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Painted Cube 3x3x3: Counting Colored Unit Cubes in a 3D Visualization

Painted Cube 3x3x3: Counting Colored Unit Cubes in a 3D Visualization
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2016-03-06 21:00
PolygonVectorCoordinate GeometrySolid GeometryGeometric Construction
①This example demonstrates a 3x3x3 cube with its surface painted, using GeoGebra's 3D view to visually distinguish unit cubes by their position (corner, edge, face-center, and interior) and support counting colored faces. ②Core mathematical concepts include solid geometry, spatial partitioning, combinatorial counting (e.g., the number of small cubes with 0, 1, 2, or 3 painted faces), and relationships among points, segments, faces, and volumes in 3D space. ③Construction highlights: The Cube command creates the main cube; Intersect, Point, and Segment commands generate vertices, edges, and intersection points; PointIn marks interior points; multiple faces (faceABCD, faceADHE, etc.) and edges (edgeAB, edgeBC, etc.) are explicitly defined to clarify the structure. ④This is suitable for middle school (grades 7-9) solid geometry lessons, as well as introductory combinatorics or spatial reasoning exercises, helping students classify and count elements in a 3D grid.

GeoGebra Commands & Parameters

  1. Intersect(yAxis, xAxis) → A
  2. Point(xAxis) → B
  3. Point(Circle[B, Distance[A, B], Segment[A, B]]) → C
  4. Cube(A, B, C) → a, D, faceCDHG, faceEFGH, edgeAD, edgeAB, edgeBC, edgeCD, edgeAE, edgeDH, edgeEH, edgeEF, E, edgeBF, edgeFG, edgeCG, edgeGH, F, G, H, faceABCD, faceADHE, faceABFE, faceBCGF
  5. Point(edgeCD) → I
  6. Point(edgeGH) → J
  7. Segment(I, J) → b
  8. Point(edgeCD) → K
  9. Point(edgeGH) → L
  10. Segment(K, L) → c
  11. Point(edgeBC) → M
  12. Point(edgeFG) → N
  13. Segment(M, N) → d
  14. Point(edgeBC) → O
  15. Point(edgeFG) → P
  16. Segment(O, P) → e
  17. PointIn(faceEFGH) → Q
  18. Segment(N, Q) → f
  19. Point(edgeEH) → R
  20. Segment(P, R) → g
  21. PointIn(faceEFGH) → S
  22. Segment(J, S) → h
  23. Point(edgeEF) → T
  24. Segment(L, T) → i
  25. Point(edgeCG) → U
  26. PointIn(faceCDHG) → V
  27. Segment(U, V) → j
  28. Point(edgeCG) → W
  29. PointIn(faceCDHG) → Z
  30. Segment(W, Z) → k
  31. Point(edgeBF) → A_1
  32. Segment(W, A_1) → l
  33. PointIn(faceBCGF) → B_1
  34. Segment(U, B_1) → m
  35. Point(zAxis) → C_1
  36. Segment(C_1, A_1) → n
  37. Point(zAxis) → D_1
  38. Segment(D_1, B_1) → p
  39. PointIn(faceABFE) → E_1
  40. Segment(T, E_1) → q
  41. PointIn(faceABFE) → F_1
  42. Segment(S, F_1) → r
  43. Segment(D_1, V) → s
  44. Segment(Z, C_1) → t
  45. Point(yAxis) → G_1
  46. Segment(R, G_1) → a_1
  47. Point(yAxis) → H_1
  48. Segment(H_1, Q) → b_1
  49. Segment(M, H_1) → c_1
  50. Segment(G_1, O) → d_1
  51. Segment(I, F_1) → e_1
  52. Segment(K, E_1) → f_1

AI Drawing Prompt Reference

Visualize a 3x3x3 cube with painted surfaces, using a 3D view to distinguish unit cubes by position (corner, edge, face center, and interior) and support counting colored faces. Define the cube using points on three axes, then generate vertices, edges, and faces. Mark interior points and explicitly define edges and faces to clarify the structure. Highlight different regions like corners, edges, and face centers using lines connecting specific points, enabling classification and counting of various types of unit cubes.
The prompt above was compiled from the original drawing commands and is for reference only.