Mathematica's banner

Mathematica

@mathemetica • 34,318 subscribers

Math isn't escape. It's the map through the madness.

Shorts

max⁡(∣x∣,∣y∣,∣z∣,∣w∣)=1 You’re witnessing a 2D projection of a 3D shadow cast by a 4D tesseract. As it rotates through the W-axis, the "inner" and "outer" cubes swap roles: a spatial inversion that feels like a glitch only because our biology is trapped in 3-space. It’s a hauntingly beautiful reminder that our "reality" is often just a lower-dimensional cross-section of a much more complex structure. Perspective is everything. This is used in high-dimensional data visualization, hypercube topologies in parallel computing networks, and exploring the geometry of extra dimensions in theoretical physics.

max⁡(∣x∣,∣y∣,∣z∣,∣w∣)=1 You’re witnessing a 2D projection of a 3D shadow cast by a 4D tesseract. As it rotates through the W-axis, the "inner" and "outer" cubes swap roles: a spatial inversion that feels like a glitch only because our biology is trapped in 3-space. It’s a hauntingly beautiful reminder that our "reality" is often just a lower-dimensional cross-section of a much more complex structure. Perspective is everything. This is used in high-dimensional data visualization, hypercube topologies in parallel computing networks, and exploring the geometry of extra dimensions in theoretical physics.

600,490 Aufrufe

Paschen’s law: V_b = (B p d) / [ln(A p d) - ln(ln(1 + 1/γ))] Pull the plunger on a syringe next to a live Tesla coil and watch physics lose its mind. The instant pressure drops, that p d term collapses; breakdown voltage craters, the field ionizes the rarefied air, and violet plasma arcs flood inside the barrel like lightning bottled in glass. No more jumping to the needle. Just pure glowing discharge channels forming in vacuum. This is the exact reason vacuum tubes, neon signs, and Crookes tubes exist.

Paschen’s law: V_b = (B p d) / [ln(A p d) - ln(ln(1 + 1/γ))] Pull the plunger on a syringe next to a live Tesla coil and watch physics lose its mind. The instant pressure drops, that p d term collapses; breakdown voltage craters, the field ionizes the rarefied air, and violet plasma arcs flood inside the barrel like lightning bottled in glass. No more jumping to the needle. Just pure glowing discharge channels forming in vacuum. This is the exact reason vacuum tubes, neon signs, and Crookes tubes exist.

382,160 Aufrufe

This isn’t CGI. This is real mathematics controlling atoms inside every processor on Earth. Watch until your brain restarts.

This isn’t CGI. This is real mathematics controlling atoms inside every processor on Earth. Watch until your brain restarts.

458,899 Aufrufe

F = q(E + v × B) Watching the Lorentz force go from a textbook abstraction to 30,000 RPM in real-time is pure dopamine for the brain. It’s all fun and games until the magnetic field flux becomes a physical blur. Most people see a "spinning toy," but we see the cross product of velocity and magnetic field vectors working in perfect, violent harmony. The torque on that armature is proof that Maxwell’s equations aren't just ink on paper: they are the literal pulse of the universe. This shows the foundational principle behind DC Electric Motors.

F = q(E + v × B) Watching the Lorentz force go from a textbook abstraction to 30,000 RPM in real-time is pure dopamine for the brain. It’s all fun and games until the magnetic field flux becomes a physical blur. Most people see a "spinning toy," but we see the cross product of velocity and magnetic field vectors working in perfect, violent harmony. The torque on that armature is proof that Maxwell’s equations aren't just ink on paper: they are the literal pulse of the universe. This shows the foundational principle behind DC Electric Motors.

201,135 Aufrufe

A donut (torus) has the exact same volume and surface area as a cylinder with radius r and height 2πR! Watch every swivel preserve both perfectly... Geometry just got delicious!

A donut (torus) has the exact same volume and surface area as a cylinder with radius r and height 2πR! Watch every swivel preserve both perfectly... Geometry just got delicious!

205,130 Aufrufe

“Physics is the law, everything else is a recommendation.” — Elon Musk

“Physics is the law, everything else is a recommendation.” — Elon Musk

94,018 Aufrufe

y[1+(y′)^2]=k The Brachistochrone problem is the ultimate "work smarter, not harder" of the physical world. While a straight line is the shortest distance, the cycloid is the fastest path because it front-loads acceleration. It’s a beautiful reminder that in a gravitational field, the most direct route is rarely the most efficient. This is the Calculus of Variations in its most visceral, satisfying form. Seeing the cycloid absolutely dust the straight line never gets old; it’s where mathematical optimization meets physical reality.

y[1+(y′)^2]=k The Brachistochrone problem is the ultimate "work smarter, not harder" of the physical world. While a straight line is the shortest distance, the cycloid is the fastest path because it front-loads acceleration. It’s a beautiful reminder that in a gravitational field, the most direct route is rarely the most efficient. This is the Calculus of Variations in its most visceral, satisfying form. Seeing the cycloid absolutely dust the straight line never gets old; it’s where mathematical optimization meets physical reality.

87,678 Aufrufe

ΔL/L = β Δm Physics doesn’t need fireworks. Drop water dead center on these dry matchsticks resting on wood and they quietly torque themselves into a flawless five-pointed star; no wires, no magnets, just raw mechanics. Capillary action drives the moisture straight into the cellulose fibers. The wood swells anisotropically (way more across the grain than along it), building internal stress that pivots every stick outward in perfect symmetry. It’s self-assembly the universe coded into everyday material. This exact hygromorphic response powers climate-responsive building skins that breathe with humidity, moisture-triggered soft robotics in medicine, and smart packaging that opens itself when conditions change. Wood doing what silicon dreams of.

ΔL/L = β Δm Physics doesn’t need fireworks. Drop water dead center on these dry matchsticks resting on wood and they quietly torque themselves into a flawless five-pointed star; no wires, no magnets, just raw mechanics. Capillary action drives the moisture straight into the cellulose fibers. The wood swells anisotropically (way more across the grain than along it), building internal stress that pivots every stick outward in perfect symmetry. It’s self-assembly the universe coded into everyday material. This exact hygromorphic response powers climate-responsive building skins that breathe with humidity, moisture-triggered soft robotics in medicine, and smart packaging that opens itself when conditions change. Wood doing what silicon dreams of.

80,402 Aufrufe

If you only knew the magnificence of the 3, 6 and 9, then you would have the key to the universe. — Nikola Tesla

If you only knew the magnificence of the 3, 6 and 9, then you would have the key to the universe. — Nikola Tesla

109,544 Aufrufe

This is a Collapsible 3D Printed Vortex, a masterpiece of "Print-in-Place" engineering. How it works: The entire structure is printed as one single piece with incredibly tight tolerances—roughly 0.2mm to 0.3mm gaps between the rings. Because the layers don't actually touch, they can slide freely, creating that hypnotic "endless" spiral effect without any assembly required! It’s the perfect blend of mathematical precision and art.

This is a Collapsible 3D Printed Vortex, a masterpiece of "Print-in-Place" engineering. How it works: The entire structure is printed as one single piece with incredibly tight tolerances—roughly 0.2mm to 0.3mm gaps between the rings. Because the layers don't actually touch, they can slide freely, creating that hypnotic "endless" spiral effect without any assembly required! It’s the perfect blend of mathematical precision and art.

77,675 Aufrufe

A hexagonal ring of squares doesn’t just look pretty— it folds into a perfect cube. This graceful transformation reveals the power of topology: how flat surfaces become strong 3D forms through clever alignment and folding.

A hexagonal ring of squares doesn’t just look pretty— it folds into a perfect cube. This graceful transformation reveals the power of topology: how flat surfaces become strong 3D forms through clever alignment and folding.

95,270 Aufrufe

In this mesmerizing y=Ax video, eigenvectors glow with brilliance! As you rotate the vector x, Ax traces the ellipse perfectly. But the eigenvector lights up—no rotation at all, just pure scaling by its eigenvalue!

In this mesmerizing y=Ax video, eigenvectors glow with brilliance! As you rotate the vector x, Ax traces the ellipse perfectly. But the eigenvector lights up—no rotation at all, just pure scaling by its eigenvalue!

68,135 Aufrufe

Bernoulli’s Equation: P + ½ρv² + ρgh = constant Mind-bending science that looks like pure sorcery! A lightweight paper loop floats and dances in a high-velocity air stream, seemingly defying gravity. Fast-moving air over and around the loop creates a low-pressure zone above and along its curved surfaces. Higher atmospheric pressure underneath pushes it upward, trapping it in the airstream. The Coanda Effect makes the air hug the curve of the loop, keeping the airflow attached and giving it remarkable stability. Gravity pulls down, while the pressure difference lifts up — creating a beautiful dynamic equilibrium. With nothing but moving air molecules, you can guide the loop wherever you want. Invisible forces at work!

Bernoulli’s Equation: P + ½ρv² + ρgh = constant Mind-bending science that looks like pure sorcery! A lightweight paper loop floats and dances in a high-velocity air stream, seemingly defying gravity. Fast-moving air over and around the loop creates a low-pressure zone above and along its curved surfaces. Higher atmospheric pressure underneath pushes it upward, trapping it in the airstream. The Coanda Effect makes the air hug the curve of the loop, keeping the airflow attached and giving it remarkable stability. Gravity pulls down, while the pressure difference lifts up — creating a beautiful dynamic equilibrium. With nothing but moving air molecules, you can guide the loop wherever you want. Invisible forces at work!

55,994 Aufrufe

“Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line.” — Benoit Mandelbrot

“Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line.” — Benoit Mandelbrot

68,194 Aufrufe

C = εr ε0 (A/d) This demo shows capacitive coupling using water as a variable dielectric (εᵣ ≈ 80). The human body completes the circuit by touching the water, acting as the second conductor with its capacitance to ground. Hand/finger movement changes the effective area A and distance d, while ripples dynamically modulate C. This varying capacitance alters an oscillator’s frequency, which is converted into audio tones or MIDI in real time. Educational fluid-based sensors and interactive MIDI controllers for bio-signal acquisition and gesture-controlled music. Ideal for physics demos and human-interface experiments.

C = εr ε0 (A/d) This demo shows capacitive coupling using water as a variable dielectric (εᵣ ≈ 80). The human body completes the circuit by touching the water, acting as the second conductor with its capacitance to ground. Hand/finger movement changes the effective area A and distance d, while ripples dynamically modulate C. This varying capacitance alters an oscillator’s frequency, which is converted into audio tones or MIDI in real time. Educational fluid-based sensors and interactive MIDI controllers for bio-signal acquisition and gesture-controlled music. Ideal for physics demos and human-interface experiments.

46,619 Aufrufe

Gauss-Lucas theorem: derivative roots lie in convex hull of P roots. Animation shows them pulling inward with each differentiation, hull shrinking. Degree-12 poly: roots converge to barycenter after 11 steps. Raw math visualization (insane).

Gauss-Lucas theorem: derivative roots lie in convex hull of P roots. Animation shows them pulling inward with each differentiation, hull shrinking. Degree-12 poly: roots converge to barycenter after 11 steps. Raw math visualization (insane).

45,230 Aufrufe

Watch points instantly snap to their nearest neighbor center; arrows show the magnetic pull in real time! Local greedy rule produces macroscopic order; perfect swirling dipoles one moment, full turbulence the next.

Watch points instantly snap to their nearest neighbor center; arrows show the magnetic pull in real time! Local greedy rule produces macroscopic order; perfect swirling dipoles one moment, full turbulence the next.

43,084 Aufrufe

"I, a universe of atoms, an atom in the universe" — Richard P. Feynman

"I, a universe of atoms, an atom in the universe" — Richard P. Feynman

41,118 Aufrufe

This is what a mixed quantum state looks like and it’s hypnotizing ρ = ∑ pₙ |ψₙ⟩⟨ψₙ|

This is what a mixed quantum state looks like and it’s hypnotizing ρ = ∑ pₙ |ψₙ⟩⟨ψₙ|

43,143 Aufrufe

“There is geometry in the humming of the strings. There is music in the spacing of the spheres.” — Pythagoras

“There is geometry in the humming of the strings. There is music in the spacing of the spheres.” — Pythagoras

40,068 Aufrufe

Videos

Anya Rossi

sweetdream.ai

SweetDream.ai•Sponsored•Livecam

Watch Anya Live

Anya is streaming live right now! Join her private show and enjoy exclusive content.

Exclusive private shows

1.2k viewers online

Private Show

Join now for exclusive access

Free preview available • Premium content

Hooke's law in motion: a dancer's joints anchor a real-time spring-mass lattice. Every extension, contraction, and twist propagates tension through the system; rendering these hypnotic, radiating dashed vectors as emergent force fields.

Hooke's law in motion: a dancer's joints anchor a real-time spring-mass lattice. Every extension, contraction, and twist propagates tension through the system; rendering these hypnotic, radiating dashed vectors as emergent force fields.

1,579,345 Aufrufe • vor 2 Monaten

Terence Tao is answering a fundamental question regarding the safety and reliability of modern AI: "How can we use a tool that is powerful, but unreliable?" W = ∑(wᵢ ⋅ xᵢ) + b AI isn’t just about “smart”; it’s about the probability of *looking* right. We’ve built systems where the weights (wᵢ) are optimized for plausibility, not veracity. This creates a “convincing mirror” that confidently serves dangerous advice in medicine or finance. The gap between “convincing” and “correct” is the most critical variable we need to solve for.

Terence Tao is answering a fundamental question regarding the safety and reliability of modern AI: "How can we use a tool that is powerful, but unreliable?" W = ∑(wᵢ ⋅ xᵢ) + b AI isn’t just about “smart”; it’s about the probability of looking right. We’ve built systems where the weights (wᵢ) are optimized for plausibility, not veracity. This creates a “convincing mirror” that confidently serves dangerous advice in medicine or finance. The gap between “convincing” and “correct” is the most critical variable we need to solve for.

580,438 Aufrufe • vor 29 Tagen

Harmonious Geometry: The Hirajoshi Wave. Watch as these gravity-defying spheres trace the hauntingly beautiful paths of the C Hirajoshi scale. Each ball is tuned to a specific frequency within this traditional Japanese pentatonic scale (C, D, Eb, G, Ab), creating a mesmerizing "Polyrhythmic Pendulum" effect. As the balls oscillate at slightly different speeds, they drift into chaotic patterns before perfectly realigning into a breathtaking visual and auditory climax. From the sharp, angular bounce to the fluid, sweeping curves of the rainbow trails, this is where physics meets fine art. Credit: project.jdm

Harmonious Geometry: The Hirajoshi Wave. Watch as these gravity-defying spheres trace the hauntingly beautiful paths of the C Hirajoshi scale. Each ball is tuned to a specific frequency within this traditional Japanese pentatonic scale (C, D, Eb, G, Ab), creating a mesmerizing "Polyrhythmic Pendulum" effect. As the balls oscillate at slightly different speeds, they drift into chaotic patterns before perfectly realigning into a breathtaking visual and auditory climax. From the sharp, angular bounce to the fluid, sweeping curves of the rainbow trails, this is where physics meets fine art. Credit: project.jdm

645,463 Aufrufe • vor 1 Monat

When four simple periodic grids in CMYK phase-shift you get this. Beat frequencies, moiré patterns, and full-spectrum hallucinations from basic trig.

When four simple periodic grids in CMYK phase-shift you get this. Beat frequencies, moiré patterns, and full-spectrum hallucinations from basic trig.

327,278 Aufrufe • vor 2 Monaten

Harvard SEAS prismatic architected metamaterial: extruded cubic cells (24 faces, 36 edges) tessellated via snapology origami. Reconfigures volume, shape, and effective stiffness on demand through hinge folding; scale-independent from nano to meter.

Harvard SEAS prismatic architected metamaterial: extruded cubic cells (24 faces, 36 edges) tessellated via snapology origami. Reconfigures volume, shape, and effective stiffness on demand through hinge folding; scale-independent from nano to meter.

241,286 Aufrufe • vor 1 Monat

λ · μ = lk(∂λ, ∂μ) + ∑ handles intersections Watching Lisa Piccirillo own the blackboard like this; intersection forms on homology modules, handle cancellations in 4-manifolds; hits different. The precision, the chalk dust, the way she just sees the topology. This is pure 4D magic: where smooth vs topological distinctions go wild, and one clever trace or form can kill a 50-year knot problem. Her Conway knot proof lives here; showing it's not smoothly slice, unlocking exotic phenomena that change how we understand spacetime surgery and manifold classification.

λ · μ = lk(∂λ, ∂μ) + ∑ handles intersections Watching Lisa Piccirillo own the blackboard like this; intersection forms on homology modules, handle cancellations in 4-manifolds; hits different. The precision, the chalk dust, the way she just sees the topology. This is pure 4D magic: where smooth vs topological distinctions go wild, and one clever trace or form can kill a 50-year knot problem. Her Conway knot proof lives here; showing it's not smoothly slice, unlocking exotic phenomena that change how we understand spacetime surgery and manifold classification.

184,610 Aufrufe • vor 1 Monat

Prime numbers are supposed to be random. This video proves they aren't. In 1963, mathematician Stanislaw Ulam discovered an unexpected order within the "randomness" of prime numbers. While doodling during a presentation, he arranged integers in a square spiral and circled the primes. To his surprise, the primes; which were thought to be scattered without reason; clustered into distinct diagonal lines. This visual phenomenon, now called the Ulam Spiral, revealed that prime numbers follow a mysterious underlying structure often linked to quadratic polynomials, proving that even mathematical "atoms" have a hidden architecture. Credit: maths.visualization

Prime numbers are supposed to be random. This video proves they aren't. In 1963, mathematician Stanislaw Ulam discovered an unexpected order within the "randomness" of prime numbers. While doodling during a presentation, he arranged integers in a square spiral and circled the primes. To his surprise, the primes; which were thought to be scattered without reason; clustered into distinct diagonal lines. This visual phenomenon, now called the Ulam Spiral, revealed that prime numbers follow a mysterious underlying structure often linked to quadratic polynomials, proving that even mathematical "atoms" have a hidden architecture. Credit: maths.visualization

190,983 Aufrufe • vor 1 Monat

A = 4πr² Exactly the surface area of a sphere, yet this shape—born from two perpendicular circles—unfolds perfectly flat without stretching, rolls in hypnotic figure-8 rhythms, and keeps its center of gravity impossibly still. The Oloid: where math whispers secrets of effortless motion, zero turbulence, and pure elegance.

A = 4πr² Exactly the surface area of a sphere, yet this shape—born from two perpendicular circles—unfolds perfectly flat without stretching, rolls in hypnotic figure-8 rhythms, and keeps its center of gravity impossibly still. The Oloid: where math whispers secrets of effortless motion, zero turbulence, and pure elegance.

245,053 Aufrufe • vor 3 Monaten

Forget matches; this is how our ancestors used the laws of thermodynamics to start fires in seconds using nothing but wood and air. The secret lies in the sudden increase in pressure. By trapping air in a narrow tube and forcing a piston down, you are doing work on the gas. In a closed system like this, that work translates directly into thermal energy. Because the movement is so fast, it is considered "adiabatic," meaning no heat escapes. The resulting flash of heat is intense enough to create a glowing ember, which is then transferred to a larger "tinder bundle" and blown into a flame.

Forget matches; this is how our ancestors used the laws of thermodynamics to start fires in seconds using nothing but wood and air. The secret lies in the sudden increase in pressure. By trapping air in a narrow tube and forcing a piston down, you are doing work on the gas. In a closed system like this, that work translates directly into thermal energy. Because the movement is so fast, it is considered "adiabatic," meaning no heat escapes. The resulting flash of heat is intense enough to create a glowing ember, which is then transferred to a larger "tinder bundle" and blown into a flame.

195,564 Aufrufe • vor 2 Monaten

∇⋅B=0 Most people think magnetic fields are just a concept in textbooks. But when you drop a magnet into this 3D field viewer, the invisible becomes crystal clear. Watch the exact moment the iron filings align; it’s like the universe is following a hidden blueprint. The way they "freeze" in 3D space is incredibly satisfying to witness. Science is basically just real-life magic.

∇⋅B=0 Most people think magnetic fields are just a concept in textbooks. But when you drop a magnet into this 3D field viewer, the invisible becomes crystal clear. Watch the exact moment the iron filings align; it’s like the universe is following a hidden blueprint. The way they "freeze" in 3D space is incredibly satisfying to witness. Science is basically just real-life magic.

111,519 Aufrufe • vor 1 Monat

A single microscopic error doesn’t just cause wear: it causes instant, explosive destruction. What you’re looking at is a 140-year-old masterpiece that still generates 80% of the world's electricity: the Steam Turbine. It’s an engineering paradox. These rotors weigh tens of tons, yet they are as delicate as a watch. Engineers limit them to 3,000 RPM because at these scales, the laws of physics are unforgiving. The smallest imbalance isn't just a vibration; it’s a wrecking ball that can lead to catastrophic failure in milliseconds. From hand-seating blades with wooden mallets to numbering every individual component, this is where raw, volcanic heat is tamed into perfect, rotational motion. Extreme power. Absolute precision.

A single microscopic error doesn’t just cause wear: it causes instant, explosive destruction. What you’re looking at is a 140-year-old masterpiece that still generates 80% of the world's electricity: the Steam Turbine. It’s an engineering paradox. These rotors weigh tens of tons, yet they are as delicate as a watch. Engineers limit them to 3,000 RPM because at these scales, the laws of physics are unforgiving. The smallest imbalance isn't just a vibration; it’s a wrecking ball that can lead to catastrophic failure in milliseconds. From hand-seating blades with wooden mallets to numbering every individual component, this is where raw, volcanic heat is tamed into perfect, rotational motion. Extreme power. Absolute precision.

195,011 Aufrufe • vor 3 Monaten

I want to know God's thoughts - the rest are mere details. — Albert Einstein

I want to know God's thoughts - the rest are mere details. — Albert Einstein

122,401 Aufrufe • vor 1 Monat

A square is constructed using its Fourier series (represented by sums of circles rotating around circles), then the higher order (smaller, faster rotation) circles are removed in turn. Credit: generomics

A square is constructed using its Fourier series (represented by sums of circles rotating around circles), then the higher order (smaller, faster rotation) circles are removed in turn. Credit: generomics

137,863 Aufrufe • vor 2 Monaten

This robot arm = your AI model. One math number decides if it learns or completely breaks. This is Gradient Descent — the core math behind all of AI you are using in your day-to-day life.

This robot arm = your AI model. One math number decides if it learns or completely breaks. This is Gradient Descent — the core math behind all of AI you are using in your day-to-day life.

134,997 Aufrufe • vor 2 Monaten

A 2-layer neural network goes from total chaos to perfectly separating left vs right classes in real time. Watch the decision boundary form live as gradient descent works its magic! Pure maths beauty in motion..

A 2-layer neural network goes from total chaos to perfectly separating left vs right classes in real time. Watch the decision boundary form live as gradient descent works its magic! Pure maths beauty in motion..

144,883 Aufrufe • vor 2 Monaten

Before we had silicon chips, we had needle and thread? In the 1960s, NASA didn’t ‘upload’ code; they sewed it. To get Apollo 11 to the moon, skilled weavers (often called ‘Little Old Ladies’) literally hand-stitched software into physical objects. By passing copper wire through tiny magnetic rings, they created Core Rope Memory. The logic was beautifully simple: wire through a ring was a ‘1’; wire around it was a ‘0’. Because the code was physically woven, it was virtually indestructible. It couldn’t be deleted, it couldn’t crash, and it survived the intense radiation of deep space with just 72 kilobytes of data: millions of times less than a single photo on your phone today. It proves sometimes the most advanced tech is actually handmade.

Before we had silicon chips, we had needle and thread? In the 1960s, NASA didn’t ‘upload’ code; they sewed it. To get Apollo 11 to the moon, skilled weavers (often called ‘Little Old Ladies’) literally hand-stitched software into physical objects. By passing copper wire through tiny magnetic rings, they created Core Rope Memory. The logic was beautifully simple: wire through a ring was a ‘1’; wire around it was a ‘0’. Because the code was physically woven, it was virtually indestructible. It couldn’t be deleted, it couldn’t crash, and it survived the intense radiation of deep space with just 72 kilobytes of data: millions of times less than a single photo on your phone today. It proves sometimes the most advanced tech is actually handmade.

144,601 Aufrufe • vor 2 Monaten

"The mathematician does not study pure mathematics because it is useful; he studies it because he delights in it, and he delights in it because it is beautiful." — Henri Poincaré

"The mathematician does not study pure mathematics because it is useful; he studies it because he delights in it, and he delights in it because it is beautiful." — Henri Poincaré

101,968 Aufrufe • vor 2 Monaten

There's a profound elegance in watching gradient descent find the signal in the noise. This is a simple MLP with sigmoid activations slowly warping to approximate f(x) = 1/(1 + x²). Watch the loss curve right around epoch 400. That sudden phase transition where the network finally "gets it" and snaps into the Witch of Agnesi shape is incredibly satisfying to watch.

There's a profound elegance in watching gradient descent find the signal in the noise. This is a simple MLP with sigmoid activations slowly warping to approximate f(x) = 1/(1 + x²). Watch the loss curve right around epoch 400. That sudden phase transition where the network finally "gets it" and snaps into the Witch of Agnesi shape is incredibly satisfying to watch.

77,441 Aufrufe • vor 1 Monat

One single still photo of a massive bird flock contains the full flapping animation of a single bird. The flock is a statistical ensemble sampling every moment of the flap cycle ergodicity turns a snapshot into motion!

One single still photo of a massive bird flock contains the full flapping animation of a single bird. The flock is a statistical ensemble sampling every moment of the flap cycle ergodicity turns a snapshot into motion!

78,790 Aufrufe • vor 2 Monaten

Watch hydrogen atom wavefunctions smoothly transition between quantum states (n,l,m) This is the actual probability density of finding the electron; not just static orbitals, but the real quantum dance as it morphs between energy levels.

Watch hydrogen atom wavefunctions smoothly transition between quantum states (n,l,m) This is the actual probability density of finding the electron; not just static orbitals, but the real quantum dance as it morphs between energy levels.

66,986 Aufrufe • vor 2 Monaten