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Binomial Expansion Cube Model

Binomial Expansion Cube Model
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2016-11-08 07:37
VectorFunctionSolid GeometryGeometric Construction
This example demonstrates the geometric interpretation of binomial expansion using a three-dimensional cube and its components. It visualizes how the terms of (a+b)^3 correspond to the volumes of sub-cubes and prisms within the cube. Core mathematical concepts include the binomial theorem and solid geometry. The construction uses GeoGebra commands such as Vector, Cube, and Prism, defining multiple vectors (vector_0 to vector_8), points (A to K), faces (faceABCD, etc.), and edges (edgeAD, etc.) to build the cube and its internal partitions. Suitable for high school or early college students to explore the algebraic-geometric connection, and useful for interactive math education or visualization.

GeoGebra Commands & Parameters

  1. Vector((-4, -4, -4)) → vector_0
  2. Vector((0, 0, 0), (-1, -1, -1)) → vector_1
  3. Vector((0, 0, 0), (1, -1, 1)) → vector_6
  4. Vector((0, 0, 0), (-1, -1, 1)) → vector_5
  5. Vector((0, 0, 0), (-1, 1, 1)) → vector_8
  6. Vector((0, 0, 0), (1, 1, 1)) → vector_7
  7. Vector((0, 0, 0), (-1, 1, -1)) → vector_4
  8. Vector((0, 0, 0), (1, 1, -1)) → vector_3
  9. Vector((0, 0, 0), (1, -1, -1)) → vector_2
  10. 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
  11. Cube(I, J, K) → b, L, faceKLPO, faceMNOP, edgeIL, edgeIJ, edgeJK, edgeKL, edgeIM, edgeLP, edgeMP, edgeMN, M, edgeJN, edgeNO, edgeKO, edgeOP, N, O, P, faceIJKL, faceILPM, faceIJNM, faceJKON
  12. Prism(Q, R, S, T, Z) → c, U, edgeQR, edgeRS, edgeST, edgeQT, edgeQZ, edgeRU, edgeSV, edgeTW, edgeUZ, edgeUV, V, edgeVW, edgeWZ, W, faceQRST, faceQRUZ, faceRSVU, faceSTWV, faceQTWZ, faceUVWZ
  13. Prism(A_1, B_1, C_1, D_1, H_1) → d, E_1, edge1, edge4, edge6, edge2, edge3, edge5, edge7, edge8, edge9, edge11, F_1, edge12, edge10, G_1, face1, face2, face4, face5, face3, face6
  14. Prism(I_1, J_1, K_1, L_1, M_1) → e, N_1, edge13, edge16, edge18, edge14, edge15, edge17, edge19, edge20, edge21, edge23, O_1, edge24, edge22, P_1, face7, face8, face10, face11, face9, face12
  15. Prism(Q_1, R_1, S_1, T_1, U_1) → f, V_1, edge25, edge28, edge30, edge26, edge27, edge29, edge31, edge32, edge33, edge35, W_1, edge36, edge34, Z_1, face13, face14, face16, face17, face15, face18
  16. Prism(A_2, B_2, C_2, D_2, E_2) → g, F_2, edge37, edge40, edge42, edge38, edge39, edge41, edge43, edge44, edge45, edge47, G_2, edge48, edge46, H_2, face19, face20, face22, face23, face21, face24
  17. Prism(I_2, J_2, K_2, L_2, M_2) → h, N_2, edge49, edge52, edge54, edge50, edge51, edge53, edge55, edge56, edge57, edge59, O_2, edge60, edge58, P_2, face25, face26, face28, face29, face27, face30