let it leave me like a long breath

please reblog my photoset etc

i've already made that joke

so i still don't have better graph grammars working but i realized that it's actually real easy to synthesize the necessary stitching geometry on polyhedra, so, i did that. theoretically i could do this for any rectified polyhedra, not just the mostly-hexagonal ones.

the real big issue is that i'm having a hard time thinking of any reasonable way to turn all those points into a coordinate space. flat, euclidean geometry is real easy to manage. nice simple distance metrics. easy ways to calculate relative angle and positioning of two arbitrary tiles. all of that goes away once you bring curvature into the picture. i was actually having similar problems trying to figure out geometry in the hyperbolic plane a while back, and it didn't really occur to me at the time that the fundamental problem was curvature, not specifically the hyperbolic curvature.

all of that is to say that to make a better graph embedding i'll need a robust polyhedra coordinate system, and that... seems difficult. i might try to patch up the graph code itself but it's not very useful at all without an embedder.

iii got rectification working!

theoretically i think that's enough for me to generate the rest of the archimedian and catalan solids (definitely the rhombic dodecahedron and the rhombic triacontahedron) but i still need to tweak the vertex positioning a little to get the relevant faces uniform. right now my rhombic dodecahedron actually has busted concave faces.

the other thing is now i need to actually focus on doing stuff with these polyhedra, instead of just generating more. i mean more is fine; i have no issue with generating more, it's just that i'd really like to get some more gamelike elements working for something

i got kleetopes working! this means i can now generate several of the catalan and archimedean solids.

( images )

the gaps are for the 3rd and 6th polyhedra, since they involve kleetopes+duals of the rhombic dodecahedron and the rhombic triacontahedron, and i haven't figured out how to generate those yet.

except...

so a kleetope is the Geometry Name for taking a polyhedra, adding a point in the center of each face, and dragging all the faces out. so like if you have a cube you add one point to each of the six faces and then pull those points out a little. the thing is, depending on how much you pull you get different kinds of polyhedra, since there's a point when all the new faces 'line up'. so for the tetrahedron, you first get the triakis tetrahedron, and then you get a cube (because each new face lines up perfectly with the new faces on the adjacent old faces to form a square), and then you get a 'stellated tetrahedron', which is just a tetrahedron but bigger, and then you get a caltrop-like shape. and these polyhedra change based on the base shape.

and for the kleetope of the octahedron, the point where all the faces line up turns the shape into a rhombic dodecahedron. and for the kleetope of the icosahedron, the point where all the faces line up turns the shape into a rhombic triacontahedron

( another image )

it turns out that the rhombic dodecahedron is catalan solid #8, and the rhombic tricontahedron is catalan solid #9, so if i can get 'real' versions of those i can generate the missing #3 and #6 polyhedra and also #8 and #9. but for those polyhedra to be the actual shape i'll need to figure out a way to cut edges and merge planar faces. plus i'll need to do the math to make sure that the faces are actually planar, and not like 0.5% off.

but you know, optimistic.

i need some kind of periodic table of the polyhedra b/c this is not a good way to display what i'm capable of generating so far.

anyway i've added pyramids and frustums (the pyramid-with-the-top-cut-off ones) as well as dipyramids (which i'm just generating by getting the dual of a prism)

earlier when i was doing tree rendering i wrote code that could handle any possible convex prismatoid, and i guess i should probably just translate that into this new data format so i can use it to generate most of these.

ultimately i want to get kleetopes working b/c those are needed for a lot of the more complex regular polyhedra, and i kind of understand how i'd need to approach those now. but also i should try to get sweep objects with arbitrary transforms working, since that's what's needed for a lot of the more organic or complex shapes i'd want to render.

please reblog my photoset etc

i've been working on rendering more complex shapes, because currently all i can really manage is prisms and prismatoid-style shapes. so i was like, let's try generating all the platonic solids, and then all the archimedian solids and the catalan solids, and go from there. there are lots of neat shapes out there that are computationally-feasible to generate, it's just, uhh i'm bad at shapes.

historically when i've rendered stuff i've just had a polygon soup, but for this i wanted to have enough information around so that i could do things like truncation or generate duals, so i'm using a wing edge data structure, and only just starting to make my way around understanding how it works. but at least i have it working well enough to render some basic polyhedra. theoretically when i get duals working i'll be able to dual the cube and the dodecahedron to get the octahedron and the icosahedron, and the first few catalan solids are all kleetopes of simple polyhedra, which means if you kleetope the platonic solids and then get the dual of that shape you've gotten an archimedian solid. turns out by repeating various klee / truncation / dual operations you can end up constructing a whole bunch of complex polyhedra

but for right now i'm just glad i got three rendering

e: got prisms rendering


this has given me some understanding of how to algorithmically generate polyhedra in this format. antiprisms might not be too tough either.

e²: got duals generating

now it's time for either antiprisms or kleetopes. with kleetopes and duals it's possible to generate many of the archemedean/catalan solids. antiprisms are another infinite family of shapes. getting something like trapezohedrons generating would be neat, and those are apparently the duals of antiprisms. diminished trapezohedrons are neater-looking but i don't think there's any easier way to generate them aside from just generating them.