▦ 生成康托尔尘Generate Cantor Dust
生成二维康托尔尘：把正方形分成 3×3 格，只保留四个角，再对每个角反复同样操作，形成粉尘状点集。导出 PNG。Generate 2D Cantor dust: split a square into 3×3 cells, keep only the four corners, and repeat on each corner to get a dust of points. Export PNG.
使用说明与常见问题How-to & FAQ
「生成康托尔尘」是什么？What is the Generate Cantor Dust tool?
生成二维康托尔尘：把正方形分成 3×3 格，只保留四个角，再对每个角反复同样操作，形成粉尘状点集。导出 PNG。Generate 2D Cantor dust: split a square into 3×3 cells, keep only the four corners, and repeat on each corner to get a dust of points. Export PNG.
如何使用「生成康托尔尘」？How do I use the Generate Cantor Dust tool?
在「生成康托尔尘」中，按需设置「宽度」、「高度」、「边距」、「线宽」、「迭代次数」等选项，然后点击「绘制」、「下载 PNG」按钮即可。In the Generate Cantor Dust: Set "Width", "Height", "Padding", "Line width", "Iterations", then click "Draw", "Download PNG".
和一维康托尔集有何关系？How does this relate to the 1D Cantor set?
康托尔尘是康托尔集与自身的笛卡尔积：横向和纵向都去掉中间三分之一，所以每个正方形分成 3×3 格后只留四个角（第 1 次迭代是 4 个方块，第 2 次是 16 个）。去掉中心、保留其余 8 格的是谢尔宾斯基地毯，不是康托尔尘。Cantor dust is the Cartesian product of the Cantor set with itself: the middle third is removed both horizontally and vertically, so each square is split 3×3 and only the four corners stay (4 squares after one iteration, 16 after two). Removing only the centre and keeping the other 8 cells gives the Sierpinski carpet instead.
迭代上限为什么较低？Why is the iteration cap lower?
方块数按 4^n 增长、边长按 3^n 缩小，6 次时已有 4096 个方块，边长只剩原来的 1/729，在常见画布上接近单个像素。The square count grows as 4^n while the side shrinks as 3^n; at 6 iterations there are 4,096 squares, each 1/729 of the original side, close to a single pixel on a typical canvas.
有哪些选项可以调整？What options can I adjust?
「生成康托尔尘」的设置：「宽度」默认 512，范围 64–2048；「高度」默认 512，范围 64–2048；「边距」默认 20，范围 0–256；「线宽」默认 2，范围 0.5–40；「迭代次数」默认 4，范围 1–6；「背景」默认 #ffffff；「线条」默认 #1e3a8a。每次生成结果都按当前设置处理。Settings in the Generate Cantor Dust — "Width" default 512, range 64–2048; "Height" default 512, range 64–2048; "Padding" default 20, range 0–256; "Line width" default 2, range 0.5–40; "Iterations" default 4, range 1–6; "Background" default #ffffff; "Line" default #1e3a8a. The result always uses the current settings.
「生成康托尔尘」免费吗？需要注册吗？Is the Generate Cantor Dust free? Do I need an account?
「生成康托尔尘」完全免费，无需注册或安装，打开网页即可使用，手机和电脑都能用；输入的内容只在你的浏览器本地处理。The Generate Cantor Dust is completely free with no sign-up or install and works on phones and desktops; what you enter is processed only in your browser.