Today's number is 78,585,627
A holyhedron is a type of 3-dimensional shape. It is a polyhedron where every face has at least one hole. The full definition excludes faces where the hole touches the outer boundary of the face and where the boundaries of two holes intersect.
In other words, each face is a planar polygon containing at least one interior hole, with holes disjoint from each other and from the boundary in a precise topological sense.
So instead of faces being simple polygons, each face becomes something closer to a "Swiss chess polygon". This makes holyhedra very different from familiar polyhedra like cubes or dodecahedra. Even though the definition sounds like a small modification, it forces the geometry to become extremely complicated extremely quickly.
Essentially, a hole in a face requires additional edges. These edges must belong to adjacent faces. Adjacent faces now inherit the hole constraint that force further subdivisions. Subdivision creates more faces, which may require more holes. So instead of a single modification, you get a recursive geometric bookkeeping problem where every local fix propagates globally.
A Brief History
The idea of a holyhedron was introduced by John H. Conway (known famously as the creator of Conway's Game of Life) in 1997 as part of a prize problem. The challenge was to construct a finite holyhedron. At the time, it was not obvious that such an object could exist at all. The requirement that every face must contain a hole creates strong global constraints: holes force extra structure, which forces more faces, which forces more holes, and so on.
Conway's original expectation was that if a solution existed, it would likely have on the order of 100 faces or so. So he offered a prize of $10,000 USD divided by the number of faces, expecting to pay out around $1,000.
That intuition turned out to be dramatically optimistic
Various Constructions
When Jade P. Vinson first heard about this problem, he was a grad student at Princeton. He spent a great deal of time working on finding a solution. Eventually, it paid off with him being the first to discover a holyhedron, one with 78,585,627 faces in 1999.
This solution was worth about $0.00012724973 USD, but it's unknown if that money was ever paid out.
In 2003, Don Hatch presented a holyhedron with 492 faces, which was about $20.33 in prize money. And in 2026, a new construction was obtained by Geby Jaff using Archivara with 476 faces.
These shapes are (unsurprisingly) hard to visualize, but here are a few images showcasing the construction of Don Hatch's 492-face shape, which is built in 9 layers. It should give good intuition for what I mean by "holes", and should instill fear for the first holyhedron discovered with over 78 million faces.
To see the full construction (with .wrl files), below is a link:
And here's a cutaway of the full construction:
78,585,627 is remarkable because it is forced. It comes from a simple-sounding rule applied relentlessly across a geometric object until intuition completely breaks down.
That's one of the recurring surprises in mathematics: constraints that look harmless locally can become overwhelming globally, and the resulting objects often tell you more about structure than about shape.
Overall, I consider this part of math fascinating, and a 78 million-faced holyhedron definitely contributes to that feeling.