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Cubic Doubling by Archimedes

Cubic Doubling by Archimedes
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2017-11-17 02:06
Solid GeometryTransformationGeometric Construction
This example demonstrates Archimedes' method of cubic doubling. Key mathematical concepts include cubes, geometric construction, and transformation. Using GeoGebra commands such as Center, Cylinder, and Dilate, it illustrates how to achieve cubic volume doubling through geometric transformations. Suitable for high school mathematics education and research.

GeoGebra Commands & Parameters

  1. Center(c) → A
  2. Cylinder(A, B, r / 2) → f, f_2, f_3, f_4
  3. Cone(C, (0, 0), (1.5 * r)) → i_3, i_2, i_3
  4. OrthogonalLine(P, xOyPlane) → i
  5. Intersect(i, xOyPlane) → D
  6. OrthogonalLine(D, x = 0, xOyPlane) → j
  7. Intersect(j, yAxis) → E
  8. Segment(P, D) → k
  9. Segment(D, E) → l
  10. Segment(M, N) → m
  11. Segment(N, E) → n
  12. Surface(((r / 2 * cos(u)) * (1 + cos(v))), ((r / 2 * sin(u)) * (1 + cos(v))), (r / 2 * sin(v)), u, 0, (2 * π), v, 0, π) → g
  13. Surface(((s * r) * cos(t)), (s * r), ((s * r) * sin(t)), s, 0, 1.5, t, 0, π) → o
  14. CurveCartesian(f1(t), f2(t), f3(t), t, 0, π) → b
  15. CurveCartesian((f4(t) * cos(t)), f4(t), (f4(t) * sin(t)), t, 0, π) → d
  16. CurveCartesian((r / 2 * sin(t)), (r / 2 * cos(t)) + r / 2, f5(t), t, 0, π) → e
  17. Point(Circle[G, Distance[F, G], Segment[F, G]]) → H
  18. Cube(F, G, H) → h, I, caraHIOL, caraJKLO, arestaFI, arestaFG, arestaGH, arestaHI, arestaFJ, arestaIO, arestaJO, arestaJK, J, arestaGK, arestaKL, arestaHL, arestaLO, K, L, O, caraFGHI, caraFIOJ, caraFGKJ, caraGHLK
  19. Dilate(h, 2^(1 / 3), Q) → h'
  20. Dilate(F, 2^(1 / 3), Q) → F'
  21. Dilate(G, 2^(1 / 3), Q) → G'