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乌拉姆螺旋线

乌拉姆螺旋线
几何画板及绘图内容加载中，请稍等……
2012-06-14 13:14
数列解析几何函数图像
这个范例演示了乌拉姆螺旋线的图形现象。核心数学概念包括质数分布和几何构造。作图要点：使用了 GeoGebra 命令 Join, Min, Sequence, Polygon, Length, Zip, RemoveUndefined；对象有 n, center, stepX, stepY, nn, listX, listY, scale, numsShow, factsShow, sprShow, A, B, C, back, D, mm, lp, prmsShow, text1, circles, spiral, pntShow, primes, numbers, cx, cy, prmsPoints, numsPoints。适合高中及以上学段的数学爱好者和研究人员。

GeoGebra 命令及参数

  1. Join({{0, 1}, Join[Sequence[Join[{Sequence[0, j, 1, m], Sequence[(-1)^(m), j, 1, m + 1]}], m, 1, n]]}) → stepX
  2. Join({{0, 0}, Join[Sequence[Join[{Sequence[(-1)^(m + 1), j, 1, m], Sequence[0, j, 1, m + 1]}], m, 1, n]]}) → stepY
  3. Min(Length[stepX], Length[stepY]) → nn
  4. Sequence(Sum[stepX, m], m, 1, nn) → listX
  5. Sequence(Sum[stepY, m], m, 1, nn) → listY
  6. Polygon((-1.2, -1.2), (1.2, -1.2), 4) → back
  7. Length(PrimeFactors[mm]) → lp
  8. Zip(Circle[(2 * cx / n, 2 * cy / n), LP[center + k] / scale / n], cx, listX, cy, listY, k, Sequence[nn - 1]) → circles
  9. Sequence(mySegm[2 * (Element[listX, k], Element[listY, k]) / n, 2 * (Element[listX, k + 1], Element[listY, k + 1]) / n], k, 1, nn - 2) → spiral
  10. RemoveUndefined(Zip[If[LP[center + k] ≟ 1, Text[center + k, (2 * (cx - 0.15) / n, 2 * (cy - 0.05) / n)]], cx, listX, cy, listY, k, Sequence[nn - 1]]) → primes
  11. RemoveUndefined(Zip[Text[center + k, (2 * (cx - 0.15) / n, 2 * (cy - 0.05) / n)], cx, listX, cy, listY, k, Sequence[nn - 1]]) → numbers
  12. RemoveUndefined(Zip[If[LP[center + k] ≟ 1, 2 * (cx, cy) / n], cx, listX, cy, listY, k, Sequence[nn - 1]]) → prmsPoints
  13. Zip(2 * (cx, cy) / n, cx, listX, cy, listY, k, Sequence[nn - 1]) → numsPoints