Tom Lowe

Mastodon @toml@mathstodon.xyz

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Here's a 600-cell inversive limit set.
This is a set that is invariant to inversions around spheres centred at the 600-cell's vertices, and stereographically projected into 3D.
The spheres must meet at a dihedral angle of π/𝑛 for some integer 𝑛. So there are a countably infinite number of such shapes, however there are three that are the key ones:
𝑛=3: the set complement of a ball packing.
𝑛=4:

10 views Here's a 600-cell inversive limit set. This is a set that is invariant to inversions around spheres centred at the 600-cell's vertices, and stereographically projected into 3D. The spheres must meet at a dihedral angle of π/𝑛 for some integer 𝑛. So there are a countably infinite number of such shapes, however there are three that are the key ones: 𝑛=3: the set complement of a ball packing. 𝑛=4:

Here's the 𝑛=∞ case.
Both have a tetrahedral rotational symmetry, that's because the cells of the 600-cell are tetrahedra.
While these are stereograpic projections of objects in 4D, they aren't themselves 4D objects. They are subsets of the 3-sphere.
I prefer to think of them natively in 3D. So rather than just a projection from a shape in 4D they are a 'conformal type' in 3D.
While a shape is

7 views Here's the 𝑛=∞ case. Both have a tetrahedral rotational symmetry, that's because the cells of the 600-cell are tetrahedra. While these are stereograpic projections of objects in 4D, they aren't themselves 4D objects. They are subsets of the 3-sphere. I prefer to think of them natively in 3D. So rather than just a projection from a shape in 4D they are a 'conformal type' in 3D. While a shape is

This limit set is different to other substitution limit sets (https://www.researchgate.net/publication/407295122_Inversive_Substitution_Systems) the blue and green tree limbs interlock a little, overlapping each other's space.
This normally doesn't happen in escape-time limit sets because the geometry is local to individual spheres. So to support it the distance estimator needs to make a new iter

6 views This limit set is different to other substitution limit sets (https://www.researchgate.net/publication/407295122_Inversive_Substitution_Systems) the blue and green tree limbs interlock a little, overlapping each other's space. This normally doesn't happen in escape-time limit sets because the geometry is local to individual spheres. So to support it the distance estimator needs to make a new iter

The above example was a tree-cluster-tree. A cluster-tree is like the Mandelbrot set, lots of dense balls that kiss in a tree structure (acyclic connectivity). A tree-cluster-tree is the same but the dense balls are trees.
Anyway, here's a similar example but with just a tree structure. The limbs are non-aligned, and there's a bit of overlapping of each set's space.
I suspect that this method cou

5 views The above example was a tree-cluster-tree. A cluster-tree is like the Mandelbrot set, lots of dense balls that kiss in a tree structure (acyclic connectivity). A tree-cluster-tree is the same but the dense balls are trees. Anyway, here's a similar example but with just a tree structure. The limbs are non-aligned, and there's a bit of overlapping of each set's space. I suspect that this method cou

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Engagement vs followers

Sep 2024

On Mastodon since

🔥 Top post: Here's a 600-cell inversive limit set. This is a set that is in · 10 likes + reposts

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1.3 / week

A lower-frequency account — each post lands with more weight.

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Recent: 1 text · 9 image · 2 video.

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Follows 260 back. A more reciprocal / networked account.

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▶ 🔥 Top post Here's a 600-cell inversive limit set. This is a set that is invariant to inversions around spheres centred at the 600-cell's vertices, and stereographically projected into 3D. The spheres must meet at a dihedral angle of π/𝑛 for some integer 𝑛. So there are a countably infini… ★ 10

The above example was a tree-cluster-tree. A cluster-tree is like the Mandelbrot set, lots of dense balls that kiss in a tree structure (acyclic connectivity). A tree-cluster-tree is the same but the dense balls are trees. Anyway, here's a… ★ 5

This was built from a basic octahedral set (grey) with two spheres added on its left hand side. To get interlocking limbs that don't join to a sponge we need to substitute each coloured sphere with the set minus its opposing sphere. So the…

This limit set is different to other substitution limit sets (https://www.researchgate.net/publication/407295122_Inversive_Substitution_Systems) the blue and green tree limbs interlock a little, overlapping each other's space. This normal… ★ 6

The least dense needs a little work to render it nicer, but this is the shape. The realtime renderer is here: https://www.shadertoy.com/view/sXlGWs (3/3) ★ 3

The 5-cell, 8-cell, 16-cell and 24-cell inversive limit sets are in the void-sponge class here: https://www.shadertoy.com/view/cslfWn The last of the regular polychora is the 120-cell. This the last as it has 600 vertices, which is a bit t… ★ 4

There is a denser 600-cell inversive limit set if we break the rules and allow half-integer orders. This is the densest that isn't the complement of a ball packing, at 𝑛=3.5. It isn't really breaking the rules, the dihedral angle between s… ★ 5 600-cells are interesting, their usual choice of 4D vertices can also be interpreted as quaternions 𝑞ᵢ, and since a rotation in 4D is specified by two quaternions, we can get all of the symmetries of the 600-cell by applying all combinatio… ★ 4

▶ Here's the 𝑛=∞ case. Both have a tetrahedral rotational symmetry, that's because the cells of the 600-cell are tetrahedra. While these are stereograpic projections of objects in 4D, they aren't themselves 4D objects. They are subsets of … ★ 7

The last method needs a tweak, the little green patches do need the transition (dull yellow) tile between them and the larger brown patch. But with that in place it seems to be solidly continuous. Here I'm stress testing it by changing the… ★ 2

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 … ★ 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 tric… ★ 3

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On Mastodon since

Sep 2024

An established account with real history on the platform.

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🗓️Joined Mastodon in 2024 — 1 year ago.

👁️Averages 5 views per post.

📤Posts about 1.3× per week.

🛡️ Audience credibility

75/100

Good Credibility score

92%

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Est. 92% real, active audience · Low fake-follower risk.
Strong engagement (~2.4% of followers engage each post) — an active, real audience.

Heuristic estimate from engagement, follower ratios, account age & growth — a screening signal, not a guarantee.

About

I'm interested in Computer Graphics, Geometry, Mathematics and Physics. I have worked in Games Development, Animation Development, and now in Robotics Development, I guess that makes me a developer. Vous pouvez me parler dans n'importe quelle langue, c'est assez facile à traduire en ligne si je ne le connais pas.

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AHAB 10%

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Tom Lowe

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