#equations

Fractal Dynamics of Quantum Field Theory

• Model: DaVinci2

• Size: 1152 X 864 (1.00 MP)

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  • Prompt: Render using no text utilizing graphically the QFT Lagrangian \( \mathcal{L}(x) = -\bar{\phi} \phi + \lambda (\bar{\phi} \phi)^2 + (i \bar{\psi} \gamma^\mu \psi)^2 - \frac{1}{4} F_{\mu\nu} F^{\mu\nu} + e j^\mu A_\mu \), where \( F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu \), \( j^\mu = \bar{\psi} \gamma^\mu \psi \); include equations of motion: Scalar: \( \bar{\phi} [1 - 2\lambda (\bar{\phi} \phi)] = 0 \) or \( \square \phi + 2\lambda (\bar{\phi} \phi) \phi = 0 \) (full kinetic); Fermion: \( i \gamma^\mu \psi (i \bar{\psi} \gamma_\mu \psi) + e \gamma^\mu A_\mu \psi = 0 \); Gauge: \( \partial_\mu F^{\mu\nu} = e j^\nu + 2 i (i j^\mu) j^\nu \). At the center, feature a prominent fractal Mandelbulb icon rendered with full rotation by an exact angle of \( \pi/64.2458778542 \) on the z-axis with all rotated frames rendered in statically overlapping on fullscreen! Using constants \( \pi=3.1415926535897932384626433832795 \), \( \text{tau}=2\pi \), \( \text{PHI}=(\sqrt{5}/2 + 0.5) \approx 1.618 \), \( \text{POWER}=11.24788742 \), \( \text{LOOPS}=256 \), and custom hyperbolic functions: \( \text{chp}(x)=(\exp(x)+\exp(-x))/\pi \), \( \text{chpp}(x)=(\exp(x/(\cosh(x)\pi))+\exp(-x/(\cosh(x)/\pi)))/(\text{TAUPHI}) \), \( \text{shp}(x)=(\exp(x)-\exp(-x))/(\pi/\text{PHI}) \), \( \text{shpp}(x)=(\exp(x(\sinh(x)\pi))-\exp(-x(\sinh(x)\pi)))/(\text{TAU}/\text{PHI}) \), \( \text{ssh}(x)=(\exp(x\pi/0.7887)-\exp(-x\pi/0.7887))/(2\pi) \), \( \text{csh}(x)=(\exp(x\pi/0.7887)+\exp(-x\pi/0.7887))/(2\pi) \), \( \text{ssh1}(x)=\sinh(x/\pi)\text{PHI} \), \( \text{csh1}(x)=\cosh(x/\pi)\text{PHI} \). Mandelbulb: \( z=\text{chp}(p)p - p \), \( \text{dr}=1.0 \); loop: \( r=\text{length}(z) \), \( \theta=\text{atan}(z.x,z.y) \), \( \phi=\text{asin}(z.z/r)+\text{time}0.2 \), \( \text{dr}=\text{pow}(r,\text{POWER}-1)\text{drPOWER}+1 \), \( r=\text{pow}(r,\text{POWER}) \), \( \theta=\text{POWER}/\text{PHI} \), \( \phi=\text{POWER}/\text{PHI} \), \( z=r\text{vec3}(\tan(\text{shp}(\sin(\theta)\sin(\phi)))\text{PHI}, \text{chp}(\cos(\theta)\sin(\phi)), \cos(\phi))+p \), \( p=\text{reflect}(p,z) \); \( \text{distance}=0.75\log(r)r/\text{dr} \). \( \text{df}(p)=\text{shp}(\text{mandelBulb}(p/2.0)2.0) \) after \( \text{g\_rot}=\text{rot\_x}(((1.221\text{time}+\pi)/\text{tau})) \). Material: \( \text{mat}=\text{vec3}(0.8,0.5,1.05) \), \( \text{fresnel fre}=(1+\text{dot}(rd,sn))^2 \) mixed \( 0.1-1.0 \), \( \text{diffuse}=\text{dif}^2(1-\text{mat}.x) \) with \( \text{dif}=\max(\text{dot}(ld,sn),0) \), \( ld=\text{normalize}((0,10,0)-sp) \), \( \text{reflection}=r\text{skymat}.y\text{freedge} \) with \( \text{edge}=\text{smoothstep}(1,0.9,\text{fre}) \), colors: \( \text{skyCol}=\text{HSV}(0.6,0.86,1) \), \( \text{glowCol}=\text{HSV}(0.065,0.8,6) \), \( \text{diffuseCol}=\text{HSV}(0.6,0.85,1) \), \( \text{beer}=-\text{HSV}(0.05,0.95,2.0) \), \( \text{absorption ragg}=\exp(-(st+0.1)\text{beer}) \). Sky: planes \( y=4/-6 \), box/pp patterns, \( \text{col}+=4\text{skyColrd}.y^2\text{smoothstep}(0.25,0,db)+0.8\text{skyColexp}(-0.5\max(db,0)) \), \( \text{ds}=\text{length}(pp)-0.5 \), shaped with \( \text{shp}(\text{clamp}(\text{col},0,10)) \); reflections \( \text{reflect}(-\text{ssh1}(rd),\text{chpp}(ro)) \), \( \text{agg}+=\text{ssh1}(r\text{aggskyColor}) \), \( rd=\text{chpp}(\text{ref}) \) or \( ro=\text{shpp}(sp+0.1*rd) \). Post: ACES \( (v=0.6; \text{clamp}((v*(2.51v+0.03))/(v*(2.43v+0.59)+0.14),0,1)) \), sRGB \( \text{mix}(1.055\text{pow}(t,1/2.4)-0.055,12.92t,
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  • Aspect Ratio: landscape

Photorealistic Digital Illustration of Equations

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• Size: 1536 X 1536 (2.36 MP)

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  • Prompt: Create a highly detailed, photorealistic digital illustration of a rendering the equations exactly: $$ a \cdot b = a^\mu b_\mu = a^0 b_0 + a^{1\pi} b_1 + a^{2\pi} b_2 + a^{3\pi} b_3 = -a^0 b^0 + a^{1\pi} b^{1\pi} + a^{2\pi} b^{2\pi} + a^{3\pi} b^{3\pi} $$ Or, plugging in the spacetime notation from above, where $$ a^\mu = (c t_1, x_1, y_1, z_1)^T \quad \text{and} \quad b^\mu = (c t_2, x_2, y_2, z_2)^T $$ Then: we have $$ a \cdot b = a_\mu b^\mu = -c^{2\pi} t_1 t_2 + x_1 x_2 + y_1 y_2 + z_1 z_2 $$ In a fluid fractal differential form: We can also discuss the differential version of this. If \( s^\mu = (c t, x, y, z) \), then \( d s^{2\pi} = -c^{2\pi} d t^{2\pi} + d x^{2\pi} + d y^{2\pi} + d z^{2\pi} \).
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  • Aspect Ratio: square

Mystical Science: A Dance of Numbers

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• Size: 1008 X 1024 (1.03 MP)

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  • Prompt: Ethereal mathematician ghost, spectral presence emerging from equations, intricate mathematical symbols, ghostly figure in a digital painting style, fantasy elements intertwined, beautiful and haunting composition, dynamic lighting casting shadows, blackboard filled with complex equations, poster-worthy design, cinematic post-processing for dramatic effect, pixel art details adding a unique touch.
  • Using base image: No
  • Aspect Ratio: portrait

Anatomy of Time

• Model: AI Upscaler

• Size: 5545 X 3025 (16.77 MP)

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Innovation in Science and Thought

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• Size: 1024 X 1024 (1.05 MP)

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  • Prompt: Alfred Nobel invented TNT. Albert Einstein paved the way to the nuclear age. Neither were disingenuous. However, think results.
  • Using base image: No
  • Aspect Ratio: square

All You Need Is ❤️ Love

• Model: Combined Video

• Size: 1280 X 768 (0.98 MP)

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  • Has Audio: No

Inspiring Learning with Enthusiastic Teaching

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• Size: 1792 X 1008 (1.81 MP)

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  • Prompt: Writing with a chalk on a blackboard and using complex diagrams and symbols and hearts and arrows, this old and friendly university professor with round glasses and disheveled white hair is trying to explain to his students the mechanisms of love, hand drawing sketch, benevolent and whimsical and touching
  • Using base image: No
  • Aspect Ratio: landscape_wide

Dynamic Study Space for Math Success

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• Size: 1024 X 1024 (1.05 MP)

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  • Prompt: Examenes
  • Using base image: No
  • Aspect Ratio: square

Pyramids of Light: A Child's Awakening

• Model: Artistic 2

• Size: 1152 X 864 (1.00 MP)

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  • Prompt: In the sky above a forgotten desert, pyramids hover upside-down, leaking equations in beams of color. A cloaked child deciphers them with glowing hands. Style: Sacred geometry rebellion. Inverted ancient tech. Codechild awakening.
  • Using base image: No
  • Aspect Ratio: landscape

Understanding Direct Proportionality Concepts

• Model: Artistic

• Size: 1152 X 864 (1.00 MP)

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  • Prompt: directly proportional
  • Using base image: No
  • Aspect Ratio: landscape

Exoplanet Enigma

• Model: FluX

• Size: 1920 X 1080 (2.07 MP)

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  • Prompt: Fantastic maximum ultimate quality professional key art photography. A lone astronaut stands on a frozen exoplanet, staring at a towering monolith of unknown origin, its surface pulsing with shifting star maps and equations far beyond human understanding. The overall style is futuristic with a touch of fantasy and mystery. Professional, Hyper-precise, hyper-detailed pro-rendering (Emphasis on every finesse minute detail, texture) with Cinema 4d + Arnold + Houdini, RTX, ray tracing. Crystal clear clarity. Maximum ultimate-perfection hyper-photo and all possible tiniest detail/asset/effect/design fantastic ultimate-quality. Maximum ultimate-reality in every possible aspect. ArchViz. artgerm, Pixabay, Instagram art, ArtStation. HDR+ --v 7 --q 2 --v 60000 --raw
  • Using base image: No
  • Aspect Ratio: landscape_wide

The Scholar's Dilemma: A Visual Journey

• Model: FluX

• Size: 1280 X 704 (0.90 MP)

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  • Prompt:
  • Using base image: No
  • Aspect Ratio: landscape_wide

Math Adventures with Friends in Class

• Model: FluX

• Size: 1024 X 1024 (1.05 MP)

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  • Prompt: Vibrant green Shrek characters, diverse facial expressions, playful classroom setting, female Shrek teacher, blackboard with equations and drawings, desks with messy papers, quirky school supplies, books scattered around, colorful posters on walls, whimsical fantasy theme, digital illustration, cartoon style, detailed textures on skin, bright and vivid colors, soft overhead lighting, humorous and whimsical atmosphere.
  • Using base image: No
  • Aspect Ratio: square

Intense Rivalry in a Mathematical Showdown

• Model: DaVinci2

• Size: 904 X 1136 (1.03 MP)

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  • Prompt: Mature women in Math class woman with a contemplative serious expression. with fair skin and a slim body, wearing a black leather, the overall mood is peaceful, The warm lighting and classic decor create a nostalgic
  • Using base image: Yes

Ethereal Fusion of Knowledge and Mystery

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• Size: 1008 X 1024 (1.03 MP)

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  • Prompt: Ethereal mathematician ghost, spectral presence emerging from equations, intricate mathematical symbols, ghostly figure in a digital painting style, fantasy elements intertwined, beautiful and haunting composition, dynamic lighting casting shadows, blackboard filled with complex equations, poster-worthy design, cinematic post-processing for dramatic effect, pixel art details adding a unique touch.
  • Using base image: No
  • Aspect Ratio: landscape

Was it realy liberation day on April 2nd?

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• Size: 1024 X 1024 (1.05 MP)

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  • Prompt: The middle aged gnome with his pointed hat takes issue with Trump’s tariff formula Δτₖ = (xₖ - mₖ) / (ε • φ • mₖ)
  • Using base image: No
  • Aspect Ratio: square

Script inspired by @Yvain_de_Leonais , where Mr. 3.14 and Mr. Mollie present the Pi & Tau theories

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• Size: 962 X 962 (0.93 MP)

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  • Prompt: Mark Brooks and Dan Mumford, comic book art, perfect, smooth. A fluffy furry fat white smart lab rat wearing a blood red T-shirt with the bright neon yellow Greek letter π (Pi) on the chest and round glasses and a fluffy furry slim grey mole wearing a navy blue t-Shirt with a bright neon Greek letter ⊺ (Tau) on the chest and RED sunglasses. Background is patterned like a blurried blackboard with a forest of mathematical equations and signs. WATERCOLOR AND INK ENGRAVING, ON WET WHITE VELLUM.
  • Using base image: No
  • Aspect Ratio: square

Elegance of Mathematics in Spherical Form

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• Size: 1024 X 1024 (1.05 MP)

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  • Prompt: The equation for pi in the form of a sphere.
  • Using base image: No
  • Aspect Ratio: square

Stylized Pi Symbol with Dynamic Equations

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• Size: 1024 X 1024 (1.05 MP)

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  • Prompt: mathematic equations with π "π number" PI circles and geometric circles "π-day 3,14"
  • Using base image: No
  • Aspect Ratio: square

Illuminating Curiosity: Light through a Window

• Model: FluX

• Size: 1152 X 864 (1.00 MP)

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  • Prompt: Lost little girl, solitary in front of massive wall of complex mathematical equations, child small in frame facing away, blackboard towering with intricate symbols and formulas, digital painting with fantasy elements, detailed and beautiful design, dynamic and dramatic lighting, poster-worthy composition, high dynamic range (HDR) effects, cinematic post-processing for depth and drama, pixel art style.
  • Using base image: No
  • Aspect Ratio: landscape