{ "openapi": "3.1.1", "info": { "title": "Mandelbrot \u0026 Julia Deep Zoom Renderer API", "version": "1.0.0", "summary": "Render the Mandelbrot or a Julia set anywhere, down to zoom 10²⁵⁰ with perturbation theory.", "description": "Every endpoint performs the same calculation as the human page and returns inputs, result, units, assumptions, sources, warnings and computedAt. MCP endpoint: /mcp." }, "servers": [ { "url": "https://prooflinefractal.usefulweb.app" } ], "paths": { "/api/v1/calculate": { "post": { "operationId": "calculate", "summary": "Render the Mandelbrot or a Julia set anywhere, down to zoom 10²⁵⁰ with perturbation theory.", "description": "Render the Mandelbrot set or a Julia set as a PNG image at any centre and zoom. Shallow views iterate in double precision; deep zooms beyond what doubles can resolve use perturbation theory with an exact BigInt reference orbit and rebasing, so a view of width 1e-22 or 1e-100 renders correctly. Centre coordinates may have hundreds of digits. The result lists the zoom factor, the method, the share of pixels inside the set and the escape-iteration statistics. Part of the Proofline math notebook.\n\nMethodology:\n- Escape time: z ← z² + c from z = 0 (Mandelbrot, c = pixel) or z = pixel (Julia, c fixed) until |z| \u003e 256 or the iteration limit; the smooth count n + 1 − log₂(log₂|z|) gives continuous colours; bounded pixels are black.\n- Deep zooms: the view centre’s orbit Zₙ is computed exactly in BigInt fixed point (precision about −log₂(pixel size) + 48 bits); each pixel iterates its offset δ in doubles, δₙ₊₁ = 2Zₙδₙ + δₙ² + δc. When |Zₙ + δₙ| \u003c |δₙ| or the reference orbit ends, the pixel continues from the start of the reference orbit with δ = Zₙ + δₙ (rebasing), which removes perturbation glitches without extra reference points.\n- Verified: escape times of single points (c = 1 escapes after 3 iterations with radius 2; c = −1 never escapes); perturbation and direct iteration agree to within 1.5 iterations on more than 99.5 % of the pixels of a shallow view; at width 1e-22 the perturbation result equals an exact 200-bit BigInt iteration of the same pixels.\n\nEach pixel shows the escape time of its centre point (no anti-aliasing); pixels that would escape after more iterations than the limit appear inside the set. Colours are a free artistic choice.", "tags": [ "apps" ], "x-mcp-tool": "proofline_fractal", "x-owner-app": "proofline", "requestBody": { "required": true, "content": { "application/json": { "schema": { "$ref": "#/components/schemas/ProoflineFractalInput" }, "examples": { "example1": { "summary": "The whole Mandelbrot set", "value": {} }, "example2": { "summary": "Deep zoom into Seahorse Valley (width 1e-22, perturbation)", "value": { "centerX": "-0.743643887037158704752191506114774", "centerY": "0.131825904205311970493132056385139", "width": "1e-22", "maxIterations": 20000, "imageWidth": 240, "imageHeight": 180 } }, "example3": { "summary": "Julia set for c = −0.8 + 0.156i", "value": { "set": "julia", "centerX": "0", "centerY": "0", "width": "3.2", "juliaRe": "-0.8", "juliaIm": "0.156", "palette": "fire", "imageWidth": 480, "imageHeight": 360 } }, "example4": { "summary": "The period-3 mini Mandelbrot on the real axis", "value": { "centerX": "-1.7548776662", "centerY": "0", "width": "0.04", "maxIterations": 3000, "palette": "ice", "imageWidth": 400, "imageHeight": 300 } } } } } }, "responses": { "200": { "description": "Calculation result envelope.", "content": { "application/json": { "schema": { "$ref": "#/components/schemas/ProoflineFractalResult" } } } }, "400": { "description": "Malformed JSON body.", "content": { "application/problem+json": { "schema": { "$ref": "#/components/schemas/Problem" } } } }, "422": { "description": "Invalid input (see errors[]).", "content": { "application/problem+json": { "schema": { "$ref": "#/components/schemas/Problem" } } } }, "429": { "description": "Rate limit exceeded (see Retry-After).", "content": { "application/problem+json": { "schema": { "$ref": "#/components/schemas/Problem" } } } } } } }, "/api/v1/tool": { "get": { "operationId": "getToolDefinition", "summary": "Inputs, output schema and examples for this tool", "responses": { "200": { "description": "Tool definition.", "content": { "application/json": { "schema": { "type": "object" } } } } } } } }, "components": { "schemas": { "Problem": { "type": "object", "description": "RFC 9457 problem details. 422 responses include per-field `errors`.", "required": [ "type", "title", "status", "detail" ], "properties": { "type": { "type": "string" }, "title": { "type": "string" }, "status": { "type": "integer" }, "detail": { "type": "string" }, "code": { "type": "string" }, "errors": { "type": "array", "items": { "type": "object", "required": [ "field", "message" ], "properties": { "field": { "type": "string" }, "message": { "type": "string" } } } } } }, "ProoflineFractalInput": { "type": "object", "additionalProperties": false, "properties": { "set": { "title": "Fractal", "type": "string", "enum": [ "mandelbrot", "julia" ], "default": "mandelbrot" }, "centerX": { "title": "Centre x (real part)", "description": "A decimal number; deep zooms need as many digits as the zoom has zeros. For a Julia set, 0 centres the picture.", "type": "string", "maxLength": 1300, "default": "-0.75" }, "centerY": { "title": "Centre y (imaginary part)", "type": "string", "maxLength": 1300, "default": "0" }, "width": { "title": "View width", "description": "Width of the view in the complex plane, e.g. 3.5 for the whole set or 1e-30 for a deep zoom (at least 1e-250).", "type": "string", "maxLength": 40, "default": "3.5" }, "juliaRe": { "title": "Julia constant c, real part", "type": "string", "maxLength": 400, "default": "-0.8" }, "juliaIm": { "title": "Julia constant c, imaginary part", "type": "string", "maxLength": 400, "default": "0.156" }, "maxIterations": { "title": "Maximum iterations", "description": "Deep zooms need more: roughly 2,000 at 1e-10, 20,000 at 1e-22.", "type": "integer", "minimum": 10, "maximum": 100000, "default": 1000 }, "imageWidth": { "title": "Image width (pixels)", "type": "integer", "minimum": 16, "maximum": 1600, "default": 640 }, "imageHeight": { "title": "Image height (pixels)", "type": "integer", "minimum": 16, "maximum": 1600, "default": 480 }, "method": { "title": "Method", "type": "string", "enum": [ "auto", "direct", "perturbation" ], "default": "auto" }, "palette": { "title": "Colours", "type": "string", "enum": [ "classic", "fire", "ice", "grey" ], "default": "classic" }, "colorDensity": { "title": "Colour density", "type": "number", "minimum": 0.05, "maximum": 50, "default": 1 } }, "required": [] }, "ProoflineFractalResult": { "type": "object", "additionalProperties": false, "required": [ "schemaVersion", "tool", "computedAt", "inputs", "summary", "result", "units", "assumptions", "sources", "warnings", "presentation", "links" ], "properties": { "schemaVersion": { "type": "string", "const": "1.0" }, "tool": { "type": "object", "additionalProperties": false, "required": [ "id", "name", "version" ], "properties": { "id": { "type": "string" }, "name": { "type": "string" }, "version": { "type": "string" } } }, "computedAt": { "type": "string", "format": "date-time" }, "inputs": { "type": "object", "description": "Normalised inputs actually used, including applied defaults." }, "summary": { "type": "string" }, "result": { "type": "object", "additionalProperties": false, "required": [ "set", "centerX", "centerY", "width", "height", "zoom", "pixelSize", "juliaRe", "juliaIm", "maxIterations", "imageWidth", "imageHeight", "method", "precisionBits", "referenceIterations", "referenceEscaped", "rebases", "inside", "escaped", "insideFraction", "minIterations", "maxEscapeIterations", "meanIterations", "image" ], "properties": { "set": { "type": "string", "enum": [ "mandelbrot", "julia" ] }, "centerX": { "type": "string" }, "centerY": { "type": "string" }, "width": { "type": "number" }, "height": { "type": "number" }, "zoom": { "type": "number" }, "pixelSize": { "type": "number" }, "juliaRe": { "type": [ "string", "null" ] }, "juliaIm": { "type": [ "string", "null" ] }, "maxIterations": { "type": "integer" }, "imageWidth": { "type": "integer" }, "imageHeight": { "type": "integer" }, "method": { "type": "string", "enum": [ "direct", "perturbation" ] }, "precisionBits": { "type": [ "integer", "null" ] }, "referenceIterations": { "type": [ "integer", "null" ] }, "referenceEscaped": { "type": [ "boolean", "null" ] }, "rebases": { "type": "integer" }, "inside": { "type": "integer" }, "escaped": { "type": "integer" }, "insideFraction": { "type": "number" }, "minIterations": { "type": [ "integer", "null" ] }, "maxEscapeIterations": { "type": [ "integer", "null" ] }, "meanIterations": { "type": [ "number", "null" ] }, "image": { "type": "object", "additionalProperties": false, "required": [ "artifactId", "filename", "mime", "bytes", "sha256", "path" ], "properties": { "artifactId": { "type": "string" }, "filename": { "type": "string" }, "mime": { "type": "string" }, "bytes": { "type": "integer" }, "sha256": { "type": "string" }, "path": { "type": "string", "description": "Download path on the same site (valid for about 2 hours)." } } } } }, "units": { "type": "object", "additionalProperties": { "type": "string" } }, "assumptions": { "type": "array", "items": { "type": "string" } }, "sources": { "type": "array", "items": { "type": "object", "additionalProperties": false, "required": [ "title" ], "properties": { "title": { "type": "string" }, "url": { "type": "string" }, "publisher": { "type": "string" }, "retrievedAt": { "type": "string" }, "note": { "type": "string" } } } }, "warnings": { "type": "array", "items": { "type": "string" } }, "presentation": { "type": "array", "items": { "type": "object" }, "description": "Display blocks used by the human page (same data)." }, "links": { "type": "object", "additionalProperties": false, "required": [ "human", "api", "openapi", "methodology" ], "properties": { "human": { "type": "string", "format": "uri" }, "api": { "type": "string", "format": "uri" }, "openapi": { "type": "string", "format": "uri" }, "methodology": { "type": "string", "format": "uri" } } } } } } } }