Penrose Mosaics: Geometric Construction and Mathematical Principles

Penrose Mosaics: Geometric Construction and Mathematical Principles
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2023-03-13 19:26
In collections geometriadinamica.es
This example demonstrates non-periodic tiling patterns using geometric transformations and mathematical concepts like the golden ratio and Fibonacci sequences. The construction involves commands such as Dardo and Cometa to generate polygons, with dynamic visualization through coordinates and transformations. Suitable for high school students learning geometry and mathematical modeling, keywords include geometric construction, golden ratio, and sequences.

GeoGebra Commands & Parameters

  1. Dardo4Conway(A, B) → poly1, C, D, a, b, c, d, e, f
  2. Cometa2Conway(D, C) → poly2, E, F, g, h, i, j, k, p
  3. Cometa1Conway(C, B) → poly3, G, H, l, m, n, q, r, s
  4. Cometa2Conway(D, F) → poly4, I, J, t, a1, b1, c1, d1, e1
  5. Cometa1Conway(G, B) → poly5, K, L, f1, g1, h1, i1, k1, p1
  6. Dardo1Conway(D, J) → poly6, M, N, j1, l1, m1, n1, q1, r1
  7. Dardo2Conway(K, B) → poly7, O, P, s1, t1, a2, b2, c2, d2
  8. Cometa3Conway(J, I) → poly8, Q, R, e2, f2, g2, h2, k2, p2
  9. Cometa4Conway(L, K) → poly9, S, T, i2, j2, l2, m2, q2, r2
  10. Dardo2Conway(T, O) → poly10, U, V, s2, t2, n2, a3, b3, c3
  11. Dardo1Conway(N, Q) → poly11, W, Z, d3, e3, f3, g3, h3, k3
  12. Cometa4Conway(L, T) → poly12, A1, B1, i3, j3, l3, m3, p3, q3
  13. Cometa3Conway(Q, I) → poly13, C1, D1, n3, r3, s3, t3, c4, d4
  14. Cometa3Conway(V, O) → poly14, E1, F1, a4, b4, e4, f4, g4, h4
  15. Cometa4Conway(N, W) → poly15, G1, H1, i4, j4, k4, l4, p4, q4
  16. Cometa3Conway(E1, S) → poly16, I1, J1, m4, n4, r4, s4, t4, c5
  17. Cometa4Conway(N, H1) → poly17, K1, L1, a5, b5, d5, e5, f5, g5
  18. Dardo1Conway(J1, H1) → poly18, M1, N1, h5, i5, j5, k5, p5, q5
  19. Dardo1Conway(F1, E1) → poly19, O1, P1, l5, m5, n5, r5, s5, t5
  20. Cometa3Conway(G, L) → poly20, Q1, R1, a6, b6, c6, d6, e6, f6
  21. Cometa3Conway(Q1, L) → poly21, S1, T1, g6, h6, i6, j6, k6, p6
  22. Cometa3Conway(C1, I) → poly22, U1, V1, l6, m6, n6, q6, r6, s6
  23. Cometa3Conway(U1, I) → poly23, W1, Z1, t6, a7, b7, c7, d7, e7
  24. Dardo4Conway(F, E) → poly24, A2, B2, f7, g7, h7, i7, k7, p7
  25. Dardo3Conway(E, G) → poly25, C2, D2, j7, l7, m7, n7, q7, r7
  26. Cometa1Conway(A2, E) → poly26, E2, F2, s7, t7, a8, b8, c8, d8
  27. Cometa2Conway(B2, A2) → poly27, G2, H2, e8, f8, g8, h8, k8, p8
  28. Cometa4Conway(F2, D2) → poly28, I2, J2, i8, j8, l8, m8, q8, r8
  29. Cometa4Conway(F2, J2) → poly29, K2, L2, n8, s8, t8, a9, c9, d9
  30. Cometa4Conway(F2, L2) → poly30, M2, N2, b9, e9, f9, g9, h9, k9
  31. Cometa2Conway(Z1, F) → poly31, O2, P2, i9, j9, l9, m9, p9, q9
  32. Cometa3Conway(P2, B2) → poly32, Q2, R2, n9, r9, s9, t9, c10, d10
  33. Dardo4Conway(Q2, B2) → poly33, S2, T2, a10, b10, e10, f10, g10, h10
  34. Cometa4Conway(C2, G) → poly34, U2, V2, i10, j10, k10, l10, p10, q10
  35. Cometa4Conway(C2, V2) → poly35, W2, Z2, m10, n10, r10, s10, t10, c11
  36. Dardo3Conway(C2, Z2) → poly36, A3, B3, a11, b11, d11, e11, f11, g11
  37. Dardo1Conway(Z1, P2) → poly37, C3, D3, h11, i11, j11, k11, p11, q11
  38. Dardo1Conway(W2, V2) → poly38, E3, F3, l11, m11, n11, r11, s11, t11
  39. Dardo1Conway(R1, Q1) → poly39, G3, H3, a12, b12, c12, d12, e12, f12
  40. Dardo1Conway(V1, U1) → poly40, I3, J3, g12, h12, i12, j12, k12, p12
  41. Dardo4Conway(M1, G1) → poly41, K3, L3, l12, m12, n12, q12, r12, s12
  42. Dardo4Conway(M1, L3) → poly42, M3, N3, t12, a13, b13, c13, d13, e13
  43. Dardo4Conway(M1, N3) → poly43, O3, P3, f13, g13, h13, i13, k13, p13
  44. Dardo4Conway(E3, T1) → poly44, Q3, R3, j13, l13, m13, n13, q13, r13
  45. Dardo4Conway(E3, R3) → poly45, S3, T3, s13, t13, a14, b14, c14, d14
  46. Dardo4Conway(E3, T3) → poly46, U3, V3, e14, f14, g14, h14, k14, p14
  47. Dardo4Conway(C3, R2) → poly47, W3, Z3, i14, j14, l14, m14, q14, r14
  48. Dardo4Conway(C3, Z3) → poly48, A4, B4, n14, s14, t14, a15, c15, d15
  49. Dardo4Conway(C3, B4) → poly49, C4, D4, b15, e15, f15, g15, h15, k15
  50. Cometa2Conway(N3, M3) → poly50, E4, F4, i15, j15, l15, m15, p15, q15
  51. Cometa4Conway(E4, M3) → poly51, G4, H4, n15, r15, s15, t15, c16, d16
  52. Cometa4Conway(E4, H4) → poly52, I4, J4, a16, b16, e16, f16, g16, h16
  53. Cometa4Conway(E4, J4) → poly53, K4, L4, i16, j16, k16, l16, p16, q16
  54. Cometa4Conway(E4, L4) → poly54, M4, N4, m16, n16, r16, s16, t16, c17
  55. Cometa2Conway(T3, S3) → poly55, O4, P4, a17, b17, d17, e17, f17, g17
  56. Cometa2Conway(B4, A4) → poly56, Q4, R4, h17, i17, j17, k17, p17, q17
  57. Cometa4Conway(O4, S3) → poly57, S4, T4, l17, m17, n17, r17, s17, t17
  58. Cometa4Conway(O4, T4) → poly58, U4, V4, a18, b18, c18, d18, e18, f18
  59. Cometa4Conway(O4, V4) → poly59, W4, Z4, g18, h18, i18, j18, k18, p18
  60. Cometa4Conway(O4, Z4) → poly60, A5, B5, l18, m18, n18, q18, r18, s18
  61. Cometa4Conway(Q4, A4) → poly61, C5, D5, t18, a19, b19, c19, d19, e19
  62. Cometa4Conway(Q4, D5) → poly62, E5, F5, f19, g19, h19, i19, k19, p19
  63. Cometa4Conway(Q4, F5) → poly63, G5, H5, j19, l19, m19, n19, q19, r19
  64. Cometa4Conway(Q4, H5) → poly64, I5, J5, s19, t19, a20, b20, c20, d20
  65. Cometa2Conway(F1, O3) → poly65, K5, L5, e20, f20, g20, h20, k20, p20
  66. Cometa2Conway(N3, F4) → poly66, M5, N5, i20, j20, l20, m20, q20, r20
  67. Cometa2Conway(W2, U3) → poly67, O5, P5, n20, s20, t20, a21, c21, d21
  68. Cometa2Conway(T3, P4) → poly68, Q5, R5, b21, e21, f21, g21, h21, k21
  69. Cometa2Conway(V1, C4) → poly69, S5, T5, i21, j21, l21, m21, p21, q21
  70. Cometa2Conway(B4, R4) → poly70, U5, V5, n21, r21, s21, t21, c22, d22
  71. Cometa2Conway(Z3, W3) → poly71, W5, Z5, a22, b22, e22, f22, g22, h22
  72. Cometa2Conway(R2, Q2) → poly72, A6, B6, i22, j22, k22, l22, p22, q22
  73. Cometa2Conway(R3, Q3) → poly73, C6, D6, m22, n22, r22, s22, t22, c23
  74. Cometa2Conway(T1, B1) → poly74, E6, F6, a23, b23, d23, e23, f23, g23
  75. Cometa2Conway(L3, K3) → poly75, G6, H6, h23, i23, j23, k23, p23, q23
  76. Cometa2Conway(G1, W) → poly76, I6, J6, l23, m23, n23, r23, s23, t23
  77. Dardo4Conway(W, Z) → poly77, K6, L6, a24, b24, c24, d24, e24, f24
  78. Dardo4Conway(W, L6) → poly78, M6, N6, g24, h24, i24, j24, k24, p24
  79. Cometa2Conway(G6, M6) → poly79, P6, Q6, l24, m24, n24, q24, r24, s24
  80. Cometa3Conway(Q6, P6) → poly80, R6, S6, t24, a25, b25, c25, d25, e25
  81. Cometa3Conway(R6, P6) → poly81, T6, U6, f25, g25, h25, i25, k25, p25
  82. Cometa3Conway(T6, P6) → poly82, V6, W6, j25, l25, m25, n25, q25, r25
  83. Cometa3Conway(V6, P6) → poly83, Z6, A7, s25, t25, a26, b26, c26, d26
  84. Dardo4Conway(H4, G6) → poly84, B7, C7, e26, f26, g26, h26, k26, p26
  85. Dardo3Conway(U, V) → poly85, D7, E7, i26, j26, l26, m26, q26, r26
  86. Dardo3Conway(D7, V) → poly86, F7, G7, n26, s26, t26, a27, c27, d27
  87. Cometa1Conway(G7, K5) → poly87, H7, I7, b27, e27, f27, g27, h27, k27
  88. Cometa4Conway(I7, H7) → poly88, J7, K7, i27, j27, l27, m27, p27, q27
  89. Cometa4Conway(I7, K7) → poly89, L7, M7, n27, r27, s27, t27, c28, d28
  90. Cometa4Conway(I7, M7) → poly90, N7, O7, a28, b28, e28, f28, g28, h28
  91. Cometa4Conway(I7, O7) → poly91, P7, Q7, i28, j28, k28, l28, p28, q28
  92. Dardo1Conway(J7, H7) → poly92, O6, R7, m28, n28, r28, s28, t28, c29
  93. Cometa1Conway(B1, U) → poly93, S7, T7, a29, b29, d29, e29, f29, g29
  94. Cometa1Conway(E7, D7) → poly94, U7, V7, h29, i29, j29, k29, p29, q29
  95. Cometa1Conway(K6, Z) → poly95, W7, Z7, l29, m29, n29, r29, s29, t29
  96. Cometa1Conway(V6, L6) → poly96, A8, B8, a30, b30, c30, d30, e30, f30
  97. Dardo3Conway(C6, B1) → poly97, C8, D8, g30, h30, i30, j30, k30, p30
  98. Dardo3Conway(T7, O7) → poly98, E8, F8, l30, m30, n30, q30, r30, s30
  99. Dardo3Conway(Z7, C1) → poly99, G8, H8, t30, a31, b31, c31, d31, e31
  100. Dardo4Conway(V6, Z7) → poly100, I8, J8, f31, g31, h31, i31, k31, p31
  101. Cometa2Conway(T7, F8) → poly101, K8, L8, j31, l31, m31, n31, q31, r31
  102. Cometa4Conway(K8, F8) → poly102, M8, N8, s31, t31, a32, b32, c32, d32
  103. Cometa4Conway(K8, N8) → poly103, O8, P8, e32, f32, g32, h32, k32, p32
  104. Cometa4Conway(K8, P8) → poly104, Q8, R8, i32, j32, l32, m32, q32, r32
  105. Cometa4Conway(K8, R8) → poly105, S8, T8, n32, s32, t32, a33, c33, d33
  106. Cometa2Conway(Z7, H8) → poly106, U8, V8, b33, e33, f33, g33, h33, k33
  107. Cometa3Conway(I8, U8) → poly107, W8, Z8, i33, j33, l33, m33, p33, q33
  108. Cometa3Conway(W8, U8) → poly108, A9, B9, n33, r33, s33, t33, c34, d34
  109. Cometa3Conway(A9, U8) → poly109, C9, D9, a34, b34, e34, f34, g34, h34
  110. Cometa3Conway(C9, U8) → poly110, E9, F9, i34, j34, k34, l34, p34, q34
  111. Dardo1Conway(D9, C9) → poly111, G9, H9, m34, n34, r34, s34, t34, c35
  112. Dardo1Conway(G5, F5) → poly112, I9, J9, a35, b35, d35, e35, f35, g35
  113. Cometa2Conway(E5, D5) → poly113, K9, L9, h35, i35, j35, k35, p35, q35
  114. Cometa3Conway(L9, K9) → poly114, M9, N9, l35, m35, n35, r35, s35, t35
  115. Dardo3Conway(K9, D5) → poly115, O9, P9, a36, b36, c36, d36, e36, f36
  116. Dardo4Conway(M9, K9) → poly116, Q9, R9, g36, h36, i36, j36, k36, p36
  117. Cometa2Conway(R9, Q9) → poly117, S9, T9, l36, m36, n36, q36, r36, s36
  118. Cometa4Conway(S9, Q9) → poly118, U9, V9, t36, a37, b37, c37, d37, e37
  119. Cometa4Conway(S9, P9) → poly119, W9, Z9, f37, g37, h37, i37, k37, p37
  120. Cometa4Conway(S9, Z9) → poly120, A10, B10, j37, l37, m37, n37, q37, r37
  121. Cometa4Conway(S9, B10) → poly121, C10, D10, s37, t37, a38, b38, c38, d38
  122. Dardo1Conway(A10, Z9) → poly122, E10, F10, e38, f38, g38, h38, k38, p38
  123. Dardo1Conway(R9, T9) → poly123, G10, H10, i38, j38, l38, m38, q38, r38
  124. Cometa2Conway(C10, B10) → poly124, I10, J10, n38, s38, t38, a39, c39, d39
  125. Cometa3Conway(J10, I10) → poly125, K10, L10, b39, e39, f39, g39, h39, k39
  126. Dardo3Conway(I10, B10) → poly126, M10, N10, i39, j39, l39, m39, p39, q39
  127. Cometa2Conway(I10, N10) → poly127, O10, P10, n39, r39, s39, t39, c40, d40
  128. Cometa4Conway(O10, N10) → poly128, Q10, R10, a40, b40, e40, f40, g40, h40
  129. Cometa4Conway(O10, R10) → poly129, S10, T10, i40, j40, k40, l40, p40, q40
  130. Cometa4Conway(O10, T10) → poly130, U10, V10, m40, n40, r40, s40, t40, c41
  131. Cometa4Conway(O10, V10) → poly131, W10, Z10, a41, b41, d41, e41, f41, g41
  132. Dardo1Conway(S10, R10) → poly132, A11, B11, h41, i41, j41, k41, p41, q41
  133. Dardo1Conway(U10, T10) → poly133, C11, D11, l41, m41, n41, r41, s41, t41
  134. Dardo1Conway(I10, P10) → poly134, E11, F11, a42, b42, c42, d42, e42, f42
  135. Cometa3Conway(L2, M10) → poly135, G11, H11, g42, h42, i42, j42, k42, p42
  136. Cometa3Conway(G11, M10) → poly136, I11, J11, l42, m42, n42, q42, r42, s42
  137. Dardo1Conway(M2, G11) → poly137, K11, L11, t42, a43, b43, c43, d43, e43
  138. Dardo1Conway(C10, J10) → poly138, M11, N11, f43, g43, h43, i43, k43, p43
  139. Dardo1Conway(W10, V10) → poly139, O11, P11, j43, l43, m43, n43, q43, r43
  140. Cometa3Conway(O11, U10) → poly140, Q11, R11, s43, t43, a44, b44, c44, d44
  141. Cometa3Conway(Q11, U10) → poly141, S11, T11, e44, f44, g44, h44, k44, p44
  142. Cometa3Conway(C11, S10) → poly142, U11, V11, i44, j44, l44, m44, q44, r44
  143. Cometa3Conway(U11, S10) → poly143, W11, Z11, n44, s44, t44, a45, c45, d45
  144. Cometa2Conway(T11, C11) → poly144, A12, B12, b45, e45, f45, g45, h45, k45
  145. Cometa4Conway(A12, C11) → poly145, C12, D12, i45, j45, l45, m45, p45, q45
  146. Cometa4Conway(A12, D12) → poly146, E12, F12, n45, r45, s45, t45, c46, d46
  147. Cometa4Conway(A12, F12) → poly147, G12, H12, a46, b46, e46, f46, g46, h46
  148. Cometa4Conway(A12, H12) → poly148, I12, J12, i46, j46, k46, l46, p46, q46
  149. Dardo1Conway(O5, D12) → poly149, K12, L12, m46, n46, r46, s46, t46, c47
  150. Dardo1Conway(V11, U11) → poly150, M12, N12, a47, b47, d47, e47, f47, g47
  151. Dardo1Conway(G12, F12) → poly151, O12, P12, h47, i47, j47, k47, p47, q47
  152. Dardo1Conway(C6, R8) → poly152, Q12, R12, l47, m47, n47, r47, s47, t47
  153. Dardo1Conway(Q8, P8) → poly153, S12, T12, a48, b48, c48, d48, e48, f48
  154. Dardo1Conway(O8, N8) → poly154, U12, V12, g48, h48, i48, j48, k48, p48
  155. Dardo1Conway(W4, V4) → poly155, W12, Z12, l48, m48, n48, q48, r48, s48
  156. Dardo1Conway(I12, H12) → poly156, A13, B13, t48, a49, b49, c49, d49, e49
  157. Dardo1Conway(R11, Q11) → poly157, C13, D13, f49, g49, h49, i49, k49, p49
  158. Dardo1Conway(T11, B12) → poly158, E13, F13, j49, l49, m49, n49, q49, r49
  159. Dardo1Conway(E5, L9) → poly159, G13, H13, s49, t49, a50, b50, c50, d50
  160. Dardo1Conway(I5, H5) → poly160, I13, J13, e50, f50, g50, h50, k50, p50
  161. Cometa3Conway(A13, G12) → poly161, K13, L13, i50, j50, l50, m50, q50, r50
  162. Cometa3Conway(K13, G12) → poly162, M13, N13, n50, s50, t50, a51, c51, d51
  163. Dardo1Conway(L13, K13) → poly163, O13, P13, b51, e51, f51, g51, h51, k51
  164. Dardo1Conway(A5, Z4) → poly164, Q13, R13, i51, j51, l51, m51, p51, q51
  165. Cometa2Conway(U4, T4) → poly165, S13, T13, n51, r51, s51, t51, c52, d52
  166. Cometa3Conway(T13, Q8) → poly166, U13, V13, a52, b52, e52, f52, g52, h52
  167. Dardo4Conway(W12, U4) → poly167, W13, Z13, i52, j52, k52, l52, p52, q52
  168. Cometa1Conway(O7, N7) → poly168, A14, B14, m52, n52, r52, s52, t52, c53
  169. Cometa3Conway(U12, E8) → poly169, C14, D14, a53, b53, d53, e53, f53, g53
  170. Dardo1Conway(D14, A14) → poly170, E14, F14, h53, i53, j53, k53, p53, q53
  171. Dardo1Conway(N7, M7) → poly171, G14, H14, l53, m53, n53, r53, s53, t53
  172. Dardo1Conway(L7, K7) → poly172, I14, J14, a54, b54, c54, d54, e54, f54
  173. Dardo1Conway(M4, L4) → poly173, K14, L14, g54, h54, i54, j54, k54, p54
  174. Dardo1Conway(S6, R6) → poly174, M14, N14, l54, m54, n54, q54, r54, s54
  175. Dardo1Conway(U6, T6) → poly175, O14, P14, t54, a55, b55, c55, d55, e55
  176. Dardo1Conway(J8, W8) → poly176, Q14, R14, f55, g55, h55, i55, k55, p55
  177. Dardo1Conway(B9, A9) → poly177, S14, T14, j55, l55, m55, n55, q55, r55
  178. Dardo1Conway(K4, J4) → poly178, U14, V14, s55, t55, a56, b56, c56, d56
  179. Dardo4Conway(U14, I4) → poly188, Q15, R15, a59, b59, d59, e59, f59, g59
  180. Dardo3Conway(M4, K14) → poly190, U15, V15, l59, m59, n59, r59, s59, t59
  181. Cometa3Conway(H4, S6) → poly192, A16, B16, g60, h60, i60, j60, k60, p60
  182. Cometa3Conway(Q15, S6) → poly193, C16, D16, l60, m60, n60, q60, r60, s60
  183. Cometa4Conway(J7, F4) → poly199, O16, P16, i62, j62, l62, m62, q62, r62
  184. Cometa4Conway(J7, V15) → poly200, Q16, R16, n62, s62, t62, a63, c63, d63
  185. Cometa3Conway(V6, J8) → poly206, E17, F17, m64, n64, r64, s64, t64, c65
  186. Cometa3Conway(E17, J8) → poly207, G17, H17, a65, b65, d65, e65, f65, g65
  187. Dardo1Conway(W6, E17) → poly208, I17, J17, h65, i65, j65, k65, p65, q65
  188. Cometa1Conway(R4, I5) → poly209, K17, L17, l65, m65, n65, r65, s65, t65
  189. Cometa4Conway(D9, K17) → poly210, M17, N17, a66, b66, c66, d66, e66, f66
  190. Dardo3Conway(I5, I13) → poly211, O17, P17, g66, h66, i66, j66, k66, p66
  191. Dardo4Conway(E14, D14) → poly225, S18, T18, m70, n70, r70, s70, t70, c71

AI Drawing Prompt Reference

Create a non-periodic 2D tiling made from repeating polygon units. Start with two reference points to generate an initial polygon and establish several key vertex points. Then repeatedly add new polygons by applying two distinct transformation steps to pairs of adjacent existing points/edges, so each new polygon shares a boundary with previously created ones. Continue extending the construction by connecting newly created endpoints back into the existing chain, producing many polygons whose vertices lie at dynamically determined coordinate positions. The final result should resemble a Penrose-style geometric mosaic: a dense arrangement of connected polygons with shared edges/vertices forming a non-periodic pattern.
The prompt above was compiled from the original drawing commands and is for reference only.