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# Conext 21 Polyhedron

"Conext 21 polyhedron" is the name given here to the solid underlying the soccer ball of the 2020 Tokyo Olympic Games (held in 2021). It is implemented in the Wolfram Language as PolyhedronData["Conext21Polyhedron"].

The official Conext 21 ball sports four 'propellers' with 4-fold symmetry and 'triskelions' nestled between the blades of these structures (left figure). However, since theis form of the ball was moulded with no stitched seams, geometric analysis of its structure is not very instructive. In contrast, "replica" versions of the ball (right figure) contain extra sewn seams that allow its panel structure to be analyzed (Kuchel 2022).

When the replica is analyzed, its structure is seen to correspond to a small rhombicuboctahedron in which "diagonal" squares are subdivided into two congruent rectangles and triangles into three congruent kites. The final result has 80 vertices, 132 edges, and 54 faces (many of while are coplanar but which expand differently upon inflation).

The 132 edges consist of 24 edges of length (corresponding to internal triangulations of the triangles of the underlying small rhombicuboctahedron, indicated in red above), 96 of length (corresponding to half the edge lengths of the underlying small rhombicuboctahedron, indicated in green), and 12 of length (corresponding to the underlying edge lengths, indicated in blue).

The skeleton of the Conext 21 polyhedron is the untraceable graph illustrated above with 56 vertices of degree 3 and 24 vertices of degree 4. It is implemented in the Wolfram Language as GraphData["Conext21Skeleton"].

Jabulani Polyhedron, Soccer Ball, Truncated Icosahedron

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## References

Kuchel, P. "Analysis of the Underlying Polyhedron in the Conext21/Tsubasa Football." Apr. 26, 2022. https://community.wolfram.com/groups/-/m/t/2519100.

## Cite this as:

Weisstein, Eric W. "Conext 21 Polyhedron." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Conext21Polyhedron.html