MathPlotAI Logo MathPlotAI

Infinite Series of Squares

Infinite Series of Squares
Loading the math board and drawing, please wait…
2019-07-24 19:08
TrigonometrySequenceCoordinate GeometryCalculusGeometric Construction
This GeoGebra example dynamically constructs and illustrates the geometric pattern of an infinite series of squares, exploring the convergence of areas and side lengths. Core mathematical concepts include series limits, geometric sequencing, trigonometric relationships, and analytic geometry. Construction employs commands such as Polygon, Sequence, Sum, IntegralBetween, and OrthogonalLine, along with points, lines, and Conic objects to model progressive square arrangements. It is suitable for high school and higher education calculus, series, and geometry instruction, serving as a visual aid for limit processes and spatial reasoning.

GeoGebra Commands & Parameters

  1. Polygon(A, B, 4) → polígono1, f, g, h, i, C, D
  2. CircleArc(D, A, C) → c
  3. Segment(D, B) → j
  4. Intersect(c, j, 1) → E
  5. OrthogonalLine(E, g) → k
  6. OrthogonalLine(E, f) → l
  7. Intersect(k, g) → F
  8. Intersect(l, f) → G
  9. Line(G, j) → m
  10. Intersect(c, m, 1) → H
  11. Line(H, k) → n
  12. Line(H, l) → p
  13. Intersect(l, n) → I
  14. Intersect(p, f) → J
  15. Line(J, m) → q
  16. Intersect(c, q, 1) → K
  17. Line(K, n) → r
  18. Line(K, p) → s
  19. Intersect(p, r) → L
  20. Intersect(s, f) → M
  21. Line(M, q) → t
  22. Intersect(c, t, 1) → N
  23. Line(N, r) → a
  24. Line(N, s) → b
  25. Intersect(a, s) → O
  26. Intersect(f, b) → P
  27. Line(P, t) → d
  28. Intersect(c, d, 1) → Q
  29. Line(Q, a) → e
  30. Line(Q, b) → f_1
  31. Intersect(b, e) → R
  32. Intersect(f_1, f) → S
  33. Line(S, d) → g_1
  34. Intersect(c, g_1, 1) → T
  35. Line(T, e) → h_1
  36. Line(T, f_1) → i_1
  37. Intersect(f_1, h_1) → U
  38. Intersect(i_1, f) → V
  39. Line(V, g_1) → j_1
  40. Intersect(c, j_1, 1) → W
  41. Line(W, h_1) → k_1
  42. Line(W, i_1) → l_1
  43. Intersect(k_1, i_1) → Z
  44. Intersect(f, l_1) → A_1
  45. Line(A_1, j_1) → m_1
  46. Intersect(c, m_1, 1) → B_1
  47. Line(B_1, k_1) → n_1
  48. Line(B_1, l_1) → p_1
  49. Intersect(l_1, n_1) → C_1
  50. Intersect(p_1, f) → D_1
  51. Line(D_1, m_1) → q_1
  52. Intersect(c, q_1) → E_1,
  53. Line(E_1, n_1) → r_1
  54. Line(E_1, p_1) → s_1
  55. Intersect(p_1, r_1) → F_1
  56. Intersect(s_1, f) → G_1
  57. Line(G_1, q_1) → t_1
  58. Intersect(c, t_1, 1) → H_1
  59. Line(H_1, s_1) → a_1
  60. Line(H_1, r_1) → b_1
  61. Intersect(b_1, s_1) → I_1
  62. Intersect(a_1, f) → J_1
  63. Line(J_1, t_1) → c_1
  64. Intersect(c, c_1, 1) → K_1
  65. Line(K_1, b_1) → d_1
  66. Line(K_1, a_1) → e_1
  67. Intersect(a_1, d_1) → L_1
  68. Intersect(e_1, f) → M_1
  69. Line(M_1, c_1) → f_2
  70. Intersect(c, f_2, 1) → N_1
  71. Line(N_1, d_1) → g_2
  72. Line(N_1, e_1) → h_2
  73. Intersect(e_1, g_2) → O_1
  74. Intersect(h_2, f) → P_1
  75. Line(P_1, f_2) → i_2
  76. Intersect(c, i_2) → Q_1,
  77. Line(Q_1, g_2) → j_2
  78. Line(Q_1, h_2) → k_2
  79. Intersect(h_2, j_2) → R_1
  80. Intersect(k_2, f) → S_1
  81. Polygon(S_1, P_1, 4) → p2, l_2, m_2, n_2, p_2, T_1, U_1
  82. Polygon(P_1, M_1, 4) → p3, q_2, r_2, s_2, t_2, V_1, W_1
  83. Polygon(M_1, J_1, 4) → p4, a_2, b_2, c_2, d_2, Z_1, A_2
  84. Polygon(J_1, G_1, 4) → p5, e_2, f_3, g_3, h_3, B_2, C_2
  85. Polygon(G_1, D_1, 4) → p6, i_3, j_3, k_3, l_3, D_2, E_2
  86. Polygon(D_1, A_1, 4) → p7, m_3, n_3, p_3, q_3, F_2, G_2
  87. Polygon(A_1, V, 4) → p8, r_3, s_3, t_3, a_3, H_2, I_2
  88. Polygon(V, S, 4) → p9, b_3, c_3, d_3, e_3, J_2, K_2
  89. Polygon(S, P, 4) → p10, f_4, g_4, h_4, i_4, L_2, M_2
  90. Polygon(P, M, 4) → p11, j_4, k_4, l_4, m_4, N_2, O_2
  91. Polygon(M, J, 4) → p12, n_4, p_4, q_4, r_4, P_2, Q_2
  92. Polygon(J, G, 4) → p13, s_4, t_4, a_4, b_4, R_2, S_2
  93. Polygon(G, B, 4) → p14, c_4, d_4, e_4, f_5, T_2, U_2
  94. Sequence(2, 14) → l2
  95. Sum(lista1) → o
  96. Intersect(h, i) → V_2
  97. Intersect(g, h) → W_2
  98. Angle(E, V_2, W_2) → α
  99. Angle(H, V_2, E) → β
  100. Angle(K, V_2, H) → γ
  101. Angle(N, V_2, K) → δ
  102. Sequence((Element[l2, i], Element[lista1, i]), i, 1, 13) → l3
  103. Intersect(h, i) → Z_2
  104. Intersect(c, f, 1) → A_3
  105. Point(f) → B_3
  106. Line(A_3, B_3) → g_5
  107. Parabola((-4, 2), y = -10) → c_5
  108. Conic(F, I, L, O, R) → d_5
  109. Tangent((-4, -4), d_5) → h_5,
  110. Midpoint(E, B_3) → C_3
  111. Midpoint(H, G) → D_3
  112. Midpoint(K, J) → E_3
  113. Midpoint(N, M) → F_3
  114. Midpoint(Q, P) → G_3
  115. Midpoint(T, S) → H_3
  116. Conic(H_3, G_3, F_3, E_3, D_3) → e_5
  117. Intersect(e_5, q_5) → I_3, J_3
  118. Tangent(I_3, e_5) → i_5,
  119. Midpoint(J_3, I_3) → K_3
  120. Line(K_3, i_5) → j_5
  121. Intersect(e_5, j_5) → L_3, M_3
  122. Intersect(Y, d_5) → N_3, O_3
  123. Intersect(d_5, p_5) → P_3, Q_3
  124. Segment(N_3, O_3) → k_5
  125. Segment(Q_3, P_3) → l_5
  126. Segment(Z_2, B) → r_5
  127. Slope(l_5) → m_6
  128. Slope(k_5) → m_7
  129. Segment(W_2, A) → m_5
  130. Intersect(r_5, m_5) → R_3
  131. Parabola((-4, 0), y = -8) → s_5
  132. IntegralBetween(s_5, (0 * x) - 4, -4, 4) → u
  133. Midpoint(Q, R) → S_3
  134. Midpoint(N, O) → T_3
  135. Midpoint(K, L) → U_3
  136. Midpoint(H, I) → V_3
  137. Midpoint(E, F) → W_3
  138. Conic(W_3, V_3, U_3, T_3, S_3) → t_5
  139. Parabola((-4, 4), y = -12) → c_6
  140. IntegralBetween(c_6, (0 * x) - 4, -4, 4) → v
  141. Intersect(e_5, g) → B_4, C_4
  142. Intersect(e_5, i) → Z_3, A_4
  143. Intersect(f, g) → D_4
  144. Segment(Z_3, D_4) → n_5
  145. Segment(D_4, C_4) → a_5
  146. CurveCartesian((-8.908 * cos(t)) + (0.803 * sin(t)), (-2.103 * cos(t)) - (3.403 * sin(t)), t, 1.03, 1.521) → w
  147. Intersect(b_4, f) → E_4
  148. Segment(E_4, Z_3) → b_5
  149. Intersect(e_5, b_4) → F_4,
  150. Segment(F_4, E_4) → f_6
  151. Vector(G_4, H_4) → v_1

AI Drawing Prompt Reference

Draw an infinite series of squares, starting with a square and then constructing a series of squares where each new square's side is perpendicular to the previous square's side and gradually gets smaller. Begin with a square and use rotation and intersection to construct a series of squares, forming an infinite series of square patterns.
The prompt above was compiled from the original drawing commands and is for reference only.