Can’t be done, MM. You can’t make a sphere using only hexagons.
The problem is that when you project a plane figure onto a sphere, you get distortion and the angles change.
If you start with a hexagon as a polar cap, you can surround it with hexagons, but since the sphere is spreading out as you lay them, the hexagon’s edges furthest from the starting point must also spread out. When you add another row of hexagons, the edges start spreading so far that there’s no room to fit the needed hexagons. You can start squeezing them together, but then you no longer have hexagons, you have funny looking 6-sided shapes.
You can see this effect easily on a globe of the earth. Towards the equator, a single block outlined by 2 adjacent parallels and 2 adjacent longitudes approximate a square. But if you trace from that square upwards, the “tops” of the squares get progressively narrower, and you end up with narrow strips near the pole.
One of the slickest tricks for covering a sphere with repeating geometric shapes comes from soccer, which uses 12 identical pentagons and 20 identical hexagons.
One of the slickest tricks for covering a sphere with repeating geometric shapes comes from soccer, which uses 12 identical pentagons and 20 identical hexagons.
The Traveller Role playing game took the soccer ball idea and expanded on it. They recombined and subdivided the hexagons & pentagons to create many more hexagons and make the pentagons much smaller. Here is a picture of that pattern projected onto an icosahedron and unfolded.
I have no idea how to go from this to a sphere. If you did there would be some variation in the size of the hexagons. If I had to do it, I would start with the soccerball layout and subdivide from there.
The Traveller Role playing game took the soccer ball idea and expanded on it. They recombined and subdivided the hexagons & pentagons to create many more hexagons and make the pentagons much smaller. Here is a picture of that pattern projected onto an icosahedron and unfolded
That’s fascinating as hell, but I must be getting old…
I stared at that thing for 10 minutes trying to figure out where the pentagons were before I realized that there were only 5 “sawtooth” points top and bottom, not 6 (the old “forest/trees” thngy…). So that gave me the 2 “polar” pentagons. As I was folding it in my mind, It took me a few more minutes to realize that the “hexagons” I was seeing at the unfolded apexes were an illusion - they’re each pentagons with a split side.
So curiously, even at the reduced size/increased counts of the polys, there are STILL only 12 pentagons as in the soccer ball, but many more hexagons (looks like 240?), and I strongly doubt that inflating the assembled icosahedron into a sphere would result in any notable distortion of any of the polys.
Thanks for posting that, Pwiecek! A very interesting exercize!
No doubt what you have written is very interesting.
I admire your culture.
Thank you for the information.
However, with Hexagon is not possible to model a golf ball?
In the golf ball there is some imperfection .......6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,5,6,6,6,6,6,6,6,6,6,5 etc. etc. etc.
...................... and I strongly doubt that inflating the assembled icosahedron into a sphere would result in any notable distortion of any of the polys..
Due to inflating being based on vertex direction, you do get a change in size of the Hexagons. Maybe better to take the icosahedron up to 40 faces? But even then, I think the sizes will differ enough to very noticeable.
I did quickly make the icosahedron based on the image posted, and inflated to sphere to check (no, not done in Hexagon).
...................... and I strongly doubt that inflating the assembled icosahedron into a sphere would result in any notable distortion of any of the polys..
Due to inflating being based on vertex direction, you do get a change in size of the Hexagons
True, but by “notable” (not to be confused with “noticeable”), I simply meant I doubted it would be great enough to worry about.
Maybe better to take the icosahedron up to 40 faces? But even then, I think the sizes will differ enough to very noticeable.
I did quickly make the icosahedron based on the image posted, and inflated to sphere to check (no, not done in Hexagon)
And a fine looking sphere it is!
There may be noticeable distortion to some hexagons as you rotate the sphere, but the general aesthetics are still very pleasing.
BTW: What was the program you used to do that? It really is a good looking sphere.
However, with Hexagon is not possible to model a golf ball? In the golf ball there is some imperfection .......6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,5,6,6,6,6,6,6,6,6,6,5 etc. etc. etc.
