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How Many Square Pyramids Fit in a Cube?

How Many Square Pyramids Fit in a Cube?
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2020-08-31 07:06
Solid GeometryTransformationGeometric Construction
① This example demonstrates the decomposition of a cube into several square pyramids, visually showing how many pyramids fill the cube and their spatial arrangement. ② Core mathematical concepts: solid geometry, volume of a pyramid, cube dissection, spatial reasoning, and geometric transformations. ③ Construction highlights: GeoGebra commands Point, Intersect, Polygon, Pyramid, Line, and Rotate are used to create vertices A, B, C, D, E, F, G, H, I, J, K, Q and objects Base_1–Base_4, faceABQ, faceACQ, faceCDQ, faceBDQ, edgeBQ, edgeAQ_1, edgeCQ; rotation and intersection build four square pyramids that combine into a complete cube. ④ Suitable for high‑school (grade 10) solid‑geometry lessons, volume calculation demos, and introductory GeoGebra modeling activities.

GeoGebra Commands & Parameters

  1. Point(xAxis) → A
  2. Intersect(xAxis, zAxis) → B
  3. Polygon(B, A, 4, xOyPlane) → Base_1, f, g, h, i, C, D
  4. Point(xAxis) → E
  5. Polygon(A, E, 4) → Base_{2}, j, k, l, m, F, G
  6. Polygon(D, G, 4, xOyPlane) → Base_{3}, n, p, q, r, H, I
  7. Polygon(G, F, 4) → Base_{4}, s, t, a, b, J, K
  8. Pyramid(Base_1, Q) → AGreen, faceABQ, faceACQ, faceCDQ, faceBDQ, edgeBQ, edgeAQ_1, edgeCQ, edgeDQ_1
  9. Pyramid(Base_{3}, Q_{3}) → Blue_{1}, face5, face7, face8, face6, edge5, edge6, edge7, edge8
  10. Pyramid(Base_{4}, Q_{4}) → Yellow_{1}, face9, face10, face12, face11, edge10, edge9, edge11, edge12
  11. Polygon(A, R, S, G) → Base_{2}', c, d, e, f_1
  12. Pyramid(Base_{2}, Q_{2}) → Red_{1}, face1, face3, face4, face2, edge1, edge2, edge3, edge4
  13. Pyramid(Base_{2}', Q) → Red_{2}, faceAQR, faceQRS, faceGQS_1, faceAGQ, edgeAQ, edgeQR, edgeQS_1, edgeGQ_1
  14. Polygon(S, T, D, G) → Base_{3}', s_1, t_1, d_1, g_1
  15. Pyramid(Base_{3}', Q) → Blue_{2}, faceQST, faceDQT, faceDGQ, faceGQS, edgeQS, edgeQT, edgeDQ, edgeGQ
  16. Line(I, E) → h_1
  17. Rotate(Yellow_{1}, 110°, h_1) → Yellow_{2}

AI Drawing Prompt Reference

Draw a cube decomposed into four square pyramids. The cube has eight vertices and faces. Four pyramids are formed from different faces, sharing a common vertex. Construct by first determining the cube's vertices and faces, then drawing vertical lines from each face center to the cube's center to create pyramid bases and apexes. Ensure the diagram clearly shows the relationship between the pyramids and the cube.
The prompt above was compiled from the original drawing commands and is for reference only.