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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.

 

 
 
Notice that in the above diagrams, the transformed triangle has the same area as the original triangle. Although the edges have been reshaped, no area is added or removed, only redistributed. Preserving the area ensures that four identical curved panels can still cover the sphere completely without leaving gaps or overlapping.
 
Now that we know how to change one face of the tetrahedron, we need to perform the same sort of transformation (from a triangle to a curved tile), on the surface of a sphere. To start, we can inscribe the tetrahedron inside the sphere, like this:
From here, we can project the edges of the tetrahedron onto the sphere, creating six great-circle-arcs (also known as geodesics) as shown in the diagram below.
Each region enclosed by these geodesics corresponds to one triangular face of the tetrahedron within the sphere. By transforming each geodesic triangle into a smooth curved tile (using a bit of AI help), we create a tiling of the surface similar to that of the 2026 FIFA World Cup ball!
Because each curved tile maintains the area of the geodesic-generated region, the four panels form a complete tiling of the sphere. 
 
I would have liked to find a better function between the points on the sphere that resemble the actual Trionda ball more accurately but didn't get the chance to dive into that. If you want to take on the challenge and are successful, please reply in the comments.
 
To see the Maple Worksheet used to generate these diagrams, check out: Trionda Ball Worksheet
geometry sphere plotting polyhedron

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