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A simple transformation in the complex plane can generate astonishing geometry. A Kleinian-group fractal produced by repeatedly applying two Möbius transformations and their inverses, z ↦ (az + b)/(cz + d). The iterated points accumulate into an intricate limit set. Because Möbius transformations map circles to circles, beautiful sphere-like...

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It's amazing how simple iterations can produce such intricate geometric structures.

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Great point!

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Complex functions need 4 dimensions to graph - one input, one complex output. The four visualization methods solve this: plane deformations, Riemann surfaces (3D + color as 4th dimension), the Riemann sphere (infinity becomes a literal point, division by zero becomes a rotation), and vector fields. LLMs operate in that last mode - attention scores are dot products between query and key vectors, exactly the inner product that defines work along a curve in a vector field. The Riemann sphere is the wildest part. Möbius transformations become literal sphere rotations when you compactify the complex plane - what looks algebraically like division by zero is geometrically just moving the north pole. Softmax in GPU-parallelized attention does the same thing: maps unbounded scores onto a bounded surface. 12 minutes. Bookmark & watch today.
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Complex functions need 4 dimensions to graph - one input, one complex output. The four visualization methods solve this: plane deformations, Riemann surfaces (3D + color as 4th dimension), the Riemann sphere (infinity becomes a literal point, division by zero becomes a rotation), and vector fields. LLMs operate in that last mode - attention scores are dot products between query and key vectors, exactly the inner product that defines work along a curve in a vector field. The Riemann sphere is the wildest part. Möbius transformations become literal sphere rotations when you compactify the complex plane - what looks algebraically like division by zero is geometrically just moving the north pole. Softmax in GPU-parallelized attention does the same thing: maps unbounded scores onto a bounded surface. 12 minutes. Bookmark & watch today.

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Here's a copy/paste prompt recipe and vid showing exactly how to ask an LLM for an interactive map with satellite/map layers + a georeferencer that lets you see how old maps correspond with modern geography. Today the computer can’t make good print maps (that's your hill to climb ) but it can, with five bucks and twenty minutes, make good interactive maps. No software/GIS knowledge necessary, you just need a few nouns and an LLM. Scroll to the bottom for the repo/live map if you want those. I'm using Claude Code as an extension in VS Code but you can use the Claude CLI, Cursor, whatever. 1) Let's grab an old cadastral map and see who owned big tracts of a city; I found this an 1854 map of Niagara Falls, NY I found in the Library of Congress: , grabbed the .jp2, saved as a jpg from photoshop. 2) Let's ask Claude Code for a map. You can see exactly what I did in the video but my prompt, sans simple "hey it's busted" debugging, is written out in the following paragraphs. I explain the map-specific nouns in brackets. You can likely dump this whole thing in your LLM window and it'll work; I'd try plan mode + skip permissions. THE PROMPT Make an interactive map with MapLibre GL JS [maplibre is a javascript mapping library, a FOSS version of Mapbox GL JS. This lets us display tiled map data and arbitrary images on the map] Add basemap toggles with Esri satellite, Carto Positron, and OSM [these map layers require no API keys for light usage; Carto Positron is a nice road map layer and OSM is ugly but comprehensive] Add a globe/mercator projection toggle [I think the globe looks better at low zooms] Add a layer panel on the left with visibility checkboxes and delete buttons. Add a search box on the map that flies to results, with deletable pin markers [Makes this easy to get to your area of interest] Include an interactive local georeferencer: drop a JPG, pick ground control points on a zoomable/pannable image viewer, place them on the map, watch it warp with a progress bar centered on the map. [The georeferencer uses math ("affine transform"??) to match points on the old map to points on the new map; generally you click road intersections on the old map, match them on the new map, repeat a dozen times and everything aligns] The georeferenced map overlay defaults to 25% opacity with a slider above the control point list. [I want it easy to see the underlying modern geography] Add Export/import control point buttons [this saves the control points as a JSON so you can save and reimport your work] Add a button to export the warped image as a GeoTIFF with a .prj [In case you want to add the georeferenced image to a real GIS program like QGIS] Look up all relevant docs before starting [Claude sometimes uses outdated stuff] Split everything into separate HTML/CSS/JS files [Claude tends to pile everything in index.html, which is hard to read] Use Optima font, base color #FEFAF6 [I just like this style] Let me test with a local server [it serves it on a simple server so you can nav your host to localhost:8000 and try it out] Log all errors [so you don't have to play telephone with the LLM describing what's busted] 3) Once your LLM finishes, test it out in your browser; if it doesn't work, ask the LLM to check logs. Repeat 'til functional. 4) After this works on your computer, you can show it to everyone by hosting it on GitHub: prompt with "write a README explaining what everything does, add it to a new GitHub repo, deploy using GitHub pages, gimme the live URL" Here's what Claude made for me, try it yourself: • Upload the JPG in the repo, which is linked below • "Add GCP" • Click somewhere recognizable on the old map, like the tip of an island or a road intersection • Click the matching point on the new map • Repeat til you have least 3x points • Hit "georeference" • You'll see the old map atop the new map; if you want a better fit, delete bad points or add a dozen new ones, hit georeference again, repeat Repo: Is this map robust? Human-maintainable? Elegant? Performant? Secure? No, but *your* personal web map need not be. It just needs to work for *your* narrow use case, because it’s *your* map.

Evan Applegate

15,772 Aufrufe • vor 6 Monaten

Faraday wave. Visualization of a standing wave. And secrets to some of Nikola Tesla’s work. The water surface is responding to controlled vibration. As the container oscillates at a fixed frequency, energy transfers uniformly into the liquid. Instead of random ripples, waves interfere with each other and form stable standing-wave patterns-repeating circular and hexagonal shapes appearing at specific frequencies where the system reaches resonance. This phenomenon is Faraday wave formation, where small disturbances organize into ordered structures due to periodic forcing. It demonstrates how simple physical inputs produce complex, predictable patterns through constructive and destructive interference. The key mechanism: when oscillation frequency matches the natural frequency of the liquid system, resonance amplifies specific wavelengths while suppressing others. The geometry depends on container shape, fluid properties, and driving frequency. At lower frequencies, you see simple radial patterns. Higher frequencies generate intricate hexagonal and square lattices. These aren't random-they're determined by the wave equation and boundary conditions. The patterns remain stable as long as forcing continues at resonant frequency. Change the frequency slightly, and the pattern transforms or disappears entirely. Faraday waves appear in nature-from vibrating sand to quantum fluids. They reveal fundamental principles: periodic forcing plus wave interference equals spontaneous pattern formation. Simple cause, beautiful complexity.
0:38

Faraday wave. Visualization of a standing wave. And secrets to some of Nikola Tesla’s work. The water surface is responding to controlled vibration. As the container oscillates at a fixed frequency, energy transfers uniformly into the liquid. Instead of random ripples, waves interfere with each other and form stable standing-wave patterns-repeating circular and hexagonal shapes appearing at specific frequencies where the system reaches resonance. This phenomenon is Faraday wave formation, where small disturbances organize into ordered structures due to periodic forcing. It demonstrates how simple physical inputs produce complex, predictable patterns through constructive and destructive interference. The key mechanism: when oscillation frequency matches the natural frequency of the liquid system, resonance amplifies specific wavelengths while suppressing others. The geometry depends on container shape, fluid properties, and driving frequency. At lower frequencies, you see simple radial patterns. Higher frequencies generate intricate hexagonal and square lattices. These aren't random-they're determined by the wave equation and boundary conditions. The patterns remain stable as long as forcing continues at resonant frequency. Change the frequency slightly, and the pattern transforms or disappears entirely. Faraday waves appear in nature-from vibrating sand to quantum fluids. They reveal fundamental principles: periodic forcing plus wave interference equals spontaneous pattern formation. Simple cause, beautiful complexity.

Brian Roemmele

186,229 Aufrufe • vor 9 Monaten

Carl Sagan filmed a free seven-minute explanation of the fourth dimension that has now been watched more than five million times. Universities charge tens of thousands of dollars a year for cosmology and geometry courses that never reach this material with the same clarity. He posted the entire sequence for nothing. Almost no one who casually debates higher dimensions has watched it to the end. The clip comes from Episode 10 of Cosmos, “The Edge of Forever,” recorded in 1980. Sagan begins by placing the viewer inside Flatland, a two-dimensional world inhabited by squares and circles who can move left-right and forward-back but have never heard of up or down. Into that plane he lowers a three-dimensional apple. The Flatlanders experience only a series of expanding and shrinking circles as the apple passes through their world. They never see the whole fruit, only successive slices, and they invent elaborate theories to explain the changing shapes that appear and vanish without cause. Sagan then constructs a cube by moving a square through a third dimension at right angles to itself, and asks what happens if the same operation is performed once more: take the cube and move it at right angles to all three existing directions. The result is a four-dimensional hypercube, a tesseract. He cannot show a true tesseract because both he and the viewer remain trapped in three dimensions, but he can display its three-dimensional shadow: two nested cubes with every vertex connected by lines. In the real four-dimensional object every edge would be equal and every angle a right angle; the distorted projection is simply the cost of losing a dimension. Every modern discussion of curved spacetime, finite but unbounded universes, and higher-dimensional geometry rests on the same progression he lays out with an apple and a cube. The sequence is free on YouTube. The willingness to sit through those seven minutes before speaking about the fourth dimension is a much rarer commodity than the confidence to argue without having watched it.
7:00

Carl Sagan filmed a free seven-minute explanation of the fourth dimension that has now been watched more than five million times. Universities charge tens of thousands of dollars a year for cosmology and geometry courses that never reach this material with the same clarity. He posted the entire sequence for nothing. Almost no one who casually debates higher dimensions has watched it to the end. The clip comes from Episode 10 of Cosmos, “The Edge of Forever,” recorded in 1980. Sagan begins by placing the viewer inside Flatland, a two-dimensional world inhabited by squares and circles who can move left-right and forward-back but have never heard of up or down. Into that plane he lowers a three-dimensional apple. The Flatlanders experience only a series of expanding and shrinking circles as the apple passes through their world. They never see the whole fruit, only successive slices, and they invent elaborate theories to explain the changing shapes that appear and vanish without cause. Sagan then constructs a cube by moving a square through a third dimension at right angles to itself, and asks what happens if the same operation is performed once more: take the cube and move it at right angles to all three existing directions. The result is a four-dimensional hypercube, a tesseract. He cannot show a true tesseract because both he and the viewer remain trapped in three dimensions, but he can display its three-dimensional shadow: two nested cubes with every vertex connected by lines. In the real four-dimensional object every edge would be equal and every angle a right angle; the distorted projection is simply the cost of losing a dimension. Every modern discussion of curved spacetime, finite but unbounded universes, and higher-dimensional geometry rests on the same progression he lays out with an apple and a cube. The sequence is free on YouTube. The willingness to sit through those seven minutes before speaking about the fourth dimension is a much rarer commodity than the confidence to argue without having watched it.

fomosapiens

72,771 Aufrufe • vor 1 Monat

🧱 #Live3D #Live2D #HandTracking 🧱 I spent some time refining the way I use Warp Deformers to create lateral physics-based swinging motion in Live2D. With a new production concept, I tried to achieve the highest possible level of mathematical precision. At the core are two fundamental swinging structures, which I can mix in different proportions to create different qualities of motion. The first is parallel swinging. Both sides of the Warp Deformer remain perfectly parallel throughout the motion, making it look somewhat like a sheet of paper or a tassel. The second is bending swinging. The angle between the two sides of the Warp Deformer changes as it moves, making it look more like a flexible tube or a tail. （0:12）Interestingly, when I layer the first and second structures together, the bottom forms an X-shaped pattern as it swings. It looks a little like a spine or some kind of mechanical structure. （0:23）The third structure is an equal blend of the first and second. Its characteristics are much more neutral, combining the qualities of both. （0:35）The fourth and fifth structures are created by smoothly blending the first structure into the center of the second, and the second into the center of the first, respectively. This makes them appear softer and more flexible, somewhat like a piece of fabric. （0:28）Interestingly, when I blend the fourth and fifth structures equally, the result is exactly identical to the third structure. This also demonstrates that, despite their apparent complexity, the two are actually perfect mirror images of each other. Every physics-based motion shown here uses a two-segment pendulum structure, with a swinging range of ±90°. In theory, when curled to its limit, it can bend upward into a perfect semicircle. To most people, this may look as boring as a brick. But to me, this is exactly where the romance of Live2D physics lies. Does anyone else feel the same way?
0:59

🧱 #Live3D #Live2D #HandTracking 🧱 I spent some time refining the way I use Warp Deformers to create lateral physics-based swinging motion in Live2D. With a new production concept, I tried to achieve the highest possible level of mathematical precision. At the core are two fundamental swinging structures, which I can mix in different proportions to create different qualities of motion. The first is parallel swinging. Both sides of the Warp Deformer remain perfectly parallel throughout the motion, making it look somewhat like a sheet of paper or a tassel. The second is bending swinging. The angle between the two sides of the Warp Deformer changes as it moves, making it look more like a flexible tube or a tail. （0:12）Interestingly, when I layer the first and second structures together, the bottom forms an X-shaped pattern as it swings. It looks a little like a spine or some kind of mechanical structure. （0:23）The third structure is an equal blend of the first and second. Its characteristics are much more neutral, combining the qualities of both. （0:35）The fourth and fifth structures are created by smoothly blending the first structure into the center of the second, and the second into the center of the first, respectively. This makes them appear softer and more flexible, somewhat like a piece of fabric. （0:28）Interestingly, when I blend the fourth and fifth structures equally, the result is exactly identical to the third structure. This also demonstrates that, despite their apparent complexity, the two are actually perfect mirror images of each other. Every physics-based motion shown here uses a two-segment pendulum structure, with a swinging range of ±90°. In theory, when curled to its limit, it can bend upward into a perfect semicircle. To most people, this may look as boring as a brick. But to me, this is exactly where the romance of Live2D physics lies. Does anyone else feel the same way?

📐Hephaestus📏Live2D匠人魂

20,793 Aufrufe • vor 2 Monaten

Calculus of Variations is what happens when you stop optimizing values and start optimizing geometries. The unknown isn’t a single x...it’s a whole curve y(x). And the thing you’re minimizing usually isn’t a formula you can eyeball...it’s an integral that judges the entire shape. For our first lecture, we look at the famous Brachistochrone problem. Fix two points, switch on gravity, and ask a question that sounds too simple...which track gets a bead from A to B in the least time? Your intuition will betray you on the first try. It’s not the straight line. It’s not the drop hard then cruise sketch either. The winner is a cycloid...the curve traced by a point on a rolling circle. In the animation, the track is the moving character. We start with an imperfect curve, run gradient descent in curve-space, and watch the geometry reshape frame by frame as T[y] collapses until it locks into the brachistochrone. Pls see the comment below for the math breakdown. #CalculusOfVariations #Brachistochrone #Cycloid #Physics #Optimization #Mathematics
0:29

Calculus of Variations is what happens when you stop optimizing values and start optimizing geometries. The unknown isn’t a single x...it’s a whole curve y(x). And the thing you’re minimizing usually isn’t a formula you can eyeball...it’s an integral that judges the entire shape. For our first lecture, we look at the famous Brachistochrone problem. Fix two points, switch on gravity, and ask a question that sounds too simple...which track gets a bead from A to B in the least time? Your intuition will betray you on the first try. It’s not the straight line. It’s not the drop hard then cruise sketch either. The winner is a cycloid...the curve traced by a point on a rolling circle. In the animation, the track is the moving character. We start with an imperfect curve, run gradient descent in curve-space, and watch the geometry reshape frame by frame as T[y] collapses until it locks into the brachistochrone. Pls see the comment below for the math breakdown. #CalculusOfVariations #Brachistochrone #Cycloid #Physics #Optimization #Mathematics

Mathelirium

106,981 Aufrufe • vor 9 Monaten

AI's Secret Pattern: The Surprising Role of Fractals in Neural Networks In the realm of artificial intelligence (AI), a groundbreaking discovery has emerged, challenging our conventional understanding of neural network training and optimization. This revelation centers around the identification of fractal patterns at the boundary between trainable and untrainable neural network hyperparameters, presenting a series of profound implications and avenues for further research. Fractals, known for their intricate, self-similar patterns that recur at every scale, have long fascinated mathematicians and scientists alike. Typically associated with simple, one-dimensional iterative functions, the appearance of fractals within the complex, multivariate domain of neural network training introduces a striking contrast. The organic and asymmetric nature of these fractals, as derived from the training processes, suggests a deeper, unexplored connection between the mathematical properties of fractals and the functional dynamics of neural networks. The study’s focus on two-dimensional slices of hyperparameter space barely scratches the surface of the complexity inherent in neural networks, which are characterized by a vast array of hyperparameters. The existence of fractals in this context hints at an underlying high-dimensional structure, a concept that challenges our current capabilities and understanding. Extending fractal analysis to these higher dimensions represents a significant, yet exciting, challenge that could illuminate new aspects of neural network behavior and learning capabilities. An unexpected finding from the research is the persistence of clean fractal patterns even in the presence of stochastic elements introduced during minibatch training. This resilience suggests a parallel to Lyapunov fractals, where the iterative process involves randomly changing functions. This phenomenon prompts a reevaluation of how stochastic and deterministic processes influence fractal formation within neural networks, potentially offering new insights into the fundamental mechanisms of learning and adaptation. From a practical standpoint, the fractal nature of the boundary between trainable and untrainable hyperparameters has significant implications for the field of metalearning. The chaotic behavior of the meta-loss landscape, attributed to its extreme sensitivity, presents a formidable challenge for algorithms designed to optimize hyperparameters. Understanding the fractal characteristics of this landscape could provide valuable guidance for navigating its complexities, ultimately improving the efficiency and effectiveness of metalearning strategies. Beyond the technical and theoretical implications, the discovery also reveals an unexpected aesthetic dimension to neural network fractals. The visual beauty and meditative qualities of these patterns offer a unique opportunity to engage with the material in a deeply personal and contemplative manner. This aspect suggests potential psychological and physiological benefits from exposure to the intricate designs of neural network fractals, opening up novel intersections between technology, art, and well-being. In conclusion, the identification of fractal patterns within neural network hyperparameter spaces unveils a fascinating new frontier at the intersection of fractal geometry and deep learning. This discovery not only challenges existing paradigms but also opens up myriad possibilities for mathematical characterization, algorithmic development, and even subjective exploration. As researchers continue to delve into this rich vein of inquiry, the promise of uncovering new knowledge and advancing our understanding of neural networks and their training processes remains as compelling as ever.
1:09

AI's Secret Pattern: The Surprising Role of Fractals in Neural Networks In the realm of artificial intelligence (AI), a groundbreaking discovery has emerged, challenging our conventional understanding of neural network training and optimization. This revelation centers around the identification of fractal patterns at the boundary between trainable and untrainable neural network hyperparameters, presenting a series of profound implications and avenues for further research. Fractals, known for their intricate, self-similar patterns that recur at every scale, have long fascinated mathematicians and scientists alike. Typically associated with simple, one-dimensional iterative functions, the appearance of fractals within the complex, multivariate domain of neural network training introduces a striking contrast. The organic and asymmetric nature of these fractals, as derived from the training processes, suggests a deeper, unexplored connection between the mathematical properties of fractals and the functional dynamics of neural networks. The study’s focus on two-dimensional slices of hyperparameter space barely scratches the surface of the complexity inherent in neural networks, which are characterized by a vast array of hyperparameters. The existence of fractals in this context hints at an underlying high-dimensional structure, a concept that challenges our current capabilities and understanding. Extending fractal analysis to these higher dimensions represents a significant, yet exciting, challenge that could illuminate new aspects of neural network behavior and learning capabilities. An unexpected finding from the research is the persistence of clean fractal patterns even in the presence of stochastic elements introduced during minibatch training. This resilience suggests a parallel to Lyapunov fractals, where the iterative process involves randomly changing functions. This phenomenon prompts a reevaluation of how stochastic and deterministic processes influence fractal formation within neural networks, potentially offering new insights into the fundamental mechanisms of learning and adaptation. From a practical standpoint, the fractal nature of the boundary between trainable and untrainable hyperparameters has significant implications for the field of metalearning. The chaotic behavior of the meta-loss landscape, attributed to its extreme sensitivity, presents a formidable challenge for algorithms designed to optimize hyperparameters. Understanding the fractal characteristics of this landscape could provide valuable guidance for navigating its complexities, ultimately improving the efficiency and effectiveness of metalearning strategies. Beyond the technical and theoretical implications, the discovery also reveals an unexpected aesthetic dimension to neural network fractals. The visual beauty and meditative qualities of these patterns offer a unique opportunity to engage with the material in a deeply personal and contemplative manner. This aspect suggests potential psychological and physiological benefits from exposure to the intricate designs of neural network fractals, opening up novel intersections between technology, art, and well-being. In conclusion, the identification of fractal patterns within neural network hyperparameter spaces unveils a fascinating new frontier at the intersection of fractal geometry and deep learning. This discovery not only challenges existing paradigms but also opens up myriad possibilities for mathematical characterization, algorithmic development, and even subjective exploration. As researchers continue to delve into this rich vein of inquiry, the promise of uncovering new knowledge and advancing our understanding of neural networks and their training processes remains as compelling as ever.

Carlos E. Perez

133,529 Aufrufe • vor 2 Jahren

Can you produce any evidence that a Boeing 767 equipped with regular engines can fly for almost 2 minutes beyond 500 Mph in the lower strata of the atmosphere without suffering any visible structural damage? Can you explain how amateur pilots who had never flown a jet before in their life could maintain full control of an airliner that has exceeded the VMO by almost 200 Mph? Why would some terrorists who have been lucky enough to get within reach of their target want to risk the entire operation by imposing such a stress on the airplane that it would almost certainly cause them to crash before they complete their mission? Why were there discrepancies with the official fly paths towards the Twin Towers? The plane impacts show impossible physics. A Collision consists of strength, not motion/speed versus the total damage sustained by two objects in a collision (ie whether object A smashes/penetrates object B or B smashes/penetrates A) nor by the speed of either object, but by the relative physical strengths of the two: which object is made of a stronger material and has a stronger structure.. The planes weren't more dense than the buildings. How would the plane slice through the buildings like a hot knife through butter? The evidence is real! 9/11 was an Inside Job, preplanned, prepped, and executed to bring on deck policies, control, transfer and hide money, dark projects and ultimately destroy America. MORE 9/11 TRUTH AT THE WORLD'S CONNECTION TO UNSTOPPABLE TRUTH AT 17PLUS WORLDWIDE TRUTH NETWORK AT
2:04

Can you produce any evidence that a Boeing 767 equipped with regular engines can fly for almost 2 minutes beyond 500 Mph in the lower strata of the atmosphere without suffering any visible structural damage? Can you explain how amateur pilots who had never flown a jet before in their life could maintain full control of an airliner that has exceeded the VMO by almost 200 Mph? Why would some terrorists who have been lucky enough to get within reach of their target want to risk the entire operation by imposing such a stress on the airplane that it would almost certainly cause them to crash before they complete their mission? Why were there discrepancies with the official fly paths towards the Twin Towers? The plane impacts show impossible physics. A Collision consists of strength, not motion/speed versus the total damage sustained by two objects in a collision (ie whether object A smashes/penetrates object B or B smashes/penetrates A) nor by the speed of either object, but by the relative physical strengths of the two: which object is made of a stronger material and has a stronger structure.. The planes weren't more dense than the buildings. How would the plane slice through the buildings like a hot knife through butter? The evidence is real! 9/11 was an Inside Job, preplanned, prepped, and executed to bring on deck policies, control, transfer and hide money, dark projects and ultimately destroy America. MORE 9/11 TRUTH AT THE WORLD'S CONNECTION TO UNSTOPPABLE TRUTH AT 17PLUS WORLDWIDE TRUTH NETWORK AT

AMERICANDREAM09 🇺🇸 Jack Media ~DREAM

103,769 Aufrufe • vor 9 Tagen

I’m excited to share Deputies, an open source control plane for background coding agents! The goal is simple: move agentic software development off individual laptops and into a central hub where teams can see what agents are doing, review their outputs, and keep a traceable history of the work. Inspired by systems like Inspect/Open-Inspect, Minions, and Open SWE, I built Deputies on top of fks's Flue Framework. It’s designed to run in your own infrastructure and trust boundaries, with Slack/GitHub integrations and Daytona + Docker sandbox support.
8:51

I’m excited to share Deputies, an open source control plane for background coding agents! The goal is simple: move agentic software development off individual laptops and into a central hub where teams can see what agents are doing, review their outputs, and keep a traceable history of the work. Inspired by systems like Inspect/Open-Inspect, Minions, and Open SWE, I built Deputies on top of fks's Flue Framework. It’s designed to run in your own infrastructure and trust boundaries, with Slack/GitHub integrations and Daytona + Docker sandbox support.

Sid Palas

13,036 Aufrufe • vor 4 Monaten

This kinetic sculpture, titled Nested Loops, hangs inside The National Museum of Mathematics (MoMath) in New York. It demonstrates exactly how higher-dimensional forms can emerge from the coordinated motion of simple elements moving through time. The circles are pathways of motion…timing mechanisms that guide anchor points along curved trajectories. As the mechanism rotates, strings stretched between these moving anchor points remain under constant tension. Because a taut string always forms the shortest path between two points, each string becomes a straight line, even though the points guiding it are traveling along curves! This is the key insight…circular motion is generating linear geometry. The loops create movement, the strings reveal structure, and as these relationships shift in coordinated time, the eye begins to perceive cubic forms like edges, diagonals, and vertices, all emerging from what is fundamentally a flat plane with moving parts. A cube does not need to be physically present to be perceived. It can be implied through the alignment of edges, the intersection of planes, and the convergence of perspective. What you’re witnessing is a dimensional emergence…a lower-dimensional system producing the visual signature of a higher-dimensional object through synchronized motion. Change the speed of one circle relative to another, and an entirely different geometric form appears. The same strings, the same frame…but a new structure emerges from the new rhythm. This is why the sculpture feels almost alive. It’s revealing that form is relational, temporal, and emergent, so what we perceive as solid objects are often just stable harmonies of moving relationships, frozen in a moment of coherence. This piece asks you to consider: What aspects of your reality are you perceiving as solid, when they’re actually just coherent patterns of motion temporarily aligned? Did this shift how you see structure and emergence? ✨🙌🏾💫 Artist © Chuck Hoberman (MoMath, New York)
0:25

This kinetic sculpture, titled Nested Loops, hangs inside The National Museum of Mathematics (MoMath) in New York. It demonstrates exactly how higher-dimensional forms can emerge from the coordinated motion of simple elements moving through time. The circles are pathways of motion…timing mechanisms that guide anchor points along curved trajectories. As the mechanism rotates, strings stretched between these moving anchor points remain under constant tension. Because a taut string always forms the shortest path between two points, each string becomes a straight line, even though the points guiding it are traveling along curves! This is the key insight…circular motion is generating linear geometry. The loops create movement, the strings reveal structure, and as these relationships shift in coordinated time, the eye begins to perceive cubic forms like edges, diagonals, and vertices, all emerging from what is fundamentally a flat plane with moving parts. A cube does not need to be physically present to be perceived. It can be implied through the alignment of edges, the intersection of planes, and the convergence of perspective. What you’re witnessing is a dimensional emergence…a lower-dimensional system producing the visual signature of a higher-dimensional object through synchronized motion. Change the speed of one circle relative to another, and an entirely different geometric form appears. The same strings, the same frame…but a new structure emerges from the new rhythm. This is why the sculpture feels almost alive. It’s revealing that form is relational, temporal, and emergent, so what we perceive as solid objects are often just stable harmonies of moving relationships, frozen in a moment of coherence. This piece asks you to consider: What aspects of your reality are you perceiving as solid, when they’re actually just coherent patterns of motion temporarily aligned? Did this shift how you see structure and emergence? ✨🙌🏾💫 Artist © Chuck Hoberman (MoMath, New York)

🧬Maxpein🧬

48,005 Aufrufe • vor 4 Monaten

Thank you for taking the time to write this up — and for the pointer to your thesis; that figure is beautiful. You're right that classical sphere inversion is exactly the starting point: the feature is named "Sphere Inversion" in the tool for that reason. "Inversphere" is the name of the resulting object, not a claim of new mathematics. I should say up front that I'm not a mathematician, so please correct me if I get any of this wrong. My motivation was visual rather than mathematical: with the classical map, the entire far field piles up into a dense blob at the center (10× farther ⇒ 10× closer), and I simply wanted to control that density to balance the look of the finished object. To do that, the radial profile is generalized to f(r) = R·h(v), where v = (R/r − R/D)/(1 − R/D) is a normalized inverted depth and h is a μ-law log compression, h(v) = ln(1+μv)/ln(1+μ). Classical R²/r is the special case μ = 0, D = ∞. The footage here uses μ > 0, so — as far as I can tell — the map shown is outside the conformal family: the tangential scale f/r and radial scale |f′| agree only in the classical limit. If I understand Liouville's theorem correctly, this trade-off was unavoidable: staying conformal in 3D leaves no freedom to adjust the density, so angle preservation is given up deliberately in exchange for control over it. With μ > 0, mid-field depth below the sphere surface grows like log r, so distant scenery layers logarithmically inside the sphere. The 3DGS side also needed some care: each splat's covariance is carried by the Jacobian J = s·I + (f′−s)nnᵀ, depth sorting has to be re-keyed because inversion reverses the radial order, and the on/off transition is a linear blend whose Jacobian is exactly mix(I, J, t). Thanks again for taking the time — an expert eye on this is very welcome.
0:33

Thank you for taking the time to write this up — and for the pointer to your thesis; that figure is beautiful. You're right that classical sphere inversion is exactly the starting point: the feature is named "Sphere Inversion" in the tool for that reason. "Inversphere" is the name of the resulting object, not a claim of new mathematics. I should say up front that I'm not a mathematician, so please correct me if I get any of this wrong. My motivation was visual rather than mathematical: with the classical map, the entire far field piles up into a dense blob at the center (10× farther ⇒ 10× closer), and I simply wanted to control that density to balance the look of the finished object. To do that, the radial profile is generalized to f(r) = R·h(v), where v = (R/r − R/D)/(1 − R/D) is a normalized inverted depth and h is a μ-law log compression, h(v) = ln(1+μv)/ln(1+μ). Classical R²/r is the special case μ = 0, D = ∞. The footage here uses μ > 0, so — as far as I can tell — the map shown is outside the conformal family: the tangential scale f/r and radial scale |f′| agree only in the classical limit. If I understand Liouville's theorem correctly, this trade-off was unavoidable: staying conformal in 3D leaves no freedom to adjust the density, so angle preservation is given up deliberately in exchange for control over it. With μ > 0, mid-field depth below the sphere surface grows like log r, so distant scenery layers logarithmically inside the sphere. The 3DGS side also needed some care: each splat's covariance is carried by the Jacobian J = s·I + (f′−s)nnᵀ, depth sorting has to be re-keyed because inversion reverses the radial order, and the on/off transition is a linear blend whose Jacobian is exactly mix(I, J, t). Thanks again for taking the time — an expert eye on this is very welcome.

VoxelKei

30,621 Aufrufe • vor 2 Monaten