The Lunar New Year is approaching and 2024 is the Year of the Dragon! This inspired me to create a visualization approximating the dragon curve in Maple Learn, using Maple. 

The dragon curve, first described by physicist John Heighway, is a fractal that can be constructed by starting with a single edge and then continually performing the following iteration process:  

Starting at one endpoint of the curve, traverse the curve and build right triangles on alternating sides of each edge on the curve. Then, remove all the original edges to obtain the next iteration. 

This process continues indefinitely, so while we can’t draw the fractal perfectly, we can approximate it using a Lindenmayer system. In fact, Maple can do all the heavy lifting with the tools found in the Fractals package, which includes the LSystem subpackage to build your own Lindenmayer systems. The subpackage also contains different examples of fractals, including the dragon curve. Check out the Maple help pages here: 

Overview of the Fractals Package  

Overview of the Fractals:-LSystem Subpackage 

Using this subpackage, I created a Maple script (link) to generate a Maple Learn document (link) to visualize the earlier iterations of the approximated dragon curve. Here’s what iteration 11 looks like: 

You can also add copies of the dragon curve, displayed at different initial angles, to visualize how they can fit together. Here are four copies of the 13th iteration: 

Mathematics is full of beauty and fractals are no exception. Check out the LSystemExamples subpackage to see many more examples. 

Happy Lunar New Year! 

A ball on a turntable can move in circles instead of falling off the edge (provided the initial conditions are appropriate). The effect was demonstrated in a video and can be simulated with MapleSim. The amination below shows a simulation of a frictionless case (falling off the table) and the case with a coefficient of friction of one.

Also demonstrated in the video: Tilting the table leads to a sideward (not a downhill) movement of the ball.

The presenter of the video noted that in the untilted state, the ball eventually drifts off the table, attributing this to slippage. This drift is also observable in the animation above, where the ball starts moving in a spiral, whereas in a Maple simulation (below to the left), the ball follows a perfect circle. Only after optimizing contact and initial conditions, MapleSim produced a result (using the same parameters) that exhibits a similar circle, with a slight difference in angular orientation after completing two revolutions about the center of the circle.

Some observations on the MapleSim model:

• The results only slightly depend on the solvers. Numerical inaccuracies do not seem to be the reason for the difference in orientation. (Edit: see update below for the reason).
• The ball bounces up and down in the MapleSim simulation (0.0025 of the balls radius). The bouncing is caused by the fact that the initial position of the ball is above the elastic equilibrium position with respect to the table (the elastic contact makes the ball sink in a bit). Adding damping in the settings of the contact element attenuates this effect and reduces the drift.
• Drift is not observable anymore if in the contact element setting for "kmu" (smoothness coefficient of sliding friction) is set to larger values (above 10 in this example). This is an indication that sliding friction occurs during the simulation.
• Choosing the equilibrium position as initial condition for the ball does not prevent initial bouncing because MapleSim sets an initial velocity for the ball that is directed away from the table. I did not manage to enforce strictly zero velocity. Maybe someone can tell why that is and how to set MapleSim to start the simulation without vertical velocity. (Edit: see update below for the reason).
• Assuming an initial velocity towards the contact to cancel the initial vertical velocity set by MapleSim substantially reduces bouncing and increases the diameter of the circle. This finally leads to a diameter that matches the Maple simulation. Therefore the initial bouncing combined with slippage seems to dissipate energy which leads to smaller circles. Depending on the contact settings and initial conditions for vertical movement the diameter of the circle varied moderately by about 15%.

In summary, MapleSim can be parametrized to simulate an ideal case without slippage. From there it should be possible to tune contact parameters to closely reproduce experiments where parameters are often not well known in advance.  

Some thoughts for future enhancement of MapleSim:

• In the model presented here, a patterned ball would have been helpful to visualize the tumbling movement of the ball. Marking two opposite sides of the ball with colored smaller spheres is a fiddly exercise that does not look nice.
• A sensor component that calculates the energy of a moving rigid body would have helped identifying energy loss of the system. Ideally the component could have two ports for the rotational and translational energy components. I see professional interest in such calculations and not only educational value for toy experiments.
• A port for slippage on the contact elements would have helped understanding where slippage occurs. Where slippage is, there is wear and this is also of interest for industrial applications.

Turntable_Paradox.msim (contains parameter sets for the above observations)

Curve sketching is an important skill for all calculus students to learn. In an era where technology is increasingly relied upon to perform mathematical computations and representations, maintaining fundamental skills such as curve sketching is more important than ever.

The new “Curve Sketching” collection is now available on Maple Learn. This collection provides background information on the process of curve sketching and opportunities to put this knowledge into practice. By starting with the “Curve Sketching Guide” and “Relationships Between Derivatives” documents, students are exposed to observational and computational strategies for drawing a function and its 1st and 2nd derivatives.

After looking through these documents, students can begin to practice sketching by observing and interpreting graphical properties with the “Sketch Derivative From Function Graph”, “Sketch Second Derivative From Function Graph”, and “Sketch Function From Derivative Graphs” activities:

Once a student has mastered extracting sketching information by graphical observation, they are ready for the next step: extracting information from a function’s definition. At this point, the student is ready to try sketching from a blank canvas with the “Sketch Curve From Function Definition” activity:

This collection also has activities for students below the calculus level. For example, the “Curve Sketching Quadratics Activity”, can be completed using only factoring strategies:

Whether you are a quadratics rookie or a calculus pro, this collection has an interactive activity to challenge your knowledge. Have fun sketching!

My friend and colleague Nic Fillion and I are writing another book, this one on perturbation methods using backward error analysis (and Maple).  We have decided to make the supporting materials available by means of Jupyter notebooks with a Maple kernel (there are some Maple worksheets and workbooks already, but going forward we will use Jupyter).

The presentation style is meant to aid reproducibility, and to allow others to solve related problems by changing the scripts as needed.

The first one is up at 

https://github.com/rcorless/Perturbation-Methods-in-Maple

Comments very welcome.  This particular method is a bit advanced in theory (but it's very simple in practice, for weakly nonlinear oscillators).  I haven't coded for efficiency and there may be some improvements possible ("may" he says, sheesh).  Comments on that are also welcome.

-r

Two solstices occur on Earth every year, around June 21^st and December 21^st, often called the “June Solstice” and the “December Solstice” respectively. These solstices occur when the sun reaches its northernmost or southernmost point relative to the equator. During a solstice, the Northern Hemisphere will either experience the most sunlight of the year or the least sunlight of the year, while the Southern Hemisphere will experience the opposite phenomenon. The hemisphere with the most sunlight experiences a summer solstice, while the other hemisphere experiences a winter solstice.

Canada is located in the Northern Hemisphere and this Thursday, December 21^st, we will be experiencing a winter solstice. As the day with the least sunlight, this will be the shortest day of the year and consequently the longest night of the year.

Here in Canada, the sun will reach its minimum elevation during the winter solstice, and it will reach its maximum elevation during the Southern Hemisphere’s summer solstice on the same day. 

How high in the sky does the sun really get during these solstices? Check out our new Maple Learn document, Winter and Summer Solstice Sun Angles to find out. The answer depends on your latitude; for instance, with a latitude of approximately 43.51°, the document helps us find that the maximum midday elevation of the sun, which occurs during a summer solstice, will be 69.99°.

But how is the latitude of a location determined in the first place? See Maple Learn’s Calculating Latitude document to find out how the star Polaris, the center of the Earth, and the equator are all connected to latitude.

Latitude is one of two geographical coordinates that are paired together to specify a position on Earth, the other being longitude. See our Calculating Longitude document to explore how you can use your local time to approximate your longitude.

Armed with these coordinates, you can describe your position on the planet and find any number of interesting facts, such as your solstice sun angles from earlier, the time for sunrise and sunset, and the position of the Moon.

Happy Winter Solstice!

A new “Sudoku Puzzle” document is now on Maple Learn! Sudoku is one of the world’s most popular puzzle games and it is now ready to play on our online platform. 

This document is a great example of how Maple scripts can be used to create complex and interactive content. Using Maple’s built-in DocumentTools:-Canvas package, anyone can build and share content in Maple Learn. If you are new to scripting, a great place to start is with one of the scripting templates, which are accessible through the Build Interactive Content help page. In fact, I built the Sudoku document script by starting from the “Clickable Plots” template.

A Sudoku puzzle is a special type of Latin Square. The concept of a Latin Square was introduced to the mathematical community by Leonard Euler in his 1782 paper, Recherches sur une nouvelle espèce de Quarrés, which translates to “Research on a new type of square”. A Latin Square is an n by n square array composed of n symbols, each repeated exactly once in every row and column. The Sudoku board is a Latin Square where n=9, the symbols are the digits from 1 to 9,  and there is an additional requirement that each 3 by 3 subgrid contains each digit exactly once. 

Mathematical research into Sudoku puzzles is still ongoing. While the theory about Latin Squares is extensive, the additional parameters and relative novelty of Sudoku means that there are still many open questions about the puzzle. For example, a 2023 paper from Peter Dukes and Kate Nimegeers examines Sudoku boards through the lenses of graph theory and linear algebra.

The modern game of Sudoku was created by a 74-year-old Indiana retiree named Howard Garnes in 1979 and published under the name “Number Place”. The game had gained popularity in Japan by the mid-1980s, where it was named “Sudoku,” an abbreviation of the phrase “Sūji wa dokushin ni kagiru,” which means “the numbers must be single”.

Today, Sudoku is a worldwide phenomenon. This number puzzle helps players practice using their logical reasoning, short-term memory, time management, and decision-making skills, all while having fun. Furthermore, research from the International Journal of Geriatric Psychiatry concluded that doing regular brain exercises, like solving a Sudoku, is correlated with better brain health for adults over 50 years old. Additionally, research published in the BMJ medical journal suggests that playing Sudoku can help your brain build and maintain cognition, meaning that mental decline due to degenerative conditions like Alzheimer’s would begin from a better initial state, and potentially delay severe symptoms. However, playing Sudoku will by no means cure or prevent such conditions.

If you are unfamiliar with the game of Sudoku, need a refresher on the rules, or want to improve your approach, the “Sudoku Rules and Strategies” document is the perfect place to start. This document will teach you essential strategies like Cross Hatching:

And Hidden Pairs:

After reading through this document, you will have all the tools you need to start solving puzzles with the “Sudoku Puzzle” document on Maple Learn. 

Have fun solving!

How much did you weigh when you were born? How tall are you? What is your current blood pressure? It is well documented that in the general population, these variables – birth weight, height, and blood pressure – are normally or approximately normally distributed. This is the case for many variables in the natural and social sciences, which makes the normal distribution a key distribution used in research and experiments. 

The Maple Learn Examples Gallery now includes a series of documents about normal distributions and related topics in the Continuous Probability Distributions subcollection.

The Normal Distribution: Overview will introduce you to the probability density function, cumulative distribution function, and the parameters of the distribution. This document also provides an opportunity for you to alter the parameters of a normal distribution and observe the resulting graphs. Try out a few real life examples to see the graphs of their distributions! For example, according to Statology, diastolic blood pressure for men is normally distributed with a mean of 80 mmHg and a standard deviation of 20 mmHg.

Next, the Normal Distribution: Empirical Rule document introduces the empirical rule, also referred to as the 68-95-99.7 rule, which describes approximately what percentage of normally distributed data lies within one, two, and three standard deviations of the distribution’s mean.

The empirical rule is frequently used to assess whether a set of data might fit a normal distribution, so Maple Learn also provides a Model Checking Exploration to help you familiarize yourself with applications of this rule. 

In this exploration, you will work through a series of questions about various statistics from the data – the mean, standard deviation, and specific intervals – before you are asked to decide if the data could have come from a normal distribution. Throughout this investigation, you will use the intuition built from exploring the Normal Distribution: Overview and Normal Distribution: Empirical Rule documents as you analyze different data sets.

Once you are confident in using the empirical rule and working with normal distributions, you can conduct your own model checking investigations in real life. Perhaps a set of quiz grades or the weights of apples available at a grocery store might follow a normal distribution – it’s up to you to find out!

A new feature has been released on Maple Learn called “collapsible sections”! This feature allows for users to hide content within sections on the canvas. You can create a section by highlighting the desired text and clicking this icon in the top toolbar: . 

“Well, when can I actually use sections?” you may ask. Let me walk you through two quick scenarios so you can get an idea.

For our first scenario, let’s say you’re an instructor. You just finished a lesson on the derivatives of trigonometric functions and you’re now going through practice problems. The question itself is not long enough to hide the answers, so you’re wondering how you can cover the two solutions below so that the students can try out the problem themselves first.

Before, you might have considered hyperlinking a solution document or placing the solution lower down on the page. But now, collapsible sections have come to the rescue! Here’s how the document looks like now:  

You can see that the solutions are now hidden, although the section title still indicates which solution it belongs to. Now, you can 1) keep both solutions hidden, 2) show one solution at a time, or 3) show solutions side-by side and compare them!

Now for the second scenario, imagine you’re making a document which includes a detailed visualization such as in Johnson and Jackson’s proof of the Pythagorean theorem. You want the focus to be on the proof, not the visualizations commands that come along with the proof. What do you do?

It’s an easy solution now that collapsible sections are available!

Now, you can focus on the proof without being distracted by other information—although the visualization commands can still be accessed by expanding the section again.

So, take inspiration and use sections to your advantage! We will be doing so as well. you may gradually notice some changes in existing documents in the Maple Learn Gallery as we update them to use collapsible sections. 

Happy document-making!

A new collection has been released on Maple Learn! The new Pascal’s Triangle Collection allows students of all levels to explore this simple, yet widely applicable array.

Though the binomial coefficient triangle is often referred to as Pascal’s Triangle after the 17th-century mathematician Blaise Pascal, the first drawings of the triangle are much older. This makes assigning credit for the creation of the triangle to a single mathematician all but impossible.

Persian mathematicians like Al-Karaji were familiar with the triangular array as early as the 10th century. In the 11th century, Omar Khayyam studied the triangle and popularised its use throughout the Arab world, which is why it is known as “Khayyam’s Triangle” in the region. Meanwhile in China, mathematician Jia Xian drew the triangle to 9 rows, using rod numerals. Two centuries later, in the 13th century, Yang Hui introduced the triangle to greater Chinese society as “Yang Hui’s Triangle”. In Europe, various mathematicians published representations of the triangle between the 13th and 16th centuries, one of which being Niccolo Fontana Tartaglia, who propagated the triangle in Italy, where it is known as “Tartaglia’s Triangle”. 

Blaise Pascal had no association with the triangle until years after his 1662 death, when his book, Treatise on Arithmetical Triangle, which compiled various results about the triangle, was published. In fact, the triangle was not named after Pascal until several decades later, when it was dubbed so by Pierre Remond de Montmort in 1703.

The Maple Learn collection provides opportunities for students to discover the construction, properties, and applications of Pascal’s Triangle. Furthermore, students can use the triangle to detect patterns and deduce identities like Pascal’s Rule and The Binomial Symmetry Rule. For example, did you know that colour-coding the even and odd numbers in Pascal’s Triangle reveals an approximation of Sierpinski’s Fractal Triangle?

See Pascal’s Triangle and Fractals

Or that taking the sum of the diagonals in Pascal's Triangle produces the Fibonacci Sequence?

See Pascal’s Triangle and the Fibonacci Sequence

Learn more about these properties and discover others with the Pascal’s Triangle Collection on Maple Learn. Once you are confident in your knowledge of Pascal’s Triangle, test your skills with the interactive Pascal’s Triangle Activity. 

Many everyday decisions are made using the results of coin flips and die rolls, or of similar probabilistic events. Though we would like to assume that a fair coin is being used to decide who takes the trash out or if our favorite soccer team takes possession of the ball first, it is impossible to know if the coin is weighted from a single trial.

Instead, we can perform an experiment like the one outlined in Hypothesis Testing: Doctored Coin. This is a walkthrough document for testing if a coin is fair, or if it has been doctored to favor a certain outcome. 

This hypothesis testing document comes from Maple Learn’s new Estimating collection, which contains several documents, authored by Michael Barnett, that help build an understanding of how to estimate the probability of an event occurring, even when the true probability is unknown.

One of the activities in this collection is the Likelihood Functions - Experiment document, which builds an intuitive understanding of likelihood functions. This document provides sets of observed data from binomial distributions and asks that you guess the probability of success associated with the random variable, giving feedback based on your answer. 

Once you’ve developed an understanding of likelihood functions, the next step in determining if a coin is biased is the Maximum Likelihood Estimate Example – Coin Flip activity. In this document, you can run as many randomized trials of coin flips as you like and see how the maximum likelihood estimate, or MLE, changes, bearing in mind that if a coin is fair, the probability of either heads or tails should be 0.5. 

Finally, in order to determine in earnest if a coin has been doctored to favor one side over the other, a hypothesis test must be performed. This is a process in which you test any data that you have against the null hypothesis that the coin is fair and determine the p-value of your data, which will help you form your conclusion.

This Hypothesis Testing: Doctored Coin document is a walkthrough of a hypothesis test for a potentially biased coin. You can run a number of trials on this coin, determine the null and alternative hypotheses of your test, and find the test statistic for your data, all using your understanding of the concepts of likelihood functions and MLEs. The document will then guide you through the process of determining your p-value and what this means for your conclusion.

So if you’re having suspicions that a coin is biased or that a die is weighted, check out Maple Learn’s Estimating collection and its activities to help with your investigation!

The Maple Conference starts tomorrow Oct. 26 at 9am EDT! It's not too late to register: https://www.maplesoft.com/mapleconference/2023/. Even if you can't attend all the presentations, registration will allow you to view the recorded videos after the conference. 

Check out the detailed conference program here: https://www.maplesoft.com/mapleconference/2023/full-program.aspx

With Halloween right around the corner, we at Maplesoft wanted to celebrate the occasion with an activity where you can carve your own pumpkin… using math! 

Halloween is said to have originated a few hundred years back in ancient Celtic festivals, specifically one called Samhain. This was celebrated from October 31st to November 1st to mark the end of harvesting season and the beginning of winter, or the "darker quarter" of the year. Since then, Halloween has evolved into a fun celebration of candy and costumes in many countries!

With that said, here’s my take on the pumpkin carving activity: 

The great thing is, if you mess up, you can always go back; unlike carving pumpkins in real life. My design is pretty simple (although cute), so let’s see what you all can impress us with!

You can also make your own original art and publish it to your channel so that anyone can see your own artistic creations. You can also attend the Maple Conference next week on October 26 and 27, an event filled with two days of presentations from members of the Maplesoft Community. Participants will also be able to see all the artwork submitted for the Art Gallery and Creative Showcase, where you can draw inspiration for your own submissions to next year’s showcase! The conference is virtual and free of charge, and you can register here.

Looking forward to seeing you there!

The Maple Conference will be starting in two weeks! The detailed agenda, which includes abstracts of invited and contributed talks, is available here: https://www.maplesoft.com/mapleconference/2023/full-program.aspx.

Please join us on October 26 and 27 for two days of presentations from our staff members and the larger Maple community, a look at our Art Gallery and Creative Showcase, opportunities for networking with other Maple enthusiasts, and more! The conference is virtual and free of charge, and you can register at https://www.maplesoft.com/mapleconference/2023/.

We look forward to seeing you at the conference!

Almost 300 years ago, a single letter exchanged between two brilliant minds gave rise to one of the most enduring mysteries in the world of number theory. 

In 1742, Christian Goldbach penned a letter to fellow mathematician Leonhard Euler proposing that every even integer greater than 2 can be written as a sum of two prime numbers. This statement is now known as Goldbach’s Conjecture (it is considered a conjecture, and not a theorem because it is unproven). While neither of these esteemed mathematicians could furnish a formal proof, they shared a conviction that this conjecture held the promise of being a "completely certain theorem." The following image demonstrates how prime numbers add to all even numbers up to 50:

From its inception, Goldbach's Conjecture has enticed generations of mathematicians to seek evidence of its legitimacy. Though weaker versions of the conjecture have been proved, the definitive proof of the original conjecture has remained elusive. There was even once a one-million dollar cash prize set to be awarded to anyone who could provide a valid proof, though the offer has now elapsed. While a heuristic argument, which relies on the probability distribution of prime numbers, offers insight into the conjecture's likelihood of validity, it falls short of providing an ironclad guarantee of its truth.

The advent of modern computing has emerged as a beacon of progress. With vast computational power at their disposal, contemporary mathematicians like Dr. Tomàs Oliveira e Silva have achieved a remarkable feat—verification of the conjecture for every even number up to an astonishing 4 quintillion, a number with 18 zeroes.

Lazar Paroski’s Goldbach Conjecture Document on Maple Learn offers an avenue for users of all skill levels to delve into one of the oldest open problems in the world of math. By simply opening this document and inputting an even number, a Maple algorithm will swiftly reveal Goldbach’s partition (the pair of primes that add to your number), or if you’re lucky it could reveal that you have found a number that disproves the conjecture once and for all.

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?