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

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.

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!

Have you ever heard of the Maurer Rose?

The Maurer Rose was demonstrated in 1987 by Peter Maurer and is created by connecting certain points on a rose curve. This creates petal-like patterns, caused by the oscillation of a sine curve.

So, how are these created? A "rose curve" is created in polar coordinates with the equation sin(nt) for a (positive integer) value of n.  To create the Maurer Rose, straight line segments are drawn connecting points on the curve at incrementing angle values.  The size of this increment (called d in our examples) leads to different patterns of lines across the curve.

This can be done in Maple Learn! One example of the Maurer Rose already exists, complete with a full interactive visualization and a more detailed overview of the Maurer Rose.

Play around with it and look below at some of the different shapes that can be created using this document! The first is created with an n value of 31 and a d value of 65, with blue and red. The second uses an n value of 4 and a d value of 133, and purple and green.

Are there any other concepts you’d like to see represented in Maple Learn’s document gallery? Please let us know in the comments below!

Have you heard of Maple Scripting before? Do you want to extend your Maple Learn documents with your Maple knowledge? Scripting is the process of using Maple to create Maple Learn documents. If you’re already used to Maple, this may be a piece of cake for you, but we wanted to start from the basics for anyone who wants to extend their Maple Learn and Maple knowledge. This process can be used for many different types of documents, from quizzes to intensive 3D visualizations.

So, let’s get started! All Maple Learn document scripting needs the DocumentTools:-Canvas package. The canvas, as you know, is that white space in a Maple Learn document. Therefore, this package is the core content of a scripted document! Always put:

with(DocumentTools:-Canvas):

At the top of your code, or put

uses DocumentTools:-Canvas:

At the start of your procedures.

Now that we’ve told Maple to use the DocumentTools:-Canvas, we need to create a canvas.

Canvases are created as variables, using the command NewCanvas. Inside NewCanvas, you will add a square-bracket list of all the content you want to see inside. For now, just know that you can add text cells with Text(“YOUR TEXT”) and a math cell with Math(YOUR MATH). On the next line, make sure to put either ShareCanvas(YOUR CANVAS VARIABLE) or ShowCanvas(YOUR CANVAS VARIABLE).  ShareCanvas creates a Maple Learn sharelink, while ShowCanvas shows the canvas directly in Maple. Note that ShowCanvas does not have every Maple Learn feature, but makes quick work of fast error checking.

canvas := NewCanvas([Text(“My first canvas”), Math(3*x+2*y)]):

ShareCanvas(canvas);

There are two more things I want to show you in this post: How to make a group have multiple cells (instead of just the one), and how to position your items on the canvas. Let’s start with group making.

To create a group with multiple cells, use the Group() command within the NewCanvas command, and separate the cells with commas, in a list. You don’t need to specify Text() or Math() when using Group().

canvas := NewCanvas([Group([“This is the first cell…”, “The second….”, “and the third.”])]):

At the end of any command/canvas element, within the brackets, you can define position=[x,y] to specify where on the canvas the object should go. You can adjust the precision pixel by pixel until you are happy with the layout.

When we put all these together, we get code that looks like this:

with(DocumentTools:-Canvas):

canvas := NewCanvas([

Group(["This is the first cell…", "The second…", "and the third."], position=[200,200]),

Math(3*x+2, position=[100,100]),

Text("This is text!", position=[400,400])]):

ShareCanvas(canvas);

And in the end, your scripted document looks like this.

We hope this helps you get started with Maple Scripting. There will be another post on even more of what we can do with Maple Scripting, and how we can make these documents even more interactive. Let us know if there’s anything specific you want to see in that post!

We've just released Maple Flow 2022.2. The update enhances the user experience in many areas, including user interaction, performance, and the interface.

Performance is a signficant focus.

• Maple Flow prioritizes the evaluation of the math you see on screen, giving you faster calculation updates for the part of the worksheet you’re working on, with more math being evaluated as you scroll down.
• We also have more users developing larger documents. Adding white space to large documents, and interacting with sections is now more response and snappier.

In response to many user requests for faster interaction, a new optional evaluation method lets you simply hit equals to evaluate math and display results.

We've also refreshed the in-product Application Gallery with a new look and many new applications (this includes a library of section properties).

You can also optionally restrict printing to the left-most column of pages, allowing you to have off-screen supporting calculations not displayed in the final report.

You'll find a complete list of enhhacements here, and you can download the update here.

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!

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.

Welcome back to another Maple Learn blog post! We know it is midterm season, and we’re here to help. Maple Learn can be used to study in many different ways, and I’m sure you’ve already tried some of them. One way is making your notes in Learn, or making your own examples, but have you taken a look at our document gallery? We have a wide range of subjects and types of documents, so let’s take a look at some documents!

I’m going to start by talking about the documents in the gallery which are content learning focused, then move into practice problems and a special document for studying.

First, let’s look at some calculus content learning documents! The calculus collection is our largest, reaching over 250 documents and still counting. The two documents I’ve picked from this category are our documents on the Fundamental Theorem of Calculus and a Visualization of Partial Derivatives. See a screenshot of the visualisations for each document below!

Are there other subjects you’d like to look at? Well, take a look at our list below!

Algebra: Double Vertical Asymptote Slider Graph

Graph Theory: Dijkstra’s Algorithm for Shortest Paths

Economics: Increase in Demand in a Market

Chemistry: Combined Gas Law Examples

Biology: Dihybrid Cross Punnett Squares

Physics: Displacement, Velocity, and Acceleration

We have many other subjects for documents, of course, but they wouldn’t fit in this post! Take a look at our entire document gallery for the others.

Another class of documents we have are the practice problems. Perfect for studying, we have practice problems ranging from practicing the four color theorem, to practicing mean, median, and mode, to even practicing dihybrid cross genotypes!

Now for, in my opinion, our most useful document for the midterm season: A study time calculator!

This document allows you to put in the amount you want to study each class over the day or week, and breaks down visually what that would look like.  

This allows you to make sure you’re taking enough time for breaks and sleep, and not overloading yourself. Feel free to customise the document to make it work better for you and your study style!

We hope you enjoyed this post, and that we could help you study! Let us know below if there’s anything else you’d want to see to support you during midterms and exams.

Have you heard the news yet? Maple Learn has had a major update! You may be wondering what this means, and what all the shiny new features are. Let’s go through them together.

First, as with many updates, we’ve improved performance with Maple Learn. Longer documents will load and perform faster, requiring less computing power for operations, and as a result your browser will be more responsive. Performance on Chromebooks is also improved.

Operations that previously would have needed to be refreshed now automatically calculate. Up until now, if you performed a menu operation on an expression and then changed the value of the expression, the result would turn orange to warn you that the result was no longer valid. You would then have to refresh manually. Now, this is no longer the case, the orange refresh button has been removed from Maple Learn, and results are never out of date.

The plot window, inline plots, and the context panel are all resizable now. This means that, for example, if you’re presenting using Maple Learn, you can enlarge the plot window to be the focus of the presentation, and shrink the context panel out of the way. Take a look at the difference, with our animation of it in action!

Sliders are also more flexible now! Bounds for sliders can be expressed in terms of variables or symbols like π. As well, you can now animate sliders, animating the graph. This allows for more interactivity in documents. See the old view on the left, and the new view on the right! Make sure to take a look at an example of the animated slider below the views as well. 

You can also now snap groups to a grid, allow them to automatically adjust their position as other groups adjust. This ensures better alignment of groups. It also allows you to easily rearrange elements of your documents.

Next, Maple Learn could handle 3D plots before, but now Maple Learn supports 3D parametric plots!

Finally, Maple Learn now has printing! This means you can print out your Maple Learn documents, with two options: to print just the canvas, or to print just the plot. This was requested by many users.

Multiple selection is also possible, allowing you to select multiple cells in a group by holding down the Ctrl/Command key while clicking and dragging.

That’s all for the updates in this version, but keep an eye out for our other updates! For more details, please take a look at our What’s New In Maple Learn page. We hope you enjoy our new features, and let us know if there are any more features you’d like to see in Maple Learn below.

Mathematical visualizations are beautiful representations of technical phenomena.  From the visual “perfection” of the golden spiral to the pattern generation of fractals, so many works of art can be boiled down to formulas and equations.  Such is the case with N.G. de Bruijn’s medallion and frieze patterns.  Given two starting values, two lines of mathematical formulae produce a recursive sequence of complex numbers.  We can associate these numbers with the four cardinal directions, following the steps on a plot to produce beautiful patterns.  The patterns are of two different types, the closed medallion or repeating frieze, depending on the starting values.

When you need a complex math visualization, Maple is a perfect place to go.  A demonstration of medallion and frieze patterns is available in the Maple Application Center, in which you can vary the starting values and watch the outcome change, along with more detailed background information.  However, there’s an even simpler way to explore this program with the help of Maple Learn.  Maple Learn has the same computational power as Maple, streamlined into an easy-to-use notebook style.  

Maple Learn includes many core features, and anything missing can be ported in through Maple.  This is done using Maple’s DocumentTools:-Canvas package.  The package contains the necessary procedures to convert Maple code into a “canvas”, which can be opened as a Maple Learn sheet.  This makes the whole document look cleaner and allows for easy sharing with friends.

The medallion and frieze document, along with the additional contextual information, is now also available in Maple Learn’s Document Gallery, home to over one thousand example documents covering calculus, geometry, physics, and more.