Gabriel’s Horn is one of the most famous examples in calculus of how infinity can behave in ways that completely defy our intuition.

The horn-shaped object is created from a very simple curve: y = 1/x for x ≥ 1 (pictured below).

Now imagine rotating this curve around the x-axis. The resulting surface stretches infinitely far to the right while becoming thinner and thinner. Visually, it resembles a long trumpet or horn that continuously narrows to a thickness of zero.

At first glance, nothing about this shape seems particularly mysterious. As x grows larger, the radius 1/x becomes smaller and smaller. It seems reasonable that both the volume contained inside the horn and the area of its surface would remain finite (or at least if the volume was finite, then the surface area would also be finite). After all, the horn gets extremely thin very quickly.

Calculus allows us to test that intuition.

To compute the volume of the horn, we use the disk method. Each slice perpendicular to the x-axis forms a circular disk of radius r = 1/x, each with an area of π*r² = π*(1/x²).

The total volume is the sum of an infinite number of these disc areas with thickness dx. As an integral,

V = π ∫₁^∞ (1/x²) dx.

This is a simple integral that converges to a value of 1. We could use the power or rule or our favourite computing software (I used Maple below).

Hence, V = π ∫₁^∞ 1/x² dx = π*1 = π. This means the horn contains only π cubic units of space, even though it extends infinitely far. 

Now let’s compute the surface area of the horn. For a surface of revolution, the surface area is

A = 2π ∫₁^∞ y √(1 + (y′)²) dx.

Since y = 1/x, we have y′ = −1/x². Substituting into the formula gives

A = 2π ∫₁^∞ (1/x) √(1 + 1/x⁴) dx.

Software like Maple can easily handle this integral. It tells us the integral diverges to infinity.

However, this is difficult to solve analytically. To understand what happens to this integral, notice that for large x, the square root term is very close to 1, since 1/x⁴ can be approximated as 0 as x grows large. This means the integrand behaves roughly like 1/x (it's actually slightly larger than 1/x). But

∫₁^∞ 1/x dx diverges, and ∫₁^∞ (1/x) √(1 + 1/x⁴) dx > ∫₁^∞ 1/x dx, so ∫₁^∞ (1/x) √(1 + 1/x⁴) dx must also diverge. As a result, the surface area of Gabriel’s Horn is infinite.

This leads to the famous, surprising conclusion:

• The horn has finite volume.
• The horn has infinite surface area.

In other words, it could be filled with a finite amount of paint, but it would require an infinite amount of paint to coat its inside surface.

Of course, real paint has thickness, so the paradox disappears in the physical world. Eventually, the horn would become thinner than the paint layer itself. But mathematically, the result is perfectly consistent.

The key idea lies in how quickly the function 1/x shrinks. The cross-sectional area of the disks scales like (1/x)² = 1/x², and the integral of 1/x² converges.

But the circumference of each slice scales like 1/x, and the integral of 1/x diverges.

So as the horn extends outward, the added volume decreases quickly enough to sum to a finite value, while the added surface area decreases too slowly and accumulates forever.

Gabriel’s Horn beautifully illustrates one of the central themes of calculus: infinite processes can produce results that feel deeply counterintuitive.

Volume and surface area seem closely related, but can behave in completely different ways when infinite limits are involved. A shape can stretch endlessly yet still contain a finite amount of space.

This strange object reminds me that mathematics isn’t just about calculating numbers, but is also about exploring the strange and fascinating consequences of simple ideas pushed to their limits.

In mathematics, us humans love to rely on intuition. It helps us make physical sense of phenomena and guide our thinking before formal reasoning is developed.

For example, approximating the derivative of a function at a point can be thought of intuitively as dividing the function’s rise by its run. As we shorten the distance we run, this ratio approaches the value of the function’s derivative at that point. See this in the demonstration from Maple Learn below.

It is impossible to fully grasp the idea of moving an infinitesimal distance, so we make it easier by asking: “If we move an extremely small distance to the right, how much do we move up?”.

Intuition is typically a beautiful tool for approximating limits, but limits tend to limit (pun intended) the utility of our intuition. A perfect example of this? The Staircase Paradox.

Consider any rectangle you’d like. In the following example, we'll use a rectangle of width 3 and length 4 for convenience, but this paradox extends to any rectangle.

The name of the game is to ask yourself: how far must we walk along the edge of the rectangle to get from the top left corner to the bottom right. Here, the distance is of course 7 units (3 units right and 4 units down). This looks like a bit of a scary fall, so let’s add some stairs.

Even with the stairs, we’re still travelling a total distance of 7 units (1.5 + 1.5 units right, 2 + 2 units down). To shorten the fall even more, we can keep adding more and more stairs.

The important thing to notice is that no matter how many stairs we add, the distance travelled is always 7.

Now you may be wondering, where exactly is the paradox? Well, imagine now we have an infinite number of stairs. Our intuition tells us that our path to the bottom becomes more like a slide instead of a staircase. The steps we take are infinitely small, so it seems like we’re just travelling in a straight line down to the bottom right corner. However, if this were the case, we would have a right triangle! Using the Pythagorean Theorem, the length of our travelled path would be sqrt(3²+4²) = 5.

In other words, our calculations from before were wrong! But... they can’t be wrong, because we saw that the total distance of 7 units travelled was independent of the number of stairs we added.

This is a consequence of something called the “Manhattan distance”, which is the distance you travel if you can only move horizontally and vertically, like navigating the grid of streets in Manhattan. No matter how small we make the steps in our staircase, we are still only moving right and down. We never actually move diagonally. So even though the staircase looks more and more like a straight line, its length is always computed using horizontal distance + vertical distance. The limit of the shapes is a diagonal line, but the limit of the lengths is not the length of that diagonal. And that’s where our intuition stumbles.

The key lesson of the Staircase Paradox is that a sequence of curves can converge to a straight line, while their lengths converge to something completely different.

This is one of the quiet but profound messages of higher mathematics: limits preserve some properties, but not all. Smoothness, shape, and position may converge nicely, while quantities like length, area, or curvature behave in more subtle ways. Mathematics has a gentle way of reminding us that how we measure something can matter just as much as what we’re measuring.

Some mathematical theorems don't just prove a statement to be true, but they reveal something beautiful that itches our intuition. Viviani's Theorem is one of those rare gems. The theorem says something profound but geometrically elegant:

For any point inside an equilateral triangle, the sum of the perpendicular distances from that point to the triangle's three sides is always equal to the height of the triangle.

Take the equilateral triangle below as an example:

By picking any point P, we can draw lines perpendicular to each edge of the triangle that touch point P. The length of these lines always add to 6, which we will discover is the height of this triangle.

To explore this more, former Maplesoft co-op student Michael Barnett made a Maple Learn document on Viviani's Theorem where this is seen in action, as shown below.

No matter where the point P moves, the sum of perpendicular distances from that point to the edges of the triangle always add to the same value: the height of the triangle.

To see why this is the case mathematically, consider the example from before:

More generally, if we let x, y, and z be the shortest distances from the point P inside the triangle to the sides AB, AC and BC, respectively, one can conclude that:

In words, the sum of these distances x+y+z is simply the height of the triangle, h, no matter where the point P lies inside the triangle.

Even after reading this proof, it may be tempting to think of cases where this would not be true. I like to think of Maple Learn as a playground for geometry, algebra and visualization to interact. It helps to convince us that even for many different cases (in fact, all cases!), this theorem holds true.

Viviani's Theorem is a reminder that mathematics isn't always about answers. Instead, it's about finding hidden harmonies that our intuition begs us to question and search for. With tools like Maple Learn, those harmonies buried in symbols and complicated definitions can be uncovered and explored.

If geometry ever felt too distant or abstract, this is your invitation to see it come alive!

Many problems in mathematics are easy to define and conceptualize, but take a bit of deeper thinking to actually solve. Check out the Olympiad-style question (from this link) below:

Former Maplesoft co-op student Callum Laverance decided to make a document in Maple Learn to de-bunk this innocent-looking problem and used the powerful tools within Maple Learn to show step-by-step how to think of this problem. The first step, I recommend, would be to play around with possible values of a and b for inspiration. See how I did this below:

Based on the snippet above, we might guess that a = 0.5 and b = 1.9. The next step is to think of some equations that may be useful to help us actually solve for these values. Since the square has a side length of 4, we know its area must be 4² = 16. Therefore, the Yellow, Green and Red areas must add exactly to 16. That is,

With a bit of calculus and Maple Learn's context panel, we can integrate the function f(x) = ax² from x = -2 to x = 2 and set it equal to this value of 8/3. This allows us to solve for the value of a.

We see that a = 1/2. Since the area of the Red section must be three times that of the Yellow (which we determined above to be 8/3), we get Red = (8/3)*3 = 8.

The last step is to find the value of b. In the figure below, we know that the line y = 4 and the curve y = bx² intersect when bx² = 4 (i.e. when x = ± 2/sqrt(b)).

Since we know the area of the red section is 8 square units, that must be the difference between the entire area underneath the horiztonal line at y = 4 and the curve y = bx² on the interval [-2/sqrt(b), 2/sqrt(b)]. We can then write the area of the Red section as an integral in terms of b, then solve for the value of b, since we know the Red area is equal to 8.

Voila! Setting a = 1/2 and b = 16/9 ≈ 1.8 guarantees that the ratio of Yellow to Green to Red area within the square is 1:2:3, respectively. Note this is quite close to our original guess of a = 0.5 and b = 1.9. With a bit of algebra and solving a couple of integrals, we were able to solve a mathematics Olympiad problem!

The need to solve quadratic equations never seems to disappear. Whether it is completing a physics problem, solving a differential equation, or performing equilibrium calculations in chemistry, quadratic equations are an integral part of all STEM-based disciplines.

Depending on the complexity of the quadratic equation, the typical 'guess-and-check' method taught in most high school classes can often be frustrating and time-consuming. Professor of mathematics Dr. Po-Shen Loh, in his new method shown here, recognizes some important properties of solutions to quadratic equations and integrates them into a more intuitive approach that students are much more likely to feel motivated by.

For example, consider the equation x^2 - 14x + 45 = 0. Most students are taught to first factor this equation by thinking of two numbers that multiply to 45 and add to -14. After trying multiple values, we would discover that those values are -5 and -9. We would use these values to factor the equation into the form (x-5)*(x-9) = 0. Setting each factor equal to zero, we would get x = 5 or x = 9. Equivalently, to solve for x more directly, we need two numbers that multiply to 45 and add to 14 (again, x = 5 and x = 9).

The only way to speed up this process of guess-and-check is to do enough similar problems until the guesses become second nature. Not to mention, this becomes exponentially more difficult as the coefficient on x^2 increases (for example, solving the equation 6x^2 + 7x - 20 = 0).

For the example above, Dr. Loh's method builds on a simple starting point:

(i) We know that the numbers (call them R and S) add to 14

(ii) We know that since the numbers add to 14, they must have a mean value of 14/2 = 7

(iii) If the two numbers have an average of 7, they must be an equal 'distance' (call this distance z) from 7

(iv) We can write the two numbers as R = 7+z and S = 7-z

(v) Since the numbers R and S multiply to 45, then (7+z)*(7-z) = 45 ⇒ 49 - z^2 = 45. In other words, z^2 = 4, so z = +2 or z = -2

(vi) The solution to the equation is then R = 7+2 = 9 and S = 7-2 = 5 (as we predicted)

We can generalize this idea for any complex coefficients a, b and c in the equation ax^2 + bx + c = 0 to actually prove the quadratic formula. However, using Dr. Loh's method on specific examples (as above) helps build intuition for why the quadratic formula works in the first place. Other proof methods such as completing the square are just as mathematically sound, but they do not utilize the mathematical instinct that makes solving a problem in mathematics so gratifying.

Although I am currently a student working for Maplesoft, I had not used Maple Learn extensively beforehand. Dr. Loh's idea of creating a more intuitive way to solve such a conventional problem inspired me to create a document in Maple Learn, linked here, outlining the steps above.

Learning new ways to solve a problem in mathematics is exciting, but it is often difficult to present in a way that is clear, visually-appealing and easy to create. Most online mathematical environments are difficult to navigate and typically lack visualizations to accompany an idea. With Maple Learn, it felt comforting to open a clean canvas where I was able to easily build a document in just a few hours that not only summarized the main ideas of this new method, but also showed the user why the method works using live animations and colour schemes (see some examples below).

I surprised myself (as well as my managers) by how quickly I was able to transfer all of my ideas into the document. I could also split related content into groups and use collapsible sections to keep the document uncluttered and easy to read.

I also took advantage of the freedom to explore other documents and directly reference them through hyperlinks.

Sometimes it can be difficult to follow a new concept without having some background information. Adding these references makes it simple for the reader to access supporting documents and ensure there are no knowledge gaps to be filled along the way. Once you make a document, you also have the option to publish it to your own gallery and make it public for others to use and learn from.

Maple Learn has been incredibly helpful for sharing the things that interest me the most. If you have something related to mathematics that excites you, try not to keep it to yourself. Consider using Maple Learn to share your ideas with the world and see your vision come to life!