Mandelbrot explorer · grok-4.7

I am building an interactive Mandelbrot set explorer.

I am planning the implementation details. The explorer will use a full-screen canvas with drag-to-pan and wheel-to-zoom centered on the cursor. It will compute the Mandelbrot set dynamically from complex coordinates with smooth escape-time coloring and a distinct color for interior points. Iteration depth will scale with zoom level to reveal finer detail. A reset button will restore the initial view, and mouse wheel events will be intercepted to prevent page scrolling.

Canvas 2D with ImageData will render the fractal. Progressive chunked rendering will keep controls responsive, and active renders will be cancelled on navigation. The escape-time formula is z = z² + c, with smooth coloring using n − log₂(log₂(|z|)). The initial view will frame the main cardioid and surrounding bulbs.

The initial view will span real coordinates from -2.5 to 1.0 and symmetric imaginary bounds to preserve aspect ratio. Iteration depth will scale with zoom using a base plus log2 factor.

The iteration depth will be capped at 2000 and calculated as 80 plus 40 times the base-2 logarithm of zoom. Rendering will proceed row by row via requestAnimationFrame, aborting if the view changes mid-frame. Using devicePixelRatio for sharper output is possible but may reduce performance.

The canvas will be sized to match the viewport, with devicePixelRatio capped at 1.5 for performance. The color gradient will be a multi-stop interpolated palette rather than a simple blue or gold scheme. Touch support is optional.

Body overflow will be hidden to prevent overscroll. Dragging will translate the existing canvas image for immediate feedback, then recompute the fractal on pointer release.

The iteration depth formula is refined to 100 times the fourth root of zoom plus 50 times the base-2 logarithm of zoom, capped at 4000. A HUD will display coordinates, iteration count, and zoom level. Pixel writes will use a Uint32Array on the ImageData buffer for speed. The smooth coloring loop will track squared magnitudes to avoid expensive square roots.

The color palette will use a continuous sinusoidal RGB function derived from the smooth iteration value, with interior points rendered in deep black. A subtle vignette is unnecessary. The UI will display a title overlay at the top-left corner.

The reset button will sit at the top-right or bottom of the viewport. The HUD will show zoom level, iteration count, and coordinates. The UI will use a dark glassmorphism style with no external fonts.

Rendering will process 8 to 20 rows per frame to keep the interface responsive. A two-pass interlaced strategy is being considered: first filling every eighth row for a quick preview, then refining the gaps.

A generation token will cancel stale renders when the user zooms or pans mid-frame. The progressive fill will use three passes: every eighth pixel first, then every fourth, then full resolution row by row. Dragging will shift the canvas image via CSS transform for immediate feedback, and zooming during a drag will be handled cleanly.

The HTML structure is now fixed: a full-screen canvas with an overlay UI containing the title, info panel, and reset button. A `setViewFromCenter` helper will compute the complex bounds from a center point and width, preserving aspect ratio when the viewport resizes.

The initial view will be computed by fitting a fixed complex rectangle into the viewport. The target bounds are [-2.3, 1.1] on the real axis and [-1.3, 1.3] on the imaginary axis, centered at (-0.6, 0). The code scales these dimensions to preserve aspect ratio while letterboxing in whichever dimension has extra space.

The initial view is refined to guarantee the full cardioid and surrounding lobes remain visible. The iteration-depth formula is being adjusted from a linear log2 term to a power-law curve so deep zooms can reach higher iteration counts and reveal finer filaments.

The iteration-depth formula is finalized as 128 times the zoom raised to the 0.18 power, plus 40 times the base-2 logarithm of zoom, capped at 12000. This yields roughly 128 iterations at zoom 1 and about 4500 at zoom 1e8. The inner Mandelbrot loop will be optimized by avoiding Math calls and using local variables.

The progressive renderer will write pixels directly into a Uint32Array-backed ImageData buffer and call putImageData periodically. Each chunk will target roughly 12 milliseconds of work, aborting early if the frame exceeds that budget so the browser stays responsive.

I am weighing endianness for the pixel buffer. Uint32Array on little-endian platforms stores pixels as ABGR, but Uint8ClampedArray is safer and slightly slower. I will verify the exact byte order before choosing.

I will test the pixel buffer endianness by reading a Uint32Array over a four-byte buffer. During drag, the last completed image will be cached so the canvas can be shifted instantly; on release, the view bounds will be adjusted by the pixel delta and the fractal recomputed. If zooming occurs while a render is still in progress, the current generation token will be cancelled and the new bounds will be applied before starting again. The zoom factor will be derived from wheel delta using exponential scaling.

Wheel zoom will use exponential scaling with a clamped factor between 0.5 and 2, applied via preventDefault and passive: false to block page scrolling. The initial view bounds will be stored after the first layout so Reset returns exactly to the original framing. On resize, the center and width will stay fixed while the height adjusts to preserve aspect ratio.

Reset will restore the initial center and a complex width fitted to the current canvas size, ensuring the framing matches the original exactly. Double-click zoom and keyboard controls are being skipped to keep the implementation focused.

A "Rendering..." status indicator will appear during active draws. The color palette is being defined as a deep-space gradient: black interior points, and exterior colors cycling through warm gold, bright cyan, and deep magenta based on the smooth iteration value.

The exterior color will be a continuous repeating gradient generated by cosine waves with phase offsets, rather than a stepped palette. Supersampling and histogram coloring are being skipped to keep rendering fast.

Pointer capture will keep panning active even when the cursor leaves the canvas. A precision warning for extreme zoom is unnecessary. The verification plan includes confirming the initial view shows the cardioid, checking for console errors, and testing that zooming into a boundary point keeps the cursor aligned.

I will expose the current view center and bounds in the DOM so the browser tool can inspect them after zooming. The verification plan targets the top of the main cardioid or the seahorse valley near (-0.75, 0.1) to confirm the cursor stays aligned during zoom.

The browser tool's scroll action may not generate wheel events on the canvas, so I need to verify whether it dispatches wheel events or simply scrolls the page. If it does not trigger wheel events, I will need an alternative way to test zooming.

Keyboard zoom will be added as a fallback if wheel events are unavailable. Verification will rely on visual screenshots rather than console evaluation.

Canvas dimensions will be set to `clientWidth * devicePixelRatio` so CSS scaling does not blur the image. A web worker was considered but rejected as overkill; chunked rendering on the main thread is sufficient for responsiveness. During a drag, the current render buffer will be cached in an offscreen canvas so panning can display the last completed frame instantly while the user moves the cursor.

The UI will sit in a bottom bar so it does not obscure the fractal. It will display the current center coordinates, zoom level, iteration count, and a rendering indicator. Touch pinch gestures are being skipped. The smooth coloring formula is being checked to avoid NaN.

The smooth coloring formula is finalized using the escape-time formula `mu = i + 1 - log(log(mag)) / log(2)`, with a bailout of 65536 to keep the magnitude above 256 and avoid undefined logarithms. The inner Mandelbrot loop will use local variables for `zx`, `zy`, `zx2`, and `zy2` to eliminate repeated Math calls. Overflow to Infinity during squaring is noted as a potential edge case.

The color function will use a precomputed 2048-entry palette for speed. It will generate a multi-octave cosine gradient in the style of Inigo Quilez.

The palette will use cosine waves with phase offsets of 0.0, 0.15, and 0.40 for a rich, varied gradient. Mapping the smooth iteration value `mu` to the palette by scaling and wrapping is being refined to avoid banding discontinuities.

The color function will use a continuous cosine gradient with a frequency of 0.045, mapped to the palette via `floor((mu * 0.045 % 1) * PAL_SIZE)` to avoid banding. A 4096-entry palette is being considered for smoothness.

The interior color will be pure black, while the exterior palette is clamped to a minimum brightness of 30 so it never becomes black. The UI will be a thin card. I am ready to write the file and test.

A coordinate mapping bug was identified: the canvas y-axis increases downward, while complex imaginary values increase upward. The `screenToComplex` function will invert the vertical axis so zoom centering and cursor alignment remain correct.

Mouse coordinates will be mapped to canvas pixels using the element’s bounding rectangle, scaled by the device pixel ratio to keep the image sharp on high-density displays. Wheel zoom steps will be kept small for finer control.

Wheel zoom will scale the view width by `Math.exp(deltaY * 0.0015)`. The delta value is multiplied by 16 for line-based scrolling and 400 for page-based scrolling so trackpads and mice produce comparable zoom steps.

The zoom factor is clamped between 0.2 and 5. The `zoomAt` function computes the new view bounds by scaling the current width and height around the cursor, using the cursor’s fractional position on the canvas to keep the point under the mouse fixed.

The zoom math was verified at the top, center, and bottom of the canvas to confirm the cursor stays fixed. Panning will convert drag distance into complex offsets using the original view width and height divided by canvas dimensions.

The panning formula is being refined to convert drag distance into complex offsets using the original view width and height divided by canvas dimensions. The vertical drag direction is being checked against the standard pointer-to-view mapping so that dragging down shifts the view toward higher imaginary values.

The panning math is now fully derived. Dragging down shifts the view origin upward by `dy / canvasHeight * originalHeight`, while dragging right shifts it left by `dx / canvasWidth * originalWidth`. This keeps the complex point under the cursor fixed during a pan.

An offscreen canvas will cache the last completed frame so panning can display it instantly while the next chunk renders. The visible canvas will receive progressive blits from that buffer. Both canvases will resize together on window resize. A help text reading "Drag to pan · Scroll to zoom" will be added to the UI. Iteration count will be clamped to a minimum of 64 and raised for very zoomed views.

Periodicity detection is being considered to accelerate interior-point rendering. Quick geometric checks for the main cardioid and period-2 bulb could bypass the full iteration loop for many pixels. A full 150-iteration render at test viewport resolution should still complete in a few seconds with progressive display.

The interior-point shortcuts are now implemented. The main cardioid is detected with `q*(q+(cx-0.25)) <= 0.25*cy^2` and the period-2 bulb with `(cx+1)^2 + cy^2 <= 0.0625`. These skip the full iteration loop for many pixels, improving initial render speed.

Chunked rendering will keep the UI responsive by yielding to the browser between frames. Reset will cancel any active render and restart from scratch. The canvas will use `touch-action: none` and `overscroll-behavior: none` to prevent native scrolling and context menus. A live DOM element will display the complex coordinates under the cursor during mouse movement.

The render state will be shown in a `#status` element. Iteration count will scale with zoom to reveal finer detail.

A two-stage render will make zooming feel instant. The first pass fills 4×4 rectangles with computed colors, then the full pixel grid is filled. This avoids stretching a static image and keeps the fractal math fresh during zoom.

The coarse pass will draw 4×4 pixel blocks by computing the center of each block and filling its pixels. The fine pass then overwrites every pixel afterward. A `paint` helper is being drafted to fill these blocks.

A manual `paint` helper was drafted to fill 4×4 pixel blocks directly into the canvas buffer. The coarse pass will use a row-based Uint32 fill for speed. A full initial render is estimated at roughly 3–10 seconds, but interior shortcuts should cut that by about 40%.

Exterior pixels escape quickly, so the initial render should finish in about half a second to two seconds. Boundary zooms will raise average iterations, making chunked rendering essential. The cardioid checks will be added next, and the loop may be optimized before writing the file. A 0.75 render scale was considered but full resolution is preferred. Keyboard shortcuts for reset and zoom at center will also be included.

The UI will stay minimal with only a Reset button, drag panning, and wheel zoom. I will attach the wheel listener to the window so any scroll attempt zooms the fractal. I need to verify that the test tool’s scroll event includes coordinates, since it may only provide a delta value.

I will implement the wheel listener and test it empirically, since the tool schema does not specify whether scroll events include coordinates. I will skip extra on-screen zoom buttons and rely on mouse wheel plus keyboard shortcuts for verification.

Arrow keys will pan the view. The help text will hint at these shortcuts. The UI will use a full black canvas with a bottom-left floating panel using CSS backdrop-filter for blur. Georgia will serve as the title font.

The title will use Palatino, Iowan Old Style, Georgia, or serif. The UI panel will be dark with a subtle white border. Buttons will use a gold accent on dark backgrounds. The cursor will switch to grab and grabbing during panning. The code is ready to be written.

The initial view bounds will be saved immediately after the first resize so Reset always restores the same framing. The DOM status text will read "Ready" when a render finishes, letting verification tools detect completion. The full application will be written before testing.

A potential bug with wheel zoom centering on the cursor was noted. Keyboard zoom at center is the most reliable way to verify that a feature stays fixed during zoom. The scroll tool likely lacks coordinate data, so cursor-centered verification may not be possible.

I will track the last pointer position from clicks so the zoom center can fall back to that coordinate if the scroll tool omits coordinates. I will also log wheel events to confirm whether they include mouse position. The next step is to test this behavior empirically.

The cursor-centered zoom logic is mathematically sound, but the verification tool only provides scroll deltas without coordinates. Clicks and drags do include coordinates, yet they do not trigger zoom. Keyboard shortcuts also lack mouse position data. This means a true before-and-after visual check of a feature staying fixed under the cursor may be impossible with the available automation.

I will try a click-then-scroll sequence to test whether the wheel event carries cursor coordinates. An on-canvas coordinate readout will be added for manual verification. Iteration counts will rise more aggressively at deep zoom so high-boundary pixels do not collapse into black interior regions.

The iteration formula now scales with zoom logarithmically. At zoom 1 it uses 200 iterations, rising to roughly 3,600 at zoom 4,096. This keeps boundary detail visible at deep magnification without collapsing into black interior regions.

The iteration ceiling is now 5,000 to keep the UI responsive. A four-pass adaptive pixel stride is planned: 8×8 blocks, then 4×4, then 2×2, then full pixels. This progressive refinement avoids changing color values while still yielding a fast initial preview.

The progressive render will use a two-pass strategy instead of four. A coarse preview pass fills 4×4 blocks, followed by a final pass that overwrites every pixel at full resolution. This keeps the math simple and limits double computation to about 6% overhead. A `renderToken` counter will be used to cancel any active render when the user triggers a new zoom or reset.

Drag panning will pause rendering until the pointer is released. A wheel event during a drag will end the pan and zoom instead. The code will include an aria-label on the reset button. I verified that no external fonts or images are loaded. The Mandelbrot mapping keeps the cardioid on the right and the main bulb on the left, with positive y pointing upward. I decided against adding a grid or axis overlay to keep the view clean.

Drag panning will use an offscreen snapshot to keep the visible canvas stable. The current image is copied to a hidden context before dragging begins, and during a drag the visible canvas is cleared and redrawn from that snapshot shifted by the pan offset. This prevents the view from jumping while the user moves the mouse. Once the drag ends, the full fractal is recomputed and progressively filled.

The first progressive chunk would wipe the panned snapshot, so a simpler strategy is to render into an offscreen ImageData buffer. Uncomputed rows will be filled by copying from the shifted snapshot, keeping the background stable during zoom.

Only completed rows will be drawn to the visible canvas using putImageData with a dirty rectangle, leaving unrendered areas black. The pan snapshot is cleared once rendering finishes. During zoom, the previous image should not be scaled; instead, the new view must be computed from scratch.

The info panel will continuously display the current view bounds. `user-select: none` will prevent text selection during panning. The reset button sits outside the canvas, so it cannot accidentally trigger a drag.

The canvas will fill the viewport with a fixed inset, while the UI panel sits above it using z-index and pointer-events. I will cap the device pixel ratio at 2 to keep rendering fast, and I may add a tweakable render scale constant for later tuning.

I will use the motion tool to capture a short video of the zoom animation and render progress. The implementation will be kept clean and comment-free.

Zoom bounds will be clamped to prevent numeric precision collapse: a minimum width of 1e-14 and a maximum width of 100. The interior region will use a very dark blue (#070714) so it does not look like an empty canvas, while the exterior minimum brightness will be set around 40.

The color palette will use a hybrid cosine gradient. Far regions will stay deep blue, while points near the boundary brighten toward white and yellow. This avoids the banded look of pure cyclic colors at extreme zoom.

The color function now uses HSL with hue shifting and lightness oscillating by iteration count. To avoid per-pixel math, the palette will be precomputed into a 1024-entry lookup table.

The palette will be precomputed at 0.5-iteration intervals into a 1024-entry lookup table. A `lookup` function will interpolate between neighboring entries using the fractional part of the iteration count, avoiding banding while keeping the color math fast.

The color function is being refined to use a visually striking sine-wave pattern with clamping to prevent out-of-range values. Colors will be based on absolute iteration counts rather than normalized ratios, so the palette stays stable when the iteration ceiling changes during zoom.

The color palette will use Inigo Quilez's cosine formula with tuned amplitude and phase offsets. A dark exponential fade will keep distant regions dim, but the minimum shade is clamped to 0.15 so the interior stays pure black and avoids looking like an empty canvas.

The shade formula is now fixed at 0.18 plus 0.82 times an exponential fade, and the palette will be finalized after a quick visual check. The controls will form a bottom-center pill bar with drag-to-pan and scroll-to-zoom hints, a "Reset view" button, and a stats line. I am beginning implementation now.

I decided to use byte-level writes for the coarse render pass rather than Uint32Array packing, avoiding endianness issues entirely.

The inner Mandelbrot loop is being optimized with a loose pre-test for the main cardioid and period-two bulb, followed by a fast iterative update using squared real and imaginary parts to avoid repeated multiplications. A logarithmic bailout estimate will be used to compute smooth iteration counts.

The smooth iteration count is computed as `i + 1 - nu`, where `nu` is derived from the logarithm of the bailout estimate. The cardioid and period-two-bulb pre-tests use exact closed-set formulas, so their boundaries are correctly classified as inside the Mandelbrot set.

I will skip the period-3 bulb test to avoid correctness risks. Wheel events will be captured on the canvas with `{passive: false}` and blocked from scrolling the page via `preventDefault` on the document.

Wheel events will be captured on the window with `preventDefault` to stop page scrolling. Zooming will only trigger when the cursor is inside the canvas bounds, checked via `getBoundingClientRect`, so UI controls like the reset button do not accidentally scroll the view.

I will add a loading class to indicate rendering progress. Safari gesture events may also be blocked. I am weighing WebGL against Canvas 2D; a fragment shader would render instantly and look more impressive.

WebGL would make zooming instant and keep controls responsive, but fragment shaders risk timing out at extreme iterations. Fragment precision is another concern: WebGL1 mediump often fails under deep zoom, and highp support varies across devices.

Canvas 2D JavaScript was chosen over WebGL for reliability and precision. Float32 limits deep zoom around 1e5–1e6 magnification, but that is sufficient to reveal new detail. A WebGL fallback would add unnecessary code risk, so the implementation will stay pure JavaScript with progressive rendering. OffscreenCanvas was also considered and rejected.

A periodicity check will be added to skip deep interior iterations. Every 20 steps the code stores the current complex coordinates; if they match the stored pair, the point is classified as inside the set immediately. This avoids thousands of extra loop cycles for points that never escape.

A small epsilon will be used for the periodicity check to avoid false positives near slow spirals. The implementation is now ready to be written and benchmarked.

Rendering will be split into 10 ms chunks so the UI stays responsive. A percentage progress indicator will appear during updates. The HUD will show crosshair coordinates on hover, and numbers will format with `toPrecision` based on zoom level. Reset view must restore the exact initial bounds.

The initial viewport bounds will be saved after the first fit calculation. Reset view will restore those exact values, or recompute them from the current canvas size if the viewport has changed since startup.

The initial viewport will be centered at (-0.6, 0) with a 3.2 by 2.4 aspect ratio, giving horizontal bounds from -2.2 to 1.0.

The initial vertical bounds are -1.2 to 1.2, which captures the full Mandelbrot set including its left spike and top antenna. The horizontal range from -2.2 to 1.0 keeps the visual center near -0.6.

The initial viewport is being tightened to x −2.3 to 1.2 and y −1.35 to 1.35, centered at (−0.55, 0), so the period-3 bulbs and leftmost filament remain fully visible. The HUD will sit at the bottom to avoid obscuring the fractal center.