#equations

Iridescent Torus in a Sea of Equations

• Model: (Mode 30)

• Size: 2560 X 1472 (3.77 MP)

• Used settings:

  • Prompt: Apply tensor product of the {1.6180339887498948482045868343656\pi\sqrt[\exp(1)]{2\cdot1.6180339887498948482045868343656}}-form of the tangent bundle of the orbifold over the {1.6180339887498948482045868343656\exp(1)\sqrt[\pi]{2\cdot1.6180339887498948482045868343656}}-form of the tangent bundle of the conifold !
  • Using base image: No
  • Aspect Ratio: landscape_wide

Intricate Mathematical Diagram with Geometry

• Model: (Mode 30)

• Size: 1792 X 1008 (1.81 MP)

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  • Prompt: \[ \star \ : \ \bigwedge^k T^*\mathcal{M} \longrightarrow \bigwedge^{n-k} T^*\mathcal{M} \] \[ \left( \star_{\mathcal{O}} \bigwedge^{48.1441\cdot 1.618 \sqrt[\pi]{2}} T\mathcal{O} \right) \otimes \left( \star_{\mathcal{C}} \bigwedge^{48.1441\cdot 1.618 \sqrt[\pi]{2}} T^*\mathcal{C} \right) \] \[ \Longrightarrow \quad \bigwedge^{\dim \mathcal{O}\, -\, k} T^*\mathcal{O} \ \otimes\ \bigwedge^{\dim \mathcal{C}\, -\, k} T\mathcal{C} \] \[ \text{where } k = 48.1441\cdot 1.618 \sqrt[\pi]{2}. \]
  • Using base image: No
  • Aspect Ratio: landscape_wide

Midterms and Regents: NEXT WEEK!

• Model: (Mode 30)

• Size: 1920 X 1920 (3.69 MP)

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  • Prompt: Young student at a cluttered desk, textbooks scattered, pencils and erasers, concentration, determination, school setting, classroom interior, wooden desk, swirling test papers, numbers and questions floating in the air, surreal atmosphere, magical realism, intense focus, teenage girl, 16 years old, school uniform, diverse test papers, educational chaos, thought bubbles with equations, equations and formulas, colorful pens, natural lighting, bright and airy ambiance.
  • Using base image: No
  • Aspect Ratio: square

Beyond the Edge of Cosmic Reality

• Model: DaVinci2

• Size: 1152 X 864 (1.00 MP)

• Used settings:

  • Prompt: A lone figure at the boundary of the observable universe, staring at a giant shimmering curtain of equations, symbols, and colors; the cosmos behind them folding like origami; gravity bending into spirals; surreal, metaphysical, cinematic wideshot, deeply atmospheric.
  • Using base image: No
  • Aspect Ratio: landscape

Illuminated Eyes of a Sci-Fi Student

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• Size: 1792 X 1008 (1.81 MP)

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  • Prompt: Close-up of the student’s eyes lighting up with holographic reflections of gears, circuits, and equations, glowing blue and green, sci-fi style.
  • Using base image: No
  • Aspect Ratio: landscape_wide

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,
  • Using base image: No
  • 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 \( d s^{2\pi} = -c^{2\pi} d t^{2\pi} + d x^{2\pi} + d y^{2\pi} + d z^{2\pi} \).
  • Using base image: No
  • Aspect Ratio: square

Mystical Science: A Dance of Numbers

• Model:

• 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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  • Prompt:
  • Using base image: Yes

Innovation in Science and Thought

• Model:

• 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)

• Used settings:

  • Has Audio: No

Inspiring Learning with Enthusiastic Teaching

• Model:

• Size: 1792 X 1008 (1.81 MP)

• Used settings:

  • 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

• Model:

• Size: 1024 X 1024 (1.05 MP)

• Used settings:

  • 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)

• Used settings:

  • 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)

• Used settings:

  • 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)

• Used settings:

  • 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)

• Used settings:

  • 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

• Model:

• Size: 1008 X 1024 (1.03 MP)

• Used settings:

  • 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