Happy Easter to all those who celebrate! One common tradition this time of year is decorating Easter eggs. So, we’ve decided to take this opportunity to create some egg-related math content in Maple Learn. This year, a blog post by Tony Finch inspired us to create a walkthrough exploring the four-point egg. The four-point egg is a method to construct an egg-shaped graph using just a compass and a ruler, or in this case, Maple Learn. Here's the final product:

The Maple Learn document, found here, walks through the steps. In general, each part of the egg is an arc corresponding to part of a circle centred around one of the points generated in this construction.

For instance, starting with the unit circle and the three red points in the image below, the blue circle is centred at the bottom point such that it intersects with the top of the unit circle, at (0,1). The perpendicular lines were constructed using the three red points, such that they intersect at the bottom point and pass through opposite side points, either (-1,0) or (1,0). Then, the base of the egg is constructed by tracing an arc along the bottom of the blue circle, between the perpendicular lines, shown in red below.

Check out the rest of the steps in the Maple Learn Document. Also, be sure to check out other egg-related Maple Learn documents including John May’s Egg Formulas, illustrating other ways to represent egg-shaped curves with mathematics, and Paige Stone’s Easter Egg Art, to design your own Easter egg in Maple Learn. So, if you’ve had your fill of chocolate eggs, consider exploring some egg-related geometry - Happy Easter!

To celebrate this day of mathematics, I want to share my favourite equation involving Pi, the Bailey–Borwein–Plouffe (BBP) formula:

This is my favourite for a number of reasons. Firstly, Simon Plouffe and the late Peter Borwein (two of the authors that this formula is named after) are Canadian! While I personally have nothing to do with this formula, the fact that fellow Canadians contributed to such an elegant equation is something that I like to brag about.

Secondly, I find it fascinating how Plouffe first discovered this formula using a computer program. It can often be debated whether mathematics is discovered or invented, but there’s no doubt here since Plouffe found this formula by doing an extensive search with the PSLQ integer relation algorithm (interfaced with Maple). This is an example of how, with ingenuity and creativity, one can effectively use algorithms and programs as powerful tools to obtain mathematical results.

And finally (most importantly), with some clever rearranging, it can be used to compute arbitrary digits of Pi!

Digit 2024 is 8

Digit 31415 is 5

Digit 123456 is 4

Digit 314159 is also 4

Digit 355556 is… F?

That last digit might look strange… and that’s because they’re all in hexadecimal (base-16, where A-F represent 10-15). As it turns out, this type of formula only exists for Pi in bases that are powers of 2. Nevertheless, with the help of a Maple script and an implementation of the BBP formula by Carl Love, you can check out this Learn document to calculate some arbitrary digits of Pi in base-16 and learn a little bit about how it works.

After further developments, this formula led to project PiHex, a combined effort to calculate as many digits of Pi in binary as possible; it turns out that the quadrillionth bit of Pi is zero! This also led to a class of BBP-type formulas that can calculate the digits of other constants like (log2)*(π^2) and (log2)^5.

Part of what makes this formula so interesting is human curiosity: it’s fun to know these random digits. Another part is what makes mathematics so beautiful: you never know what discoveries this might lead to in the future. Now if you’ll excuse me, I have a slice of lemon meringue pie with my name on it ðŸ˜‹

References

BBP Formula (Wikipedia)

A Compendium of BBP-Type Formulas

The BBP Algorithm for Pi

On International Women’s Day we celebrate the achievements of women around the world. One inspiring story of women in STEM is that of Sophie Germain (1776-1831), a French mathematician and physicist who made groundbreaking strides in elasticity theory and number theory. She learned mathematics from reading books in her father’s library, and while she was not permitted to study at the École Polytechnique, due to prejudice against her gender, she was able to obtain lecture notes and decided to submit work under the name Monsieur LeBlanc. Some prominent mathematicians at the time, including Joseph-Louis Lagrange and Carl Friedrich Gauss, with whom Germain wrote, recognized her intellect and were supportive of her work in mathematics.

Sophie Germain is remembered as a brilliant and determined trailblazer in mathematics. She was the first woman to win a prize from the Paris Academy of Sciences for her contributions in elasticity theory and was among the first to make significant contributions toward proving Fermat’s Last Theorem. Among her many accomplishments, one special case of Fermat’s Last Theorem she was able to prove is when the exponent takes the form of what is now known as a Sophie Germain prime: a prime, p, such that 2p+1 is also a prime. The associated prime, 2p+1, is called a safe prime.

To mark International Women’s Day, I’ve created a document exploring the Ulam spiral highlighting Sophie Germain primes and safe primes, as an adaptation of Lazar Paroski’s Ulam spiral document. The image below displays part of the Ulam spiral with Sophie Germain primes highlighted in red, safe primes highlighted in blue, primes that are both a Sophie Germain prime and safe prime highlighted in purple, and primes that are neither in grey.

The document also contains small explorations of these types of prime numbers. For instance, one interesting property of safe primes is that they must either be 5, 7 or take the form 12k-1 for some k≥1. This can be shown from the fact a safe prime q must equal 2p+1 for some prime, p (a Sophie Germain prime), by definition. Then, since q and p are prime, for q > 7 we can determine through contradiction that q ≡ 3 (mod 4) and q ≡ 5 (mod 6), to conclude q ≡ 11 (mod 12) ≡ -1 (mod 12). And so, q = 12k-1 for some k≥1. The Maple Learn document can be found here along with its Maple script. The document also includes a group where you can test whether some positive integer of your choice, n, is a Sophie Germain prime or a safe prime. Alternatively, given n, a button press will display the next Sophie Germain prime greater than n, using Maple’s NextSafePrime command in the number theory package.

In mathematics, there is no shortage of interesting rabbit holes to dive into; many of which are the result of past and present women in mathematics, like Sophie Germain, who have persevered despite their hardships.

## Volunteer code reviewers needed for Maple Transac...

by: rcorless 530Maple Transactions frequently gets submissions that contain Maple code. The papers (or videos, or Maple documents, or Jupyter notebooks) that we get are, if the author wants a refereed submission, sent to referees by a fairly usual academic process. We look for well-written papers on topics of interest to the Maple community.

But we could use some help in reviewing code, for some of the submissions. Usually the snippets are short, but sometimes the packages involved are more substantial.

If you would be interested in having your name on the list of potential code reviewers, please email me (or Paulina Chin, or Jürgen Gerhard) and we will gratefully add you. You might not get called on immediately---it depends on what we have in the queue.

Thank you very much, in advance, for sharing your expertise.

Rob

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: