MathPlotAI Logo MathPlotAI

Tetrahedral Sphere Packing Fun Loop

Tetrahedral Sphere Packing Fun Loop
Loading the math board and drawing, please wait…
2019-08-07 05:10
Solid GeometryGeometric Construction
This example demonstrates the packing of spheres within a tetrahedron and then repeating the process, related to the methane (CH4) molecular structure. It involves core mathematical concepts such as solid geometry, spheres, tetrahedra, midpoints, and symmetry transformations. The construction utilizes GeoGebra commands/objects like Point, Rotate, Mirror, Polygon, Tetrahedron, Midpoint, OrthogonalPlane, Intersect, Polynomial, Translate, Sphere, Ray, Sequence, and If. Suitable for high school and above students studying solid geometry, spatial visualization, and geometric constructions.

GeoGebra Commands & Parameters

  1. Point(xAxis) → Q
  2. Rotate(A, 120°) → A'
  3. Mirror(A', xAxis) → A''
  4. Polygon(A', A'', A) → t1, a, a', a''
  5. Tetrahedron(t1, true) → b, B, faceAA'B, faceA'A''B, faceAA''B, edgeA'B, edgeAB, edgeA''B
  6. Midpoint(edgeAB) → C
  7. OrthogonalPlane(C, edgeAB) → p
  8. Intersect(zAxis, p) → D
  9. Polynomial({M, N, F}) → s
  10. Translate(S, Vector[(0, 0, Ds(t))]) → S'
  11. Sphere(S', Ra(t)) → i
  12. Intersect(zAxis, i) → G, H
  13. Sphere(I, r) → c
  14. Sphere(J, r) → d
  15. Sphere(K, r) → e
  16. Sphere(S, r) → g
  17. Translate(I, Vector[(f(t) * I)]) → I'
  18. Ray(O, I) → k
  19. Sphere(S', r + Ra(t)) → l
  20. Intersect(k, l) → L,
  21. Sequence(Rotate[Sphere[L, r], (k * 120°), zAxis], k, 0, 2) → l2
  22. Sphere(I', r + Ra(t)) → h
  23. If(t ≥ 0, H + (0, 0, r), H + (0, 0, r) + (0, 0, f_1(t))) → E
  24. Sphere(E, r) → j
  25. Polynomial({P, Q, R, T, U}) → g_1
  26. Rotate(l2, ((4 / 5 * t) * 120°)) → l2'
  27. Tetrahedron(Polygon[Sequence[Rotate[L, (k * 120°), zAxis], k, 0, 2]], true) → m, V, face2, face1, face3, edge1, edge3, edge2
  28. Rotate(m, ((4 / 5 * t) * 120°)) → m'

AI Drawing Prompt Reference

Draw a tetrahedral sphere packing. Start by taking a point Q on the x-axis. Rotate point A 120 degrees around the x-axis to get A'. Mirror A' to get A''. Use A', A'', and A to form a triangle t1, and construct a tetrahedron b with t1 as the base. Find the midpoint C of edge AB, construct a plane p perpendicular to AB with C as the origin, and find the intersection D with the z-axis. Construct a polynomial s with D as a parameter, translate point S along the z-axis to get S', and construct a sphere i centered at S' with radius Ra(t), intersecting the z-axis at G and H. Then construct spheres c, d, e, and g centered at G and H with radius r.
The prompt above was compiled from the original drawing commands and is for reference only.