Constructing Regular Octagons

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2019-12-31 16:40

This example demonstrates the construction of regular octagons in Islamic geometric patterns. Key mathematical concepts include angles, vectors, and geometric construction. Using GeoGebra commands such as Intersect, ConstructionStep, and Point, regular octagons can be constructed. Suitable for middle school and above students to learn geometric construction and symmetry.

GeoGebra Commands & Parameters

Intersect(xAxis, yAxis) → A
ConstructionStep() → stap
Point(xAxis) → B
Polygon(A, B, 8) → veelhoek1, f, D, E, F, G, H, g, h, i, j, k, l, m, C
Polygon(D, C, 8) → veelhoek2, n, J, K, L, M, N, p, q, r, s, t, a, b, I
Polygon(L, K, 8) → veelhoek3, c, P, Q, R, S, T, d, e, f_1, g_1, h_1, i_1, j_1, O
Polygon(R, Q, 8) → veelhoek4, k_1, V, W, Z, A_1, B_1, l_1, m_1, n_1, p_1, q_1, r_1, s_1, U
Polygon(B, A, 8) → veelhoek5, t_1, D_1, E_1, F_1, G_1, H_1, a_1, b_1, c_1, d_1, e_1, f_2, g_2, C_1
Polygon(J, I, 8) → veelhoek6, h_2, J_1, K_1, L_1, M_1, N_1, i_2, j_2, k_2, l_2, m_2, n_2, p_2, I_1
Polygon(P, O, 8) → veelhoek7, q_2, P_1, Q_1, R_1, S_1, T_1, r_2, s_2, t_2, a_2, b_2, c_2, d_2, O_1
Polygon(V, U, 8) → veelhoek8, e_2, V_1, W_1, Z_1, A_2, B_2, f_3, g_3, h_3, i_3, j_3, k_3, l_3, U_1
Polygon(F, E, 8) → veelhoek9, m_3, D_2, E_2, F_2, G_2, H_2, n_3, p_3, q_3, r_3, s_3, t_3, a_3, C_2
Polygon(N, M, 8) → veelhoek10, b_3, J_2, K_2, L_2, M_2, N_2, c_3, d_3, e_3, f_4, g_4, h_4, i_4, I_2
Polygon(T, S, 8) → veelhoek11, j_4, P_2, Q_2, R_2, S_2, T_2, k_4, l_4, m_4, n_4, p_4, q_4, r_4, O_2
Polygon(B_1, A_1, 8) → veelhoek12, s_4, V_2, W_2, Z_2, A_3, B_3, t_4, a_4, b_4, c_4, d_4, e_4, f_5, U_2
Midpoint(j) → C_3
Midpoint(k) → D_3
Midpoint(l) → E_3
Midpoint(m) → F_3
Midpoint(f) → G_3
Midpoint(g) → H_3
Midpoint(h) → I_3
Midpoint(i) → J_3
Segment(C_3, F_3) → g_5
Segment(D_3, G_3) → h_5
Segment(E_3, H_3) → i_5
Segment(F_3, I_3) → j_5
Segment(G_3, J_3) → k_5
Segment(H_3, C_3) → l_5
Segment(I_3, D_3) → m_5
Segment(J_3, E_3) → n_5
Segment(C_3, F_3) → p_5
Midpoint(s) → K_3
Midpoint(p) → L_3
Midpoint(a) → M_3
Midpoint(b) → N_3
Midpoint(q) → O_3
Midpoint(r) → P_3
Midpoint(t) → Q_3
Midpoint(i_2) → R_3
Midpoint(f_2) → S_3
Midpoint(k_2) → T_3
Midpoint(l_2) → U_3
Midpoint(m_2) → V_3
Midpoint(n_2) → W_3
Midpoint(p_2) → Z_3
Midpoint(g_2) → A_4
Midpoint(a_1) → B_4
Midpoint(b_1) → C_4
Midpoint(c_1) → D_4
Midpoint(d_1) → E_4
Midpoint(e_1) → F_4
Segment(M_3, L_3) → q_5
Segment(N_3, O_3) → r_5
Segment(I_3, P_3) → s_5
Segment(L_3, K_3) → t_5
Segment(O_3, Q_3) → a_5
Segment(P_3, M_3) → b_5
Segment(K_3, N_3) → c_5
Segment(Q_3, I_3) → d_5
Segment(G_3, D_4) → e_5
Segment(B_4, E_4) → f_6
Segment(C_4, F_4) → g_6
Segment(D_4, S_3) → h_6
Segment(E_4, A_4) → i_6
Segment(F_4, G_3) → j_6
Segment(S_3, B_4) → k_6
Segment(A_4, C_4) → l_6
Segment(O_3, T_3) → m_6
Segment(R_3, U_3) → n_6
Segment(S_3, V_3) → p_6
Segment(T_3, W_3) → q_6
Segment(U_3, Z_3) → r_6
Segment(V_3, O_3) → s_6
Segment(W_3, R_3) → t_6
Segment(Z_3, S_3) → a_6
Line(E_3, H_3) → b_6
Line(K_3, L_3) → c_6
Intersect(b_6, c_6) → G_4
Segment(H_3, G_4) → d_6
Segment(L_3, G_4) → e_6
Line(C_3, H_3) → f_7
Line(M_3, L_3) → g_7
Line(C_4, A_4) → h_7
Line(W_3, R_3) → i_7
Line(E_4, A_4) → j_7
Line(U_3, R_3) → k_7
Intersect(i_7, h_7) → H_4
Intersect(g_7, k_7) → I_4
Intersect(f_7, j_7) → J_4
Segment(H_3, J_4) → l_7
Segment(A_4, J_4) → m_7
Segment(A_4, H_4) → n_7
Segment(L_3, I_4) → p_7
Segment(H_4, R_3) → q_7
Segment(R_3, I_4) → r_7
Intersect(m_5, n_5) → K_4
Intersect(m_5, p_5) → L_4
Intersect(n_5, p_5) → M_4
Intersect(h_5, n_5) → N_4
Intersect(h_5, p_5) → O_4
Intersect(i_5, p_5) → P_4
Intersect(h_5, i_5) → Q_4
Intersect(h_5, j_5) → R_4
Intersect(i_5, j_5) → S_4
Intersect(i_5, k_5) → T_4
Intersect(j_5, k_5) → U_4
Intersect(j_5, l_5) → V_4
Intersect(k_5, l_5) → W_4
Intersect(k_5, m_5) → Z_4
Intersect(l_5, m_5) → A_5
Intersect(l_5, n_5) → B_5
PolyLine(K_4, L_4, U_4, V_4, W_4, Z_4, A_5, B_5, K_4, M_4, N_4, O_4, P_4, Q_4, R_4, S_4, T_4) → s_7
PolyLine(C_3, L_4, H_3, V_4, I_3, Z_4, J_3, B_5, C_3, D_3, N_4, E_3, P_4, F_3, R_4, G_3, T_4) → t_7
Midpoint(C_3, G_3) → D_5
Polygon(C_3, L_4, K_4, B_5) → vierhoek1, a_7, b_7, c_7, d_7
Sequence(Polygon[Rotate[C_3, (nn * 45°), D_5], Rotate[L_4, (nn * 45°), D_5], Rotate[K_4, (nn * 45°), D_5], Rotate[B_5, (nn * 45°), D_5]], nn, 0, 7) → l1
Polygon(L_3, G_4, H_3, J_4, A_4, H_4, R_3, I_4) → veelhoek13, e_7, f_8, g_8, h_8, i_8, j_8, k_8, l_8
Vector(A, I) → hot
Vector(A, F) → vert
Sequence(Translate[Sequence[Translate[l1, Vector[(nn * hot)]], nn, 0, 3], Vector[(mm * vert)]], mm, -1, 1) → l2
Sequence(Translate[Sequence[Polygon[Translate[L_3, Vector[(nn * hot)]], Translate[G_4, Vector[(nn * hot)]], Translate[H_3, Vector[(nn * hot)]], Translate[J_4, Vector[(nn * hot)]], Translate[A_4, Vector[(nn * hot)]], Translate[H_4, Vector[(nn * hot)]], Translate[R_3, Vector[(nn * hot)]], Translate[I_4, Vector[(nn * hot)]]], nn, 0, 2], Vector[(mm * vert)]], mm, 0, 1) → l3
Polygon(B, H_1, I, C) → vierhoek2, b_8, m_8, i_9, c_8
Sequence(Translate[Sequence[Polygon[Translate[B, Vector[(nn * hot)]], Translate[H_1, Vector[(nn * hot)]], Translate[I, Vector[(nn * hot)]], Translate[C, Vector[(nn * hot)]]], nn, 0, 2], Vector[(mm * vert)]], mm, 0, 1) → l4

AI Drawing Prompt Reference

Draw an Islamic geometric pattern composed of multiple regular octagons that connect around a central point to form a symmetrical star shape. The pattern centers on the intersection of two perpendicular axes, with multiple octagons extending outward, each side of which intersects the sides of adjacent octagons, creating a complex geometric design. Maintain symmetry and geometric precision in the pattern.

The prompt above was compiled from the original drawing commands and is for reference only.