Tom Lowe

I'm most interested in this third type as it seems to be a bit closer to natural fluvial landscapes of ridges and valleys. I can now add one of these types to different generating spheres of the icosahedral set that I showed. A white version of the ridged structure for the northern pole mountains, a green version of the tree-tree on the left for some canopy, and a brown version of the shell-tree on the right. Blog: https://tglad.blogspot.com/2026/04/inversive-global-landscapes.html (3/3)

@lisyarus@mastodon.gamedev.place @johncarlosbaez@mathstodon.xyz There's a pretty nice trick for cheap Euler fluid dynamics solutions in 2D using the contour tangent of various scalar fields. I did a post about it: https://tglad.blogspot.com/2019/12/2d-eddie-currents.html

With the overlap limit set placed between them we have a continuous landscape. It also removes ambiguity of which child set has priority at the overlap. The overlap constraints leave four free spheres that can be adjusted on this overlap area, so it can look more like the brown or the green landscape if needed. Perhaps by default it should average the two sets prior to solving the dihedral angle constraints between the spheres. As far as I can work out, the smaller green disks don't need to use the overlap set against the large brown one, as they remain continuous. But you can see the green-brown overlap sets between smaller green and brown disks. The number of combinations of overlap sets between pairs of substitution sets at different orientations suggests that it would be better for them to be auto-generated as an interpolation of the two neighbouring sets, rather than being manually specified. blog: https://tglad.blogspot.com/2026/04/improving-global-landscapes.html (4/4)

This image shows what's happening with the substitutions. The top sphere of the parent set (left) substitutes in the green (right) set, and the top sphere of the green set (right) substitutes in the golden coloured parent set (left). It is of course that large bottom green sphere that makes the green limit set so hilly. Variations include substituting a different child sphere, or a grandchild sphere, or substituting in a third set. (2/4)

The second problem is that the terrain patches are all separated disks which is also unnatural. That's because you can't apply a substitution to two neighbouring spheres, or it causes discontinuities. This is solvable but it is a bit tricky. You need to create an inversive limit set specifically for the overlap region between the neighbouring spheres. You then need to constrain the overlap limit set spheres in a specific way. If set A constrains sphere ids 𝑎ᵢ to its parent and child set B constraints sphere ids 𝑏ᵢ to its parent, then the overlap set needs to contrain its sphere ids 𝑎ᵢ to those on B and ids 𝑏ᵢ to those on A. Here is a basic flat icosahedral limit set with a green mountainous set on top and a brown hilly set on the left. The subtitution spheres are neighbours so the patches overlap and you can see the discontinuity between them. This happend regardless of whether you switch to the brown or green set at the overlap. (3/4)

@johncarlosbaez@mathstodon.xyz I interpreted it that she is looking straight at us and in the mirror she is looking at the customer therefore we are the customer. But if the perspective doesn't add up then I guess we are the next customer along.

@runevision@mastodon.gamedev.place wonderful

@lisyarus@mastodon.gamedev.place @johncarlosbaez@mathstodon.xyz animating the stream function is likely non-physical but each time point is physical so as long as the animation is slow relative to the motion of particles it'd be close enough.

@metin as an aside the boiling frog effect is most likely fake. Nobody has ever reproduced it since the original experiments. Frogs jump out of hot water, even if its temperature is raised very slowly.

@divbyzero@mathstodon.xyz "Every Riemannian manifold that is topologically a sphere has infinitely many closed geodesics" If I think of the C(0) version where you have geodesics over a cube for example, then you probably get a general closed curve whenever the initial gradient is rational. For a cuboid you'd scale the set of gradients accordingly. So that seems to fit with the more general case. The word ergodic probably features in there somewhere too.

This arrangement retains six spheres for adjusting the style of terrain in each substitutions set. They fall into three categories: The first is either hierarchical hills (tree-tree) or craters (shell-shell). The second is a shell-tree, which is a combination of hierarchical hills and craters. Hills or craters can be inside hills or craters, but they don't overlap each other's edges. I'm not sure the class of the last type, I think it is still a shell-tree. It uses overlapping hills and craters to form a landscape primarily out of connecting ridges and valleys. Examples left, middle and right here.. (2/3)

@runevision Brilliant. That's exactly what I was pondering about a week or so ago: https://tglad.blogspot.com/2026/03/fluvial-landscapes.html I didn't get far, but this looks great so I'll enjoy reading about it.

@r_flash@mastodon.r-flash.eu it looks like water ripples rendered with reflectivity. Very cool.

@OscarCunningham@mathstodon.xyz that is a mad unicode symbol, totally visually confusing

@tao it reminds me of another interesting fact: on average your friends have more friends than you do. This has been verified empirically using Facebook data (and I had a go at modelling it mathematically here: https://tglad.blogspot.com/2023/12/friends.html) In both cases the person feels lesser than their peers due to a sort of hidden bias.

@tao@mathstodon.xyz sounds like it would help to quantify the roughness of the function. Smooth analytic functions are easier to extremise than fractals, which are easier than even rougher functions. Comes down to the asymptotic behaviour of its power spectrum. They typically require different optimisers. On the other hand, this roughness may very in space and with scale, making the problem harder. It may even vary non-smoothly with these!

@annierau I tend to agree about hippos and motorbikes, and even smashing animals into oncoming traffic - though the flying bear entry is still up on Wikipedia.

@tao when I was young 'clever' was one of the best regarded traits. Leonardo Da Vinci and Newton were clever people. Now people seem to value 'smart' and clever has been relegated to things like life hacks or making a pun.

@blinry@chaos.social @RedGlow@mastodon.gamedev.place if you reduce the size of the planets it'll probably look more and more complicated. Rather like with magnets: https://fractalfoundation.org/OFCA/magpendattractor.jpg

@tapirgirl@flipping.rocks closest animal to a snowman ;)

@tao@mathstodon.xyz nice feature