#fractal

Eternal Spin of a Fractal Dreamscape

• Model: DaVinci2

• Size: 1920 X 1080 (2.07 MP)

• Used settings:

  • Prompt: A mesmerizing single static image emerges from the ray-marched Mandelbulb scene with seamless full-360° z-axis rotation capture. Each frame increments by exactly π/64.45788754 radians and the fixed camera at 0.6·vec3(0, -12.75, 5.87), preserving the tan(TAU/6) FOV where TAU ≈ 4.957 (2π·0.7887). All rotational scenes are baked-in statically rendered on top of each other in fullscreen ! The fractal's intricate, hyperbolic-warped tendrils—distorted via custom chp/shp/ssh functions, PHI-scaled powers (11.24788742^LOOPS=256 iterations), and time-frozen φ offset (0.2·asin(z.z/r))—unfurl in violet-magenta glows (mat=vec3(0.8,0.5,1.05)), kissed by HSV(0.6,0.85,1) diffuse from the ld=(0,10,0) key light. Fresnel edges (smoothstep(1,0.9,(1+dot(rd,sn))^2)) blend 0.1-1.0 mixes of reflection (r·skymat.y·fre·edge) against skyCol=HSV(0.6,0.86,1) planes at y=±4/6, etched with box/pp noise (ds=length(pp)-0.5, shaped by shp(clamp(col,0,10))) and exp(-0.5·max(db,0)) falloff + 4·skyCol·rd.y²·smoothstep(0.25,0,db). Subtle beer absorption (exp(-(st+0.1)·-HSV(0.05,0.95,2.0))) adds volumetric haze, aggregated via ssh1(r·agg·skyColor) in 64-bounce refractions (reflect(-ssh1(rd), chpp(ro)); rd=chpp(ref) or ro=shpp(sp+0.1·rd)).Post-processed through ACES tonemapping (v=0.6; clamp((v*(2.51v+0.03))/(v*(2.43v+0.59)+0.14),0,1)) and sRGB gamma (mix(1.055·pow(t,1/2.4)-0.055, 12.92t, step(t,0.0031308))), the grid pulses with glowCol=HSV(0.065,0.8,6) auras against the rotated g_rot=rot_x(((1.221·time+π)/tau)) baseline, df(p)=shp(mandelBulb(p/2.0)*2.0). No artifacts, no text—pure, tolerance-0.00001 precision (max 784.0 length, 487 marches) frozen in eternal spin, seed 1924139471 anchoring the chaos.
  • Upscale & Enhance: 1
  • Aspect Ratio: landscape_wide

Swirling Vibrance in Purple and Yellow

• Model: Artistic 2

• Size: 2496 X 3312 (8.27 MP)

• Used settings:

  • Prompt: Optical art, trippy fractal with zentangle elements
  • Using base image: Yes (Effect strength - 70%)
  • Base image resolution: Default
  • Upscale & Enhance: 0

Vibrant Fractal Patterns in Dynamic Colors

• Model: AIVision

• Size: 1024 X 1024 (1.05 MP)

• Used settings:

  • Prompt: In the mesmerizing world of fractals—where geometry, art, and science merge with fantasy—each pattern unfolds an endless journey of discovery. Every level reveals new and breathtaking shapes, like a high-quality masterpiece born from the imagination of visionary artists and the brilliance of fine jewelry.
  • Upscale & Enhance: 0
  • Aspect Ratio: square

Fractal Dynamics of Quantum Field Theory

• Model: DaVinci2

• Size: 1152 X 864 (1.00 MP)

• Used settings:

  • 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,
  • Upscale & Enhance: 0
  • Aspect Ratio: landscape

Ethereal Fractal Glow: A Symmetrical Journey

• Model: AIVision

• Size: 1536 X 1536 (2.36 MP)

• Used settings:

  • Prompt: An image of a highly detailed, symmetrical 3D fractal structure exhibiting translucent, refractive qualities, internal amber-tinted glow from Beer-Lambert absorption using high-contrast ethereal vibrancy achieved via ray marching with tolerance 0.00001, max ray length 748.0, up to 748 marches, and 16 bounces for reflections and refractions, starting from camera at (1.256,2.47855874,-7.5) looking at origin with FOV tan(tau/16.61) where tau=2*pi. The distance field df(p) = shp(mandelBulb(p/z1)*z1) with z1=2.0, where shp(x) = (exp(x)-exp(-x))/(pi/PHI) and PHI=(sqrt(5)/2 + 0.5)≈1.618.The mandelBulb(p) function iterates with power=11.24788742 and loops=16: initialize z = chp(p)*p - p where chp(x)=(exp(x)+exp(-x))/pi; dr=1.0; for each loop, r=length(z), bail if r>2; theta=atan(z.x,z.y); phi=acos(z.z/r) + asinh(time)*0.2; dr = pow(r,power-1)*dr*power +1; r=pow(r,power); theta*=power/PHI; phi*=power/PHI; z = r * vec3(tan(shp(sin(theta)*sin(phi)))/(PHI*tau), chp(cos(theta)*sin(phi))/(PHI*tau), (cos(phi)*sin(phi))/(PHI*tau)) + p; p=reflect(p,z); return 0.75*log(r)*r/dr.Incorporate custom hyperbolic functions for distortions: chpp(x)=(exp(x/(cosh(x)*pi))+exp(-x/(cosh(x)/pi)))/(TAU*PHI) with TAU=(2*pi)*0.7887≈4.951; shpp(x)=(exp(x*(sinh(x)*pi))-exp(-x*(sinh(x)*pi)))/(TAU/PHI); ssh(x)=(exp(x*pi/0.7887)-exp(-x*pi/0.7887))/(2*pi); csh(x)=(exp(x*pi/0.7887)+exp(-x*pi/0.7887))/(2*pi); ssh1(x)=sinh(x/pi)*PHI; csh1(x)=cosh(x/pi)*PHI. Use these in skyColor with reflections as reflect(-ssh1(rd), chpp(ro)), in rendering aggregation as agg += ssh1(ragg*skyColor(ro,rd)), and ray updates as rd=chpp(ref) or ro=shpp(sp + initt*rd) with initt=0.1.Material properties: mat=vec3(0.8,0.5,1.05) for diffuse, specular, refractive index; Fresnel fre=1+dot(rd,sn), fre*=fre, mix(0.1,1,fre); diffuse col += diffuseCol * dif*dif *(1-mat.x) with dif=max(dot(ld,sn),0), ld=normalize(lightPos-sp), lightPos=(0,10,0); reflection col += rsky*mat.y*fre*vec3(1)*edge with edge=smoothstep(1,0.9,fre); colors from HSV: skyCol=HSV(0.6,0.86,1), glowCol=HSV(0.065,0.8,6), diffuseCol=HSV(0.6,0.85,1). Inside traversal flips dfactor=-1, applies absorption ragg *= exp(-(st+initt)*beer), and refracts with index 1/mat.z when inside.Normals computed via finite differences: nor.x = df(pos+eps.xyy)-df(pos-eps.xyy) etc., with eps=(0.0005,0). Sky includes ray-plane intersections tp=(dot(ro,p.xyz)+p.w)/dot(rd,p.xyz) for planes at y=4 and y=-6, with box(pp,vec2(6,9))-1 for patterns, col += 4*skyCol*rd.y*rd.y*smoothstep(0.25,0,db) + 0.8*skyCol*exp(-0.5*max(db,0)), and similar for circular ds=length(pp)-0.5, clamped and shaped with shp(clamp(col,0,10)).Match the visual style of a Shadertoy-generated "Inside the Mandelbulb II" fractal art piece, capturing a static frame of the animated, lucky-bug emergent symmetry from reflections and custom coordinate remaps for asymmetric flaring in protrusions, explosive tan-amplified edges, and harmonic golden-ratio scalings.
  • Upscale & Enhance: 1
  • Aspect Ratio: square

Fractal Mask in a Dreamy Sky Landscape

• Model: AIVision

• Size: 1536 X 1536 (2.36 MP)

• Used settings:

  • Prompt: Create a highly detailed, symmetrical 3D fractal structure resembling a modified Mandelbulb, rendered as an abstract organic form resembling a crystalline, glassy mask or face with intricate, swirling coral-like protrusions and self-similar details, floating against a procedural gradient blue sky background with soft cyan-to-deep blue tones and subtle plane-based depth elements like top and bottom horizons with box-shaped patterns and exponential glow falloff. The central shape features two large, spiral red-orange eyes formed by hyperbolic distortions and a downward-curving dark blue mouth evoking a surprised or melancholic expression, with vibrant pink and orange hues for the main body exhibiting translucent, refractive qualities, internal amber-tinted glow from Beer-Lambert absorption using vector beer = -HSV(0.05, 0.95, 2.0), and high-contrast ethereal vibrancy achieved via ACES tone mapping approximation (v *= 0.6; clamp((v*(2.51*v+0.03))/(v*(2.43*v+0.59)+0.14), 0,1)) followed by sRGB gamma correction (mix(1.055*pow(t,1/2.4)-0.055, 12.92*t, step(t,0.0031308))). Generate the fractal using ray marching with tolerance 0.00001, max ray length 20.0, up to 48 marches, and 5 bounces for reflections and refractions, starting from camera at (0,2,5) looking at origin with FOV tan(tau/6) where tau=2*pi, incorporating global time-animated rotation around x-axis by (1.221*time + pi)/tau. The distance field df(p) = shp(mandelBulb(p/z1)*z1) with z1=2.0, where shp(x) = (exp(x)-exp(-x))/(pi/PHI) and PHI=(sqrt(5)/2 + 0.5)≈1.618, applied after rotating p by transpose(inverse(g_rot)). The mandelBulb(p) function iterates with power=11.24788742 and loops=3: initialize z = chp(p)*p - p where chp(x)=(exp(x)+exp(-x))/pi; dr=1.0; for each loop, r=length(z), bail if r>2; theta=atan(z.x,z.y); phi=asin(z.z/r) + time*0.2; dr = pow(r,power-1)*dr*power +1; r=pow(r,power); theta*=power/PHI; phi*=power/PHI; z = r * vec3(tan(shp(sin(theta)*sin(phi)))*PHI, chp(cos(theta)*sin(phi)), cos(phi)) + p; p=reflect(p,z); return 0.75*log(r)*r/dr. Incorporate custom hyperbolic functions for distortions: chpp(x)=(exp(x/(cosh(x)*pi))+exp(-x/(cosh(x)/pi)))/(TAU*PHI) with TAU=(2*pi)*0.7887≈4.951; shpp(x)=(exp(x*(sinh(x)*pi))-exp(-x*(sinh(x)*pi)))/(TAU/PHI); ssh(x)=(exp(x*pi/0.7887)-exp(-x*pi/0.7887))/(2*pi); csh(x)=(exp(x*pi/0.7887)+exp(-x*pi/0.7887))/(2*pi); ssh1(x)=sinh(x/pi)*PHI; csh1(x)=cosh(x/pi)*PHI. Use these in skyColor with reflections as reflect(-ssh1(rd), chpp(ro)), in rendering aggregation as agg += ssh1(ragg*skyColor(ro,rd)), and ray updates as rd=chpp(ref) or ro=shpp(sp + initt*rd) with initt=0.1. Material properties: mat=vec3(0.8,0.5,1.05) for diffuse, specular, refractive index; Fresnel fre=1+dot(rd,sn), fre*=fre, mix(0.1,1,fre); diffuse col += diffuseCol * dif*dif *(1-mat.x) with dif=max(dot(ld,sn),0), ld=normalize(lightPos-sp), lightPos=(0,10,0); reflection col += rsky*mat.y*fre*vec3(1)*edge with edge=smoothstep(1,0.9,fre); colors from HSV: skyCol=HSV(0.6,0.86,1), glowCol=HSV(0.065,0.8,6), diffuseCol=HSV(0.6,0.85,1). Inside traversal flips dfactor=-1, applies absorption ragg *= exp(-(st+initt)*beer), and refracts with index 1/mat.z when inside. Normals computed via finite differences: nor.x = df(pos+eps.xyy)-df(pos-eps.xyy) etc., with eps=(0.0005,0). Sky includes ray-plane intersections tp=(dot(ro,p.xyz)+p.w)/dot(rd,p.xyz) for planes at y=4 and y=-6, with box(pp,vec2(6,9))-1 for patterns, col += 4*skyCol*rd.y*rd.y*smoothstep(0.25,0,db) + 0.8*skyCol*exp(-0
  • Upscale & Enhance: 1
  • Aspect Ratio: square

Surreal Crystalline Fractal Mask Artistry

• Model: Ideogram

• Size: 1312 X 736 (0.97 MP)

• Used settings:

  • Prompt: Generate a highly detailed, surreal 3D fractal artwork in the style of a modified Mandelbulb hybrid, rendered as an abstract, crystalline mask-like organic form with 5-lobe rotational symmetry and intricate, swirling coral-like protrusions emerging from a central glassy structure evoking an alien eye with stalk extensions, floating against a procedural gradient sky background transitioning from soft cyan (#00FFFF) to deep blue (#000080) with subtle plane-based depth elements at horizons y=4 and y=-6 featuring box-shaped patterns via box(pp, vec2(6,9))-1 and exponential glow falloff exp(-0.5*max(db,0)), incorporating self-similar recursive details from ray marching with tolerance 0.00001, max ray length 20.0, up to 48 marches, and 5 bounces for reflections and refractions. The central form features a glossy orange iris with dark pupil void from orbit trap sphere at origin (radius 0.1), surrounded by five radiating mushroom-like stalks with bumpy textures from sphere folds (minR²=0.2-0.3, fixedR²=1.0-1.2), vibrant pink-orange hues (HSV(0.065,0.8,6) for glow, HSV(0.6,0.85,1) for diffuse) exhibiting translucent refractive qualities with internal amber-tinted glow via Beer-Lambert absorption ragg *= exp(-(st+initt)*beer) where beer = -HSV(0.05, 0.95, 2.0) and initt=0.1, high-contrast ethereal vibrancy via ACES tone mapping v *= 0.6; clamp((v*(2.51*v+0.03))/(v*(2.43*v+0.59)+0.14), 0,1) followed by sRGB gamma mix(1.055*pow(t,1/2.4)-0.055, 12.92*t, step(t,0.0031308)). Use the merged distance field df(p) = shp(mandelBulb(p/z1)*z1) with z1=2.0, where shp(x) = (exp(x)-exp(-x))/(pi/PHI) and PHI=(sqrt(5)/2 + 0.5)≈1.618, applied after rotating p by transpose(inverse(g_rot)) with x-axis animation (1.221*time + pi)/tau where tau=2*pi. The mandelBulb(p) iterates with power n≈6-11.24788742 and loops=3: initialize z = chp(p)*p - p where chp(x)=(exp(x)+exp(-x))/pi; dr=1.0; for each loop, r=length(z), bail if r>2; theta=atan(z.x,z.y); phi=asin(z.z/r) + time*0.2; dr = pow(r,power-1)*dr*power +1; r=pow(r,power); theta*=power/PHI; phi*=power/PHI; z = r * vec3(tan(shp(sin(theta)*sin(phi)))*PHI, chp(cos(theta)*sin(phi)), cos(phi)) + p; p=reflect(p,z); return 0.75*log(r)*r/dr. Incorporate pre-folding with hyperbolic distortion: z' = chp(z) · z - z, then integer power fold for each axis i: z_i'' = {2f - z_i' if z_i' > f; -2f - z_i' if z_i' < -f; z_i' otherwise}, f≈1.2-1.5, then z_i''' = shp(z_i'') = (exp(z_i'') - exp(-z_i'')) / (pi / PHI). Follow with Amazing Box fold u' = s · clamp(u, -l, l) - (s - 1) · u for u∈{x,y,z}, s≈1.8-2.2, l≈1.0, then sphere fold r²=||z||², z' = z · μ where μ = {r/m if r² < m; r/r² if m ≤ r² < r; 1 otherwise}, then modulate z'' = shpp(z') = (exp(z' · (sinh(z') · pi)) - exp(-z' · (sinh(z') · pi))) / (TAU / PHI) with TAU=(2*pi)*0.7887≈4.951, dr' = |s| · pow(r^{n-1}, PHI) · dr + 1. Post-transform: z' = k · R · z + t, k≈1.1-1.3, R=transpose(inverse(g_rot)), t≈(0,0,0.2). Custom hyperbolic functions: chpp(x)=(exp(x/(cosh(x)*pi))+exp(-x/(cosh(x)/pi)))/(TAU*PHI); shpp(x)=(exp(x*(sinh(x)*pi))-exp(-x*(sinh(x)*pi)))/(TAU/PHI); ssh(x)=(exp(x*pi/0.7887)-exp(-x*pi/0.7887))/(2*pi); csh(x)=(exp(x*pi/0.7887)+exp(-x*pi/0.7887))/(2*pi); ssh1(x)=sinh(x/pi)*PHI; csh1(x)=cosh(x/pi)*PHI. Use in skyColor with reflections reflect(-ssh1(rd), chpp(ro)), rendering aggregation agg += ssh1(ragg*skyColor(ro,rd)), ray updates rd=chpp(ref) or ro=shpp(sp + initt*rd). Materials: mat=vec3(0.8,0.5,1.05) for diffuse, specular, refracti
  • Upscale & Enhance: 0
  • Aspect Ratio: landscape_wide
  • Ideogram Style: Realistic

Fractal Dynamics of Quantum Field Theory

• Model: AIVision

• Size: 1536 X 1536 (2.36 MP)

• Used settings:

  • Prompt: Render 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,\text{s
  • Upscale & Enhance: 1
  • Aspect Ratio: square

Realist Fusion of cow

• Model: ProVideo

• Size: 1660 X 1244 (2.07 MP)

• Used settings:

  • Mode: Cine
  • Generated From: Image
  • Duration: 5 seconds

Intricate 3D Fractal with Ethereal Details

• Model: AIVision

• Size: 1536 X 1536 (2.36 MP)

• Used settings:

  • Prompt: An image of a highly detailed, symmetrical 3D fractal structure resembling a modified Mandelbulb, rendered as an abstract organic form resembling a crystalline, glassy rendition intricate, swirling coral-like protrusions and self-similar details, floating against a procedural gradient blue sky background with soft cyan-to-deep blue tones and subtle plane-based depth elements like top and bottom horizons with box-shaped patterns and exponential glow falloff. The central shape features two large, spiral red-orange eyes formed by hyperbolic distortions and a downward-curving dark blue mouth evoking a surprised or melancholic expression, with vibrant pink and orange hues for the main body exhibiting translucent, refractive qualities, internal amber-tinted glow from Beer-Lambert absorption using vector beer = -HSV(0.05, 0.95, 2.0), and high-contrast ethereal vibrancy achieved via ACES tone mapping approximation (v *= 0.6; clamp((v*(2.51*v+0.03))/(v*(2.43*v+0.59)+0.14), 0,1)) followed by sRGB gamma correction (mix(1.055*pow(t,1/2.4)-0.055, 12.92*t, step(t,0.0031308))).Generate the fractal using ray marching with tolerance 0.00001, max ray length 20.0, up to 48 marches, and 5 bounces for reflections and refractions, starting from camera at (0,2,5) looking at origin with FOV tan(tau/6) where tau=2*pi, incorporating global time-animated rotation around x-axis by (1.221*time + pi)/tau. The distance field df(p) = shp(mandelBulb(p/z1)*z1) with z1=2.0, where shp(x) = (exp(x)-exp(-x))/(pi/PHI) and PHI=(sqrt(5)/2 + 0.5)≈1.618, applied after rotating p by transpose(inverse(g_rot)).The mandelBulb(p) function iterates with power=11.24788742 and loops=3: initialize z = chp(p)*p - p where chp(x)=(exp(x)+exp(-x))/pi; dr=1.0; for each loop, r=length(z), bail if r>2; theta=atan(z.x,z.y); phi=asin(z.z/r) + time*0.2; dr = pow(r,power-1)*dr*power +1; r=pow(r,power); theta*=power/PHI; phi*=power/PHI; z = r * vec3(tan(shp(sin(theta)*sin(phi)))*PHI, chp(cos(theta)*sin(phi)), cos(phi)) + p; p=reflect(p,z); return 0.75*log(r)*r/dr.Incorporate custom hyperbolic functions for distortions: chpp(x)=(exp(x/(cosh(x)*pi))+exp(-x/(cosh(x)/pi)))/(TAU*PHI) with TAU=(2*pi)*0.7887≈4.951; shpp(x)=(exp(x*(sinh(x)*pi))-exp(-x*(sinh(x)*pi)))/(TAU/PHI); ssh(x)=(exp(x*pi/0.7887)-exp(-x*pi/0.7887))/(2*pi); csh(x)=(exp(x*pi/0.7887)+exp(-x*pi/0.7887))/(2*pi); ssh1(x)=sinh(x/pi)*PHI; csh1(x)=cosh(x/pi)*PHI. Use these in skyColor with reflections as reflect(-ssh1(rd), chpp(ro)), in rendering aggregation as agg += ssh1(ragg*skyColor(ro,rd)), and ray updates as rd=chpp(ref) or ro=shpp(sp + initt*rd) with initt=0.1.Material properties: mat=vec3(0.8,0.5,1.05) for diffuse, specular, refractive index; Fresnel fre=1+dot(rd,sn), fre*=fre, mix(0.1,1,fre); diffuse col += diffuseCol * dif*dif *(1-mat.x) with dif=max(dot(ld,sn),0), ld=normalize(lightPos-sp), lightPos=(0,10,0); reflection col += rsky*mat.y*fre*vec3(1)*edge with edge=smoothstep(1,0.9,fre); colors from HSV: skyCol=HSV(0.6,0.86,1), glowCol=HSV(0.065,0.8,6), diffuseCol=HSV(0.6,0.85,1). Inside traversal flips dfactor=-1, applies absorption ragg *= exp(-(st+initt)*beer), and refracts with index 1/mat.z when inside.Normals computed via finite differences: nor.x = df(pos+eps.xyy)-df(pos-eps.xyy) etc., with eps=(0.0005,0). Sky includes ray-plane intersections tp=(dot(ro,p.xyz)+p.w)/dot(rd,p.xyz) for planes at y=4 and y=-6, with box(pp,vec2(6,9))-1 for patterns, col += 4*skyCol*rd.y*rd.y*smoothstep(0.25,0,db) + 0.8*skyCol*exp(-0.5*max(db
  • Upscale & Enhance: 1
  • Aspect Ratio: square

Mesmerizing Fractal Waves in Blue and Gold

• Model: DaVinci2

• Size: 1416 X 840 (1.19 MP)

• Used settings:

  • Prompt: abstract vector fractal, wave function, Zentangle, 3d shading
  • Using base image: Yes (Effect strength - 60%)
  • Base image resolution: 1MP
  • Upscale & Enhance: 0

Surreal Crystalline Fractal Mask Creation

• Model: DaVinci2

• Size: 1152 X 864 (1.00 MP)

• Used settings:

  • Prompt: Generate a highly detailed, surreal 3D fractal artwork in the style of a modified Mandelbulb hybrid, rendered as an abstract, crystalline mask-like organic form with 5-lobe rotational symmetry and intricate, swirling coral-like protrusions emerging from a central glassy structure evoking an alien eye with stalk extensions, floating against a procedural gradient sky background transitioning from soft cyan (#00FFFF) to deep blue (#000080) with subtle plane-based depth elements at horizons y=4 and y=-6 featuring box-shaped patterns via box(pp, vec2(6,9))-1 and exponential glow falloff exp(-0.5*max(db,0)), incorporating self-similar recursive details from ray marching with tolerance 0.00001, max ray length 20.0, up to 48 marches, and 5 bounces for reflections and refractions. The central form features a glossy orange iris with dark pupil void from orbit trap sphere at origin (radius 0.1), surrounded by five radiating mushroom-like stalks with bumpy textures from sphere folds (minR²=0.2-0.3, fixedR²=1.0-1.2), vibrant pink-orange hues (HSV(0.065,0.8,6) for glow, HSV(0.6,0.85,1) for diffuse) exhibiting translucent refractive qualities with internal amber-tinted glow via Beer-Lambert absorption ragg *= exp(-(st+initt)*beer) where beer = -HSV(0.05, 0.95, 2.0) and initt=0.1, high-contrast ethereal vibrancy via ACES tone mapping v *= 0.6; clamp((v*(2.51*v+0.03))/(v*(2.43*v+0.59)+0.14), 0,1) followed by sRGB gamma mix(1.055*pow(t,1/2.4)-0.055, 12.92*t, step(t,0.0031308)). Use the merged distance field df(p) = shp(mandelBulb(p/z1)*z1) with z1=2.0, where shp(x) = (exp(x)-exp(-x))/(pi/PHI) and PHI=(sqrt(5)/2 + 0.5)≈1.618, applied after rotating p by transpose(inverse(g_rot)) with x-axis animation (1.221*time + pi)/tau where tau=2*pi. The mandelBulb(p) iterates with power n≈6-11.24788742 and loops=3: initialize z = chp(p)*p - p where chp(x)=(exp(x)+exp(-x))/pi; dr=1.0; for each loop, r=length(z), bail if r>2; theta=atan(z.x,z.y); phi=asin(z.z/r) + time*0.2; dr = pow(r,power-1)*dr*power +1; r=pow(r,power); theta*=power/PHI; phi*=power/PHI; z = r * vec3(tan(shp(sin(theta)*sin(phi)))*PHI, chp(cos(theta)*sin(phi)), cos(phi)) + p; p=reflect(p,z); return 0.75*log(r)*r/dr. Incorporate pre-folding with hyperbolic distortion: z' = chp(z) · z - z, then integer power fold for each axis i: z_i'' = {2f - z_i' if z_i' > f; -2f - z_i' if z_i' < -f; z_i' otherwise}, f≈1.2-1.5, then z_i''' = shp(z_i'') = (exp(z_i'') - exp(-z_i'')) / (pi / PHI). Follow with Amazing Box fold u' = s · clamp(u, -l, l) - (s - 1) · u for u∈{x,y,z}, s≈1.8-2.2, l≈1.0, then sphere fold r²=||z||², z' = z · μ where μ = {r/m if r² < m; r/r² if m ≤ r² < r; 1 otherwise}, then modulate z'' = shpp(z') = (exp(z' · (sinh(z') · pi)) - exp(-z' · (sinh(z') · pi))) / (TAU / PHI) with TAU=(2*pi)*0.7887≈4.951, dr' = |s| · pow(r^{n-1}, PHI) · dr + 1. Post-transform: z' = k · R · z + t, k≈1.1-1.3, R=transpose(inverse(g_rot)), t≈(0,0,0.2). Custom hyperbolic functions: chpp(x)=(exp(x/(cosh(x)*pi))+exp(-x/(cosh(x)/pi)))/(TAU*PHI); shpp(x)=(exp(x*(sinh(x)*pi))-exp(-x*(sinh(x)*pi)))/(TAU/PHI); ssh(x)=(exp(x*pi/0.7887)-exp(-x*pi/0.7887))/(2*pi); csh(x)=(exp(x*pi/0.7887)+exp(-x*pi/0.7887))/(2*pi); ssh1(x)=sinh(x/pi)*PHI; csh1(x)=cosh(x/pi)*PHI. Use in skyColor with reflections reflect(-ssh1(rd), chpp(ro)), rendering aggregation agg += ssh1(ragg*skyColor(ro,rd)), ray updates rd=chpp(ref) or ro=shpp(sp + initt*rd). Materials: mat=vec3(0.8,0.5,1.05) for diffuse, specular, refracti
  • Upscale & Enhance: 0
  • Aspect Ratio: landscape

The Blue Mandala

• Model: DaVinci2

• Size: 1728 X 1296 (2.24 MP)

• Used settings:

  • Prompt: Blueprint-style technical drawing on aged drafting paper, using fine blue ink lines with variable thickness. Intricate fractal structures—based on recursive geometries like Mandelbrot sets and Koch snowflakes—are layered to create a sense of spatial depth. Foreground elements are crisp and highly detailed, while background patterns become progressively smaller and slightly faded, simulating depth with a blueprint drafting aesthetic. The composition resembles generative algorithmic design fused with architectural schematics, emphasizing precision, repetition, and spatial clarity. Main Style: Ink Sub Styles: Blueprint, Fractal, Abstract
  • Upscale & Enhance: 1
  • Aspect Ratio: landscape

Mesmerizing Blue and Silver Fractal Art

• Model: DaVinci2

• Size: 1920 X 1080 (2.07 MP)

• Used settings:

  • Prompt: A highly detailed digital rendering of an abstract, symmetrical fractal structure resembling a surreal, organic face floating against a gradient blue sky background, generated using a modified Mandelbulb fractal algorithm viewed from the inside with ray marching. Incorporate precise mathematical details: Define constants pi = 3.1415926535897932384626433832795, tau = 2*pi, TAU = (2*pi)*0.7887, PHI = (sqrt(5)*0.5 + 0.5) ≈1.618 golden ratio, POWER = 11.24788742 for exponentiation, LOOPS = 3 iterations, TOLERANCE = 0.00001, MAX_RAY_LENGTH = 20.0, MAX_RAY_MARCHES = 48, NORM_OFF = 0.0005, MAX_BOUNCES = 5. Custom hyperbolic functions: chp(x) = (exp(x) + exp(-x))/pi, chpp(x) = (exp(x/(cosh(x)*pi)) + exp(-x/(cosh(x)/pi)))/(TAU*PHI), shp(x) = (exp(x) - exp(-x))/(pi/PHI), shpp(x) = (exp(x*(sinh(x)*pi)) - exp(-x*(sinh(x)*pi)))/(TAU/PHI), ssh(x) = (exp(x*pi/0.7887) - exp(-x*pi/0.7887))/(2*pi), csh(x) = (exp(x*pi/0.7887) + exp(-x*pi/0.7887))/(2*pi), ssh1(x) = sinh(x/pi)*PHI, csh1(x) = cosh(x/pi)*PHI. The Mandelbulb distance estimator mandelBulb(p): Initialize z = chp(p)*p - p, dr=1.0; for i=0 to LOOPS-1, r=length(z), theta=atan(z.x,z.y), phi=asin(z.z/r) + optional time*0.2 for animation; dr = r^(POWER-1) * dr * POWER + 1; r = r^POWER, theta *= POWER/PHI, phi *= POWER/PHI; z = r * vec3(tan(shp(sin(theta)*sin(phi)))*PHI, chp(cos(theta)*sin(phi)), cos(phi)) + p; p = reflect(p,z). Return distance 0.75 * log(r) * r / dr. Overall distance function df(p) = shp(mandelBulb(p/2.0)*2.0) after applying rotation matrix g_rot = rot_x(((1.221*time + pi)/tau)). Render with ray marching from camera at 0.6*vec3(0,2,5) looking at origin, FOV tan(TAU/6), incorporating bounces for reflection (reflect(rd,sn)), refraction (refract(rd,sn,1.0/mat.z or inverse)), fresnel fre=1+dot(rd,sn) squared and mixed 0.1-1.0, diffuse dif=max(dot(ld,sn),0)^2 * (1-mat.x) with ld to light at (0,10,0), material mat=(0.8,0.5,1.05), beer absorption exp(-(st+0.1)* -HSV(0.05,0.95,2.0)). Sky background: Procedural with planes at y=4 and y=-6, box bounds, exponential falloff, colored HSV(0.6,0.86,1.0). Colors: Glow HSV(0.065,0.8,6.0), diffuse HSV(0.6,0.85,1.0), post-processed with ACES tonemapping aces_approx(v) = clamp((v*(2.51v+0.03))/(v*(2.43v+0.59)+0.14),0,1) after *0.6, and sRGB gamma mix(1.055*t^(1/2.4)-0.055,12.92*t,step(t,0.0031308)). The structure features two large spiral-eyed voids as eyes, a curved dark blue mouth-like opening at the bottom, elaborate branching tendrils and crystalline edges with subtle particle specks dissipating at sides, ethereal pinkish-orange glow, edge fresnel effects, hyper-realistic yet fantastical Shadertoy-inspired 3D art in 16:9 aspect ratio with sharp details and no text or artifacts.
  • Upscale & Enhance: 1
  • Aspect Ratio: landscape_wide

Vibrant Fractal Landscape of Swirling Colors

• Model: Ideogram

• Size: 1968 X 1104 (2.17 MP)

• Used settings:

  • Prompt: import numpy as np import matplotlib.pyplot as plt from matplotlib.colors import LinearSegmentedColormap from mpmath import mp, mpc, mpf def mandelbrot_mp(width, height, max_iter, center_real=' -0.75035', center_imag='0.07109', zoom=mpf('5.009e633')): mp.dps = 700 # Digits for 10^633 precision; increase to 800 if artifacts center = mpc(center_real, center_imag) scale = mpf(4) / zoom # Extent ~4/zoom for symmetric framing half_scale = scale / mpf(2) # Precompute linspaces as lists for mpmath compatibility x_list = [mpf(center.real) + half_scale * (mpf(2*i)/ (width-1) - 1) for i in range(width)] y_list = [mpf(center.imag) + half_scale * (mpf(2*i)/ (height-1) - 1) for i in range(height)] iter_count = np.zeros((height, width), dtype=float) for row in range(height): cy = y_list[row] for col in range(width): cx = x_list[col] c = mpc(cx, cy) z = mpc(0) it = 0 while abs(z) <= mpf(2) and it < max_iter: z = z**2 + c it += 1 if it == max_iter: iter_count[row, col] = 0 # Inside: black else: # Smooth fractional iter mu = it - mp.log(mp.log(abs(z))) / mp.log(2) iter_count[row, col] = float(mu) # Norm with cycles for deep gradient layers (pink/red bulbs) raw_norm = np.log1p(np.abs(iter_count)) / np.log1p(max_iter) norm = (raw_norm * 5) % 1.0 # 5 cycles for recursion hues return norm # Parameters from sample (test low first!) width, height = 400, 400 # Start low; ramp to 800+ max_iter = 10000 # Test; set to 2146123 for full center_real = '-0.75035' # Paste exact long string here if found center_imag = '0.07109' zoom = mpf('5.009e633') # Or smaller like 1e10 for testing # Compute (slow—patience!) fractal = mandelbrot_mp(width, height, max_iter, center_real, center_imag, zoom) # Colormap: blue far → green/purple spirals → pink/red strawberry bulbs colors = ['#00008b', '#228b22', '#4b0082', '#ff69b4', '#b22222', '#ffff00', '#ff00ff'] cmap = LinearSegmentedColormap.from_list('deep_strawberry', colors[::-1], N=256) # Reverse for pink-high iter # Plot plt.figure(figsize=(10, 10)) half_s = float(2 / zoom) # Approx for extent center_r, center_i = float(center_real), float(center_imag) plt.imshow(fractal, origin='lower', cmap=cmap, extent=[center_r - half_s, center_r + half_s, center_i - half_s, center_i + half_s]) plt.axis('off') plt.show()
  • Upscale & Enhance: 1
  • Aspect Ratio: landscape_wide
  • Ideogram Style: Auto

Elegant Fractal Design in Gold and Teal

• Model: DaVinci2

• Size: 1280 X 720 (0.92 MP)

• Used settings:

  • Prompt: [Main Subject]: "A single, incredibly complex, ultra-fractal quantum energy eigenstate." [Core Aesthetic Modifiers - Fractal & Recursion Amplified by 16.45788754*π]: "Beyond ultra-fractal, transcendent fractal density, infinite recursive detail, hyper-complex interwoven structures." "Self-similar patterns at every conceivable scale, hyper-dimensional fractal geometry, manifesting from a 16.45788754*pi-th order derivative." "Dynamic, chaotic yet profoundly coherent fractal energy flows, shimmering quantum foam with limitless intricate detail." "Exhibiting super-roughness and non-Euclidean visual geometry." [Structure & Form]: "Central radiant core, emitting and absorbing energy in a complex spiral fractal pattern, appearing as a stable, self-sustaining light construct." "Interconnecting tendrils of luminous energy branching outwards and inwards, forming a symmetrical (or fractally symmetrical) arrangement." "Feathery, iridescent filigrees, resembling Lévy flights or a fractional Brownian motion surface, iterated to extreme orders." [Energy & Interaction Modifiers]: "Luminous ethereal light, glowing with vibrant cosmic colors (deep blues, electric cyans, magenta, purple, gold accents)." "High-energy plasma-like textures, subtle holographic effects, shimmering particles." "Reflecting profound energy, fundamental mathematical precision, and overwhelming, mind-bending complexity." [Environmental / Background Modifiers]: "Deep cosmic void background, filled with intricate fractal nebulae and distant stardust, subtly forming recursive patterns." "Scattered abstract mathematical symbols like 'π', '2π', 'ħ', 'E', and 'ψ' integrated faintly into the fractal energy field and background, indicating the governing equations." [Atmospheric / Quality Modifiers]: "Hyper-realistic quality, cinematic lighting, ultra-high definition, ethereal glow, photorealistic rendering." "Surreal, abstract, conceptual art style, at the bleeding edge of theoretical physics visualization." "Evokes a sense of profound cosmic mystery and the ultimate nature of reality." Use as a core the exact maths: -\frac{\hbar^(16.45788754\pi)}{16.45788754\pim} \frac{d^(16.45788754\pi) \psi}{dx^(16.45788754\pi)} = E \psi
  • Upscale & Enhance: 0
  • Aspect Ratio: landscape_wide

Galactic Spiral: A Vivid Fractal Journey

• Model: AIVision

• Size: 1024 X 1024 (1.05 MP)

• Used settings:

  • Prompt: [Main Subject]: "A single, incredibly complex, ultra-fractal quantum energy eigenstate." [Core Aesthetic Modifiers - Fractal & Recursion Amplified by 16.45788754*π]: "Beyond ultra-fractal, transcendent fractal density, infinite recursive detail, hyper-complex interwoven structures." "Self-similar patterns at every conceivable scale, hyper-dimensional fractal geometry, manifesting from a 16.45788754*pi-th order derivative." "Dynamic, chaotic yet profoundly coherent fractal energy flows, shimmering quantum foam with limitless intricate detail." "Exhibiting super-roughness and non-Euclidean visual geometry." [Structure & Form]: "Central radiant core, emitting and absorbing energy in a complex spiral fractal pattern, appearing as a stable, self-sustaining light construct." "Interconnecting tendrils of luminous energy branching outwards and inwards, forming a symmetrical (or fractally symmetrical) arrangement." "Feathery, iridescent filigrees, resembling Lévy flights or a fractional Brownian motion surface, iterated to extreme orders." [Energy & Interaction Modifiers]: "Luminous ethereal light, glowing with vibrant cosmic colors (deep blues, electric cyans, magenta, purple, gold accents)." "High-energy plasma-like textures, subtle holographic effects, shimmering particles." "Reflecting profound energy, fundamental mathematical precision, and overwhelming, mind-bending complexity." [Environmental / Background Modifiers]: "Deep cosmic void background, filled with intricate fractal nebulae and distant stardust, subtly forming recursive patterns." "Scattered abstract mathematical symbols like 'π', '2π', 'ħ', 'E', and 'ψ' integrated faintly into the fractal energy field and background, indicating the governing equations." [Atmospheric / Quality Modifiers]: "Hyper-realistic quality, cinematic lighting, ultra-high definition, ethereal glow, photorealistic rendering." "Surreal, abstract, conceptual art style, at the bleeding edge of theoretical physics visualization." "Evokes a sense of profound cosmic mystery and the ultimate nature of reality." Use as a core the exact maths: -\frac{\hbar^(16.45788754\pi)}{16.45788754\pim} \frac{d^(16.45788754\pi) \psi}{dx^(16.45788754\pi)} = E \psi
  • Upscale & Enhance: 0
  • Aspect Ratio: square

Cosmic Fractal Spiral in Vibrant Colors

• Model: AIVision

• Size: 1536 X 1536 (2.36 MP)

• Used settings:

  • Prompt: Exact mathematical details to visualize:Quantum Gaussian Wave Packet Identity: The identity is: \psi(x,t) = \frac{1}{\sigma^{1/2}\pi^{1/4}}\frac{1}{\sqrt{1+\frac{i\hbar t}{m\sigma^2}}} \times \exp \left(-\frac{1}{2\left(\sigma^2+\frac{i\hbar t}{m} \right)}\left(x^2-2i\sigma^2\bar{p}/\hbar\left(x-\frac{\bar{p}}{2m}t\right) \right) \right) Derivation: Let p_0 = \bar{p}/\hbar and \gamma = \sigma^2 + i \hbar t / m. LHS exponent E_L = -\frac{\sigma^2}{2} (p_0 - k)^2 + i k x - i \frac{\hbar k^2 t}{2 m} = c + b k + a k^2, where a = -\gamma / 2, b = \sigma^2 p_0 + i x, c = -\sigma^2 p_0^2 / 2. Complete the square: a k^2 + b k + c = a (k - k_0)^2 + (c - b^2/(4a)), with k_0 = (\sigma^2 p_0 + i x) / \gamma. Constant term simplifies to -\frac{i \hbar t \sigma^2 p_0^2}{2 m \gamma} + \frac{i \sigma^2 p_0 x}{\gamma} - \frac{x^2}{2 \gamma}, matching RHS.Flame Fractal Generation: Defined by N functions f_i: \mathbb{R}^2 \to \mathbb{R}^2. Affine part: \begin{pmatrix} x' \ y' \end{pmatrix} = \begin{pmatrix} a_i & b_i \ d_i & e_i \end{pmatrix} \begin{pmatrix} x \ y \end{pmatrix} + \begin{pmatrix} c_i \ f_i \end{pmatrix}. Then f_i(x, y) = \sum_j w_{ij} v_j(x', y'), with \sum w_{ij} = 1. Key variations:Swirl: v(x, y) = (x \sin r^2 - y \cos r^2, x \cos r^2 + y \sin r^2), r^2 = x^2 + y^2. Bubble: v(x, y) = \frac{4}{r^2 + 4} (x, y). Julia: v(x, y) = r^{-1/2} (\cos(\theta/2 + k \pi), \sin(\theta/2 + k \pi)), \theta = \atan2(y, x). Spherical: v(x, y) = \frac{1}{r^2} (x, y). Horseshoe: v(x, y) = \frac{1}{r} (x - y)(x + y). Iteration: Start random (x, y), color=0. For M~10^7: Pick i by p_i (\sum p_i=1), (x,y)=f_i(x,y), color=(color + c_i)/2. Bin hits, render log(1+hits), gamma correction density^0.25, HSV palette.Shared Themes: Visualize Gaussian blurs exp(-r^2/(2\sigma^2)) in fractals akin to wave spreading; complex exponentials like swirl ~ z exp(i r^2) paralleling quantum exp(i (k x - \hbar k^2 t / (2m))). Background gradients from purple to blue, foreground spirals in red-green-yellow, vertical composition for teardrop flow.
  • Upscale & Enhance: 1
  • Aspect Ratio: square

Trippy Fractal Art with Intricate Details

• Model: DaVinci2

• Size: 1920 X 1080 (2.07 MP)

• Used settings:

  • Prompt: Optical art, trippy fractal with zentangle embellishments
  • Upscale & Enhance: 1
  • Aspect Ratio: landscape_wide

Trippy Fractal Art with Intricate Details

• Model: DaVinci2

• Size: 1920 X 1080 (2.07 MP)

• Used settings:

  • Prompt: Optical art, trippy fractal with zentangle embellishments
  • Upscale & Enhance: 1
  • Aspect Ratio: landscape_wide