Have you ever wondered who the students are that help create Maplesoft’s family of products?

In this blog, we thought that it was fitting to introduce ourselves and give the MaplePrimes community some insight into the students who are committed to helping Maplesoft improve its products and who believe that Math Matters!

I’ll begin. My name is Jack Thomson and I’m in my second year of the Mathematics (Waterloo) and Business Administration (Laurier) Double Degree. This term I am the Product Management Co-op at Maplesoft where I will be helping support the development of Maplesoft's academic market products, including Maple Learn and Maple Calculator. My favorite areas of math are statistics and probability. These areas are my favorite since I like to be able to draw conclusions from data and predict the future with past trends. I am also fascinated with probability since it allows us to make more educated decisions about real-life events. This ties into my belief of why Math Matters, since it is hidden in every aspect of life and helps us understand the world around us. Besides my love for the world of mathematics, I love the outdoors, more specifically, mountain biking, backcountry camping, and skiing. I also enjoy taking photos, watching Formula 1, playing hockey, and improving my skills in the kitchen.

Continue reading below to find out more about my fellow Co-op students!

Development:

I’m Zhengmao (he/him), and I’m a third year in Software Engineering at the University of Waterloo. I’ll be working until the end of April here at Maplesoft as a Software Developer, where I’ll be working to fix bugs, add new features, and improve existing ones for our Maple Learn as well as Maple Calculator products. By the way, if you ever have any suggestions or ideas about them, don’t hesitate to reach out to me!

I’ve always been curious about working at a math company because I’ve always been so interested in math. In fact, Maplesoft is the only company I’ve consistently applied to every time I’ve gone through the Co-op application cycle! However, there’s not really any particular reason why I enjoy the subject. I find math to be beautiful in and of itself, almost like an art, and I find the kinds of math that are more discrete or algebraic tend to be a little nicer. As long as there aren’t decimals, I’m pretty happy. So, my ideal kind of math is just that: ideal! Exact values, unrealistic ideas, and as few numbers as possible. In terms of my university career, I’ve always enjoyed linear algebra much more than calculus.

Overall, I’m quite excited for this term at Maplesoft. I’ve never worked in web or mobile development before, so I’m looking forward to learning a lot of new things!

Content Creation:

Hi, I’m Paige (she/her). I am a second-year Honors Mathematics student at the University of Waterloo. This term, I am creating content for the Maple Learn document gallery. My favorite area of math is calculus because I love visualizing functions. Math matters because it is a universal language. All the math concepts we know are naturally occurring; people have observed and documented them, but no one invented them. Because of this, people from a wide range of cultures have come to the same conclusions (ex: defining pi). Math is universally understandable, which is why it can be used to connect everyone on earth (and maybe on other planets too!?!?!?!). In my free time, I like doing hand embroidery, playing video games, and cuddling my cat Licky.

My name is Laura (she/her) and I’m a second year in the math program at the University of Waterloo. This term, I am working as a ‘Math Content Developer’ at Maplesoft; I’ll be creating and scripting documents for Maple Learn’s Example Gallery, updating older content, and handling customer requests. My favorite areas of math are probability, since I find questions like the Birthday Problem interesting, and biostatistics because I enjoy learning about biology and how biological experiments can be analyzed mathematically. I believe math matters because mathematics is essential to sending equipment and people into outer space; we will never meet aliens without using math.

Quality Assurance:

Hey, my name is Stefan, I'm 19 years old and currently studying Biochemistry in second year at the University of Waterloo. I am a QA analyst here at Maplesoft, working on Maple 2023. Outside of school, a hobby of mine is making digital art. My favorite area of math is definitely calculus & analysis because I found learning the fundamental theorem really intuitive and engaging. Expanding on that, I believe math matters because of its many applications in other fields such as the use of calculus in the research and design aspects of Biochemistry.

Hey! My name is Steven Mou, I'm in CFM at UWaterloo and I'm going into my 2B term after this term. I'm one of the four QA Analyst interns and I'll be testing all things related to Maple. My favorite area of math is anything related to algebra. I just find being able to manipulate variables while maintaining the integrity of the final product, to be very fascinating. I believe math matters because our lives are pretty much completely founded by math; anything from the technology that we use to the logic that is the foundation of our thoughts. I like playing sports, dabbling with different recipes in the kitchen, and discussing any shows that I happen to always finish too quickly.

Hi, my name is Aidan and I'm a 3rd-year mathematical physics student working as a quality assurance analyst for this Co-op term. As a Co-op student working in QA, I will mostly be running tests and reporting bugs to help ensure that Maple 2023 as well as Maple Flow are ready for release. My favorite area of math is vector calculus because as I started learning it I found it very interesting in the ways it applies to things we use in our everyday life. It also combines Linear Algebra and Calculus in a way that I never would have expected before learning about it. I think that math matters because I feel as though everything you can interact with can be described and predicted mathematically and that amazes me.

Hello! My name is Sebastian, I am currently in my second year of physics and astronomy at UWaterloo, and for this Co-op, at Maplesoft, I am working as a quality assurance analyst. In this position, I will be performing tests on Maple and Maple Flow to ensure that when they are released they function as they should and are ready for consumers to use with ease. When I am not focused on my academics I enjoy spending my time playing soccer (also watching it), listening to music, and watching movies. My favorite area of math is calculus because of the interesting and complex problems it provides, and because it is an essential tool needed to understand how the universe works. I believe that math matters because, as teachers always remind us, it is all around us. Math provides the foundation for everything we know and have come to appreciate in our lives, so since it is seen so often in our lives, I believe we should put in the effort to understand it and grasp how cool it is.

When introduced to geometry, one of the first things we learn is the definition of the word “polygon”. A polygon is a closed 2-dimensional shape with at least 3 straight sides and angles. A regular polygon is a polygon with congruent sides and equal angles. A regular polygon with n sides has Schläfli symbol {n}. I’m interested in mathematical history, so when I learned that the idea of higher-dimensional spaces was invented in the middle of the nineteenth century I decided to research more about Ludwig Schläfli and the notation he came up with to describe his ideas.

In general, the Schläfli symbol is a notation of the form {p, q, r, ...} for regular polytopes. Polytopes are geometric objects with flat sides. This week, I will be focusing on 3-dimensional polytopes, also called polyhedra.

Similar to regular polygons, regular polyhedra are 3-dimensional shapes whose faces are all the same regular polygon. A regular polyhedron’s Schläfli symbol is of the form {p, q}, where p is the number of edges each face has and q is the number of faces that meet at each of the polyhedron’s vertices.

Below are two regular polyhedra: a cube (also known as a hexahedron) and a great stellated dodecahedron. The cube is one of five Platonic solids, and the great stellated dodecahedron is one of four Kepler-Poinsot polyhedra – all of these can be represented by Schläfli symbols. The cube has Schläfli symbol {4, 3}, since squares have 4 equal sides, and each vertex of a cube is created by the vertices of 3 squares meeting.

Can you figure out the Schläfli symbol for the great stellated dodecahedron?

The great stellated dodecahedron has the Schläfli symbol {5/2, 3}. This is because great stellated dodecahedrons are regular star polygons. As a result, the first number in their Schläfli symbol is an irreducible fraction whose numerator represents a number of sides and whose denominator corresponds to a turning number. The particular fraction 5/2 corresponds to a pentagram – a regular star polygon with 5 points – and great stellated dodecahedrons are composed of 12 of these pentagrams, where 3 pentagrams meet at each vertex of the shape.

One notable example of a regular polytope in pop culture is the tesseract, which has the Schläfli symbol {4, 3, 3}. This is an extension of the cube’s Schläfli symbol, {4, 3}, and the last number indicates that there are three cubes folded together around every edge. Below are two representations of a tesseract: one that uses a Schlegel diagram (left) and one from the 2012 movie Avengers (right).

Try out our Regular Polyhedra Visualization Using Schlafli Symbol Notation! In this document, you can test out your own Schläfli symbols for regular polyhedra. If they are valid Schläfli symbols, you’ll be provided with a 3-D visualization of the shape. If they are invalid, you can check out the logic for finding the specifications for regular polyhedra and this document, which provides all the 3-D regular polyhedra for you to try out.

Happy Lunar New Year to everyone in the MaplePrimes community, as we enter the Year of the Rabbit. The rabbit symbolizes longevity, positivity, auspiciousness, wittiness, cautiousness, cleverness, deftness and self-protection!

To celebrate, one of our Maple Learn content developers, Laura Layton, made a Lunar New Year Color by Number:

In this puzzle, your goal is to simplify the modulo equations in each square, and then fill in the square with the color that corresponds to the answer.

I hope you have fun solving this puzzle and revealing the hidden images and I wish everyone good health and happiness in the coming year!

Last week, one of our Maple Learn developers, Valerie McKay-Crites, published a Maple Learn document, based on the very popular Maple application by Highschool Teacher, Jason Schattman called "Just Move It Over There, Dear!".

In the Maple application, Schattman explains the math behind moving a rectangular sofa down a hallway with a 90-degree turn. In the 3D Moving Sofa Problem Estimate, Valerie uses Schattman’s math to determine the largest rectangular sofa that can be taken down a flight of stairs and down a hallway with a 90-degree turn. Both applications reminded me of how interesting the Moving Sofa Problem is, which inspired me to write a blog post about it!

If you’ve ever been tasked with moving a rectangular sofa around a 90-degree turn, you might wonder:

What is the largest sofa that can make the move?

Following these steps as outlined in Schattman’s "Just Move It Over There, Dear!", will guarantee that the sofa will make the turn:

1. Measure the width of the hallway (h)
2. Measure the length (L) and width (w) of the sofa.
3. If L + 2w is comfortably less than triple the width of the hall, you'll make it!

When we work out the math exactly, we see that if the sofa's length plus double its width is less than , the sofa will make the turn!

This problem is easy if we only consider rectangular sofas, however, the problem becomes significantly more complex if we consider sofas of different shapes and areas. In mathematics, this problem is known as the Moving Sofa Problem, and it is unsolved. If we look at a hallway with a 90-degree turn and legs of width 1 m (i.e. h = 1 above), the largest known sofa that can make the turn is Gerver’s Sofa which has an area of 2.2195 m², this area is known as the Sofa Constant. Gerver’s Sofa, created in 1992, was constructed with 18 curve sections:

Check out this GIF of the sofa moving through the turn. It provides some insight into why Gerver’s sofa is such an interesting shape:

What is fascinating is that no mathematician has yet to prove that Gerver’s sofa is the sofa with the largest area capable of making the 90-degree turn.

The Moving Sofa Problem, is a great example of how math is embedded in our everyday lives. So, don’t stop being curious about the math around you as it can be fascinating and sometimes unproven!

If you are curious to learn more about the moving sofa problem check out this video by Numberphile, featuring Dan Romik from UC Davis: https://www.youtube.com/watch?v=rXfKWIZQIo4&t=1s

This is about functionality introduced in Maple 2022, which however is still not well known: Integral Vector Calculus and parametrization using symbolic (algebraic) vector notation. Four new commands were added to the Physics:-Vectors package, implementing the parametrization of curves, surfaces and volumes, as well as the computation of path, surface and volume vector integrals. Those are integrals where the integrand is a scalar or vector function. The computation is done from any description (algebraic, parametric, vectorial) of the region of integration - a path, surface or volume.
 
There are three kinds of line or path integrals:

NOTE Jan 1: Updated the worksheet linked below; it runs in Maple 2022.
Download Integral_Vector_Calculus_and_Parametrization.mw

Edgardo S. Cheb-Terrab
Physics, Differential Equations and Mathematical Functions, Maplesoft

With the winter solstice speeding towards us, we thought we’d create some winter themed documents. Now that they’re here, it’s time to show you all! You’ll see two new puzzle documents in this post, along with three informative documents, so keep reading.

Let’s start with the tromino tree!

First, what’s a tromino? A tromino is a shape made from three equal sized squares, connected to the next along one full edge. In this puzzle, your goal is to take the trominos, and try to fill the Christmas tree shape.

There’s a smaller and larger tree shape, for different difficulties. Try and see how many ways you can fill the trees!

Next, we’ll look at our merry modulo color by numbers.

In this puzzle, your goal is to solve the modulo problems in each square, and then fill in the square with the color that corresponds to the answer. Have fun solving the puzzle and seeing what the image is in the end!

Snowballs are a quintessential part of any winter season, and we’ve got two documents featuring them.

The first document uses a snowball rolling down a hill to illustrate a problem using differential equations. Disclaimer: The model is not intended to be realistic and is simplified for ease of illustration. This document features a unique visualization you shouldn’t miss!

Our second document featuring snowballs talks about finding the area of a 2-dimensional snowman! Using the formula for the area of a circle and a scale factor, the document walks through finding the area in a clear manner, with a cute snowman illustration to match!

The final document in this mini-series looks at Koch snowflakes, a type of fractal. This document walks you through the steps to create an iteration of the Koch snowflake and contains an interactive diagram to check your drawings with!

I hope you’ve enjoyed taking a look at our winter documents! Please let us know if there’s any other documents you’d like to see featured or created.

This command should have been in Physics on day one. Being more familiar with functional differentiation, and Physics:-Fundiff was the first Physics command that ever existed, I postponed writing LagrangeEquations year after year. In general, however, functional differentiation is seen as a more advanced topic. So there is now a new command, Physics:-LagrangeEquations, taking advantage of functional differentiation on background, and distributed for everybody using Maple 2022.2 within the Maplesoft Physics Updates. This is the first version of its help page.

Download LagrangeEquations.mw

Edgardo S. Cheb-Terrab
Physics, Differential Equations and Mathematical Functions, Maplesoft

Welcome back to another Maple Learn blog post! Today we’re going to talk about the gift-wrapping algorithm, used to find the convex hull of a set of points. If you’re not sure what that means yet, don’t worry! We’re going to go through it with four Maple Learn documents; two which are background information on the topic, one that is a visualization for the gift-wrapping algorithm, and another that goes through the steps. Each will be under their own heading, so feel free to skip ahead to your skill level!

Before we can get into the gift-wrapping algorithm we need to define a few terms. Let’s start by defining polygons and simple polygons.

Polygon: A closed shape created by joining a series of line segments.

Simple polygon: A polygon without holes and that does not intersect itself.

So, what are convex and concave polygons? Well, there are three criteria that define a convex polygon. A polygon that is not convex is called concave. The criteria are…

1. Any line segment connecting any two points within the polygon stays within the polygon.
2. Any line intersects a polygon’s boundary at most twice.
3. All interior angles are less than 180 degrees or pi radians.

Because the criteria are equivalent, if any one is missing, the shape is concave. AKA, all three criteria must be present for a shape to be convex. Most “regular shapes”, such as trapezoids, are convex polygons!

A shape that satisfies convex criteria but not the criteria for being a polygon is called a convex set.

As mentioned at the start of this post, the gift-wrapping algorithm is used to find the convex hull of a set of points. Now that we know what convex polygons and convex sets are, we can define the convex hull!

Convex hull: The convex set of a shape or several shapes that fully contains the object and has the smallest possible area.

Why was the convex polygon important? Well, the convex hull of a set of points is always a convex polygon. Some of the points in the set are the vertices of said polygon, and are called extreme points. You can find the convex hull of either concave or convex polygons.

This document amazed me when I tried it for the first time. Here, you can generate a set of points with the “Generate Another” button, and then press the “Visualize” button. The document then calculates the perimeter of the convex hull of the set of points! The set can be further customized below the buttons, by changing the number of points. The other option below it allows you to slow down or speed up the visualization. Pretty cool, huh? It’s like it’s thinking!

Try the document out a few times, or watch the gif below to get a quick idea of it.

This final document walks you through the steps of how to use the gift wrapping algorithm. It is a simple loop of 4 steps, with one set-up step. Unlike the other documents in this post, I won’t be delving too far into the math behind the steps. I want to encourage you to check this one out yourself, as it’s really quite a fun problem to solve once you have some time!

I hope you check out the documents in this post. Please let us know below if there’s any other documents you’d like to see featured!

A failing slinky is another intriguing physics phenome that can be easily reproduced with MapleSim.

The bottom of a vertically suspended slinky does not move when the top is released until the slinky is fully collapsed.

To model this realistically in MapleSim, it is necessary to

• Establish a stretched equilibrium state at the start of the fall
• Avoid penetration of windings when windings collapse (i.e. get into contact)

The equilibrium state is achieved with the snapshot option. Penetration is avoided with the Elasto Gap component. Details can be found in the attached model.

A good overview of “Slinky research” is given here. The paper provides a continuous description of the collapse process (using an inhomogenous wave equation combined with contact modeling!!!) and introduces a finite time for the collapse of all windings. Results for a slinky are presented that collapses after 0.27s. The attached model has sufficient fidelity to collapse at the same time.

Real Slinkies also feature a torsional wave that precedes the compression wave and disturbs an ideal collapse. This can be seen on slowmo footage and advanced computer models. With a torsion spring constant at hand (are there formulas for coil springs?), it could also be modeled with MapleSim.

Falling_slinky.msim

An example of uniform motion along a generalized coordinate using the Draghilev method. (This post was inspired by school example in one of the forums.)
The equations used in the program are very simple and, I think, do not require any special comments. DM is a procedure that implements the Draghilev method with "partial parameterization".
DM_V.mw

When K = 1, parameterization is carried out by changing the angle of rotation of the wheel. That is, uniform rolling is carried out.

For K = 4, the coordinate corresponding to the position of the slider is parametrized.

When K = 6, the slider moves with acceleration, according to a given equation. Hence, we have carried out the parameterization with respect to “time”.

With the help of such techniques, we can obtain the calculation of the kinematics of both lever mechanisms and various types of manipulators.

Greetings, fellow educators, researchers, engineers, students, and folx who love mathematics! 

I believe in the importance of mathematics as a structure to our society, as a gateway to better financial decision making, and as a crucial subject to teach problem solving. I also believe in the success of all students, through self-discovery and creativity, while working with others to create their own knowledge. Consequently, I’ve designed my examples in the Maple Learn gallery to suit these needs. Many of my documents are meant to be “stand-alone” investigations, summary pages, or real-world applications of mathematical concepts meant to captivate the interest of students in using mathematics beyond the basic textbook work most curricula entail. Thus, I believe in the reciprocal teaching and learning relationship, through the independence and creativity that technology has afforded us. The following is an example of roller coaster track creation using functions. Split into a five part investigation, students are tasked to design the next roller coaster in a theme park, while keeping in mind the elements of safety, feasibility, and of course fun!

Common elements we take for granted such as having a starting and ending platform that is the same height (since most coasters begin and end at the same location), boarding the coaster on a flat surface, and smooth connections between curves translate into modeling with functions. 

Aside from interning with Maplesoft, I am an educator, researcher, student, financial educator, and above all, someone who just loves mathematics and wishes to share that joy with the whole world. As a practicing secondary mathematics and science teacher in Ontario, Canada, I have the privilege of taking what I learned in my doctorate studies and applying it to my classrooms on a daily basis. I gave this assignment to my students and they really enjoyed creating their coasters as it finally gave them a reason to learn why transformations of quadratics, amongst other functions, were important to learn, and where a “real life” application of a piecewise function could be used. 

Having worked with the Ontario and International Baccalaureate mathematics curricula for over a decade, I have seen its evolution over time and in particular, what concepts students struggled to understand, and apply them to the “real world.” Concurrently, working with international mathematics curricula as part of my collaboration with Maplesoft, I have also seen trends and emergent patterns as many countries’ curricula have evolved to incorporate more mathematical literacy along with competencies and skills. In my future posts, you will see Maple Learn examples on financial literacy since working as a financial educator has allowed me to see just how ill prepared families are towards their retirement and how we can get lost amongst a plethora of options provided by mass media. Hence, I have 2 main goals I dedicate to a lifelong learning experience; financial literacy and greater comprehension of mathematics topics in the classroom. 

Welcome back to another Maplesoft blog post! Today, we’re looking at how math appears in nature. Many people know that there’s math within the mysteries of nature, but don’t know exactly what’s going on. Today we’ll talk about some of the examples but remember that there’s always more.

Let’s start with a well-known example: The Fibonacci sequence! This is a recursive sequence, made by adding the previous two terms together to make the next term. The Fibonacci sequence starts with 0, then 1. So, when modelling this sequence, you get “0, 1, 1, 2, 3, 5, 8,” and so on.

Now, where can this sequence be seen? Well, the sequence forms a spiral. This spiral can be seen in fingerprints:

Image: Andrea Greengard/Mindful Living Network

Eggs:

Image: Andrea Greengard/Mindful Living Network

And, in some cases, spiral galaxies. For more examples of the Fibonacci sequence, check out a blog on examples of the Fibonacci Sequence by Andrea Greengard!

Image: Andrea Greengard/Mindful Living Network

Another interesting intergalactic math fact is that celestial bodies are typically spherical, such as stars and planets. As well, orbits tend towards spherical, often being ellipses. It’s fascinating to see how many spheres there are in nature!

Moving away from spirals in nature, another example of math in nature, although there are many more, is the Hardy-Weinburg Equilibrium.  When in Hardy-Weinburg Equilibrium, a population’s allele and genotype frequencies, in the absence of certain evolutionary factors, stay constant through generations. The Hardy-Weinburg Equilibrium is used to predict genotypes from phenotypes of certain populations, as one example. Come check out our documents on this topic for more details, both on the Hardy-Weinburg Equilibrium and some practice examples.

Image: Maplesoft

In the end, math is incredibly ingrained in nature. We can use mathematical formulas and patterns to predict how plants will grow, or population genetics, and much more! Please let us know if there’s any examples you’d like to see in more depth, and we can see if writing a blog post on it is possible, or even a Maple Learn document for the gallery!

Download Vectorial_ODEs_and_integration_constants.mw

Edgardo S. Cheb-Terrab
Physics, Differential Equations and Mathematical Functions, Maplesoft

The first day of Maple Conference 2022 is coming up on November 2 and it's not too late to register! Please go to our conference home page and click on the "Register Now" button. This is a free virtual event open to all.

The schedule is available on the conference agenda page.

Come join us to see recent developments in research, education and applications, find out about new and upcoming features in our products, talk to Maplesoft staff and other members of the Maple community and view (and vote on) Maple and Maple Learn artwork.

We hope to see you at the conference!

Physics is a very diverse field with a vast array of different branches to focus on. One of the most interesting areas of physics is optics - the study of light.

It's common to think of light as some super-fast form of matter that just bounces around at 300,000 km/s and never slows down. However, light can actually slow down when it moves through different substances. Imagine dropping a baseball from the air into a deep pool of water. It would slow down, right? Well, what happens for light isn't too different.

We call the air or the water in the previous example 'mediums' (or media). Light moves through each of these mediums differently. For example, light moves close to the speed of light in vacuum, 299 792 458 m/s, in air, but it moves considerably slower in water, closer to 225 000 000 m/s. Take a look at Indices of Refraction for more details on how we can quantify this change in speed and Dispersion for some information on the role that the wavelength of light plays.

So light slows down when it enters a medium with a higher refractive index. It also speeds up when it moves from a higher refractive index to a lower one. But did you know that it also bends? Unlike in the example of the baseball falling into the pool, light that changes speeds moving between mediums will also change direction.

Snell's Law is our way of determining how much light bends between mediums. Try playing around with the values of the indices of refraction and the incident angle and see what effect that has on the refracted ray. Is there a combination of parameters for which the refracted ray disappears? The answer can be found in Critical Angle and Total Internal Reflection.

Want to learn about how principles from optics can be applied in the real world? See Fiber Optics - Main Page for information on one of optics' most impactful applications.