration from Math3ma’s “The Tensor Product Demystified”.
Do you have suggestions of topics or blogs 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)
While browsing the math blogosphere on Twitter, I found myself diving into the wonderful 3D printing posts. Back in 2014, Evelyn Lamb wrote a post in this blog called “The Revolution Will Be 3D Printed”. Inspired by the title, I was curious to explore how 3D printing has evolved in the last few years and some of its uses in the classroom.
3D printing gives us a way to bring an abstract concept to life and interact with it in new and exciting ways. This field has been around since the 1980s and has kept growing ever since, but why did it become so popular? As mentioned by Ki Karou in “Back to the Future: 3D Printing and the Future of Math Education”,
“Although the field of 3D printing has its roots in the 1980s, it surged in popularity recently thanks to decreased costs (printers can be found in the thousand dollar range) and people’s astonishingly creative uses of the devices. News articles regularly come out with new uses for 3D printers, from the 3D printed car to the 3D printed wrench used in the space station, and even the exciting possibility of printing human organs.” – Ki Karou
In Mike Lawler’s blog post, “Ten 3D Printing math projects to help students explore math“, he shares a collection of projects that explore a wide range of mathematical ideas with some interesting applications. Some exciting examples include seeing the relationship between geometry and shadows (inspired byHenry Segerman), using tilings as cookie cutters (inspired by Laura Taalman) and revising the volume of a sphere.
As a fan of differential equations, I was excited to read the article, “Riding the “Wave” of Affordable 3D Printing”, where Nate Barlow, Colin Huber, and Olivier Montmayeur share the applications of 3D printing in Partial Differential Equations (PDEs). Here they provide examples of problems in which 3D printing lends itself nicely to classroom explorations and also how it might be useful to introduce ideas in research groups.
“Currently, we are using 3D prints in lectures for demos, for workshops, and also for homework problems. When introducing our research to new group members or talking at conferences, 3D prints have become the perfect tool for showing the difference between a linearly stable, convectively unstable, and absolutely unstable response to a localized initial disturbance”.
“For the past couple of decades, mathematicians have increasingly relied on digital imaging to see complex shapes. But certain characteristics and symmetries are just not obvious until you look at a physical representation. A digital rendering, even one you can rotate, is, after all, a just a series of 2-D images. When trying to study a shape in 4-D space, much less 3-D, even more is lost.” – Luke Whelan
Also, 3D printing has been used to replace or support coral reef systems in that have been affected by bleaching or by other weather events, manufacturing medical equipmentand surgical devices that are adapted to patients needs, and many more applications.
Another important benefit of 3D printing is to make math accessible to those who have visual impairments. As Chelsea Cook, a physicist, shares in her 2013 TedTalk, using 3D models made mathematics come to life in her Multivariable Calculus class.
“When I took Multivariable Calculus a couple of years ago, I developed a partnership with Chris Williams from the dreams laboratory. We worked together to create three-dimensional models of the shapes and graphics needed for the coursework. Again, the models made the math come alive. I could perceive every detail of a model much clearer than a tactile graphic. I made the shape work for me to reach my goal.” – Chelsea Cook
The STEM:IT curriculum created by VariQuest, whose project-based lessons include coding and 3D design and provide instructors poster templates, 3D print files, and step-by-step guides.
Robo, which designs 3D printers for classrooms, teamed up with MyStemKits.com to offer K–12 educators printers with more than 240 built-in 3D STEM lessons.
Whether you are an enthusiast, teacher, or researcher there are many wonderful resources out there to get started. MakerGirl, a non-profit organization that inspires girls to be active in STEM, mentions that are many apps, toys, and websites that promote STEM learning. In a recent post, they highlight some of their favorites including Morphi,Thinkercad, Toybox which provide ways to learn 3D design. Also, there are manygeometry labsandmakerspacesaround the world that can serve as a resource. Are you ready to join the revolution?
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)
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. Zenlyas described by Alex Sitton),and to visualize the similarities between a human’s and a chimp’s genome (as described by Martin Krzywinski).
Hilbert Curvesis 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 eventualreportstudied the distribution and mechanism of near-loopsin 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.
“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)
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.
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.
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.
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!
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.
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.
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.
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.
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).
Opinions expressed on these pages were the views of the writers and did not necessarily reflect the views and opinions of the American Mathematical Society.
