Space. The final frontier. A frontier we wouldn’t stand a chance of exploring if it weren’t for the work of one Albert Einstein and his theories of special relativity. After all, how are we supposed to determine at what speed an alien spaceship is traveling towards Earth if we can’t understand how Newtonian physics break down at high velocities? That is precisely the question that this Maple MathApp asks. Using the interactive tool, you can see how the relative velocities change depending on your reference point. Just what you need for the next time you see a UFO rocketing through the sky!

But what if you don’t have the MathApp on hand when the aliens visit? (So rare to travel anywhere without a copy of Maple on you, I know, but it could happen.) You’ll have to just learn more about special relativity so that you can make those calculations on the fly. And luckily, we have just what you need to do that: our new Maple Learn collection on modern physics, created by Lazar Paroski. Still not quite sure how to wrap your head around the whole thing? Check out this document on the postulates of special relativity, which explains and demonstrates some of the fundamentals of special relativity with lively animations.

Once you’ve gotten familiar with the basics, it’s time to get funky. This document on time dilation shows how two observers looking at the same event from different frames of reference can measure different times for that event. And of course once you start messing with time, everything gets weird. For an example, check out this document on length contraction, which explains how observers in different frames of reference can measure different lengths for the same moving object. Pretty wild stuff.

So now, armed with this collection of documents, we’ll all be ready for the next time the aliens come down to Earth—ready to calculate the relative speed of their UFOs from the perspectives of various observers. That’ll show ‘em!

Disability Pride Month happens every July to celebrate people with disabilities, combat the stigma surrounding disability, and to fight to create a world that is accessible to everyone. Celebrating disability pride isn’t necessarily about being happy about the additional difficulties caused by being disabled in an ableist society: as disabled blogger Ardra Shephard puts it, “Being proud to be disabled isn’t about liking my disability… [It] is a rejection of the notion that I should feel ashamed of my body or my disability. It’s a rejection of the idea that I am less able to contribute and participate in the world, that I take more than I give, that I have less inherent value and potential than the able-bodied Becky next to me.” The celebration started in the US to commemorate the passing of the Americans with Disabilities Act, which prohibits discrimination based on disability, and since then it has spread around the world.

So what does any of this have to do with us here in the math community? Well, while it’s easy to think of mathematics as an objective field of study that contains no barriers, the institutions and tools used to teach math are not always so friendly. For an obvious example, if there's a few steps leading up to your math classroom and you use a wheelchair, that's going to be a challenge. And that's just scratching the surface—there are countless ways to be disabled, many of which are invisible, and many of which make a typical classroom environment very challenging to learn in for a variety of different reasons. As well, it can be difficult for prospective mathematicians to ask for accommodations, because of both the stigma against disability and the systemic barriers to receiving the proper accommodations. Just ask Daniel Reinholz, a disabled math professor at San Diego State University, whose health forced them to drop out of several engineering courses during their undergraduate degree: “Throughout it all, I never had a notion that I could receive accommodation or support, or that I deserved it. (Even though I’ve never really fit into the “right” category of disabled to be accommodated, so who knows what difference it really would have made.)” While Daniel was lucky enough to find a path to mathematics that worked for them, not all disabled people currently have that path available to them. As math professor Allison Miller puts it in her AMS blog post about disability in math, “Success in mathematics should not depend on whether someone’s needs happen to mesh sufficiently well with institutional structures and spaces that have been designed to serve only certain kinds of minds and bodies.”

While we can’t make systemic changes on our own, we here at Maplesoft can still do our part to make tools for math that are something everyone can use and enjoy. As such, we’re excited to share that Maple Learn is now compatible with the screen reader NVDA. By using this screen reader, and with our extensive keyboard shortcuts that negate the need for a mouse, individuals with low or no vision can now use Maple Learn to help them explore mathematics. All you need to do is select “Enable Accessibility” from the hamburger menu, and you’ll be ready to go! Maple Learn also includes the colour palette CVD, which is designed to be accessible to colourblind users. To learn how to access the colour palettes, check out this How-To document.

There is still more work for us to be done to ensure that we’re doing our part to make math accessible to everyone. Not only are there still ways in which we’re working to improve the accessibility of our products, but we all as a math community need to strive towards recognizing the barriers we may have previously overlooked and finding ways to provide accommodation for all mathematicians. One organization, called Sines of Disability, is already working towards that very goal. They are a community of disabled mathematicians dedicated to dismantling the systemic ableism present in mathematics. For this Disability Pride Month, consider taking the time to check out these resources and learn more about this issue.

Can’t seem to find the mistake in your math? Instead of painfully combing through each line, let the new “Check my work” operation in Maple Learn help! Now in Maple Learn, you can type out a solution to a variety of math problems, and let Maple Learn check your work! Additionally, by signing on to Maple Learn and the Maple Calculator app, you can take a photo of your handwritten math, import it into Maple Learn, and check your work with the click of a button.

Whether you’re solving a system of linear equations or an algebra problem, computing an integral or a partial derivative, “Check my work” can help. Maple Learn will tell you which steps are “Ok” and which steps to double-check. If you get a step wrong, Maple Learn will point out which line has an error, then proceed to check whether the rest of your work followed the right procedure.

Here’s an example of a solution to a system of linear equations written out by hand. All I had to do was snap a picture in the Maple Calculator app, and Maple Learn instantly had my equation set ready to go in the Cloud Expressions menu. Then, I just clicked “Check my work” in the Context Panel.

Maple Learn identified that I was trying to solve a system with 3 equations, checked my steps, and concluded my solution set was correct.

What happens if you make a mistake? Here’s an example of evaluating an improper integral with a u-substitution that involves a limit. This time, I directly typed my steps into Maple Learn and pressed “Check my work” in the Context Panel. Check my work recognized the substitution step and noted what step was incorrect; can’t forget to change the limits of integration! After pointing out where my mistake was, Maple Learn continued to evaluate the rest of my steps while taking my error into account. It confirmed that the rest of the process was correct, even though the answer wasn’t.

After making my change in Maple Learn and checking again, I’ve found the correct value.

Checking your work has never been easier with Maple Learn. Whether you want to type your solution directly in Maple Learn or import math with Maple Calculator, the new “Check my work” feature has you covered. Visit the how-to document for more examples using this new feature and let us know what you think!

What was established in 1788 in Prussia, is derived from the Latin word for “someone who is going to leave”, and can be prepared for using the many capabilities of Maple Learn? Why, it’s the Abitur exam! The Abitur is a qualification obtained by German high school students that serves as both a graduation certificate and a college entrance exam. The exam covers a variety of topics, including, of course, mathematics.

So how can students prepare for this exam? Well, like any exam, writing a previous year’s exam is always helpful. That’s exactly what Tom Rocks Math does in his latest video—although, with him being a math professor at Oxford University, I’d wager a guess that he’s not doing it as practice for taking the exam! Instead, with his video, you can follow along with how he tackles the problems, and see how the content of this particular exam differs from what is taught in other countries around the world.

Oh, but what’s this? On question 1 of the geometry section, Tom comes across a problem that leaves him stumped. It happens to the best of us, even university professors writing high school level exams. So what’s the next step?  Well, you could use the strategy Tom uses, which is to turn to Maple Learn. With this Maple Learn document, you can see how Maple Learn allows you to easily add a visualization of the problem right next to your work, making the problem much easier to wrap your head around. What’s more, you don’t have to worry about any arithmetic errors throwing your whole solution off—Maple Learn can take care of that part for you, so you can focus on understanding the solution! And that’s just what Tom does. In his video, after he leaves his attempt at the problem behind, he turns to this document to go over the full solution, making it easy for the viewer (and any potential test-takers!) to understand where he went wrong and how to better approach problems like that in the future.

So to all you Abitur takers out there—that’s just one problem that can be transformed with the power of Maple Learn. The next time you find yourself getting stuck on a practice problem, why not try your hand at using Maple Learn to solve it? After that, you’ll be able to fly through your next practice exam—and that’ll put you one step ahead of an Oxford math professor, so it’s a win all around!

We have just released updates to Maple and MapleSim.

Maple 2023.1 includes improvements to the math engine, Plot Builder, import/export, and more.  We recommend that all Maple 2023 users install this update.

This update is available through Tools>Check for Updates in Maple, and is also available from the Maple 2023.1 download page, where you can find more details.

In particular, please note that this update includes fixes to problems with Quantifier Elimination and Group  Theory, and improves performance after a period of inactivity, all of which were reported on MaplePrimes. Thanks for the feedback!

At the same time, we have also released an update to MapleSim, which contains a variety of improvements to MapleSim and its add-ons. You can find more information on the MapleSim 2023.1 download page.

Sometimes, it’s the little things. Those little improvements that make a good tool even better. Sometimes, it’s as simple as an easy shorthand notation that allows you to create and label points on a graph with a single command. Just to pick a totally random example.

Okay, maybe it’s not totally random. Maybe this new point notation is one of our newest features in Maple Learn, and maybe it’s now easy and quick for you to create labeled points to your heart’s content. Maybe you could learn more about all the ins and outs of this new feature by checking out the how-to document.

But I can’t make any guarantees, of course.

That said, if this hypothetical scenario were true, you would also be able to see it in action in our new document on the proof of the triangle inequality.

With this document, you can explore a detailed (and interactive!) visualization of the proof using Euclidean geometry. You can adjust the triangles to see for yourself that the sum of the lengths of any two sides must be greater than the third side, read through the explanation to see the mathematical proof, and challenge yourself with the questions it leaves you to answer. And those points on those triangles? Labeled. Smoothly and easily. I wonder how they might have done it?

We hope you enjoy the new update! Let us know what other features you want to see in Maple Learn, and we’ll do our best to turn those dreams into reality.

This is a reminder that we are seeking presentation proposals for the Maple Conference.

Details on how to submit your proposal can be found on the Call for Participation page. Applications are due July 11, 2023.

We would love to hear about your work and experiences with Maple! Presentations about your work with Maple Learn are also welcome.

We’re now coming to the end of Pride Month, but that doesn’t mean it’s time to stop celebrating! In keeping with our celebration of queer mathematicians this month, we wanted to take some time to highlight the works of LGBT+ mathematicians throughout history. While it’s impossible to say how some of these individuals would have identified according to our modern labels, it’s still important to recognize that queer people have always existed, and have made and continue to make valuable contributions to the field of mathematics. It’s challenging to find records of LGBT+ people who lived in times when they would have been persecuted for being themselves, and because of that many contributions made by queer individuals have slipped through the cracks of history. So let’s take the time to highlight the works we can find, acknowledge the ones we can’t, and celebrate what the LGBT+ community has brought to the world of mathematics.

If you ask anyone to name a queer mathematician, chances are—well, chances are they won’t have an answer, because unfortunately the LGBT+ community is largely underrepresented in mathematics. But if they do have an answer, they’ll likely say Alan Turing. Turing (1912-1954) is widely considered the father of theoretical computer science, largely due to his invention of the Turing machine, which is a mathematical model that can implement any computer algorithm. So if you’re looking for an example of his work, look no further than the very device you’re using to read this! He also played a crucial role in decoding the Enigma machine in World War II, which was instrumental in the Allies’ victory. If you want to learn more about cryptography and how the field has evolved since Turing’s vital contributions, check out these Maple MathApps on Vigenère ciphers, password security, and RSA encryption. And as if that wasn’t enough, Turing also made important advances in the field of mathematical biology, and his work on morphogenesis remains a key theory in the field to this day. His mathematical model was confirmed using living vegetation just this year!

In 1952, Turing’s house was burgled, and in the course of the investigation he acknowledged having a relationship with another man. This led to both men being charged with “gross indecency”, and Turing was forced to undergo chemical castration. He was also barred from continuing his work in cryptography with the British government, and denied entry to the United States. He died in 1954, from what was at the time deemed a suicide by cyanide poisoning, although there is also evidence to suggest his death may have been accidental. Either way, it’s clear that Turing was treated unjustly. It’s an undeniable tragedy that a man whose work had such a significant impact on the modern era was treated as a criminal in his own time just because of who he loved.

Antonia J. Jones (1943-2010) was a mathematician and computer scientist. She worked at a variety of universities, including as a Professor of Evolutionary and Neural Computing at Cardiff University, and lived in a farmhouse with her partner Barbara Quinn. Along with her work with computers and number theory, she also wrote the textbook Game Theory: Mathematical Models of Conflict. If you want to learn more about that field, check out this collection of Maple Learn documents on game theory. As a child, Jones contracted polio and lost the use of both of her legs. This created a barrier to her work with computers, as early computers were inaccessible to individuals with physical disabilities. Luckily, as the technology developed and became more accessible, she was able to make more contributions to the field of computing. And that’s especially lucky for banks who like having their money be secure—she then exposed several significant security flaws at HSBC! That just goes to show you the importance of making mathematics accessible to everyone—who knows how many banks’ security flaws aren’t being exposed because the people who could find them are being stopped by barriers to accessibility?

James Stewart (1941-2014) was a gay Canadian mathematician best known for his series of calculus textbooks—yes, those calculus textbooks, the Stewart Calculus series. I’m a 7th edition alumni myself, but I have to admit the 8th edition has the cooler cover. To give you a sense of his work, here’s an example of an optimization problem that could have come straight from the pages of Stewart Calculus. Questions just like this have occupied the evenings of high school and university students for over 25 years. I suspect not all of those students really appreciate that achievement, but nonetheless his works have certainly made an impact! Stewart was also a violinist in the Hamilton Philharmonic Orchestra, and got involved in LGBT+ activism. In the early 1970s, a time where acceptance for LGBT+ people was not particularly widespread (to put it lightly), he brought gay rights activist George Hislop to speak at McMaster University. Stewart is also known for the Integral House, which he commissioned and had built in Toronto. Some may find the interior of the house a little familiar—it was used to film the home of Vulcan ambassador Sarek in Star Trek: Discovery!

Agnes E. Wells (1876-1959) was a professor of mathematics and astronomy at Indiana University. She wrote her dissertation on the relative proper motions and radial velocities of stars, which you can learn more about from this document on the speed of orbiting bodies and this document on linear and angular speed conversions. Wells was also a woman’s rights activist, and served as the chair of the National Woman’s Party. In her activism, she argued that the idea of women “belonging in the home” overlooked unmarried women who needed to earn a living—and women like her who lived with another woman as their partner, although she didn’t mention that part. There is a long-standing prejudice against women in mathematics, and it’s the work of women like Wells that has helped our gradual progress towards eliminating that prejudice. To be a queer woman on top of that only added more barriers to Wells’ career, and by overcoming them, she helped pave the way for all queer women in math.

Now, there is a fair amount of debate as to whether or not our next mathematician really was LGBT+, but there is sufficient possibility that it’s worth giving Sir Isaac Newton a mention. Newton (1642-1727) is most known for his formulation of the laws of gravity, his invention of calculus (contended as it is), his work on optics and colour, the binomial theorem, his law of temperature change… I could keep going; the list goes on and on. It’s unquestionable that he had a significant impact on the field of mathematics, and on several other fields of study to boot. While we can’t know how Newton may have identified with any of our modern labels, we do know that he never married, nor “had any commerce with women”^[a], leading some to believe he may have been asexual. He also had a close relationship with mathematician Nicolas Fatio de Duillier, which some believe may have been romantic in nature. In the end, we can never say for sure, but it’s worth acknowledging the possibility. After all, now that more and more members of the LGBT+ community are feeling safe enough to tell the world who they are, we’re getting a better sense of just how many people throughout history were forced to hide. Maybe Newton was one of them. Or maybe he wasn’t, but maybe there’s a dozen other mathematicians who were and hid it so well we’ll never find out. In the end, what matters more is that queer mathematicians can see themselves in someone like Newton, and we don’t need historical certainty for that.

So there you have it! Of course, this is by no means a comprehensive list, and it’s important to recognize who’s missing from it—for example, this list doesn’t include any people of colour, or any transgender people. Sadly, because of the historical prejudices and modern biases against these groups, they often face greater barriers to entry into the field of mathematics, and their contributions are frequently buried. It’s up to us in the math community to recognize these contributions and, by doing so, ensure that everyone feels like they can be included in the study of mathematics.

Some texts distinguish between unary and binary negation signs, using short dashes for unary negation and a longer dash for binary subtraction. How important is this distinction to users of Maple?

Some earlier versions of Maple used to have short dashes for negation (in some places). Maple 2023 has apparently abandoned the short dash for unary negation, and all such signs are now a long dash.

How about math books? Do all texts make this short-long distinction? The typesetters for my 2001 Advanced Engineering Math book also opted for all long dashes and that book was set from the LaTeX exported from Maple 20+ years ago. But I also have texts in my library that use a short dash for unary negation, on the grounds that -a, the additive inverse of "a" is a complete symbol unto itself, the short dash being part of the symbol for that additive inverse.

Apparently, this issue bugs me. Am I making a tempest in a teapot?

We all know that math is beautiful in and of itself—but sometimes students might need a little convincing. What better way to do that then sprucing up your math with a little colour? With Maple Learn, plot colours are fully customizable. We have several colour palettes to choose from—want your document to evoke the delicate tones of springtime? Looking for a palette that’s colourblind friendly? Or maybe you’re just nostalgic for the colours of Maple V? All these options and more are available for making your graphs colourful and coordinated. But maybe you’re the kind of person who wants to go against the grain, and you laugh in the face of predetermined colour coordination. Don’t worry, we’ve got you covered too! With our colour selector, you can also choose your own custom colours. The full colour spectrum is right at your fingertips. To learn more about how to customize the colours on your document, check out this How-To guide.

And of course, the potential for colour inevitably leads to the potential for art. Our Maple Learn Art Gallery has plenty of fun and colourful works you can admire and contemplate (and maybe even draw inspiration from!). One of our most recent and most colourful additions is this document showcasing the history of the rainbow pride flag, in honour of June being Pride Month. You can use the slider to move through time, letting you see how the colours on the flag evolved and read about the meanings behind them. And, thanks to the colour selector, the colours match the precise shades used for the original flags! That’s the magic of hexadecimal colours for you.

Hold on—the magic of hexadecimal colours, I hear you ask? What an enticing concept. If only we had some kind of document, perhaps one made in Maple Learn, that explained how hexadecimal colours worked and included an interactive example so that you could easily see how the red, green, and blue colour values blend together to create any given colour… Too bad we don’t!

Just kidding. Of course we do.

If all these colours have inspired you, be sure to check out our Call for Creative Works for the upcoming Maple Conference! Maybe your colourful creation could be this year’s winner.

If you've seen Paulina's announcement then you know that we are once again holding a virtual Maple Conference this year.  As well, we are once again going to have a virtual gallery featuring artwork and creative projects submitted by the Maple community!

Last year we had a number of great submissions to our Maple Art Gallery and our Maple Learn Creative Showcase.  These were our excellent prize winners.

From left to right we have A visualization of all the primitive roots of 10037 created by Simon Plouffe, winner of the Judge’s Choice, Mother’s Day Rose created with Maple plots by Greg Wheaton, winner of the People’s Choice, and Mona Lisa in Maple Learn created by Paul DeMarco (with help from Leonardo DaVinci), the winner of the People’s Choice for the Maple Learn Showcase.

This year we are expanding the Gallery into two collections to encourage more people to submit.  They are

• The Art Gallery - A small gallery to highlight high effort, mathematically interesting works (with stricter criteria)

• The Creative Works Showcase - A larger showcase for nearly any interesting visual works created with Maplesoft products like Maple Learn and Maple

Feel free to submit nearly anything cool for the Creative Works Showcase, if we find it particularly impressive we might even ask you to let us consider it for the gallery.  Also, do not be intimidated by the title "Art Gallery" we're looking for anything that has taken some artistic effort and tells a mathematical story.

For more information on critera and how to submit, please visit our Call for Creative Works.  The important deadline to know is the September 14th deadline for submission of works with virtual gallery reception and awards ceremony durring the conference October 26-27.

I look forward to seeing all the submissions for the Maple community again this year!

We are happy to announce another Maple Conference this year, to be held October 26 and 27, 2023!

It will be a free virtual event again this year, and it will be an excellent opportunity to meet other members of the Maple community and get the latest news about our products. More importantly, it's a chance for you to share the work work you've been doing with Maple and Maple Learn. There are two ways to do this.

First, we have just opened the Call for Participation. We are inviting submissions of presentation proposals on a range of topics related to Maple, including Maple in education, algorithms and software, and applications. We also encourage submission of proposals related to Maple Learn. 

You can find more information about the themes of the conference and how to submit a presentation proposal at the Call for Participation page. Applications are due July 11, 2023.

Presenters will have the option to submit papers and articles to a special Maple Conference issue of the Maple Transactions journal after the conference.

The second way in which to share your work is through our Maple Art Gallery and Creative Works Showcase. Details on how to submit your work, due September 14, 2023, are given on the Web page.

Registration for attending the conference will open later this month. Watch for further announcements in the coming weeks.

I encourage all of you here in the Maple Primes community to consider joining us for this event, whether as a presenter or attendee!

Paulina Chin
Contributed Program Chair

Happy Pride Month, everyone! June is a month for recognizing and celebrating the LGBT+ community. It was started to mark the anniversary of the Stonewall riots, which were a landmark event in the fight for LGBT+ rights. We celebrate Pride Month to honour those who have fought for their rights, acknowledge the struggles the LGBT+ community continues to face to this day, and celebrate LGBT+ identities and culture.

This Pride Month, I want to give a special shoutout to those in the math community who also identify as LGBT+. As a member of the LGBT+ community myself, I’ve noticed a fair amount of stigma against being queer in math spaces—and surprisingly often coming from within the community itself. It’s one thing for us to make jokes amongst ourselves about how none of us can sit in chairs properly (I don’t even want to describe how I’m sitting as I write this), but the similar jokes I’ve heard my LGBT+ friends making about being bad at math are a lot more harmful than they might realize. And of course it isn’t just coming from within the community—many people have a notion (whether conscious or unconscious) that all LGBT+ people are artistically inclined, not mathematical or scientific. Obviously, that’s just not true! So I want to spend some time celebrating queerness in mathematics, and I invite you to do the same.

One of the ways we’re celebrating queerness in math here at Maplesoft is with new Pride-themed Maple Learn documents, created by Miles Simmons. What better way to celebrate Pride than with trigonometry? This document uses sinusoidal transformations to mimic a pride flag waving in the wind. You can adjust the phase shift, vertical shift, horizontal stretch, and vertical stretch to see how that affects the shape of the flag. Then, you can watch the animation bring the flag to life! It’s a great way to learn about and visualize the different ways sinusoidal waves can be transformed, all while letting your colours fly!

For more trigonometry, you can also check out this fun paint-by-numbers that can help you practice the sines, cosines, and tangents of special angles. And, as you complete the exercise, you can watch the Pride-themed image come to life! Nothing like adding a little colour to your math practice to make it more engaging.

If you’re looking for more you can do to support LGBT+ mathematicians this Pride Month, take a look at Spectra, an association for LGBT+ mathematicians. Their website includes an “Outlist” of openly LGBT+ mathematicians around the world, and contact information if you want to learn more about their experiences. The Fields Institute has also hosted LGBT+Math Days in the past, which showcases the research of LGBT+ mathematicians and their experiences of being queer in the math community. Blog posts like this one by Anthony Bonato, a math professor at Toronto Metropolitan University, and interviews like this one with Autumn Kent, a math professor at the University of Wisconsin-Madison, can also help allies in mathematics to understand the experiences of their queer colleagues and how to best support them. Math is everywhere and for everyone—so let’s make sure that the systems we use to teach and explore math are for everyone too!

Happy Pride! 🏳️‍🌈

We've just launched Maple Flow 2023!

The new release offers many enhancements that help you calculate and write reports faster, resulting in polished technical documents. Let me describe a few of my favorite new features below.

You can now change the units of results inline in the canvas, without taking your hands off the keyboard. You can still use the Context Panel, but the new method is faster and enhances the fluid workflow that Flow exemplifies.

You can also enter a partial unit inline; Flow will automatically insert more units to dimensionally balance the system.

This is useful when results are returned in base dimensions (like time, length and mass) but you want to rescale to higher-level derived units. For an energy analysis, for example, you might guess that the result should contain units of Joules, plus some other units, but you don't know what those other units are; now, you can request that the result contains Joules, and Flow fills the rest in automatically.

The new Variables Palette lists all the user-defined variables and functions known to Flow at the point of the cursor. If you move your grid cursor up or down, the variables palette intelligently removes or adds entries.

You can now import an image by simply dragging it from a file explorer into the canvas.

This is one of those small quality-of-life enhancement that makes Flow a pleasure to use.

You can now quickly align containers to create ordered, uncluttered groups.

We've packed a lot more into the new release - head on over here for a complete rundown. And if you're tempted, you can get a trial here.

We have a lot more in the pipeline - the next 12 months will be very exciting. Let me know what you think!

On this day 181 years ago, Christian Doppler first presented the effect that would later become known as the Doppler effect. In his paper “On the coloured light of the binary stars and some other stars of the heavens”, he proposed (with a great deal of confidence and remarkably little evidence) that the observed frequency of a wave changes if either the source or observer is moving. Luckily for Doppler, he did turn out to be right! Or at least, right about the effect, not right about supernovas actually being binary stars that are moving really fast. The effect was experimentally confirmed a few years later, and it’s now used in a whole variety of interesting applications.

To learn more about how the Doppler effect works, take a look at this Maple MathApp. You can adjust the speed of the jet to see how the frequency of the sound changes, and add an observer to see what they perceive the sound as. You can even break the sound barrier, although the poor observer might not like that so much!

For Maple users, you can also check out the MathApp on the relativistic Doppler effect. You’ll find it in the Natural Sciences section, under Astronomy and Earth Sciences. Settle in to watch those colours come to life!

But wait, I mentioned interesting applications, didn’t I? And don’t worry, I’m not just here to talk about sirens moving past you or figuring out the speed of stars (although admittedly, that one is pretty interesting too). No, I’m talking about robots. Some robots make use of the Doppler effect to help monitor their own speed, by bouncing sound waves off the floor and measuring the frequency of the reflected wave. A large change in frequency means that robot is zooming!

The Doppler effect is also used in the medical field—Doppler ultrasonography uses the Doppler effect to determine and visualize the movement of tissues and body fluids like blood. It works by bouncing sound waves off of moving objects (like red blood cells) and measuring the result. The difference in frequency tells you the speed and direction of the blood flow, in accordance with the Doppler effect! Pretty neat, if you ask me.

And like any good scientific phenomena, the Doppler effect can be used for both work and pleasure. The Leslie speaker is a type of speaker invented in the 1940s that modifies the sound by rotating a baffle chamber, or drum, in front of the loudspeakers. The change in frequency dictated by the Doppler effect causes the pitch to fluctuate, creating a distinct sound that I can only describe as “woobly”. The speaker can be set to either “chorus” or “tremolo”, depending on how much woobliness the user wants. It was typically used with the Hammond organ, and you can hear it in action here!

You know who else uses the Doppler effect? Bats. Since they rely on echolocation to get around, they need some way to account for the fact that the returning sound waves won’t be at the same pitch that they were sent out at. This fantastic video explains it far better than I ever could, and involves putting bats on a swing, which I think should be enough of a recommendation all on its own.

That’s it for our little foray into the Doppler effect, although there’s still a lot more that could be said about it. Try checking out those Maple MathApps for inspiration—who knows, maybe you’ll find a whole new use for this fascinating effect!