Maplesoft Blog

The Maplesoft blog contains posts coming from the heart of Maplesoft. Find out what is coming next in the world of Maple, and get the best tips and tricks from the Maple experts.

A salesperson wishes to visit every city on a map and return to a starting point. They want to find a route that will let them do this with the shortest travel distance possible. How can they efficiently find such a route given any random map?

Well, if you can answer this, the Clay Mathematics Institute will give you a million dollars. It’s not as easy of a task as it sounds.

The problem summarized above is called the Traveling Salesman Problem, one of a category of mathematical problems called NP-complete. No known efficient algorithm to solve NP-complete problems exists. Finding a polynomial-time algorithm, or proving that one could not possibly exist, is a famous unsolved mathematical problem.

Over years of research, many advances have been made in algorithms that can solve the problem, not in perfectly-efficiently time, but quickly enough for many smaller examples that you can hardly notice. One of the most significant Traveling Salesman Problem solutions is the Concorde TSP Solver. This program can find optimal routes for maps with thousands of cities.

Traveling Salesman Problems can also be used outside of the context of visiting cities on a map. They have been used to generate gene mappings, microchip layouts, and more.

The power of the legendary Concorde TSP Solver is available in Maple. The TravelingSalesman command in the GraphTheory package can find the optimal solution for a given graph. The procedure offers a choice of the recently added Concorde solver or the original pure-Maple solver.

To provide a full introduction to the Traveling Salesman Problem, we have created an exploratory document in Maple Learn! Try your hand at solving small Traveling Salesman examples and comparing different paths. Can you solve the problems as well as the algorithm can?


we have recieved lots of great sumissions, but we want your great submission and now you have more time.


The deadline for submissions to the Art Gallery and Showcase for the 2023 Maple Conference is rapidly approaching. We really want to see your art! It doesn't have to be incredibly impressive or sophisticated, we just want to see what our community can create! If you've been working on something or have a great idea, you still have a few days to get it together to submit.

A penrose tiling mosaic of that famous Windows 95 background

Submission can be made by email to but be sure to visit the visit our Call for Creative Works for details on the format of the submission.


What are planes? Are they aircraft that soar through the sky, or flat surfaces you'd come across in your geometry textbook? By definition yes, but they can be so much more. In the world of math, observing a system of equations with three variables allows us to plot them as planes in ℝ3. As we plot planes, these geometric entities can start intersecting, creating captivating visualizations. However, the intersection of planes is not just an abstract mathematical concept present only in the classroom. Throughout our daily lives, we come into contact with intersecting planes everywhere. Have you ever noticed the point where two walls and the floor in your room converge? That’s an intersection in its simplest form! And the line where the pages of a book are bound together? Another everyday intersection!

Room image:, Book image: 

However, just seeing plane intersections is but a tiny piece of the puzzle. After all, how can we delve into the intriguing properties of these intersections without quantifying them? Enter the focus of Maple Learn's newest collection: Intersection of Planes. Not sure about how you can identify the different scenarios that three planes can form in ℝ3? Check out the eight documents that provide complete walk-throughs for solving each individual case that three planes can form! With cases ranging from three parallel and distinct planes to three planes forming a triangular prism to three planes intersecting in a line, you’ll gain a mastery of understanding the intersection of planes by the time you’re finished with the examples.


Once you’ve gained an understanding of how to identify and solve the cases that three planes can form, it’s time to test your knowledge! This quiz-like document takes you through the steps of solving for the intersection of three planes with guiding questions and comprehensive feedback. Once you successfully find the intersection or identify the case, you’ll be provided with an interactive 3D plot that allows you to see what the math you’ve been doing looks like. This opportunity to solve any of the 16 different possible systems of equations allows you to prove that you’re on another level!

With your newfound mastery of solving systems of equations, check out similar documents in the recently added linear algebra collection! Try calculating the volume of a parallelepiped or deriving the formula for the distance between a point and a plane

What are you waiting for? Gear up and join us on Maple Learn today! Whether you're diving deep into the world of linear algebra or merely dabbling, there’s a world of discovery waiting for you.

Jill is walking on some trails after a long and stressful day at work. Suddenly, her stress seems to be lifted off her shoulders as her attention gets drawn to nature's abundant beauty. From the way the flowers blossom to the way the leaves grow on their stems, it is stunning.

When many think of mathematics, what comes to mind is often numbers, equations, and calculations. While these aspects are essential to math, they only scratch the surface of a profoundly creative discipline. Mathematics is much more than mere numerical manipulation. It is a rich and intricate realm that influences everything from art and science to philosophy and technology.

Just as Jill was stunned by the beauty of nature, you too can be amazed by the beauty of math! The golden ratio is one math concept that garners a reputation for being particularly beautiful, perhaps due to its presence in different parts of nature. You can explore it through some Maple Learn documents.

Check out the Fibonacci sequence and golden ratio document to better understand the golden ratio and its relationships. Perhaps, once you have a good grasp on the basics, you would like to check out the golden spiral document. Notice how the spiral that results resembles the outline of a nautilus shell and the arms of a spiral galaxy!

The spiral generated in the maple learn document on the golden spiral. A nautilus shell whose shape resembles the golden spiral generated in the maple learn document.A spiral galaxy whose arms resemble the spiral generated in the Maple Learn document on the golden spiral.

Nautilus shell image: -- Spiral galaxy image:

Next, you may want to understand why the golden ratio is considered the most irrational number. You can do that by checking out the most irrational number document. Or you could explore this golden angle document to see how the irrationality of the number can be used to reproduce structures found in nature, such as the arrangement of seeds in a sunflower's centre!

An image generated in the golden angle Maple Learn document where points are packed around the center of a circle using the golden angle. The points are tightly packed around the center.The previous image is superimposed on top of an image of the center of a sunflower. The superimposed image matches the seeds' packing in the sunflower's center.

Sunflower image:

That's all for this post! No worries, though. Maple Learn has hundreds of documents that can aid you in exploring the abundant beauty of math. Enjoy!

It’s finally here. The mystical treasure that has long been rumoured to lie deep within the labyrinthine halls of Maple Learn is within your grasp at last. The ordeals ahead are treacherous, and most who have ventured in have never returned… but, armed with nothing but your wits and your curiosity, you know you’re prepared to conquer the trials that await you. Can you be the first to uncover the secrets of Maple Learn?

A screenshot of the start screen of 'The Treasure of Maple Learn', which consists of colourful squares spelling the word 'START'.

A screenshot of the first room of the Treasure of Maple Learn. The text reads, 'A distant howl echoes through the dark, misty forest as you tread carefully past the towering trees. Many have walked this path seeking the legendary treasure within Maple Learn, but few have returned. You stop in front of a huge stone door, carved with ancient symbols.'

Surprise! We here at Maplesoft have decided to become game developers. Okay, maybe not really, but we do have one game that we’re excited to be sharing with you all. Introducing: The Treasure of Maple Learn. This series of documents mimics the style of a text-based adventure game, and takes you through a series of puzzles that challenge you to discover for yourself all of what Maple Learn has to offer. It was originally created by myself and a team of other co-op students during the 2021 Maplesoft Hackathon, and I’m very excited to be releasing this updated and polished version of the game. Finally, it’s time to set out on your quest to discover the legendary treasure that lurks within Maple Learn… if you dare.

If you don’t dare, don’t worry, we have other options. You can also check out our new video on Getting Started with Maple Learn, which takes you through everything you need to know to become a Maple Learn expert. And if that’s not enough learning Maple Learn for you, we also have an extensive collection of How-To documents. Want an in-depth look at how to use the plot window? How about an exploration of how to work with linear algebra? Or maybe you want to unleash your artistic side? We’ve got you covered.

So if you’re just getting started with Maple Learn and are looking for a tutorial, you’ve got options—we’ve got a quick video overview, tons of collections of in-depth documentation, and a quest through the treacherous depths of Maple Learn to uncover the secrets that lie within. Pick your poison! (But maybe watch out for literal poison in that labyrinth.)

The concept of “Maple Learn art” debuted on the MaplePrimes blog in December 2021.  Since then, we’ve come a long way with new Maple Learn features and ever-growing creative minds.  Creating art using mathematical expressions and shapes is a great way to hone both your mathematical skills and your creativity, and is the perfect break from a bout of studying or the like.

I started my own Maple Learn art journey over one year ago.  Let’s see how one’s art can improve over time using new and advanced features!

Art with Shapes, March 2022

This pi-themed pie is simple and cute, but could use some additional features:

Adding Shaded() around Maple Learn shape commands colors them in!

Fun fact: I hand-picked all of the coordinates for that pi symbol.  It was an arduous but rewarding process.  Nowadays, I recommend a new method.  When you create a table in Maple Learn with two number columns, the values are plotted as points.  These points can be clicked and dragged across the plot window, and the table updates automatically to display the new coordinates.  How can you use this to make art?

  1. Create a table as described above.
  2. Move the points with your mouse to create an outline of the desired shape.
  3. Use the coordinates from your table in your geometry command.

Let’s apply these techniques in a newer piece: a full recreation of the spaghetti emoji!

Art with Shapes, August 2023

Would you look at that?!  Fully-shaded colors, a background, and lines of spaghetti noodles that weren’t painstakingly guesstimated combine to create a wonderfully improved piece of art.

Art with Animation, March 2022

Visit the document to see its animation.  Animation is an invaluable feature in Maple Learn, frequently utilized to observe how changing variables affect functions or model a concept.  We’ve harnessed its power for animated artwork!  This animation is cute, using parametric functions and more to change the image as the animation variable changes.  Like the previous piece, it’s missing a background, and the leaves overlap the stem awkwardly in some places.

Art with Animation, August 2023


This piece has a simple background made with a large black square, but it enhances the overall effect.

The animation here comes from piecewise functions, which display different values based on a given criterion.  In this case, the criterion is the current value of the animation variable.

There are 32 individual polygons in this image (including 8 really tiny ones along the edges!) and 8 rainbow colors.  Each color is associated with a different piecewise function, and displays four random squares in that color in each frame of the animation.

This image isn’t that much more advanced than the animated flower, but I think the execution has vastly improved.

Whether you’ve been following these blog posts since December 2021 or are new to Maple Learn, we hope you give Maple Learn art a try.

And don’t forget that Maple is also a goldmine of artistic potential.  Maple’s bountiful collection of packages such as Fractals, ColorTools, plottools and more are great places to start for math that is as aesthetically pleasing as it is informative.

This week, our staff participated in a series of art challenges using either Maple Learn or Maple itself, each featuring a suggested theme and suggested mathematical content.  Check out the challenges and some of our employees’ entries below, and try out a challenge for yourself!


Tuesday’s Art Theme: Pasta

Mathematical Content: Shapes

Example: Lazar Paroski’s spiraling take on spaghetti


Wednesday’s Art Theme: Nature

Mathematical Content: Fractals

Example: John May’s Penrose tiling landscape (in Maple!)


Thursday’s Art Theme: Disco

Mathematical Content: Animation

Example: Paulina Chin’s disco ball (in Maple!)


Friday’s Art Theme: Space

Mathematical Content: Color

Example: that’s today!  Who knows what our staff will create…?


We hope these prompts have inspired you! If you create some art you’re really proud of, consider submitting it to be featured in the 2023 Maple Conference’s Creative Works Showcase.

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.


Screenshot of a Maple Learn document. The right side shows a paused animation of an observer, a moving car, and a moving bird. The left side shows calculations for the relative speeds.

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.


Screenshot of a Maple Learn document, showing a paused animation of two observers, one inside a moving car, and one outside. Light inside the car is moving up and down.Screenshot of a Maple Learn document, showing a paused animation of two observers, one inside a moving bus and one outside. There is light moving back and forth inside the bus.

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.