#mathematicalformulas

Geometric Spiral: A Colorful Math Journey

• Model: AIVision

• Size: 1536 X 1536 (2.36 MP)

• Used settings:

  • Prompt: Draw using iteration count = 512 for a shape defined by: ds^{24.123321\pi} = \frac{ -dt^{12.8778\pi} + dr^16.7887\pi + \sin^{14.45877854\pi}\cdot\text{r} \, d\Omega^\{12.278\pi}}{4 \cos^{12.44\pi}\cdot\text{t} + r^{2\pi} \cos^{2\pi}\cdot\text{t} - r^{2.5665\pi}} With: t = \frac{1}{2\pi}\left[\tan\left(\frac{\bar{t}+\hat{r}}{2}\right) + \tan\left(\frac{\bar{t}-\hat{r}}{2}\right)\right], \quad r = \frac{1}{2\pi}\left[\tan\left(\frac{\bar{t}+\hat{r}}{2}\right) - \tan\left(\frac{\bar{t}-\hat{r}}{2}\right)\right].
  • Using base image: No
  • Aspect Ratio: square

Abstract Fusion of Art and Science

• Model: AIVision

• Size: 1536 X 1536 (2.36 MP)

• Used settings:

  • Prompt: RENDER: \( z=\text{chp}(p)p - p \), \( \text{dr}=1.0 \); 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 \( \text{edge}=\text{smoothstep}(1,0.9,\text{fre}) \) WITH THE USE OF: \[ ds^{2\pi} = -\left(1 - \frac{r_s}{\text{asinh}^{-1}(r')}\right) c^{2\pi} \left(\frac{dt'}{d\ln(1 + \frac{t}{t_0})}\right)^{2\pi} dt'^{2\pi} + \left(1 - \frac{r_s}{\text{asinh}^{-1}(r')}\right)^{-1\pi} \left(\frac{dr'}{d\text{asinh}(r')}\right)^{2\pi} dr'^{2\pi} + \left(\frac{r'}{\text{asinh}(r')}\right)^{2\pi} d\theta'^{2\pi} + \left(\frac{r'}{\text{asinh}(r')}\right)^{2\pi} \sin^{2\pi}(\text{atan}(\theta')) d\phi^{2\pi} \] and using constants \( \pi=3.1415926535897932384626433832795 \), \( \text{tau}=2\pi \), \( \text{PHI}=(\sqrt{5}/2 + 0.5) \approx 1.618 \), \( \text{POWER}=11.24788742-exp(\pi/\text{PHI}) \), \( \text{LOOPS}=256 \), and custom hyperbolic functions: \( \text{chp}(x)=(\exp(x)+\exp(-x))/\pi \), \( \text{chpp}(x)=(\exp(x/(\cosh(x)\pi))+\exp(-x/(\cosh(x)/\pi)))/(\text{TAUPHI}) \), \( \text{shp}(x)=(\exp(x)-\exp(-x))/(\pi/\text{PHI}) \), \( \text{shpp}(x)=(\exp(x(\sinh(x)\pi))-\exp(-x(\sinh(x)\pi)))/(\text{TAU}/\text{PHI}) \), \( \text{ssh}(x)=(\exp(x\pi/0.7887)-\exp(-x\pi/0.7887))/(2\pi) \), \( \text{csh}(x)=(\exp(x\pi/0.7887)+\exp(-x\pi/0.7887))/(2\pi) \), \( \text{ssh1}(x)=\sinh(x/\pi)\text{PHI} \), \( \text{csh1}(x)=\cosh(x/\pi)\text{PHI} \).
  • Using base image: No
  • Aspect Ratio: square

Quantum musings on Time

• Model: DaVinci2

• Size: 1864 X 1016 (1.89 MP)

• Used settings:

  • Prompt: Create a textured pencil sketch of the concept of time and time travel with abstract representations of space, clocks, clockwork and pieces of Swiss watch inner parts and mechanisms. Also add in some mathematical equations related to the concept of time and space, as well quantum mechanics. Allow for a larger space of math equations and focus less on watches. Deconstruct the clockwork pieces and separate them by piece and by layers. Show more diagrams than clocks and watches. Add in black hole diagrams and other astrophysical phenomena. The aspect ratio should be 16:9. Add an abstract geometric border to the image. Focus on diagrams and equations.
  • Using base image: Yes

Cosmic Bee in a Galaxy of Math and Beauty

• Model: AIVision

• Size: 1000 X 1000 (1.00 MP)

• Used settings:

  • Prompt: The image is a double exposure of a Bumble Bee flying into the space. The background contains mathematical equations and graphs. The image has a rough texture and is hyper-detailed with a muted palette of brown, crimson, gold, blue, and black. There are sharp lines, dripping paint and ink, and a glowing and sparkling effect.
  • Using base image: No
  • Aspect Ratio: square

Which Came First? Fish or Fibonacci?

• Model: AIVision

• Size: 1536 X 1536 (2.36 MP)

• Used settings:

  • Prompt: Koi fish immersed in mathematical wonder, thick oil paint texture, scratchy lines, cerulean background, top-level view, more Koi fish swim around the other, Orange and silvery white, wild patterns and curves beneath the fish, on the bottom under about 8” of water, fishstill clearly visible , valid mathematical formulas, white sparkling text, surreal and ethereal, Fibonacci spiral composition, 3-D effect, tree-filtered soft light, star sparkle on leaf, detailed wood painting, realistic with a touch of fantasy, by a blend of Salvador Dali and René Magritte, Artstation.
  • Using base image: No
  • Aspect Ratio: square

Floating Equations

• Model: AIVision

• Size: 1536 X 1536 (2.36 MP)

• Used settings:

  • Prompt: Mathematical formulas environment, surreal world, floating equations, everyting is made of algebra, soft colors, abstract shapes, futuristic setting, glowing symbols, intricate patterns, digital art, detailed and intricate, young beautiful female dressed in formulas, confused expression, central focus, fantasy concept, artistically rendered, soft ambient lighting, ArtStation.
  • Using base image: No
  • Aspect Ratio: square

Cosmic Vibrance: Geometry of the Universe

• Model: AIVision

• Size: 1792 X 1024 (1.84 MP)

• Used settings:

  • Prompt: impact coding capability principle Interconnections energy methods explores symbolize allows principles mirrors structures existence semantics meaning showcasing programming processes mirroring operations inseparable concept cyclical function Relationships Encapsulation manifest functions Echoes
  • Using base image: No
  • Aspect Ratio: landscape