Prompt: The Ancient Wisdom of the Dragons was what brought Atlantis into knowledgeable blooming prosperity.
The homosapientic stupidity, greed and decadence was what brought it all down...
Prompt: The supersymmetric action in 4D supergravity as a seamless, abstract geometric manifold in curved spacetime, rendered at 8K resolution with iridescent metallic gradients transitioning from deep sapphire blues and silvers for bosonic fields to vibrant emerald greens and golds for fermionic interactions, evoking quantum foam and holographic duality. Core instructions: Depict graphically, using no text, the full action \( S[e,\psi] = S[e] + S_f[e,\psi] + S_I[e,\psi] \), where the exact, completely detailedly concised maths is: \[ S[e,\psi] = S[e] + S_f[e,\psi] + S_I[e,\psi] = \int dx\, e\, e^a \wedge e^b \wedge F^{cd} \epsilon_{abcd} + \frac{1}{6} \int dx \, \theta^a \wedge e^b \wedge e^c \wedge e^d \epsilon_{abcd} + \int dx\, (\bar{\psi} \gamma_5 \gamma_a \psi)\, (\bar{\psi} \gamma_5 \gamma^a \psi) \, , \] with \(\theta^a \equiv \frac{i}{2} \left( \bar{\psi} \gamma^a D_\mu \psi - \overline{D_\mu \psi} \gamma^a \psi \right) dx^\mu \), all indices a,b,c,d=0,1,2,3 in the orthonormal frame bundle, e^a as coframe 1-forms (vielbeins), F^{cd} = dA^{cd} + A^{c e} \wedge A^{e d} the curvature 2-form of the spin connection, D_μ the covariant derivative along coordinate 1-forms dx^μ, ψ a Majorana spinor field, γ^a Dirac matrices in curved space, ε_{abcd} the Levi-Civita symbol with ε_{0123}=+1, and integrals over the 4-manifold with oriented volume form e = e^0 ∧ e^1 ∧ e^2 ∧ e^3. Visualize the first term S[e] as a swirling vortex of interlocking tetrahedral frames (symbolizing ε_{abcd} contraction) threaded by golden flux tubes (F^{cd} curvature) piercing a lattice of silver vierbein arrows (e^a, e^b) emanating from a central black hole singularity, representing the Einstein-Cartan Chern-Simons topological term. Overlay the fermionic torsion term S_f[e,ψ] as twisting helical ribbons (θ^a 1-forms derived from ψ bilinears) coiling around the vierbeins e^b,c,d into a knotted 4-simplex lattice with emerald sparks at intersection nodes, illustrating the 1/6 prefactor via sixfold symmetric bulbous expansions. Integrate the interaction S_I[e,ψ] as pulsating wave interference patterns of dual green scalar densities (ψ-bar γ5 γ_a ψ and conjugate), forming self-dual chiral currents that ripple across the manifold, modulating the geometry with fractal-like spinor foam bubbles where |ψ|^2 > threshold, colored by pseudoscalar density via smooth escape-time analogy (iterate bilinear up to 500 steps, hue H = 120° * iter / max, S=0.8, V=1). Ensure the entire composition flows as a unified holographic projection on a de Sitter boundary, with anti-aliased edges via Gaussian smoothing, subtle gravitational lensing distortions, and a faint cosmic microwave glow fading to void black; no equations, labels, or text visible; ultra-sharp filaments on torsion helices and current waves; aspect ratio 16:9; in the style of mathematical physics visualization.
Prompt: He has fuse with the universe and become one intelligence, being with infinity. He dare consume the fruits of the knowledge of good and evil and gain the forbidden God consciousness. now he is equal with the gods.... and gods do not kill gods but you are not the same, billions of you are expandable...as you are replicas producing replicas...machines to build, sustain the archonic civilization and continue the cultural memory...
Prompt: A highly detailed 3D render of an object with power P=11.24788742, fixed iterations LOOPS=64, initialized as z = chp(p)*p - p where chp(x)=(exp(x)+exp(-x))/π, shp(x)=(exp(x)-exp(-x))/π, chpp(x)=(exp(x/(cosh(x)π))+exp(-x/(cosh(x)/π)))/(2π Φ), shpp(x)=(exp(x sinh(x) π)-exp(-x sinh(x) π))/(2π Φ), ssh1(x)=sinh(x/π)/Φ, csh1(x)=cosh(x/π)/Φ, Φ=(1+√5)/2 golden ratio, τ=2π*0.7887; iteration: r=||z||, if r>2 continue, θ=asin(z_z/r)+0.2t animated, φ=atan(z_x,z_y), dr = r^{P-1} dr P +1, r=r^P, θ=θ P/Φ, φ=φ P/Φ, z += r * (tan(shp(sinθ sinφ)) Φ, chp(cosθ sinφ), cosφ) + p, p=reflect(p,z), final DE=0.75 log(r) r / dr scaled by shp(DE *2); ray-marched with max marches=96, tol=10^{-5}, bounces=8, refraction index 1.05, Beer absorption exp(-(t+0.1) * -HSV(0.05,0.95,2)), diffuse HSV(0.6,0.85,1), glow HSV(0.065,0.8,6), sky HSV(0.6,0.86,1) with warped reflections via ssh1, chpp, fract(clamp(0.125 / |reflected cross| * skyCol, -120,16.547)); rotated by rot_x((1.221 t + π)/τ), camera at (0,2,5)*0.6, FOV tan(τ/6), ACES tone-mapped, sRGB gamma; central bulbous form with pink core, orange lobes, black voids, cyan shell, rainbow tunnel background.
Prompt: **"Spaceship based on upon the following maths..."**: This immediately tells me you want a complex, possibly fractal-like, and intricately structured object. The "based upon maths" implies a non-organic, possibly generated or calculated appearance.
**Constants (\(\pi\), \(\text{tau}\), \(\text{PHI}\), \(\text{POWER}\), \(\text{LOOPS}\))**: These suggest a generative process, iterative refinement, and perhaps a sense of precision and complexity. The presence of PHI (the golden ratio) often hints at aesthetically pleasing, naturally occurring or fractal patterns.
**Custom hyperbolic functions (\(\text{chp}(x)\), \(\text{shp}(x)\), etc.)**: These are strong indicators of non-Euclidean geometry, curvature, twisting, and potentially organic yet structured forms. Hyperbolic functions often produce flowing, complex, and sometimes branching shapes. The specific forms like `chpp` and `shpp` with their nested hyperbolic functions and divisions by constants like `TAU/PHI` further emphasize extreme complexity and unique, perhaps alien, geometric properties.
**Mandelbulb formula (\(z=\text{chp}(p)p - p\), \(\text{dr}=1.0\); loop: \(r=\text{length}(z)\)...)**: This is the most crucial part. The Mandelbulb is a well-known 3D fractal. This directly tells me the desired image should exhibit:
**Fractal characteristics**: Self-similarity at different scales, infinite detail, recursive patterns.
**3D complexity**: The "bulb" implies a volumetric shape, not just a 2D pattern.
**Iterative generation**: The "loop" and power functions `pow(r,POWER)` describe how the fractal grows and forms its intricate surface.
**Specific transformations**: `\(\theta=\text{POWER}/\text{PHI}\)`, `\(z=r\text{vec3}(\tan(\text{shp}(\sin(\theta)\sin(\phi)))\text{PHI}, \text{chp}(\cos(\theta)\sin(\phi)), \cos(\phi))+p\)` indicate sophisticated rotations, trigonometric operations, and mapping of coordinates, which would result in highly sculptural and intertwined forms.
**Distance estimation**: `\(\text{distance}=0.75\log(r)r/\text{dr}\)` is a technique used in raymarching fractals to render surfaces, implying smooth yet incredibly detailed structures.
**Material properties (\(\text{mat}=\text{vec3}(0.8,0.5,1.05)\), \(\text{fresnel}\), \(\text{diffuse}\), \(\text{reflection}\))**: These terms describe how light interacts with the spaceship's surface.
**Specific `vec3` for `mat`**: Suggests a base color or material property.
**Fresnel**: Implies a metallic or reflective surface where reflectivity changes with the viewing angle.
**Diffuse**: Indicates some scattering of light.
**Reflection**: Explicitly states a desire for reflections, making the surface appear glossy or metallic.
**Colors (\(\text{skyCol}=\text{HSV}(0.6,0.86,1)\), \(\text{glowCol}=\text{HSV}(0.065,0.8,6)\), etc.)**: These provide a very clear color palette.
**SkyCol (blue/purple)**: Suggests a cosmic or abstract background.
**GlowCol (orange/yellow)**: Indicates emissive elements, perhaps engines or internal lighting.
**DiffuseCol (similar to skyCol)**: Reinforces the main color theme.
**Beer/Absorption**: Implies volumetric light absorption or scattering, possibly through nebulae or translucent parts of the spaceship, adding depth and atmospheric effects.
**Sky / Environment (\(y=4/-6\), box/pp patterns, `col+=4skyColrd.y^2...` )**: Describes the background and lighting.
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.
Prompt: Draw: A highly detailed, surreal 3D rendering of a towering, infinitely recursive crystalline spire emerging from a chaotic sea of bifurcating layers, symbolizing the logistic recurrence's bounded growth exploding into period-doubling cascades that accumulate at a universal scaling constant around four-point-six-six-nine, with sharp needle-like protrusions representing tangent bifurcations and chaotic tongues where positive divergence rates dominate, textured with self-similar folds of stability islands in negative exponent zones interspersed by superstable curves plunging to negative infinity, the structure alternating between two growth parameters in a repeating symbolic sequence like AB or AAB to force oscillations, colored in a radiant teal-to-violet gradient where bright glowing edges highlight the average logarithmic derivative sums over thousands of iterations after transient warm-up, evoking ergodic mixing and multiplicative sensitivity in a two-dimensional parameter plane sliced into volumetric depth with soft volumetric lighting casting shadows that trace the renormalization fixed points and coexisting attractors, intricate visibly etched and glowing along the surfaces and floating ethereally in the space—such as the core iteration \( x_{n+1} = r x_n (1 - x_n) \) carved into the base, the Lyapunov exponent \( \lambda = \lim_{N \to \infty} \frac{1}{N} \sum_{n=1}^N \log |r (1 - 2x_n)| \) spiraling up the spire, fixed point solutions \( x^* = 1 - \frac{1}{r} \) branching off spikes, bifurcation condition \( |f'(x^*)| = 1 \) at edges, period-doubling product \( \prod_{i=1}^k f'(x_i) = -1 \) in layered folds, Feigenbaum constant \( \delta \approx 4.669 \) inscribed on recursive crystals, and forced map \( x_{n+1} = r_n x_n (1 - x_n) \) with \( r_n \) switching via sequence S—high-resolution, intricate details on every fractal iteration, no additional text or symbols, cinematic composition.
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.
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
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
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 \( d s^{2\pi} = -c^{2\pi} d t^{2\pi} + d x^{2\pi} + d y^{2\pi} + d z^{2\pi} \).
Prompt: 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,0)), and similar for circular ds=length(pp)-0.5, clamped and shaped with shp(clamp(col,0,10)).
Dream Level: is increased each time when you "Go Deeper" into the dream. Each new level is harder to achieve and
takes more iterations than the one before.
Rare Deep Dream: is any dream which went deeper than level 6.
Deep Dream
You cannot go deeper into someone else's dream. You must create your own.
Deep Dream
Currently going deeper is available only for Deep Dreams.
