Rotation sweeps旋转扫掠播放器
Continuous sweeps of 3, 4, 5 and 6 rays, a filled equilateral triangle, and a filled square. Each ray has length 1; each polygon has circumradius 1. The slider measures path progress, preserving the angular backtracking of the 4-, 5- and 6-ray motions.
Areas are numerical estimates, not proven minima.
连续扫掠：3、4、5、6 射线，以及实心正三角形、正方形。射线臂长 1；多边形外接圆半径 1。滑块表示路径进度，保留 4、5、6 射线的真实回转。
面积为数值估计，并非已证明的最小值。
On (default): one centre loop, with a net lifted turn of 360°/n. Off: n loops, a literal 360° turn. Here n is the number of rays or polygon sides. Switching modes converts the progress and preserves the current pose.
开启（默认）：中心闭环走一遍，净提升角增加 360°/n；关闭：重复 n 遍转满 360°。n 是射线数或多边形边数。切换自动换算进度，当前姿态不变。
Mouse wheel to zoom; drag the view to pan.鼠标滚轮缩放；按住并拖动可平移视图。
The outline is a display approximation with a numerical safety margin, not an area certificate. The last reported area digits are not rigorously certified.
轮廓为带数值余量的显示近似，不是面积证书；面积末位未严格认证。
Research notes研究摘要
Five rays: a smaller, spiky branch五射线：更小的带刺分支
Select “5 rays” to play the new candidate. The candidate menu also offers the previous local control without acute tips and the previously published version. Each new centre loop has a net turn of 72°, including about 10.8050557° of cumulative backward rotation; five loops complete 360°.
选择“五射线”，即可播放新方案；在方案菜单中可切换原本地无急尖角方案和公开旧版。新方案每个中心闭环净转 72°，其中累计回转约 10.8050557°；重复五遍转满 360°。
| Five-ray candidate五射线方案 | Area ≈面积 ≈ | Acute tips急尖角 |
|---|---|---|
| Previous local control原本地方案 | 2.2054128 | 0 |
| New spiky branch新带刺分支 | 2.2053945 | 12 |
The area decreases by about 0.0000183. The 12 acute tips have opening angles of about 0.2660616°. They were detected at several boundary refinement levels and checked against tangent directions from the source motion, not merely counted in an image.
面积减少约 0.0000183；12 个急尖角的开角约 0.2660616°。它们在多档边界加密下均被检出，并与源运动的切线方向核对；不是单靠图片数出来的。
The five-ray search can produce a lower-area spiky branch. The earlier absence of acute tips was not a geometric prohibition: the search branch matters. Whether an exact global minimizer exists, must have spikes or infinitely many tips, or has a fractal boundary remains unproved.
五射线原来未见尖刺，不能解释为几何上禁止尖刺：搜索分支会影响结果。但精确全局最优解是否存在，以及“全局最优必须带刺／有无限尖刺／是分形”仍未证明。
“Acute tips” counts interior angles below 30°; a count of 0 does not mean the entire boundary is smooth. Areas and angles are numerical results with uncertified final digits. This page uses an independently cross-checked, 32-interval representative; later local refinements of about 5×10−8 are not presented as a major breakthrough.
“急尖角”统计阈值为内角小于 30°；0 不代表整个边界处处光滑。面积和尖角均为数值结果，末位未经严格认证；本页使用经独立算法复核的 32 段代表候选，不把约 5×10−8 的后续局部精修当作主要突破。
Infinite spikes and the large-n limit无限尖刺与射线数的极限
A fixed motion with finitely many analytic pieces has a swept boundary with a finite analytic decomposition, not hidden infinitely nested fractal levels. Other admissible continuous motions can produce infinitely many spikes while retaining a boundary of dimension 1, or fractal boundary pieces of dimension greater than 1. All three boundary classes can approach the area infimum arbitrarily closely; this does not determine which class attains it exactly.
固定的有限分段解析运动，其真实扫掠边界只有有限解析分解，不藏着无限分形层级。另一方面，合法连续运动可以产生无限尖刺，且边界维数仍为 1；也可以产生维数大于 1 的分形片。三类边界都能任意逼近面积下确界，不能据此判断哪一类精确达到最优。
For n equally spaced unit rays, let An be the infimum of the swept area needed for a continuous full turn. One can rigorously prove An → π, without requiring monotone angles or restricting spikes, path length, or the number of motion pieces. The proof gives a lower bound valid for every motion; it does not simply replace dense rays with a filled disk.
对等角分布、每臂长 1 的 n 射线，记连续转满一圈所需面积的下确界为 An。可以严格证明 An → π，无须角度单调，也不限制尖刺、路程或运动分段数。证明给出对所有运动都成立的下界，而非把稠密射线直接当成实心圆盘。
This is a statement about the infimum: it does not assert that every feasible sweep has area tending to π, or that the optimal boundary for a fixed n is known.
这是关于面积下确界的结论，并不是说任意可行扫掠的面积都会趋于 π，也不意味着已知固定 n 时最优边界的形状。