Parametric Snowflake – A Geometric Study of Piecewise Tangent Functions
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① This example demonstrates how to create a symmetric snowflake-shaped curve using parametric equations, blending mathematical functions with natural aesthetics. ② Core mathematical concepts include parametric curves, piecewise functions, the tangent function, and the median value theorem, highlighting the interplay between function graphs and geometric construction. ③ Construction highlights: using GeoGebra's CurveCartesian command with parameter 'a' to control scaling and rotation, while piecewise-defined tangent functions generate periodic petal-like undulations that form the complete snowflake outline. ④ Suitable for high school to undergraduate mathematics, this example is ideal for teaching parametric equations, function periodicity, piecewise function graphing, and as a creative bridge between mathematics and art, inspiring students to explore the visual potential of functions.
GeoGebra Commands & Parameters
CurveCartesian((4 / 7 * sin(20 / 11 - (318 * t))) + (3 / 13 * sin(19 / 11 - (317 * t))) + (3 / 5 * sin(21 / 16 - (316 * t))) + (1 / 6 * sin(17 / 5 - (315 * t))) + (2 / 9 * sin(20 / 19 - (314 * t))) + (5 / 9 * sin(35 / 9 - (313 * t))) + (7 / 12 * sin(9 / 8 - (310 * t))) + (5 / 16 * sin(33 / 8 - (309 * t))) + (5 / 11 * sin(31 / 11 - (308 * t))) + (4 / 7 * sin(3 / 8 - (307 * t))) + (4 / 11 * sin(9 / 8 - (306 * t))) + (7 / 8 * sin(21 / 11 - (305 * t))) + (2 / 3 * sin(55 / 13 - (304 * t))) + (5 / 9 * sin(17 / 7 - (303 * t))) + (3 / 10 * sin(3 / 13 - (302 * t))) + (4 / 11 * sin(60 / 17 - (301 * t))) + (6 / 11 * sin(48 / 11 - (300 * t))) + (9 / 19 * sin(1 / 6 - (299 * t))) + (4 / 5 * sin(19 / 11 - (298 * t))) + (7 / 13 * sin(25 / 8 - (297 * t))) + (7 / 11 * sin(19 / 7 - (296 * t))) + (1 / 2 * sin(1 - (295 * t))) + (4 / 9 * sin(24 / 11 - (294 * t))) + (1 / 3 * sin(7 / 2 - (291 * t))) + (6 / 17 * sin(15 / 13 - (290 * t))) + (11 / 17 * sin(32 / 7 - (288 * t))) + (3 / 8 * sin(33 / 8 - (287 * t))) + (4 / 7 * sin(15 / 7 - (286 * t))) + (4 / 5 * sin(48 / 11 - (284 * t))) + (6 / 7 * sin(10 / 7 - (283 * t))) + (6 / 7 * sin(20 / 11 - (282 * t))) + (3 / 8 * sin(11 / 7 - (281 * t))) + (5 / 7 * sin(23 / 6 - (280 * t))) + (1 / 21 * sin(19 / 12 - (279 * t))) + (4 / 9 * sin(1 / 5 - (278 * t))) + (5 / 8 * sin(5 / 9 - (276 * t))) + (9 / 10 * sin(2 / 3 - (274 * t))) + (5 / 8 * sin(5 / 11 - (273 * t))) + (1 / 6 * sin(9 / 2 - (272 * t))) + (12 / 25 * sin(29 / 12 - (271 * t))) + (7 / 13 * sin(59 / 15 - (270 * t))) + (5 / 7 * sin(23 / 9 - (269 * t))) + (3 / 4 * sin(9 / 2 - (268 * t))) + (5 / 11 * sin(37 / 9 - (267 * t))) + (10 / 11 * sin(11 / 7 - (266 * t))) + (1 / 3 * sin(3 / 7 - (264 * t))) + (7 / 9 * sin(33 / 17 - (262 * t))) + (5 / 8 * sin(9 / 8 - (261 * t))) + (5 / 8 * sin(38 / 13 - (260 * t))) + (11 / 21 * sin(36 / 13 - (259 * t))) + (3 / 11 * sin(1 / 29 - (258 * t))) + (8 / 15 * sin(31 / 8 - (257 * t))) + (2 / 5 * sin(3 / 13 - (256 * t))) + (1 / 2 * sin(47 / 10 - (255 * t))) + (1 / 10 * sin(33 / 10 - (254 * t))) + (2 / 5 * sin(1 / 2 - (253 * t))) + (4 / 7 * sin(33 / 7 - (252 * t))) + (6 / 17 * sin(3 / 8 - (250 * t))) + (5 / 7 * sin(25 / 9 - (249 * t))) + (7 / 9 * sin(35 / 8 - (248 * t))) + (2 / 7 * sin(81 / 20 - (247 * t))) + (5 / 8 * sin(25 / 6 - (244 * t))) + (5 / 16 * sin(11 / 21 - (243 * t))) + (11 / 13 * sin(167 / 42 - (242 * t))) + (11 / 15 * sin(18 / 5 - (241 * t))) + (13 / 14 * sin(37 / 11 - (240 * t))) + (1 / 4 * sin(20 / 9 - (239 * t))) + (9 / 14 * sin(52 / 15 - (238 * t))) + (9 / 14 * sin(17 / 14 - (237 * t))) + (6 / 13 * sin(69 / 17 - (236 * t))) + (5 / 8 * sin(74 / 21 - (235 * t))) + (7 / 15 * sin(76 / 25 - (234 * t))) + (10 / 11 * sin(15 / 8 - (232 * t))) + (5 / 11 * sin(5 / 9 - (230 * t))) + (1 / 8 * sin(8 / 3 - (229 * t))) + (5 / 9 * sin(2 / 7 - (227 * t))) + (4 / 13 * sin(32 / 9 - (226 * t))) + (2 / 3 * sin(45 / 11 - (225 * t))) + (1 / 30 * sin(53 / 15 - (223 * t))) + (7 / 11 * sin(4 / 11 - (222 * t))) + (10 / 19 * sin(31 / 13 - (221 * t))) + sin(13 / 7 - (219 * t)) + (9 / 14 * sin(33 / 7 - (216 * t))) + (2 / 3 * sin(19 / 9 - (215 * t))) + (3 / 5 * sin(27 / 11 - (214 * t))) + (9 / 11 * sin(43 / 10 - (210 * t))) + (5 / 7 * sin(13 / 8 - (209 * t))) + (5 / 9 * sin(21 / 5 - (208 * t))) + (2 / 7 * sin(14 / 9 - (206 * t))) + (9 / 8 * sin(23 / 7 - (205 * t))) + (18 / 13 * sin(11 / 9 - (203 * t))) + (7 / 4 * sin(47 / 12 - (201 * t))) + (10 / 7 * sin(8 / 9 - (200 * t))) + (7 / 10 * sin(6 / 11 - (199 * t))) + (5 / 3 * sin(7 / 6 - (198 * t))) + (19 / 11 * sin(11 / 6 - (196 * t))) + (15 / 8 * sin(9 / 8 - (195 * t))) + (8 / 17 * sin(9 / 7 - (192 * t))) + (8 / 3 * sin(39 / 10 - (191 * t))) + (23 / 10 * sin(2 / 7 - (188 * t))) + (3 / 4 * sin(3 / 5 - (187 * t))) + (7 / 12 * sin(50 / 11 - (185 * t))) + (57 / 29 * sin(4 - (184 * t))) + (9 / 8 * sin(6 / 7 - (183 * t))) + (9 / 7 * sin(15 / 13 - (182 * t))) + (5 / 13 * sin(16 / 7 - (181 * t))) + (18 / 7 * sin(5 / 14 - (180 * t))) + (17 / 9 * sin(35 / 12 - (179 * t))) + (5 / 4 * sin(5 / 7 - (178 * t))) + (22 / 23 * sin(3 / 4 - (176 * t))) + (3 / 8 * sin(48 / 13 - (175 * t))) + (15 / 11 * sin(13 / 11 - (174 * t))) + (25 / 17 * sin(23 / 5 - (173 * t))) + (18 / 11 * sin(19 / 8 - (172 * t))) + (11 / 16 * sin(5 / 3 - (170 * t))) + (39 / 38 * sin(15 / 7 - (169 * t))) + (7 / 6 * sin(36 / 11 - (166 * t))) + (15 / 11 * sin(11 / 6 - (163 * t))) + (17 / 13 * sin(3 - (162 * t))) + (11 / 9 * sin(20 / 7 - (161 * t))) + (9 / 7 * sin(35 / 9 - (160 * t))) + (7 / 6 * sin(3 / 2 - (159 * t))) + (8 / 7 * sin(9 / 10 - (158 * t))) + (12 / 25 * sin(13 / 5 - (156 * t))) + (6 / 13 * sin(25 / 13 - (154 * t))) + (9 / 13 * sin(7 / 8 - (152 * t))) + (23 / 10 * sin(33 / 14 - (151 * t))) + (8 / 11 * sin(36 / 11 - (150 * t))) + (15 / 7 * sin(26 / 7 - (149 * t))) + (6 / 5 * sin(53 / 12 - (148 * t))) + (14 / 11 * sin(3 / 2 - (147 * t))) + (9 / 8 * sin(4 / 3 - (146 * t))) + (5 / 8 * sin(18 / 13 - (145 * t))) + (15 / 7 * sin(3 / 8 - (143 * t))) + (5 / 8 * sin(5 / 6 - (142 * t))) + (6 / 7 * sin(35 / 9 - (139 * t))) + (16 / 13 * sin(1 / 2 - (138 * t))) + (9 / 4 * sin(7 / 2 - (137 * t))) + (20 / 9 * sin(15 / 8 - (135 * t))) + (11 / 8 * sin(9 / 4 - (134 * t))) + sin(19 / 10 - (133 * t)) + (22 / 7 * sin(48 / 11 - (132 * t))) + (23 / 14 * sin(1 - (131 * t))) + (19 / 9 * sin(27 / 8 - (130 * t))) + (19 / 5 * sin(20 / 7 - (129 * t))) + (18 / 5 * sin(76 / 25 - (128 * t))) + (27 / 8 * sin(4 / 5 - (126 * t))) + (37 / 8 * sin(3 / 8 - (125 * t))) + (62 / 11 * sin(11 / 3 - (124 * t))) + (49 / 11 * sin(7 / 6 - (123 * t))) + (21 / 22 * sin(23 / 12 - (122 * t))) + (223 / 74 * sin(11 / 3 - (121 * t))) + (11 / 5 * sin(19 / 5 - (120 * t))) + (13 / 4 * sin(33 / 13 - (119 * t))) + (27 / 8 * sin(22 / 5 - (117 * t))) + (24 / 7 * sin(13 / 7 - (114 * t))) + (69 / 17 * sin(18 / 17 - (113 * t))) + (10 / 9 * sin(2 / 7 - (112 * t))) + (133 / 66 * sin(12 / 7 - (111 * t))) + (2 / 5 * sin(47 / 24 - (110 * t))) + (13 / 5 * sin(11 / 6 - (108 * t))) + (16 / 7 * sin(39 / 11 - (105 * t))) + (11 / 5 * sin(25 / 9 - (104 * t))) + (151 / 50 * sin(19 / 7 - (103 * t))) + (19 / 7 * sin(12 / 5 - (101 * t))) + (26 / 7 * sin(101 / 25 - (99 * t))) + (43 / 21 * sin(41 / 14 - (98 * t))) + (13 / 3 * sin(31 / 9 - (97 * t))) + (10 / 13 * sin(1 - (95 * t))) + (17 / 7 * sin(39 / 10 - (93 * t))) + (145 / 48 * sin(3 - (92 * t))) + (37 / 6 * sin(47 / 13 - (91 * t))) + (5 / 6 * sin(36 / 13 - (89 * t))) + (9 / 4 * sin(3 / 7 - (87 * t))) + (48 / 13 * sin(26 / 17 - (86 * t))) + (7 / 3 * sin(28 / 19 - (82 * t))) + (31 / 6 * sin(8 / 7 - (81 * t))) + (36 / 7 * sin(12 / 7 - (80 * t))) + (38 / 9 * sin(25 / 9 - (79 * t))) + (17 / 2 * sin(37 / 14 - (76 * t))) + (16 / 3 * sin(19 / 20 - (75 * t))) + (81 / 16 * sin(4 / 5 - (74 * t))) + (67 / 10 * sin(19 / 15 - (73 * t))) + (40 / 11 * sin(32 / 11 - (72 * t))) + (71 / 13 * sin(21 / 20 - (71 * t))) + (68 / 15 * sin(46 / 15 - (70 * t))) + (52 / 15 * sin(27 / 10 - (69 * t))) + (57 / 14 * sin(7 / 8 - (67 * t))) + (7 / 4 * sin(42 / 13 - (66 * t))) + (39 / 11 * sin(43 / 21 - (65 * t))) + (30 / 11 * sin(33 / 8 - (64 * t))) + (7 / 5 * sin(20 / 7 - (63 * t))) + (4 / 7 * sin(13 / 14 - (62 * t))) + (39 / 10 * sin(16 / 9 - (61 * t))) + (7 / 6 * sin(137 / 34 - (59 * t))) + (16 / 13 * sin(107 / 27 - (58 * t))) + (26 / 27 * sin(17 / 5 - (57 * t))) + (4 / 3 * sin(9 / 14 - (56 * t))) + (46 / 11 * sin(5 / 3 - (55 * t))) + (11 / 6 * sin(13 / 4 - (54 * t))) + (19 / 4 * sin(17 / 5 - (53 * t))) + (19 / 7 * sin(43 / 11 - (52 * t))) + (25 / 12 * sin(30 / 7 - (51 * t))) + (15 / 7 * sin(5 / 11 - (50 * t))) + (53 / 5 * sin(21 / 13 - (49 * t))) + (62 / 13 * sin(67 / 15 - (48 * t))) + (122 / 9 * sin(48 / 13 - (47 * t))) + (20 / 13 * sin(1 - (46 * t))) + (7 / 6 * sin(32 / 7 - (43 * t))) + (12 / 7 * sin(13 / 25 - (42 * t))) + (11 / 17 * sin(9 / 10 - (40 * t))) + (11 / 9 * sin(2 - (39 * t))) + (4 / 3 * sin(19 / 7 - (38 * t))) + (12 / 5 * sin(47 / 11 - (37 * t))) + (10 / 7 * sin(12 / 7 - (36 * t))) + (108 / 17 * sin(3 / 4 - (35 * t))) + (25 / 9 * sin(19 / 5 - (34 * t))) + (7 / 13 * sin(22 / 5 - (33 * t))) + (9 / 4 * sin(13 / 11 - (32 * t))) + (181 / 15 * sin(25 / 11 - (31 * t))) + (202 / 11 * sin(57 / 13 - (29 * t))) + (2 / 11 * sin(26 / 7 - (28 * t))) + (129 / 13 * sin(38 / 15 - (25 * t))) + (13 / 6 * sin(1 / 8 - (24 * t))) + (77 / 13 * sin(11 / 8 - (23 * t))) + (19 / 6 * sin(15 / 7 - (22 * t))) + (18 / 7 * sin(29 / 10 - (21 * t))) + (9 * sin(13 / 5 - (18 * t))) + (342 / 7 * sin(11 / 6 - (17 * t))) + (3 / 5 * sin(49 / 11 - (15 * t))) + (38 / 3 * sin(19 / 7 - (14 * t))) + (994 / 9 * sin(25 / 8 - (13 * t))) + (22 / 9 * sin(49 / 12 - (10 * t))) + (97 / 9 * sin(1 / 14 - (8 * t))) + (559 / 7 * sin(47 / 14 - (7 * t))) + (19 / 13 * sin(5 / 6 - (6 * t))) + (3 * sin(57 / 17 - (4 * t))) + (28 / 5 * sin(1 - (3 * t))) + (10 / 3 * sin(22 / 7 - (2 * t))) + (1507 / 3 * sin(29 / 8 - t)) - (1407 / 13 * sin((5 * t) + 8 / 11)) - (15 / 2 * sin((9 * t) + 2 / 5)) - (1193 / 9 * sin((11 * t) + 28 / 27)) - (209 / 15 * sin((12 * t) + 2 / 5)) - (116 / 15 * sin((16 * t) + 40 / 39)) - (1105 / 33 * sin((19 * t) + 1 / 3)) - (45 / 13 * sin((20 * t) + 7 / 6)) - (91 / 46 * sin((26 * t) + 4 / 7)) - (43 / 16 * sin((27 * t) + 12 / 11)) - (46 / 13 * sin((30 * t) + 14 / 9)) - (29 / 10 * sin((41 * t) + 3 / 14)) - (31 / 11 * sin((44 * t) + 15 / 14)) - (22 / 7 * sin((45 * t) + 10 / 7)) - (7 / 8 * sin((60 * t) + 22 / 15)) - (54 / 53 * sin((68 * t) + 5 / 4)) - (214 / 15 * sin((77 * t) + 5 / 9)) - (54 / 11 * sin((78 * t) + 1 / 13)) - (47 / 6 * sin((83 * t) + 5 / 11)) - (1 / 2 * sin((84 * t) + 8 / 7)) - (2 / 3 * sin((85 * t) + 4 / 9)) - (7 / 3 * sin((88 * t) + 7 / 6)) - (15 / 4 * sin((90 * t) + 1 / 6)) - (35 / 6 * sin((94 * t) + 17 / 18)) - (77 / 26 * sin((96 * t) + 2 / 7)) - (64 / 11 * sin((100 * t) + 34 / 23)) - (13 / 6 * sin((102 * t) + 14 / 11)) - (19 / 7 * sin((106 * t) + 5 / 6)) - (13 / 6 * sin((107 * t) + 10 / 11)) - (42 / 13 * sin((109 * t) + 8 / 7)) - (69 / 35 * sin((115 * t) + 10 / 21)) - (12 / 7 * sin((116 * t) + 17 / 16)) - (8 / 3 * sin((118 * t) + 5 / 9)) - (1 / 6 * sin((127 * t) + 17 / 12)) - (13 / 7 * sin((136 * t) + 8 / 7)) - (7 / 10 * sin((140 * t) + 7 / 5)) - (15 / 7 * sin((141 * t) + 19 / 14)) - (6 / 11 * sin((144 * t) + 5 / 16)) - (3 / 2 * sin((153 * t) + 9 / 14)) - (6 / 5 * sin((155 * t) + 3 / 10)) - (3 / 8 * sin((157 * t) + 10 / 11)) - (20 / 11 * sin((164 * t) + 19 / 14)) - (7 / 5 * sin((165 * t) + 7 / 6)) - (8 / 13 * sin((167 * t) + 20 / 13)) - (7 / 8 * sin((168 * t) + 3 / 7)) - (5 / 14 * sin((171 * t) + 16 / 13)) - (22 / 7 * sin((177 * t) + 3 / 13)) - (23 / 8 * sin((186 * t) + 7 / 8)) - (13 / 7 * sin((189 * t) + 11 / 9)) - (9 / 5 * sin((190 * t) + 32 / 21)) - (27 / 28 * sin((193 * t) + 1)) - (5 / 12 * sin((194 * t) + 1 / 2)) - (44 / 43 * sin((197 * t) + 6 / 5)) - (5 / 11 * sin((202 * t) + 1 / 5)) - (8 / 7 * sin((204 * t) + 1 / 23)) - (16 / 15 * sin((207 * t) + 7 / 10)) - (1 / 2 * sin((211 * t) + 2 / 5)) - (5 / 8 * sin((212 * t) + 3 / 5)) - (10 / 13 * sin((213 * t) + 6 / 5)) - (21 / 16 * sin((217 * t) + 4 / 3)) - (11 / 5 * sin((218 * t) + 24 / 25)) - (2 / 3 * sin((220 * t) + 5 / 9)) - (13 / 10 * sin((224 * t) + 7 / 8)) - (17 / 8 * sin((228 * t) + 1 / 9)) - (3 / 7 * sin((231 * t) + 14 / 9)) - (5 / 12 * sin((233 * t) + 9 / 11)) - (3 / 5 * sin((245 * t) + 4 / 7)) - (2 / 3 * sin((246 * t) + 15 / 11)) - (3 / 8 * sin((251 * t) + 4 / 7)) - (2 / 9 * sin((263 * t) + 19 / 20)) - (1 / 2 * sin((265 * t) + 13 / 11)) - (3 / 8 * sin((275 * t) + 3 / 2)) - (17 / 35 * sin((277 * t) + 9 / 13)) - (3 / 7 * sin((285 * t) + 3 / 11)) - (9 / 10 * sin((289 * t) + 25 / 19)) - (4 / 9 * sin((292 * t) + 20 / 13)) - (12 / 25 * sin((293 * t) + 5 / 4)) - (3 / 5 * sin((311 * t) + 9 / 8)) - (33 / 32 * sin((312 * t) + 1 / 2)), (3 / 7 * sin(24 / 11 - (318 * t))) + (5 / 12 * sin(3 - (317 * t))) + (5 / 14 * sin(21 / 16 - (316 * t))) + (9 / 19 * sin(31 / 9 - (315 * t))) + (2 / 9 * sin(13 / 6 - (314 * t))) + (3 / 5 * sin(9 / 7 - (312 * t))) + (2 / 5 * sin(49 / 12 - (311 * t))) + (1 / 13 * sin(30 / 7 - (310 * t))) + (4 / 13 * sin(19 / 12 - (309 * t))) + (1 / 3 * sin(32 / 7 - (307 * t))) + (5 / 8 * sin(22 / 5 - (306 * t))) + (4 / 11 * sin(25 / 11 - (305 * t))) + (8 / 15 * sin(9 / 8 - (304 * t))) + (1 / 8 * sin(35 / 9 - (303 * t))) + (3 / 5 * sin(51 / 25 - (302 * t))) + (2 / 5 * sin(9 / 8 - (301 * t))) + (4 / 7 * sin(2 / 7 - (300 * t))) + (2 / 7 * sin(50 / 11 - (299 * t))) + (3 / 13 * sin(35 / 8 - (297 * t))) + (5 / 14 * sin(14 / 5 - (295 * t))) + (8 / 13 * sin(47 / 14 - (294 * t))) + (2 / 9 * sin(25 / 8 - (293 * t))) + (8 / 17 * sin(136 / 45 - (291 * t))) + (2 / 7 * sin(17 / 7 - (290 * t))) + (3 / 5 * sin(8 / 7 - (288 * t))) + (3 / 13 * sin(19 / 8 - (286 * t))) + (6 / 11 * sin(10 / 19 - (285 * t))) + (9 / 10 * sin(121 / 40 - (283 * t))) + (8 / 5 * sin(21 / 5 - (282 * t))) + (1 / 10 * sin(87 / 25 - (281 * t))) + (7 / 13 * sin(22 / 7 - (279 * t))) + (3 / 7 * sin(8 / 5 - (278 * t))) + (4 / 5 * sin(3 / 14 - (277 * t))) + (7 / 10 * sin(19 / 13 - (276 * t))) + (1 / 5 * sin(6 / 13 - (274 * t))) + (7 / 10 * sin(20 / 9 - (273 * t))) + (1 / 3 * sin(9 / 4 - (272 * t))) + (4 / 13 * sin(47 / 11 - (271 * t))) + (18 / 17 * sin(22 / 7 - (269 * t))) + (1 / 7 * sin(31 / 9 - (268 * t))) + (7 / 10 * sin(43 / 17 - (267 * t))) + (8 / 11 * sin(24 / 7 - (266 * t))) + (5 / 8 * sin(13 / 6 - (264 * t))) + (9 / 10 * sin(17 / 13 - (262 * t))) + (4 / 11 * sin(31 / 8 - (261 * t))) + (1 / 5 * sin(66 / 19 - (260 * t))) + (1 / 10 * sin(23 / 5 - (259 * t))) + (3 / 10 * sin(66 / 19 - (255 * t))) + (1 / 8 * sin(6 / 7 - (253 * t))) + (9 / 13 * sin(16 / 5 - (252 * t))) + (3 / 7 * sin(8 / 9 - (251 * t))) + (4 / 11 * sin(30 / 13 - (250 * t))) + (7 / 11 * sin(66 / 19 - (247 * t))) + (1 / 19 * sin(2 - (246 * t))) + (1 / 4 * sin(16 / 7 - (245 * t))) + (8 / 17 * sin(41 / 10 - (244 * t))) + (15 / 16 * sin(2 / 11 - (240 * t))) + (5 / 7 * sin(19 / 18 - (239 * t))) + (1 / 6 * sin(5 / 12 - (238 * t))) + (5 / 11 * sin(16 / 17 - (236 * t))) + (3 / 10 * sin(25 / 12 - (235 * t))) + (8 / 17 * sin(16 / 7 - (233 * t))) + (5 / 8 * sin(47 / 12 - (231 * t))) + (9 / 11 * sin(11 / 8 - (230 * t))) + (3 / 11 * sin(33 / 7 - (229 * t))) + (9 / 10 * sin(20 / 7 - (226 * t))) + (4 / 9 * sin(39 / 14 - (225 * t))) + (4 / 9 * sin(10 / 9 - (224 * t))) + (6 / 7 * sin(19 / 13 - (222 * t))) + (7 / 9 * sin(29 / 7 - (221 * t))) + (8 / 11 * sin(33 / 8 - (220 * t))) + (16 / 9 * sin(2 / 7 - (219 * t))) + (25 / 14 * sin(1 / 8 - (218 * t))) + (8 / 11 * sin(5 / 9 - (217 * t))) + (9 / 11 * sin(11 / 10 - (216 * t))) + (21 / 13 * sin(27 / 7 - (215 * t))) + (3 / 7 * sin(1 / 12 - (213 * t))) + (13 / 9 * sin(15 / 16 - (212 * t))) + (23 / 8 * sin(1 / 8 - (210 * t))) + sin(32 / 11 - (209 * t)) + (9 / 13 * sin(1 / 9 - (208 * t))) + (7 / 9 * sin(33 / 10 - (206 * t))) + (2 / 3 * sin(9 / 4 - (205 * t))) + (3 / 4 * sin(1 / 2 - (204 * t))) + (3 / 13 * sin(11 / 17 - (203 * t))) + (3 / 7 * sin(31 / 12 - (202 * t))) + (19 / 12 * sin(17 / 8 - (201 * t))) + (7 / 8 * sin(75 / 19 - (200 * t))) + (6 / 5 * sin(21 / 10 - (198 * t))) + (3 / 2 * sin(7 / 5 - (194 * t))) + (28 / 27 * sin(3 / 2 - (193 * t))) + (4 / 9 * sin(16 / 5 - (192 * t))) + (22 / 13 * sin(13 / 6 - (189 * t))) + (18 / 11 * sin(19 / 10 - (188 * t))) + sin(7 / 6 - (187 * t)) + (16 / 7 * sin(13 / 11 - (186 * t))) + (9 / 5 * sin(11 / 9 - (184 * t))) + (16 / 11 * sin(2 / 5 - (183 * t))) + (10 / 13 * sin(10 / 3 - (182 * t))) + (9 / 7 * sin(38 / 9 - (181 * t))) + (45 / 13 * sin(8 / 9 - (180 * t))) + (7 / 9 * sin(35 / 8 - (179 * t))) + (2 / 3 * sin(35 / 8 - (176 * t))) + (10 / 7 * sin(6 / 19 - (175 * t))) + (40 / 13 * sin(15 / 7 - (174 * t))) + (20 / 13 * sin(1 / 2 - (173 * t))) + (3 / 11 * sin(20 / 7 - (171 * t))) + (17 / 16 * sin(50 / 11 - (169 * t))) + (2 / 9 * sin(1 / 31 - (168 * t))) + (4 / 9 * sin(7 / 2 - (165 * t))) + (1 / 12 * sin(26 / 17 - (164 * t))) + (21 / 22 * sin(27 / 26 - (163 * t))) + (13 / 12 * sin(17 / 8 - (162 * t))) + (19 / 14 * sin(39 / 10 - (160 * t))) + (18 / 11 * sin(5 / 7 - (159 * t))) + (3 / 5 * sin(15 / 14 - (158 * t))) + (11 / 9 * sin(35 / 8 - (157 * t))) + (5 / 8 * sin(30 / 7 - (156 * t))) + (3 / 2 * sin(28 / 11 - (155 * t))) + (4 / 5 * sin(5 / 11 - (151 * t))) + (25 / 19 * sin(11 / 10 - (150 * t))) + (10 / 11 * sin(11 / 14 - (148 * t))) + (13 / 9 * sin(7 / 4 - (147 * t))) + (7 / 13 * sin(19 / 6 - (146 * t))) + (1 / 5 * sin(37 / 14 - (145 * t))) + (11 / 8 * sin(42 / 13 - (144 * t))) + (20 / 11 * sin(32 / 9 - (143 * t))) + (2 / 3 * sin(22 / 5 - (141 * t))) + (10 / 11 * sin(9 / 7 - (140 * t))) + (8 / 7 * sin(23 / 9 - (138 * t))) + (5 / 2 * sin(9 / 19 - (137 * t))) + (7 / 5 * sin(193 / 48 - (136 * t))) + (5 / 8 * sin(67 / 66 - (135 * t))) + (8 / 7 * sin(7 / 15 - (134 * t))) + (13 / 6 * sin(13 / 7 - (133 * t))) + (19 / 7 * sin(16 / 5 - (132 * t))) + (16 / 7 * sin(39 / 11 - (131 * t))) + (28 / 17 * sin(69 / 35 - (130 * t))) + (84 / 17 * sin(7 / 8 - (129 * t))) + (114 / 23 * sin(10 / 9 - (128 * t))) + (29 / 11 * sin(1 / 7 - (127 * t))) + (63 / 10 * sin(65 / 32 - (124 * t))) + (74 / 17 * sin(37 / 16 - (121 * t))) + (31 / 16 * sin(35 / 11 - (120 * t))) + (19 / 5 * sin(23 / 12 - (119 * t))) + (82 / 27 * sin(27 / 7 - (118 * t))) + (49 / 11 * sin(8 / 3 - (117 * t))) + (29 / 14 * sin(63 / 16 - (116 * t))) + (9 / 13 * sin(35 / 8 - (114 * t))) + (29 / 19 * sin(5 / 4 - (113 * t))) + (13 / 7 * sin(20 / 7 - (112 * t))) + (9 / 7 * sin(11 / 23 - (111 * t))) + (19 / 8 * sin(27 / 26 - (110 * t))) + sin(4 / 7 - (109 * t)) + (119 / 40 * sin(22 / 5 - (108 * t))) + (7 / 5 * sin(47 / 46 - (107 * t))) + (5 / 3 * sin(1 / 6 - (106 * t))) + (2 * sin(14 / 5 - (105 * t))) + (7 / 3 * sin(10 / 3 - (104 * t))) + (3 / 2 * sin(15 / 4 - (103 * t))) + (19 / 11 * sin(3 / 4 - (102 * t))) + (74 / 17 * sin(13 / 10 - (99 * t))) + (98 / 33 * sin(26 / 11 - (98 * t))) + (36 / 11 * sin(13 / 3 - (97 * t))) + (43 / 12 * sin(26 / 25 - (96 * t))) + (13 / 2 * sin(3 / 13 - (95 * t))) + (6 / 7 * sin(24 / 7 - (94 * t))) + (16 / 5 * sin(6 / 5 - (93 * t))) + (5 / 7 * sin(9 / 14 - (92 * t))) + (55 / 12 * sin(27 / 14 - (90 * t))) + (15 / 11 * sin(14 / 3 - (88 * t))) + (7 / 3 * sin(7 / 10 - (87 * t))) + (11 / 4 * sin(2 / 9 - (86 * t))) + (13 / 4 * sin(35 / 12 - (84 * t))) + (26 / 9 * sin(38 / 9 - (83 * t))) + (7 / 2 * sin(5 / 7 - (82 * t))) + (31 / 8 * sin(27 / 8 - (78 * t))) + (91 / 6 * sin(35 / 8 - (77 * t))) + (37 / 5 * sin(7 / 10 - (76 * t))) + (70 / 13 * sin(17 / 11 - (73 * t))) + (76 / 25 * sin(56 / 19 - (70 * t))) + (19 / 8 * sin(17 / 8 - (68 * t))) + (59 / 13 * sin(42 / 17 - (67 * t))) + (28 / 17 * sin(49 / 13 - (64 * t))) + (9 / 7 * sin(79 / 17 - (63 * t))) + (1 / 8 * sin(7 / 11 - (62 * t))) + (39 / 8 * sin(49 / 15 - (61 * t))) + (53 / 18 * sin(33 / 8 - (59 * t))) + (9 / 7 * sin(41 / 9 - (58 * t))) + (8 / 7 * sin(65 / 14 - (57 * t))) + (10 / 11 * sin(16 / 7 - (56 * t))) + (68 / 13 * sin(42 / 13 - (55 * t))) + (21 / 10 * sin(7 / 8 - (54 * t))) + (6 / 7 * sin(41 / 14 - (53 * t))) + (31 / 11 * sin(55 / 12 - (51 * t))) + (59 / 17 * sin(27 / 7 - (50 * t))) + (124 / 9 * sin(37 / 11 - (49 * t))) + (24 / 11 * sin(3 / 5 - (48 * t))) + (65 / 6 * sin(12 / 5 - (47 * t))) + (11 / 7 * sin(49 / 11 - (45 * t))) + (13 / 25 * sin(11 / 13 - (42 * t))) + (7 / 4 * sin(5 / 8 - (40 * t))) + (43 / 42 * sin(2 / 5 - (39 * t))) + (20 / 9 * sin(4 / 7 - (38 * t))) + (19 / 8 * sin(4 / 11 - (37 * t))) + (5 / 4 * sin(15 / 4 - (36 * t))) + (1 / 5 * sin(11 / 13 - (34 * t))) + (12 / 7 * sin(23 / 5 - (32 * t))) + (409 / 34 * sin(39 / 10 - (31 * t))) + (10 / 7 * sin(5 / 2 - (30 * t))) + (180 / 11 * sin(3 - (29 * t))) + (23 / 8 * sin(53 / 12 - (26 * t))) + (71 / 8 * sin(56 / 13 - (25 * t))) + (12 / 5 * sin(10 / 21 - (24 * t))) + (10 / 3 * sin(34 / 9 - (22 * t))) + (27 / 16 * sin(12 / 11 - (21 * t))) + (49 / 6 * sin(13 / 7 - (20 * t))) + (69 / 2 * sin(19 / 14 - (19 * t))) + (475 / 9 * sin(3 / 10 - (17 * t))) + (68 / 13 * sin(57 / 28 - (16 * t))) + (40 / 17 * sin(1 / 6 - (15 * t))) + (77 / 13 * sin(29 / 11 - (12 * t))) + (4954 / 39 * sin(15 / 4 - (11 * t))) + (1075 / 11 * sin(4 - (5 * t))) + (191 / 24 * sin(5 / 4 - (4 * t))) + (84 / 17 * sin(2 / 7 - (3 * t))) - (12 / 5 * sin((74 * t))) - (4 / 5 * sin((166 * t))) - (1523 / 3 * sin(t + 12 / 11)) - (25 / 3 * sin((2 * t) + 17 / 18)) - (13 / 8 * sin((6 * t) + 1 / 9)) - (5333 / 62 * sin((7 * t) + 9 / 7)) - (56 / 9 * sin((8 * t) + 5 / 12)) - (65 / 8 * sin((9 * t) + 2 / 5)) - (106 / 9 * sin((10 * t) + 1 / 8)) - (1006 / 9 * sin((13 * t) + 11 / 7)) - (67 / 8 * sin((14 * t) + 6 / 5)) - (25 / 8 * sin((18 * t) + 15 / 11)) - (40 / 11 * sin((23 * t) + 1 / 16)) - (4 / 7 * sin((27 * t) + 6 / 5)) - (41 / 8 * sin((28 * t) + 7 / 12)) - (8 / 5 * sin((33 * t) + 5 / 6)) - (137 / 17 * sin((35 * t) + 4 / 5)) - (29 / 12 * sin((41 * t) + 22 / 15)) - (25 / 9 * sin((43 * t) + 6 / 7)) - (12 / 25 * sin((44 * t) + 16 / 11)) - (31 / 6 * sin((46 * t) + 4 / 3)) - (19 / 5 * sin((52 * t) + 16 / 13)) - (19 / 11 * sin((60 * t) + 8 / 17)) - (16 / 7 * sin((65 * t) + 6 / 13)) - (25 / 12 * sin((66 * t) + 11 / 13)) - (8 / 9 * sin((69 * t) + 4 / 11)) - (25 / 7 * sin((71 * t) + 7 / 5)) - (11 / 10 * sin((72 * t) + 3 / 2)) - (14 / 5 * sin((75 * t) + 7 / 9)) - (107 / 14 * sin((79 * t) + 3 / 4)) - (67 / 8 * sin((80 * t) + 2 / 11)) - (161 / 27 * sin((81 * t) + 5 / 11)) - (55 / 18 * sin((85 * t) + 3 / 7)) - (161 / 40 * sin((89 * t) + 1 / 21)) - (32 / 7 * sin((91 * t) + 38 / 25)) - sin((100 * t) + 19 / 20) - (27 / 5 * sin((101 * t) + 2 / 13)) - (26 / 9 * sin((115 * t) + 1 / 44)) - (17 / 11 * sin((122 * t) + 1 / 16)) - (87 / 22 * sin((123 * t) + 2 / 3)) - (37 / 8 * sin((125 * t) + 9 / 11)) - (10 / 7 * sin((126 * t) + 8 / 7)) - (7 / 8 * sin((139 * t) + 3 / 5)) - (3 / 7 * sin((142 * t) + 5 / 6)) - (71 / 36 * sin((149 * t) + 5 / 16)) - (7 / 6 * sin((152 * t) + 1 / 9)) - (63 / 25 * sin((153 * t) + 29 / 19)) - (27 / 20 * sin((154 * t) + 8 / 15)) - (8 / 15 * sin((161 * t) + 12 / 13)) - (5 / 3 * sin((167 * t) + 13 / 10)) - (17 / 25 * sin((170 * t) + 3 / 5)) - (10 / 9 * sin((172 * t) + 3 / 8)) - (5 / 7 * sin((177 * t) + 5 / 8)) - (1 / 2 * sin((178 * t) + 7 / 6)) - (34 / 13 * sin((185 * t) + 5 / 8)) - (11 / 13 * sin((190 * t) + 38 / 39)) - (25 / 19 * sin((191 * t) + 11 / 8)) - (11 / 12 * sin((195 * t) + 18 / 19)) - (51 / 26 * sin((196 * t) + 2 / 7)) - (14 / 9 * sin((197 * t) + 4 / 11)) - (19 / 12 * sin((199 * t) + 1)) - (19 / 11 * sin((207 * t) + 11 / 8)) - (6 / 11 * sin((211 * t) + 1 / 20)) - (11 / 7 * sin((214 * t) + 1 / 14)) - (7 / 13 * sin((223 * t) + 8 / 11)) - (3 / 5 * sin((227 * t) + 12 / 13)) - (4 / 5 * sin((228 * t) + 29 / 19)) - (11 / 10 * sin((232 * t) + 2 / 7)) - (1 / 6 * sin((234 * t) + 7 / 11)) - sin((237 * t) + 60 / 59) - (5 / 11 * sin((241 * t) + 7 / 8)) - (1 / 2 * sin((242 * t) + 8 / 7)) - (7 / 15 * sin((243 * t) + 15 / 16)) - (5 / 8 * sin((248 * t) + 2 / 3)) - (1 / 3 * sin((249 * t) + 4 / 11)) - (2 / 3 * sin((254 * t) + 8 / 7)) - (10 / 19 * sin((256 * t) + 14 / 11)) - (4 / 9 * sin((257 * t) + 8 / 11)) - (3 / 4 * sin((258 * t) + 3 / 7)) - sin((263 * t) + 2 / 7) - (3 / 10 * sin((265 * t) + 1 / 28)) - (1 / 2 * sin((270 * t) + 1)) - (12 / 13 * sin((275 * t) + 5 / 8)) - (1 / 4 * sin((280 * t) + 16 / 13)) - (1 / 10 * sin((284 * t) + 5 / 8)) - (13 / 25 * sin((287 * t) + 3 / 7)) - (9 / 13 * sin((289 * t) + 3 / 5)) - (22 / 23 * sin((292 * t) + 17 / 13)) - (9 / 11 * sin((296 * t) + 17 / 11)) - (3 / 7 * sin((298 * t) + 12 / 11)) - (5 / 6 * sin((308 * t) + 1 / 2)) - (7 / 15 * sin((313 * t) + 1 / 3)), t, 0, (2 * pi)) → a
AI Drawing Prompt Reference
Draw a symmetric snowflake-shaped closed curve defined by parametric equations, featuring six-fold rotational symmetry and a star-like outline. The curve is composed of numerous superimposed sine waves, creating densely serrated petal edges that radiate outward from the center, with subtle irregular undulations on each petal. The overall appearance blends natural snowflake aesthetics with mathematical fractal-like complexity. Centered at the origin, the curve spans a full period, with amplitude and frequency variations controlling petal density and shape, resulting in an intricate, artistic, and highly detailed symmetric pattern.
The prompt above was compiled from the original drawing commands and is for reference only.