Prompt: Einführung riemannscher Flächen, auf denen mehrdeutige Funktionen wie der Logarithmus (unendlich viele Blätter) oder die Wurzelfunktion (zwei Blätter) „eindeutig“ werden. Fibonacci-Zahlen für eine fraktale, dreidimensionale Muschel, welche auch die Laplacegleichung erfüllen, bzw. äquivalent dazu die Cauchy-Riemannschen Differentialgleichungen. Auf ihren Flächen werden durch die Lagen ihrer Singularitäten und die Topologie dieser Flächen (Zahl der Schnitte u. a.) beschrieben. Das topologische „Geschlecht“ der Riemannflächen wird durch g = w / 2 − n + 1 {\displaystyle g=w/2-n+1} gegeben, wobei in den w {\displaystyle w} Verzweigungspunkten der Fläche n {\displaystyle n} Blätter aneinandergeheftet sind. Für g > 1 {\displaystyle g>1} hat die riemannsche Fläche ( 3 g − 3 ) {\displaystyle (3g-3)} Parameter (die „Moduln“).
Prompt: Einführung riemannscher Flächen, auf denen mehrdeutige Funktionen wie der Logarithmus (unendlich viele Blätter) oder die Wurzelfunktion (zwei Blätter) „eindeutig“ werden. Fibonacci-Zahlen für eine fraktale, dreidimensionale Muschel, welche auch die Laplacegleichung erfüllen, bzw. äquivalent dazu die Cauchy-Riemannschen Differentialgleichungen. Auf ihren Flächen werden durch die Lagen ihrer Singularitäten und die Topologie dieser Flächen (Zahl der Schnitte u. a.) beschrieben. Das topologische „Geschlecht“ der Riemannflächen wird durch g = w / 2 − n + 1 {\displaystyle g=w/2-n+1} gegeben, wobei in den w {\displaystyle w} Verzweigungspunkten der Fläche n {\displaystyle n} Blätter aneinandergeheftet sind. Für g > 1 {\displaystyle g>1} hat die riemannsche Fläche ( 3 g − 3 ) {\displaystyle (3g-3)} Parameter (die „Moduln“).
Prompt: Seine geometrische Begründung der Funktionentheorie mit der Einführung riemannscher Flächen, auf denen mehrdeutige Funktionen wie der Logarithmus (unendlich viele Blätter) oder die Wurzelfunktion (zwei Blätter) „eindeutig“ werden, legte er bereits in seiner Dissertation vor, die nach Angabe von Dedekind schon im Herbst 1847 in Berlin abgeschlossen war (in Diskussionen mit Eisenstein soll er seinen Differentialgleichungs-Zugang zur Funktionentheorie gegenüber der mehr formalen Einstellung Eisensteins vertreten haben). Komplexe Funktionen sind „harmonische Funktionen“ (das heißt, sie erfüllen die Laplacegleichung bzw. äquivalent dazu die Cauchy-Riemannschen Differentialgleichungen) auf diesen Flächen und werden durch die Lage ihrer Singularitäten und die Topologie dieser Flächen (Zahl der Schnitte u. a.) beschrieben. Das topologische „Geschlecht“ der Riemannflächen wird durch g = w / 2 − n + 1 {\displaystyle g=w/2-n+1} gegeben, wobei in den w {\displaystyle w} Verzweigungspunkten der Fläche n {\displaystyle n} Blätter aneinandergeheftet sind. Für g > 1 {\displaystyle g>1} hat die riemannsche Fläche ( 3 g − 3 ) {\displaystyle (3g-3)} Parameter (die „Moduln“).
Seine Beiträge zu diesem Gebiet sind zahlreich. Sein berühmter Riemannscher Abbildungssatz besagt, dass jedes einfach zusammenhängende Gebiet in der komplexen Zahlenebene C entweder zu ganz C oder dem Innern des Einheitskreises „biholomorph“ äquivalent ist (das heißt, es gibt eine analytische Abbildung, auch in umgekehrter Richtung). Die Verallgemeinerung des Satzes in Bezug auf riemannsche Flächen ist das berühmte Uniformisierungstheorem, um das sich im 19. Jahrhundert u. a. Henri Poincaré und Felix Klein bemühten. Auch hier sind strenge Beweise erst mit der Entwicklung ausreichender mathematischer Werkzeuge – in diesem Fall aus der Topologie – gegeben worden.
Prompt: A vibrant scene set in a mystical, intricate maze filled with lush greenery and ornamental plants. The maze's design features golden paths intertwined with deep emerald foliage. At the center, a diverse group of eight characters stands confidently, each representing a unique cultural or historical background.
1. **Character Details**:
- **Elderly Man**: Long white beard, wise expression, draped in flowing robes of gold and white, holding a staff.
- **Muscular Man**: Chest exposed, wearing rugged clothing, a determined look, with short, dark hair and a confident stance.
- **Young Woman**: Curly hair, wearing a flowing dress, exuding grace and strength, standing tall among the group.
- **Another Young Man**: Short hair, casual attire with a belt and tools visible, displaying a thoughtful demeanor.
- **Elderly Man with Long Hair**: Dreadlocks, wearing tribal clothing adorned with beads and jewelry, holding a walking stick, appearing wise and grounded.
- **Armored Figure**: Dressed in intricate, historical armor, displaying authority and grace, with a stern expression.
- **Man in Traditional Garb**: Clothed in rich fabrics with vibrant patterns, standing proudly, showcasing cultural heritage.
- **Another Man**: Wearing simple, functional clothing yet carries an aura of purpose and resilience.
Each character is uniquely styled to reflect their individuality, showcasing a blend of traditional attire and
Prompt: A vibrant scene set in a mystical, intricate maze filled with lush greenery and ornamental plants. The maze's design features golden paths intertwined with deep emerald foliage. At the center, a diverse group of eight characters stands confidently, each representing a unique cultural or historical background.
1. **Character Details**:
- **Elderly Man**: Long white beard, wise expression, draped in flowing robes of gold and white, holding a staff.
- **Muscular Man**: Chest exposed, wearing rugged clothing, a determined look, with short, dark hair and a confident stance.
- **Young Woman**: Curly hair, wearing a flowing dress, exuding grace and strength, standing tall among the group.
- **Another Young Man**: Short hair, casual attire with a belt and tools visible, displaying a thoughtful demeanor.
- **Elderly Man with Long Hair**: Dreadlocks, wearing tribal clothing adorned with beads and jewelry, holding a walking stick, appearing wise and grounded.
- **Armored Figure**: Dressed in intricate, historical armor, displaying authority and grace, with a stern expression.
- **Man in Traditional Garb**: Clothed in rich fabrics with vibrant patterns, standing proudly, showcasing cultural heritage.
- **Another Man**: Wearing simple, functional clothing yet carries an aura of purpose and resilience.
Each character is uniquely styled to reflect their individuality, showcasing a blend of traditional attire and
Prompt: Depict seven vingtage televison sets. On top sets a Buddha with focused eyes. A cndle is burning in one of the empty monitors. Framed in a boxy television set. The central arrangement features a the number '23' with image interference. The remaining pictures rendered in white on two stacked dark surfaces on the right side of the pile. Another monitor shows the empty circle Enso. The base television is dark and seemingly burnt out, with charred debris spilling onto the floor, suggesting a recent malfunction or destruction.
Prompt: Depict in the tradition of Sumi-e a composition of dispersed parts of the empty circle Enso shown in seven vingtage televison sets.. The topmost screen shows a Buddha's face with focused eyes, framed in a boxy television set. The central arrangement features a the number '23' rendered in white on two stacked dark surfaces on the right side of the pile. The base television is dark and seemingly burnt out, with charred debris spilling onto the floor, suggesting a recent malfunction or destruction.
Prompt: Depict a medieval japanese landscape with waterfalls and the typical flora of the highlands. Misty morning,
A lonely wayfarer makes his way through stony ground along a sinuous rill in cold rain. Depicted as a charcoal drawing with dark green and darkblue shading.
In the tradution of Sumi-e Tusche. Enso. Hanging roll
Prompt: Depict a medieval japanese landscape with waterfalls and the typical flora of the highlands. Pines, crouching bushes...
A lonely wayfarer makes his way through stony ground, depicted as an in drawing with dark green and darkblue shading.
In the tradition of Sumi-e Tusche.
Prompt: Depict a medieval japanese landscape with waterfalls and the typical flora of the highlands.
A lonely wayfarer makes his way through stony ground, depicted as a charcoal drawing with dark green and darkblue shading.
In the tradution of Sumi-e Tusche. Enso. Hanging roll
Prompt: Create an enchanting landscape dominated by deep blues and vibrant colors. In the foreground, a serene ocean features rolling waves, with a figure standing in the water, showcasing a muscular build and bronze skin tone. The figure gazes towards a mystical island at the horizon, surrounded by lush tropical plants and elegantly blooming lotus flowers in shades of pink, blue, and white.
To the left, depict two deities: one female with with intricate skin tattoes, and the other male, regal with long hair, holding a tibetan flag, both carved in detail to reflect artistry. In the background, a majestic waterfall cascades down a rocky cliff, with a golden pagoda peeking from among the greenery.
Above, a star-filled sky reveals a cosmic scene with swirling galaxies and sparkling stars. A UFO hovers overhead, with colorful beams radiating down, while lightning flashes in the distance. In the further background, a small boat with a child gliding across the water. The overall atmosphere should convey a sense of wonder and harmony between the natural and cosmic elements.
Prompt: **Prompt:** Create a detailed illustration of a serene scene featuring a woman and an elderly man in a small, traditional rowboat. The woman, with a curvy body shape, has an intricate array of colorful tattoos covering her skin, including spirals and mandalas. She has long, dark hair adorned with small flowers and is wearing minimal clothing, sitting gracefully while gazing upward. Her expression is one of tranquility and connection with nature. The man, with a long, white beard and aged features, is dressed in a thick, hooded, gray cloak, and sits behind her, holding a wooden oar. The background is a lush, green marsh with tall grass, and a small bird stands near the water's edge. The atmosphere is enhanced by a gray sky, with rain falling gently, and an artistic overlay of celestial diagrams, including moons and geometric patterns, fills the upper section of the image, adding a mystical element to the scene.
Prompt: Create an image of a fantastical winter scene on a snow-covered ship deck. In the foreground, there is a large, bearded man wearing a long, ornate fur coat with a thick collar and intricate patterns, exuding a commanding presence. He stands shoulder to shoulder with an elegant woman in a flowing, embellished outfit, with a midriff-baring top and a long, swirling skirt, decorated with ornate designs. Her long, dark hair cascades down her back, and she wears intricate jewelry.
Surrounding them, several figures exhibit excitement and urgency. On the left, one man, in a fur hat and thick wool clothing, gestures animatedly, while another, adorned with elaborate tribal decorations and a horned headdress, holds a lantern, his expression intense. A third figure, partially hidden behind the large man, appears action-oriented, and two more figures in warm clothing carry rifles, seemingly preparing for an adventure or confrontation.
Above, the sky is filled with ethereal motifs, including swirling snowflakes and shooting stars, creating a magical quality. Two angels hover above, their wings spread wide, adding a celestial element. The ship’s sails and rigging are prominent in the background, casting a silhouette against a backdrop filled with falling snow.
The overall color palette is a range of cool blues and whites, suggesting a wintry atmosphere, with hints of silver in the characters' clothing and decorations, enhancing the otherworldly theme of the scene.
Prompt: Create a surreal and vibrant scene featuring four distinct characters on a stage. On the left, a whimsical figure resembling a circus master with a long beard and exaggerated facial features, wearing a top hat and holding a forked spear, dressed in a vintage brown outfit with rolled-up sleeves and knee-high socks.
Next to him, a young female character stands confidently, with long dark hair, wearing a beige military-style dress that buttons up the front, paired with knee-high socks and stylish ankle boots. Her expression is stoic and her posture is assertive.
In the center, a bright yellow figure with a long white beard, wearing a yellow outfit that consists of a buttoned shirt and trousers, holds a cane with a red cone hat perched atop his head. He has sunglasses on, and there’s a playful and approachable quality to his demeanor.
On the right, a futuristic alien character with a bright green skin tone and large, round goggles, wears a form-fitting suit adorned with intricate mechanical buttons and gears. This character stands with an air of mystery, holding a staff.
The background features an expansive cosmic scene, with a large, mystical eye at the center of a pyramid. The stage itself has a checkerboard floor, with elegant columns and red curtains framing the scene, illuminated by soft golden lanterns.
Prompt: **Prompt:** Create a surrealistic composition featuring five distinct characters sitting together in an open field under a dramatic, cloudy sky. In the center, a regal figure with a long, white beard and a whimsically large green hat adorned with golden ornaments sits on an ornate throne. He is draped in vibrant, flowing robes with layers of colorful bead necklaces. To his left, a figure wearing a colorful harlequin-inspired outfit and a vintage gas mask with an exaggerated clown hat sits with a playful demeanor. Next to him, a character with an elephant's head wearing a formal suit, conveying a sense of wisdom and dignity. To the central figure's right, a person dressed in a classic diving suit with a golden helmet and mesh-covered eye openings exudes a sense of mystery. Finally, another figure in elegant blue robes with intricate patterns and a similar diving helmet to the previous character completes the group. The lush green forest in the background is bathed in twilight, enhancing the dreamy atmosphere, while a few birds soar above the distant hills. Include soft silhouettes of foliage in the foreground, adding depth to the composition.
Prompt: **Prompt:** Create a surrealistic composition featuring five distinct characters sitting together in an open field under a dramatic, cloudy sky. In the center, a regal figure with a long, white beard and a whimsically large green hat adorned with golden ornaments sits on an ornate throne. He is draped in vibrant, flowing robes with layers of colorful bead necklaces. To his left, a figure wearing a colorful harlequin-inspired outfit and a vintage gas mask with an exaggerated clown hat sits with a playful demeanor. Next to him, a character with an elephant's head wearing a formal suit, conveying a sense of wisdom and dignity. To the central figure's right, a person dressed in a classic diving suit with a golden helmet and mesh-covered eye openings exudes a sense of mystery. Finally, another figure in elegant blue robes with intricate patterns and a similar diving helmet to the previous character completes the group. The lush green forest in the background is bathed in twilight, enhancing the dreamy atmosphere, while a few birds soar above the distant hills. Include soft silhouettes of foliage in the foreground, adding depth to the composition.
Prompt: **Prompt:** Create a stylized illustration of five characters standing in a wooden courtroom setting with orange curtains in the background. The first character is a young man with olive skin, medium-length black hair, and a confident expression, wearing a black suit with a white shirt and a dark tie, holding a gun at his side. The second character is a petite older woman with gray hair styled in a short bob, wearing round glasses and a green dress, positioned behind the first character. The third character is a young woman with fair skin and short, curly platinum blonde hair, wearing a white dress and holding an assault rifle, standing in the center with a determined look. The fourth character is a muscular middle-aged man with salt-and-pepper hair and a stern expression, dressed in a gray suit and holding a handgun, standing confidently next to the third character. The final character is a tall, slender humanoid figure with dark purple skin, large pointed ears, and elongated features, standing slightly apart from the others, exuding an air of mystery. The courtroom scene includes wooden elements such as chairs and a backdrop that add to the dramatic atmosphere.
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.