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Exploring Nets, Vertices, Edges, and Faces: Euler Formula for Solids

Exploring Nets, Vertices, Edges, and Faces: Euler Formula for Solids
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2022-10-18 16:07
Solid GeometryGeometric Construction
This example demonstrates the construction of 3D solid nets using GeoGebra and applies the Euler formula (V - E + F = 2) to verify topological relationships in polyhedra. Key mathematical concepts include polyhedral topology, Euler's formula, net reconstruction, and geometric transformations. Construction involves using commands like Intersect, Ray, Point, Circle, Polygon, Prism, Pyramid, and Net, along with objects such as vertices, edges, and faces. Suitable for middle school students to understand solid geometry and topology, or for mathematical modeling exercises.

GeoGebra Commands & Parameters

  1. Intersect(xAxis, yAxis) → A
  2. Ray(A, (x(A), y(A), 40)) → h
  3. Point(yAxis) → B
  4. Circle(A, B) → c
  5. Point(c) → C
  6. Ray(C, (x(C), y(C), 40)) → g
  7. Sequence(Rotate[C, ((t * 2) * pi / n), A], t, 1, n) → vert
  8. Polygon(vert) → poly
  9. OrthogonalLine(C, xOyPlane) → f
  10. Point(g) → E
  11. Prism(poly, Segment[C, E]) → a, F, face3, face4, face5, face6, face2, face13, face15, faceFGHIJK, edge1, edge2, G, edge3, edge4, edge5, edge6, edge13, edge16, edgeFG, edgeGH, edgeHI, edgeIJ, H, edgeJK, edgeFK, edge14, edge17, I, J, K, L, M, face1
  12. Point(h) → D
  13. Pyramid(poly, D) → b, face7, edge8, edge9, edge10, edge11, edge12, edge15, edge18, face9, face10, face11, face12, face8, face14, face16, edge7
  14. Net(a, (-open)) → prismNet, N, W, Z, A_1, B_1, C_1, D_1, E_1, F_1, G_1, H_1, O, I_1, J_1, K_1, L_1, M_1, N_1, O_1, P_1, Q_1, R_1, P, S_1, face17, face18, face19, face20, face21, face22, face23, face24, face34, Q, face35, edge19, edge20, edge21, edge22, edge23, edge24, edge25, edge26, edge27, R, edge28, edge29, edge30, edge31, edge32, edge33, edge34, edge35, edge36, edge37, S, edge38, edge39, edge40, edge41, edge42, edge43, edge44, edge45, edge70, edge71, T, edge72, edge73, edge74, edge75, edge76, edge77, edge78, edge79, edge80, edge81, U, V
  15. Net(b, (-open)) → pyrNet, T_1, E_2, F_2, G_2, H_2, I_2, J_2, K_2, face25, face26, face27, U_1, face28, face29, face30, face31, face32, face33, edge46, edge47, edge48, edge49, V_1, edge50, edge51, edge52, edge53, edge54, edge55, edge56, edge57, edge58, edge59, W_1, edge60, edge61, edge62, edge63, edge64, edge65, edge66, edge67, edge68, edge69, Z_1, A_2, B_2, C_2, D_2
  16. Random(1, 2) → choice
  17. Random(3, 8) → nRand

AI Drawing Prompt Reference

Draw a 3D solid net that demonstrates how to unfold and fold it into a polyhedron, verifying Euler's formula V-E+F=2. Start by creating a circle, then extend rays from its center outward. Rotate these rays and connect them to form a polygon. Build a prism by extending the polygon upward and a pyramid by extending upward from the center of the polygon. Finally, draw the unfolded nets of both the prism and pyramid to show the relationships between their faces and edges.
The prompt above was compiled from the original drawing commands and is for reference only.