Beehive Hexagon Surface Area Minimum

GeoGebra Commands & Parameters

1. Point(xAxis) → A
2. Rotate(A, 60°, zAxis) → A'
3. Rotate(A, (-(60°)), zAxis) → A'_1
4. Rotate(A'_1, (-(60°)), zAxis) → B
5. Rotate(B, (-(60°)), zAxis) → B'
6. Rotate(B', (-(60°)), zAxis) → B''
7. Polygon(B, A'_1, A, A', B'', B') → poly1, b, a'_1, a, a', b'', b'
8. OrthogonalLine(B, xOyPlane) → f
9. Point(f) → H
10. Prism(poly1, H) → c, C, faceB'B''FG, faceBB'GH, faceCDEFGH, edgeBH, edge1, edgeAD, edgeA'E, edgeB''F, edgeB'G, edgeCH, D, edgeCD, edgeDE, edgeEF, edgeFG, edgeGH, E, F, G, face2, face1, faceAA'ED, faceA'B''FE
11. Point(edgeBH) → I
12. Plane(I, C, G) → d
13. Intersect(zAxis, d) → J
14. Rotate(I, 120°, zAxis) → I'
15. Rotate(I', 120°, zAxis) → I''
16. Polygon(I, C, J, G) → q1, i, c_1, j, g
17. Polygon(I', C, J, E) → q2, i', c_2, j_1, e
18. Polygon(I'', E, J, G) → q3, i'', e_1, j_2, g_1
19. Polygon(G, B', B, I) → q4, g_2, h, b_1, i_1
20. Polygon(G, B', B'', I'') → q5, g_3, k, l, m
21. Polygon(B'', I'', E, A') → q6, n, p, e_2, q
22. Polygon(A', E, I', A) → q7, r, e_3, s, a_1
23. Polygon(A, I', C, A'_1) → q8, a_2, t, c_3, f_1
24. Polygon(A'_1, C, I, B) → q9, h_1, c_4, i_2, b_2
25. PointIn(q9) → Pointq9
26. Angle(I, C, J) → β
27. OrthogonalLine(K, xAxis) → k_1
28. OrthogonalLine(K, k_1) → l_1