Every four years, the world comes together to watch one of the most anticipated sporting events in history: the FIFA World Cup.

Behind all the anticipation, venue planning, and media fanfare, there are many artists and researchers who devote themselves to designing a new FIFA World Cup ball to be rolled out for the public eye (pun intended).

This post presents an overview of the geometric ideas behind the design of the FIFA 2026 "Trionda" ball, using Maple to visualize and explore these concepts in depth. The ideas presented here were inspired by this Scientific American Article. For more information and facts about the 2026 Trionda ball, as well how the shape of the ball impacts play on the pitch, I suggest you check it out!

FIFA ball designs are often inspired by one of the 5 Platonic solids. A Platonic solid is a convex polyhedron with each face being the same regular polygon with the same number of faces meeting at each corner.

This year, the Trionda ball was constructed from the simplest of these shapes, the tetrahedron, consisting of 4 triangles, with 3 faces meeting at each corner. Of the five Platonic solids, this shape has the fewest faces, making it the least sphere-like. Turning such a simple polyhedron into a smooth ball is therefore a surprisingly challenging geometric problem.

So how can we turn our pointy tetrahedron into something that rolls? Rather than trying to transform the entire tetrahedron at once, we can start by redesigning a single triangular face. The goal is to create a curved triangle that will fit perfectly with three identical copies of itself while covering the surface of a sphere.

 

Today is Pi approximation day (22/7) and I will use that as an excuse to share my new favourite expression for Pi:

And this isn't even an approximation! Recently, continuous mathematics has found its own equivalent to the digital hardware NAND gate. In his paper “All elementary functions from a single binary operator”, Andrzej Odrzywołek demonstrated that a single functional primitive can generate the entire standard continuous spectrum of operations. In other words, every single button on a scientific calculator, from addition and subtraction to sines, cosines, and logarithms, can be built using just this one function.

This Maple Worksheet explores how the 'Exp-Minus-Log' ("EML") operator, when paired solely with the constant 1, can be systematically nested to construct basic arithmetic, constants, and complex transcendental functions within Maple.

In essence, he discovered that the binary operator EML, along with the constant 1, forms a basis for the set of standard scientific-calculator operations.

This means that functions like ,  and operations like  or  can be creating by composing EML with itself in clever ways. Some constants and functions are trivial to represent, such as EML(1,1) = e or EML(x, 1) = exp(x), others however, are not...

With a quick one-command tweak, you can get Maple to use the property of the extended reals that

And then with a simple argument about standard branches, you can construct the natural logarithm for real numbers, which immediately leads the constant zero:

You can then expand the tools in your toolbox by creating subtraction with EML, ln(x), and exp(x)

Which then expands the toolbox further by allowing for the construction unary minus from the constant 0 (since -x = 0 - x), and then addition (since a+b=a-(-b))

Since we've constructed addition, subtraction, zero and one, we can technically construct every integer! It would not be very pleasant, and by no means optimal... but you could! Here's 7 for example:

The next step to building all the standard functions is multiplication and inversion. And these use the classic trick by using the fact that x=exp(ln(x)) can help simplify:

These are compositions of exp, addition, ln, and unary minus (all functions constructed previously), which means they can be made with only EML:

It's at this point that I will leave the derivation of division (a/b) and exponentiation (a^b) as exercises for the reader, so I can skip to something a little more complex...

To go beyond the basic operators, you'll need to step into the complex domain by constructing the imaginary constant i. To do this, take ln(-1) = -i*Pi (by using the standard branch) and combine it with Euler's formula

And once again the expression on the left-hand side is made up of operations that were all previously defined, so you can compose EML to get a new constant:

And finally, it's possible to break down the expression for Pi from the start, since it's the product i*ln(-1)

By successfully extracting the mathematical constants i and Pi, I think this demonstrates the complete constructive capability of the EML operator in the complex domain. While the resulting syntax trees become exponentially deep and unoptimized for human readability, they prove that continuous operations do not require a massive, distinct toolbox. Future applications of this uniform binary structure could dramatically simplify symbolic regression and machine learning optimization models. 
Ultimately, the EML operator reveals the remarkable truth that the vast complexity of scientific mathematics can be distilled down to a single, beautiful building block.

Isn't math awesome?