Would these be very printable in a half-space model? Or too much scale + thickness distortion? There do exist horospheres represented at constant scale in that model (parallel to the boundary), so one should be able to make a slab-like tiling near one of these. Admittedly, the horospheres look much more interesting when centered on the boundary, since centering them infinitely far from it in the half-space model just generates normal flat Euclidean tilings.
The biggest mindfuck is how you can even tile hyperbolic honeycombs. They're analogous to pseudogons, with infinite sides but never converging to a single ideal point.
Absolutely fasscinating. I find hyperbolic space particularly interesting. I knew about {5,3,4} and by implication its dual {4,3,5}, but had never heard of {6,3,3} and its dual. There is also a relationship between {5,3,4} and the 120-cell that interests me: one fills negatively curved space with dodecahedra, the other fills positively curved space with dodecahedra.
Excellent work Henry. Wikipedia now has a picture of the {6,3,3} honeycomb, but just a single perspective. en.wikipedia.org/wiki/Hexagonal_tiling_honeycomb
Any chance we could get the {4,3,5} hyperbolic honeycomb 3-D file? I see all of the other honeycombs in this video uploaded on thingiverse plus the {3,5,3}, but not the {4,3,5}.
New things: Hyperbolic honeycombs. These sculptures are joint work with Roice Nelson and show various tilings of 3-dimensional hyperbolic space.
Would these be very printable in a half-space model? Or too much scale + thickness distortion?
There do exist horospheres represented at constant scale in that model (parallel to the boundary),
so one should be able to make a slab-like tiling near one of these. Admittedly, the horospheres
look much more interesting when centered on the boundary, since centering them
infinitely far from it in the half-space model just generates normal flat Euclidean tilings.
The biggest mindfuck is how you can even tile hyperbolic honeycombs. They're analogous to pseudogons, with infinite sides but never converging to a single ideal point.
Absolutely fasscinating. I find hyperbolic space particularly interesting. I knew about {5,3,4} and by implication its dual {4,3,5}, but had never heard of {6,3,3} and its dual. There is also a relationship between {5,3,4} and the 120-cell that interests me: one fills negatively curved space with dodecahedra, the other fills positively curved space with dodecahedra.
The 120-cell is {5,3,3}, so they are very similar.
It’s name simultaneously sounds like a synth pop song, surprisingly good indie sci-fi game, AND a mind numbingly complicated mathematical concept.
wooo, these are really cool! How about [7,3,3]? can it be mapped the same way as [6,3,3]?
The {6,3,3} honeycomb is so beautiful, since it shows how euclidian planes can tile H3.
Excellent work Henry. Wikipedia now has a picture of the {6,3,3} honeycomb, but just a single perspective.
en.wikipedia.org/wiki/Hexagonal_tiling_honeycomb
Very interesting! Would 4-3-4 be like a regular grid made out of cubes? 4 sides, 3 around each vertex and 4 cubes on each edge?
Yep, that’s the Minecraft grid.
This is so amazing!
Any chance we could get the {4,3,5} hyperbolic honeycomb 3-D file? I see all of the other honeycombs in this video uploaded on thingiverse plus the {3,5,3}, but not the {4,3,5}.
Cubes, dodecahedron, and horohoneycombs…
The {6,3,3} tiling is made of horospheres
These would make good honey dippers.
bubble wands :)
Mikey Bergman non Euclidean bubbles
Could you please spell the name of these symbols? Thank you.
Is there a corresponding honeycomb with the symbol (3,7,3)?
Yes. See figure 18 of arxiv.org/abs/1511.02851
As a german, your pronounciation of Schläfli is completely distracting me. ä is pronounced /ɛ/. Like the e in "let" or "men".
Pretty much everyone I've come across says it like Henry said in this video, but I see your point
The links no longer work 🫠