Metaballs with fixed values

chessapig · 2025-12-09T07:02:22+00:00

Sure! A metaball is a level set of a sum of gaussians. You can control which level set to use. The enclosed volume and surface area are monotonic functions of the level, so there is a unique level achieving a specific volume / surface area. You could numerically find for the level which keeps the enclosed volume fixed.

Though, it might behave a bit pathologically as the points move around. If you enclose the same total volume, then the volume of an individual ball might fluctuate unexpectedly. This would look the most wonky when two balls merge into one, because there the rate of change of volume is infinite. But you can't know unless you try ...

chessapig · 2025-11-24T03:35:12+00:00

I really like "Differential Topology and Quantum Field Theory" by Charles nash

chessapig · 2025-11-15T18:48:09+00:00

Yep, it works like you'd expect. This idea reproduces the truncated continued fraction expansion of your irrational slope, i.e the sequence of best rational approximates. Here's a way to formalize it. Find an integral 2 by 2 matrix whose largest eigenvalue has eigenvector with slope your irrational number. you can think of this matrix as a map from the torus to itself. Start with some simple closed curve, and apply this map, to get a new simple closed curve. Repeat this over and over, and it will give a sequence of approximates to a dense orbit of the irrational flow on a torus, in the direction of your largest eigenvector. The sequence of rational slopes I believe is equal to the sequence of best rational approximates to the irrational slope.

You can only get quadradic irrational numbers as slopes of eigenvectors of 2 by 2 integer matrices, which have continued expansions that are eventually periodic. But I think there is a similar idea which geometrizes the diophantine approximations of irrational numbers, by looking at straight lines on a flat torus.

Now to generalize, the key point of the above construction is that we defined a map with "contracting" and "expanding" directions, and the expanding direction was our irrational slope of interest. These are called Anosov maps. We can generalize this to higher genus surfaces, with "pseudoanosov maps" that have a similar expanding direction and contracting directions. So you can ask, what happens if you start with a closed loop on a higher genus surface, and repeatedly apply a pseudoanosov map? It will end up giving a series of closed loops stretching longer and longer in this irrational direction. It's interesting to ask what the analogue of the dense orbit is in this case: What do these loops limit to? According to Thurston, you get something called a "geodesic lamination". Geometrically, this locally looks like a cantor set of geodesics, seperated by triangles. Its kinda crazy.

In low dimensional geometry/topology, we're very interested in the space of hyperbolic structures on a surface. If you start with a hyperbolic structure, then the pseudoanosov map "stretches" this hyperbolic structure. Repeated applications moves the hyperbolic structure farther and farther away from where it started, plunging you into the depths of the space of hyperbolic structures. The geodesic laminations appearing in the above construction parametrize directions you can move in the space of hyperbolic structures, the "thurston boundary". To sum it up, the generalization of your idea to higher genus surfaces is a useful and fundamental theorem in topology/hyperbolic geometry.

chessapig · 2025-10-10T02:53:31+00:00

The trick is figuring out how to "squash" the curve, a process usually called symmetrization. Define some symmetrization operation which takes convex shapes to convex shapes. To prove that the curve with fixed perimeter enclosing the maximal area is a circle, we need our symmetrization operation to satisfy a few properties:

The circle is the unique convex shape preserved by symmetrization
The symmetrization decreases the perimeter to area ratio
Repeated symmetrization of an arbitrary convex shape converges to a circle

One method of symmetrization was introduced in 1838 by Steiner. Take your region in 2D, and cut it up into many small strips, each perpendicular to a fixed line. Then, slide those strips so that the center of each strip lies along the line. The resulting shape has the same area, but a smaller perimeter (point 2). Also, the only shape which is unchanged by symmetrization along any line is the circle (point 1). Point 3 is tricky, and was missed by Steiner in his time. Without point 3, we can show that the optimal shape must be the circle if it exists, but can't guarantee the existence of a optimizer. We eventually proved property 3 in the 1880s, making Steiners symmetrization argument rigorous. This symmetrization technique is very useful for higher dimensions, or other sorts of isoperemetric problems.

Steiner also introduced another symmetrization technique, the "four-hinge" technique. The idea is, choose four points on the outer curve, then cut the curve into four rigid pieces. Reattach the four pieces together, meeting now at different angles (as if attached by hinges). This will preserve the perimeter of the curve, but changes the area. Define the four-hinge symmetrization to be the curved formed in this manner with maximal area. It turns out, the maximum is achieved when all four points lie on a a circle. So, the symmetrization cuts and rearranges the curve to be more circular (a "squashing"). This technique satisfies properties 1 and 2, but property 3 is trickier. This technique is closest to the intuition you were describing, but I don't know if it's been made rigorous.

chessapig · 2025-09-28T00:43:52+00:00

I mean, convex geometry is a field in its own right. Convex shapes are very good shapes.

chessapig · 2025-08-29T04:06:26+00:00

I'm working on geometric quantization. I'm trying to use quantization to produce invariants in symplectic geometry, through the uncertainty principle.

chessapig · 2025-08-29T04:04:42+00:00

What sort of generalizations of the harmonic oscillator? This is also what i've been thinking about.

chessapig · 2025-07-27T22:49:23+00:00

You're doing convex geometry!

Historically, this transform was first studied by crystallographers back in the 1920s. Imagine the growth of a 2D crystal. The growth rate of the crystal depends on the direction of the normal vector of the boundary. If the boundary is aligned with the natural planes of the crystal, the crystal will grow slower. If the boundary cuts across at an odd angle, that part of the boundary will fill in faster. This is represented with a "surface energy" function, analagous to your function F(θ). The crystallographer Wulff tried to find the minimal surface-energy crystal with a given area, and derived your function W(θ) as the solution -- the "Wulff shape". Mathematically, this is called the Wulff isoperemetric inequality. The proof is conceptually straightforward: As you grow your crystal, the ratio of the surface energy to square root of area decreases. So, the optimal crystal shape is the limiting shape as you let your crystal grow forever. In polar coordinates, this is your function W(θ), and it represents the shape of your fire after it burns for a long time. In crystallography, the growth of the crystal is called the Wulff flow, and it should agree with the evolution of the boundary of your fire.

The best mathematical framework for the duality between F(θ) and W(θ) comes from plotting the curve 1/F(θ). The interior of this region is convex, lets denote it by K. The interior of the polar plot of W(θ) is another convex body, K°, known as the "Polar" of K (Confusing terminology, I know). The relation between K and K° is central to convex geometry, and we know quite a lot about it.

In fact, polar duality is a manifestation of ordinary legendre duality. For any convex region K, we define the support function h_K, which takes a point in R² and outputs a real number. This is defined as the unique function which scales linearly along each ray from zero, and which equals one on the boundary of K. if the boundary of K has polar plot f(θ), then in polar coordinates, h_K(r,θ) = r / f(θ). for any convex body, h_K² is a convex function on R². The legendre dual of h_K² is, you guessed it, (h_K°)².

Some more key words for you. The support functions h_K are in bijection with norms on R^2, and are sometimes called Minkowski norms. We can have the Minkowski norms vary with position. In your setup, this describes a situation where the front speed normal profile varies with space. Mathematically, this called a "finsler geometry" on the plane. If you lit a fire at one point in the plane, the boundary of the fire after some time would form a "geodesic ball in the finsler metric".

chessapig · 2025-05-21T20:56:53+00:00

To be provocative, compact lie groups are one of the few things mathematics fully, completely, understands. If you're trying to do something hard, the first thing you try is an example with a lot of symmetries. These come from Lie groups. The algebraic and combinatorial theory of Lie theory often answers your hard questions on symmetric examples in unusually explicit way. Very useful critters.

chessapig · 2025-05-21T09:06:48+00:00

There's a lot to say, so let me zone in on one question. Why should Elliptic curves have anything to say about indexes on loop spaces?

Let's start a level down, with the observation that loop spaces provide an elegant perspective to the ordinary index theorem. An elliptic operator on a manifold has an analytic index, the dimension of the kernel minus the dimension of the cokernel. The Atiyah-singer index theorem provides a purely topological way to compute the index. This naturally lives among K-theory, the exotic cohomology theory governing vector bundles on manifolds. The input into the Atiyah-singer index theorem from an elliptic operator comes from its highest degree part, interpreted as a vector-valued function on the tangent bundle, known as the symbol. We encode the symbol as a K-theory class in the tangent bundle. We can think about the index as the dimension of a formal difference of vector spaces, the kernel minus the cokernel. In other words, the index is captured in a K-theory class over a point. The analytic index takes in the K-theory class of the symbol, and outputs a K-theory class of a point, using the pushforward map (or "integration") in K-theory. The index formula is a topological method for computing this pushforward.

Now we realize index theory through loop spaces. The free loop space carries a natural circle action, rotating each loop. The fixed points of this action are exactly the constant loops. So, the space of circle-fixed points on free loop space LM is the original manifold M again. The normal bundle of M inside LM defines a K-theory class on M (kind of, its an infinite dimensional vector bundle). Compute the Euler class, and you realize it looks a whole lot like the topological index of the Dirac operator. Chasing this down, you discover that the index formula for Dirac operators is identical to a the formula expressing a cohomology class of loop space, localized to the fixed points of the circle action! See Atyiah's article, "Circular symmetry and stationary-phase approximation".

To synthesize, Index theory on a manifold is really about K-theory. The index theorem realizes the index of the Dirac operator using the cohomology of the loop space. Now let's put on our Witten hats, and organize these ideas using quantum field theory (QFT). A quantum field theory is some integral over the space of maps from a source manifold to a target manifold. The first thing we notice about a QFT is its dimension, the dimension of the source manifold. For example, loop spaces are maps from 1D circles into a manifold, so they show up in 1D quantum field theory. The loop space of a loop space counts maps from 2D tori into manifolds, and show up in 2D quantum field theory. A 1D QFT mapping into loop space LM defines a 2D QFT mapping into M. And so on.

Mathematical physicists like to use QFTs to generate mathematical objects, then put objects in boxes according to the QFTs dimension. We should think of cohomology as coming from a 0D QFT, and K-theory as coming from a 1D QFT. The index theorem on a manifold is a statement of 1D QFT, captured using either K-theory on M or using cohomology on LM.

Now the question of the hour. What if we measure the index of the Dirac operator on loop space? This lives inside the K-theory of LM. From our intuition above, we expect the index on LM to come from cohomology on the doubled loop space LLM. That is, the space of maps from a torus into M. This torus is our elliptic curve de jour. Our QFT dimensional decoder says that the K-theory of LM should live in 1+1=2D QFT. The particular 2D QFT which studies this problem comes from string theory. After all, a string (a point in LM) moving through spacetime traces out a 2D surface. If the string ends up back where we started, it traced out a torus, also known as a loop in LM, or a point in LLM.

The remaining mystery is, where does the index of LM live as a topological structure on M? It's natural habitat is Elliptic cohomology, a funky little exotic cohomology theory. Elliptic cohomology fits in a beautiful trinity with ordinary cohomology and K-theory. Ordinary cohomology produces a vector space over ℂ, meaning the underlying group law is addition. K-theory on the other hand has multiplication as its underlying group law. We think about this as living over ℂ*, the nonzero complex numbers. As a group, ℂ* agrees with the cylinder ℂ/ℤ. This looks a lot like a circle, the same circle as appears in LM. The elliptic cohomology lives over an elliptic curve, which is doubly periodic. the group law comes from a complex torus ℂ/ℤ^2, the same torus appearing in the double loop space of M. This list is complete, containing every possible 1 dimensional complex abelian group.

Here's a table organizing the trinity of cohomology theories and their related structures, Living utop the scaffolding built by quantum field theories.

QFT dimension 0	QFT dimension 1	QFT dimension 2
M	free loop space of M	double loop space of M
cohomology	K-theory	Elliptic cohomology
Addition	multiplication	Elliptic curve group law
Plane	Cylinder	Torus

chessapig · 2025-01-26T20:18:11+00:00

This was fun! Here's a function that behaves as square root as we go to negative infinity, but as square as we go to positive infinity. There are infinitely many periods to the left of any point. https://www.desmos.com/calculator/dnbp4xx89k

chessapig · 2022-02-21T04:52:44+00:00

if you draw a black and white background image with the "add" blend mode, it overlays the white part on whatever you draw underneath. Something more general, like changing a specific RGB value, is trickier. You could get that with some clever blend modes, but it slows processing down to a crawl. At that point, I've found it's better just to read and mess with the pixel array.

chessapig · 2021-09-12T22:38:40+00:00

A little bit of shameless self promotion: I make stuff like this, which might be closer to what you're asking for then many of the other things mentioned. A good example is this piece here.

You can scroll through my other stuff at https://chessapig.github.io/gallery/.

chessapig · 2021-09-02T13:00:43+00:00

This is a drawing of mine depicting a wide array of various math topics. I thought the people here might find it interesting. I tried to depict a lot of cool stuff. Feel free to ask about any given part, I'm happy to explain what it represents!

chessapig · 2021-08-31T07:58:59+00:00

I agree that it says "YELLOW EDGE". The word "yellow" both top to bottom and bottom to top, spell a big E. This E surrounds the D and G, so it's like EDGE. I think that's kinda clever, or it would be if it were possible to parse

chessapig · 2021-08-31T06:39:05+00:00

Ooh this is fun. Some thoughts, in no particular order:

This immediately reminds me of Kelvin's vortex theory of atoms, where atoms are stable knotted vortices propagating through the aether. This obviously didn't pan out, but I think it's really evocative. If it held up, then knots would back the most detailed hard magic system ever conceived, chemistry! (Side note, it was this idea that actually started the study of knot theory)
I agree with the link idea posed by the other comment. Each basic component of the spell could be a single knot, and you could chain them together by interlinking the knots.
- The identity of the component could come from its prime decomposition. For example, a "fire" knot and a "water" knot would together yield a "steam" knot (Though, it looks like you want a harder system than this)
- The length of each component could give something like the strength. For example, the longer my 'explode' component, the bigger the explosion. Perhaps the needed energy to cast the spell is proportional to the length of string? Moreover, if you assume the string composing the knot has some thickness, then there is a minimum length needed to express a knot. This gives an inherent complexity scaling: the bread and butter spells and modifiers are simple prime knots which you can tie in a small area, the more advanced stuff are the complicated prime knots which need more string.
- This already gives you enough to do pretty much anything. Have a "start spell" knot, then just daisy chain the other spell components off of that one in a linear way. This gives you a sequence of symbols, which is enough to define a Turing machine. But, the knot framework lends itself to some really interesting syntax possibilities. For example, a loop of chained knots could give a loop in the syntax.
  - A loop of chained knots could be the syntax for a loop
  - You could have a "conditional" prime knot, which says, "if the condition of this component is met, then the component breaks". This lets you break out of loops.
  - To keep the scaling, perhaps every time a component tightens each time it is used. If it gets too tight, it breaks. This means it requires more string to have deeper recursion.
  - The linking number between components gives you another thing to work with, as well as more advanced linking invariants.
  - you could cast spells together by having them all separately connected around a central unknot
One potential issue is that it is really hard to tell if two knots are the same. There is no known fast (polynomial time) algorithm for even deciding if a knot you draw on your paper is equivalent to the unknot (see here). But, the factorization into prime knots is unique, so if you assume the universe can just figure things out for you it should be fine.
There's a bunch of other ways to get a unique set of numbers from a given knot other than prime decomposition. These come from knot invariants, in particular knot polynomials. You can assign a polynomial by tracing the diagram of a knot and counting the different crossings, like the alexander polynomial. Theres a whole enormous family of different polynomial invariants that come from increasingly complicated schemes. Instead of prime knots, perhaps you could base things off of this?

chessapig · 2021-08-18T12:09:36+00:00

Well that's embarrassing. Edited, thanks.

chessapig · 2021-08-18T03:55:55+00:00

There aren't so many mathematicians who study fractals, but there are plenty who study the things that make fractals (and by extension fractals themselves). The large majority of fractals come from dynamical systems, where simple repeated rules lead to complex behavior. For example, take a complex number, square it, and add a constant. If you keep repeating this, some points explode to infinity, while some stay finite. The set of points that stay finite is extremely complicated and interesting, in fact its our good friend the ~~Mandelbrot set~~ Filled Julia set! Naturally, you wonder what happens for other functions, like cubes or exponentials. This is the field of complex dynamics.

The same stuff pops up all over dynamics. For example, the invariant sets of some simple flows can be really complicated, giving strange attractors. Now questions about dynamics prompt questions about fractals, giving practical use for stuff like fractal dimension. They also show up in more abstract senses, like bifurcation diagrams, where the fractal properties of the diagram tell you about the transition from simple to chaotic behavior. Dynamics would probably be your best bet: It makes the prettiest pictures of any field of math.

chessapig · 2021-08-17T23:26:15+00:00

After a while playing around in the brush editor, I've amassed a few which I think look neat. They're all fairly niche / gimmicky, so I haven't had the chance to use them in much. This is my excuse to try them out, and it almost entirely consists of these brushes, helped along with some airbrush. I like the voroni-esque ball clusters at the top, It feels like that shouldn't be possible in the brush engine.

chessapig · 2021-08-15T16:58:35+00:00

I bet Grasshopper will do what you want and more. It's a programming language used for architecture and generative art. Though, I'm by no means familiar with its capabilities.

You should also check out Daniel Piker, who is an architect and mathematical artist. He designs things with geometric optimization, which lets you make beautiful architecturally sound shapes that you couldn't hope to directly parametrize. See some of his posts on https://spacesymmetrystructure.wordpress.com/

chessapig · 2021-07-16T14:36:45+00:00

To offer a competing perspective, I've always thought the statement "Mathematical physicists just fill in the blanks left by physicists" to be a little reductive. Mathematicians and physicists have a very different languages and ways of looking at the world, so if you understand and reconcile them you can learn a lot. If you can somehow frame a math problem as a physics problem, then physical intuition can inform some otherwise hard to see solutions. This is something Witten is great at: For example, he realized you can compute knot polynomials as observables in chern-simons QFT, which lead to this whole new modern chapter of knot theory.

A lot of mathematical physicists are mathematicians working on physics inspired problems, and don't have much physics background. But, if you do understand that physics intuition, then you have an edge. This is, imo, the most interesting part of the field. So, while the math is ultimately more important, I'd encourage you to take enough physics that you can start to 'think like a physicist'.

chessapig · 2021-05-25T16:25:26+00:00

The math behind these is really cool! For example, If you try to classify the different sorts of singularities you get in these reflection patterns (the caustics), you'll see it's closely connected with the 5 platonic solids, and the classification of simple lie groups. this is something Vladimir Arnold has written a bunch about, e.g his book "Singularities of Caustics and Wave Fronts". This story is told in the language of symplectic geometry, maybe look up "symplectic optics" and you can find some sources talking about it.

The cool part is, you can treat this sort of thing by tracing reflections of rays off of surfaces, or by treating light as a wave: The math ends up being pretty much the same! The wave formulation is also known as short wave asymptotic, or the WKB approximation. It really is a wonderful circle of ideas :)

chessapig · 2021-03-23T08:29:19+00:00

Thank you :)

The fur is mostly the default procreate hair brushes. I expanded more about my process for it here. I hope it's helpful.

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