New 3D Print
New 3D Print
I'm really proud of this one as it was quite an effort. And it turned out so much better than I expected. The walls are thin enough to be slightly translucent, so it's lovely in certain lighting. It's also kind of pricey but there is no markup, so if you order one you will pay my cost.
http://shpws.me/Pg8i
To understand the shape, check out the following two Visual Insight posts by John Baez.
https://blogs.ams.org/visualinsight/2014/08/01/733-honeycomb/
https://blogs.ams.org/visualinsight/2014/08/14/733-honeycomb-meets-plane-at-infinity/
Which one to tackle next? There are a ton of choices...
https://en.wikipedia.org/wiki/Template:Regular_honeycomb_table
http://shpws.me/Pg8i
I'm really proud of this one as it was quite an effort. And it turned out so much better than I expected. The walls are thin enough to be slightly translucent, so it's lovely in certain lighting. It's also kind of pricey but there is no markup, so if you order one you will pay my cost.
http://shpws.me/Pg8i
To understand the shape, check out the following two Visual Insight posts by John Baez.
https://blogs.ams.org/visualinsight/2014/08/01/733-honeycomb/
https://blogs.ams.org/visualinsight/2014/08/14/733-honeycomb-meets-plane-at-infinity/
Which one to tackle next? There are a ton of choices...
https://en.wikipedia.org/wiki/Template:Regular_honeycomb_table
http://shpws.me/Pg8i
It looks a bit like a diaper (the big holes are for torso and legs)... :D
ReplyDeleteCool!
ReplyDeleteIt looks a bit like you could start from an Apolonian gasket on the sphere, and then make each circle a sphere, perhaps not with the centre on the large sphere, until the spheres intersect the others at the corect angle.
Gerard Westendorp, it is connected to the Apollonian gasket, though it isn't exactly one because the circles in the limit set on the boundary are not tangent. For {p,3,3} honeycombs, as p goes to infinity, the limit set on the ball boundary approaches the gasket. Here's the limiting honeycomb:
ReplyDeletehttps://en.wikipedia.org/wiki/Template:Regular_honeycomb_table#/media/File:Hyperbolic_honeycomb_i-3-3_poincare_vc.png
The gasket also jumps out in the dual {3,3,infinity}:
https://en.wikipedia.org/wiki/Template:Regular_honeycomb_table#/media/File:Hyperbolic_honeycomb_3-3-i_poincare_cc.png
What might also be really pretty is to paint the heptagons with 14 alternating (7.3.2) triangles. Of course the 3D print is usually in only one color, but as a picture.
ReplyDeleteGerard Westendorp But wait, those triangles should not be (7,3,2), but (7,x,2), since the heptagons meet in conical points. I can't figure out yet what x is.
ReplyDeleteGerard Westendorp, interesting observation! The heptagons are a different size than a geodesic {7,3} tiling, which determines the dihedral angle of the honeycomb cells, so I suspect you are correct. I'm guessing x will be some irrational number related to tetrahedral symmetry.
ReplyDeleteI should be able to render a picture today like you describe and will post it here. A thought I had for a single-color 3D print is to render alternating triangles at different thicknesses to make them stand out.
photos.google.com - New photo by Roice Nelson
ReplyDeleteWow!
ReplyDeleteI just today got a reel of wax filament, so I can print stuff, burn it out, and cast it, and I'm considering that this cast in brass would be spectacular.
ReplyDeleteJohn Bump, happy to share the .stl file with you if you want to give it a shot. Just email me at roice3@gmail.com.
ReplyDeleteMaximizing detail was the hardest part with shapeways and I pushed their limits (0.7mm wall thickness near the boundary). I’m able to adjust that however, so I could tune to brass casting requirements if you tell me what they are.
I'm not sure yet! Let me try a couple. The first thing I'm going to try has a 0.6mm wall thickness. (and that is affected by pour temperature and whether I'm doing it using a vacuum flask or not, so it may take me a little while to determine.)
ReplyDeleteJohn Bump - if you ever make this shape out of brass please show us a picture!
ReplyDeleteJohn Bump, here is a link to the stl definition file.
ReplyDeletehttps://www.dropbox.com/s/vbx27podxku4wdc/733_1M_scaled_2.stl.zip?dl=0
Thanks! I may have to shift my attention back to lost-PLA casting as wax printing is tricky. Will return to this idea, though.
ReplyDeleteBeautiful. Does it have closed chambers or is every side of every heptagon accessible?
ReplyDeleteArnaud Chéritat, every {7,3} cell of the honeycomb meets the boundary in a circle and is accessible. No closed chambers!
ReplyDeleteYou can see each heptagon from two directions. If you take a sharp light and move it around in the back while looking into a large cell, you can light up single heptagons at a time (if the lit open cell shares a heptagon with the large cell, which many do).
I keep wondering whether this is a closed manifold that would leave lots of trapped powder, or whether in the limit it has no volume.
ReplyDeleteRegarding Shapeways limitations, you can exceed their limits if you're willing to accept print failures using their "print it anyway" option. .5mm is often just fine. Using that option won't let you sell directly to the public, but I've found that they'll often accept models that exceed the limit without using that feature, but you are then subject to the whim of the human reviewer. When you've proven to yourself that a model in violation to that rule will still print successfully, and you still want to offer it to the public, it can be worth more than one attempt to get it past their review.
Gerard Westendorp, I've been wanting to investigate your observation about the (7,x,2) triangles and finally thought about it this morning. I've convinced myself that x = arctan( sqrt(2) ), which is a little under 55 degrees. This is half of the "edge central angle" of a tetrahedron, described on Wikipedia.
ReplyDeletehttps://en.wikipedia.org/wiki/Tetrahedron
I think the same phenomenon happens in spherical geometry, say with the {5,3,3} 120-cell. Instead of {5,3,2) triangles there, we also get (5,x,2) triangles, with x as above. In other words, I think the value for x comes down to the vertex figure of the honeycomb.
Let me know if this sounds right to you.
May I ask did you have a 3D printer at hand or you used shapeway's cloud platform to print this?
ReplyDeleteLiang Zhao, I used shapeways. To get a nice result with this model, one will need a printer having .7mm resolution (finer than a typical home 3D printer).
ReplyDeleteRoice Nelson Thanks, hope shapeways is available in China ...
ReplyDeleteIf not and you have access to another printer, I'm happy to send you the model file. Let me know via email if you'd like it!
ReplyDeleteRoice Nelson Thanks! Is the code generating the model file included in your honeycomb project? If so I think I can have a try understanding the code first and build the model on my own. The math behind it is more fascinating ... If I failed to make that I'll ask you for a copy.
ReplyDelete