There is no such animal...
- said the man who saw a giraffe for the first time in his life at the zoo.
This was the favorite saying of my former teacher and mentor at our university when he came across something that seemed unbelievable. This is what he said when he took in his hands the geometric construction that was much later called the Szilasi polyhedron.
Now I am in a similar situation. I would like to share this far from ordinary story with my readers.
Can I Solve This Unsolved Math Problem?
On the 20th of January, 2025
I came across a YouTube video and then a related website: https://github.com/HackerPoet/NeighborlyPolyhedra.
In short, the author attempts to find an ordinary polyhedron other than the tetrahedron and the Szilassi-polyhedron whose any two faces are adjacent.
We assume that our curious readers have also watched the video above, so are aware of the problem the importance and difficulty of the problem.
Here, we will attempt to present the most striking result on the subject, for which the data used – and slightly modified – has been downloaded from here: https://github.com/HackerPoet/NeighborlyPolyhedra/issues/3.
This shape is axially symmetrical, and therefore consists of six pairs of tiles, each pair of which is congruent
and has the same circumference when viewed from outside the surface. Congruent tiles are marked with the same color. The visibility of the tiles can be switched on and off individually. We recommend our readers to inspect the tiles one by one to check for self-intersection and a few at a time to check for non-collision.
Note that...
- sheets 5 and 6 are self-intersecting;
- if we display the polyhedron vertices, we can see the two unwanted self-intersection points; although the common edges of sheets 4 and 5 are elsewhere, they have two edges that intersect.
- The same is true for sheets 3 and 6. This is also the consequence of the only error.
- The applet gives the possibility to change the construction data. This option displays an input field with six points. These points are the vertices of the dual shape of the construction. If one of these dual points is P=(a,b,c), then – in this case – the equation a x +b y +c z =100 gives the plane of a given face of the construction.
Let's hope that one day we will get a glimpse of the real giraffe in the zoo of mathematics.