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5-Point Substitution Fractal

5-Point Substitution Fractal
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2021-12-28 20:12
SequenceMatrix
This example demonstrates the generation process and self-similar structure of a 5-point substitution fractal. Key mathematical concepts include fractal geometry, matrix transformations, iterative sequences, and geometric constructions. The construction utilizes GeoGebra commands Sequence and Sort, along with objects like s1, n, L, l0, l1, l2, l3, l4, and button1. It is suitable for high school and above students to understand the construction of fractal patterns and the application of matrices in geometric transformations.

GeoGebra Commands & Parameters

  1. Sequence((Mod[u, 4], floor(u / 4)), u, 0, 15) → L, {(0, 0), (1, 0), (2, 0), (3, 0), (0, 1), (1, 1), (2, 1), (3, 1), (0, 2), (1, 2), (2, 2), (3, 2), (0, 3), (1, 3), (2, 3), (3, 3)}
  2. Sort(Sample[L, 5]) → l0
  3. Sequence(Translate[Dilate[l0, 1 / 4], Element[l0, u]], u, 1, 5) → l1, {{(0, 3.75), (0.25, 3.25), (0.5, 3), (0.75, 3.25), (0.75, 3.5)}, {(1, 1.75), (1.25, 1.25), (1.5, 1), (1.75, 1.25), (1.75, 1.5)}, {(2, 0.75), (2.25, 0.25), (2.5, 0), (2.75, 0.25), (2.75, 0.5)}, {(3, 1.75), (3.25, 1.25), (3.5, 1), (3.75, 1.25), (3.75, 1.5)}, {(3, 2.75), (3.25, 2.25), (3.5, 2), (3.75, 2.25), (3.75, 2.5)}}
  4. Sequence(Translate[Dilate[l1, 1 / 4], Element[l0, u]], u, 1, 5) → l2, {{{(0, 3.9375), (0.0625, 3.8125), (0.125, 3.75), (0.1875, 3.8125), (0.1875, 3.875)}, {(0.25, 3.4375), (0.3125, 3.3125), (0.375, 3.25), (0.4375, 3.3125), (0.4375, 3.375)}, {(0.5, 3.1875), (0.5625, 3.0625), (0.625, 3), (0.6875, 3.0625), (0.6875, 3.125)}, {(0.75, 3.4375), (0.8125, 3.3125), (0.875, 3.25), (0.9375, 3.3125), (0.9375, 3.375)}, {(0.75, 3.6875), (0.8125, 3.5625), (0.875, 3.5), (0.9375, 3.5625), (0.9375, 3.625)}}, {{(1, 1.9375), (1.0625, 1.8125), (1.125, 1.75), (1.1875, 1.8125), (1.1875, 1.875)}, {(1.25, 1.4375), (1.3125, 1.3125), (1.375, 1.25), (1.4375, 1.3125), (1.4375, 1.375)}, {(1.5, 1.1875), (1.5625, 1.0625), (1.625, 1), (1.6875, 1.0625), (1.6875, 1.125)}, {(1.75, 1.4375), (1.8125, 1.3125), (1.875, 1.25), (1.9375, 1.3125), (1.9375, 1.375)}, {(1.75, 1.6875), (1.8125, 1.5625), (1.875, 1.5), (1.9375, 1.5625), (1.9375, 1.625)}}, {{(2, 0.9375), (2.0625, 0.8125), (2.125, 0.75), (2.1875, 0.8125), (2.1875, 0.875)}, {(2.25, 0.4375), (2.3125, 0.3125), (2.375, 0.25), (2.4375, 0.3125), (2.4375, 0.375)}, {(2.5, 0.1875), (2.5625, 0.0625), (2.625, 0), (2.6875, 0.0625), (2.6875, 0.125)}, {(2.75, 0.4375), (2.8125, 0.3125), (2.875, 0.25), (2.9375, 0.3125), (2.9375, 0.375)}, {(2.75, 0.6875), (2.8125, 0.5625), (2.875, 0.5), (2.9375, 0.5625), (2.9375, 0.625)}}, {{(3, 1.9375), (3.0625, 1.8125), (3.125, 1.75), (3.1875, 1.8125), (3.1875, 1.875)}, {(3.25, 1.4375), (3.3125, 1.3125), (3.375, 1.25), (3.4375, 1.3125), (3.4375, 1.375)}, {(3.5, 1.1875), (3.5625, 1.0625), (3.625, 1), (3.6875, 1.0625), (3.6875, 1.125)}, {(3.75, 1.4375), (3.8125, 1.3125), (3.875, 1.25), (3.9375, 1.3125), (3.9375, 1.375)}, {(3.75, 1.6875), (3.8125, 1.5625), (3.875, 1.5), (3.9375, 1.5625), (3.9375, 1.625)}}, {{(3, 2.9375), (3.0625, 2.8125), (3.125, 2.75), (3.1875, 2.8125), (3.1875, 2.875)}, {(3.25, 2.4375), (3.3125, 2.3125), (3.375, 2.25), (3.4375, 2.3125), (3.4375, 2.375)}, {(3.5, 2.1875), (3.5625, 2.0625), (3.625, 2), (3.6875, 2.0625), (3.6875, 2.125)}, {(3.75, 2.4375), (3.8125, 2.3125), (3.875, 2.25), (3.9375, 2.3125), (3.9375, 2.375)}, {(3.75, 2.6875), (3.8125, 2.5625), (3.875, 2.5), (3.9375, 2.5625), (3.9375, 2.625)}}}
  5. Sequence(Translate[Dilate[l2, 1 / 4], Element[l0, u]], u, 1, 5) → l3, {{{{(0, 3.984375), (0.015625, 3.953125), (0.03125, 3.9375), (0.046875, 3.953125), (0.046875, 3.96875)}, {(0.0625, 3.859375), (0.078125, 3.828125), (0.09375, 3.8125), (0.109375, 3.828125), (0.109375, 3.84375)}, {(0.125, 3.796875), (0.140625, 3.765625), (0.15625, 3.75), (0.171875, 3.765625), (0.171875, 3.78125)}, {(0.1875, 3.859375), (0.203125, 3.828125), (0.21875, 3.8125), (0.234375, 3.828125), (0.234375, 3.84375)}, {(0.1875, 3.921875), (0.203125, 3.890625), (0.21875, 3.875), (0.234375, 3.890625), (0.234375, 3.90625)}}, {{(0.25, 3.484375), (0.265625, 3.453125), (0.28125, 3.4375), (0.296875, 3.453125), (0.296875, 3.46875)}, {(0.3125, 3.359375), (0.328125, 3.328125), (0.34375, 3.3125), (0.359375, 3.328125), (0.359375, 3.34375)}, {(0.375, 3.296875), (0.390625, 3.265625), (0.40625, 3.25), (0.421875, 3.265625), (0.421875, 3.28125)}, {(0.4375, 3.359375), (0.453125, 3.328125), (0.46875, 3.3125), (0.484375, 3.328125), (0.484375, 3.34375)}, {(0.4375, 3.421875), (0.453125, 3.390625), (0.46875, 3.375), (0.484375, 3.390625), (0.484375, 3.40625)}}, {{(0.5, 3.234375), (0.515625, 3.203125), (0.53125, 3.1875), (0.546875, 3.203125), (0.546875, 3.21875)}, {(0.5625, 3.109375), (0.578125, 3.078125), (0.59375, 3.0625), (0.609375, 3.078125), (0.609375, 3.09375)}, {(0.625, 3.046875), (0.640625, 3.015625), (0.65625, 3), (0.671875, 3.015625), (0.671875, 3.03125)}, {(0.6875, 3.109375), (0.703125, 3.078125), (0.71875, 3.0625), (0.734375, 3.078125), (0.734375, 3.09375)}, {(0.6875, 3.171875), (0.703125, 3.140625), (0.71875, 3.125), (0.734375, 3.140625), (0.734375, 3.15625)}}, {{(0.75, 3.484375), (0.765625, 3.453125), (0.78125, 3.4375), (0.796875, 3.453125), (0.796875, 3.46875)}, {(0.8125, 3.359375), (0.828125, 3.328125), (0.84375, 3.3125), (0.859375, 3.328125), (0.859375, 3.34375)}, {(0.875, 3.296875), (0.890625, 3.265625), (0.90625, 3.25), (0.921875, 3.265625), (0.921875, 3.28125)}, {(0.9375, 3.359375), (0.953125, 3.328125), (0.96875, 3.3125), (0.984375, 3.328125), (0.984375, 3.34375)}, {(0.9375, 3.421875), (0.953125, 3.390625), (0.96875, 3.375), (0.984375, 3.390625), (0.984375, 3.40625)}}, {{(0.75, 3.734375), (0.765625, 3.703125), (0.78125, 3.6875), (0.796875, 3.703125), (0.796875, 3.71875)}, {(0.8125, 3.609375), (0.828125, 3.578125), (0.84375, 3.5625), (0.859375, 3.578125), (0.859375, 3.59375)}, {(0.875, 3.546875), (0.890625, 3.515625), (0.90625, 3.5), (0.921875, 3.515625), (0.921875, 3.53125)}, {(0.9375, 3.609375), (0.953125, 3.578125), (0.96875, 3.5625), (0.984375, 3.578125), (0.984375, 3.59375)}, {(0.9375, 3.671875), (0.953125, 3.640625), (0.96875, 3.625), (0.984375, 3.640625), (0.984375, 3.65625)}}}, {{{(1, 1.984375), (1.015625, 1.953125), (1.03125, 1.9375), (1.046875, 1.953125), (1.046875, 1.96875)}, {(1.0625, 1.859375), (1.078125, 1.828125), (1.09375, 1.8125), (1.109375, 1.828125), (1.109375, 1.84375)}, {(1.125, 1.796875), (1.140625, 1.765625), (1.15625, 1.75), (1.171875, 1.765625), (1.171875, 1.78125)}, {(1.1875, 1.859375), (1.203125, 1.828125), (1.21875, 1.8125), (1.234375, 1.828125), (1.234375, 1.84375)}, {(1.1875, 1.921875), (1.203125, 1.890625), (1.21875, 1.875), (1.234375, 1.890625), (1.234375, 1.90625)}}, {{(1.25, 1.484375), (1.265625, 1.453125), (1.28125, 1.4375), (1.296875, 1.453125), (1.296875, 1.46875)}, {(1.3125, 1.359375), (1.328125, 1.328125), (1.34375, 1.3125), (1.359375, 1.328125), (1.359375, 1.34375)}, {(1.375, 1.296875), (1.390625, 1.265625), (1.40625, 1.25), (1.421875, 1.265625), (1.421875, 1.28125)}, {(1.4375, 1.359375), (1.453125, 1.328125), (1.46875, 1.3125), (1.484375, 1.328125), (1.484375, 1.34375)}, {(1.4375, 1.421875), (1.453125, 1.390625), (1.46875, 1.375), (1.484375, 1.390625), (1.484375, 1.40625)}}, {{(1.5, 1.234375), (1.515625, 1.203125), (1.53125, 1.1875), (1.546875, 1.203125), (1.546875, 1.21875)}, {(1.5625, 1.109375), (1.578125, 1.078125), (1.59375, 1.0625), (1.609375, 1.078125), (1.609375, 1.09375)}, {(1.625, 1.046875), (1.640625, 1.015625), (1.65625, 1), (1.671875, 1.015625), (1.671875, 1.03125)}, {(1.6875, 1.109375), (1.703125, 1.078125), (1.71875, 1.0625), (1.734375, 1.078125), (1.734375, 1.09375)}, {(1.6875, 1.171875), (1.703125, 1.140625), (1.71875, 1.125), (1.734375, 1.140625), (1.734375, 1.15625)}}, {{(1.75, 1.484375), (1.765625, 1.453125), (1.78125, 1.4375), (1.796875, 1.453125), (1.796875, 1.46875)}, {(1.8125, 1.359375), (1.828125, 1.328125), (1.84375, 1.3125), (1.859375, 1.328125), (1.859375, 1.34375)}, {(1.875, 1.296875), (1.890625, 1.265625), (1.90625, 1.25), (1.921875, 1.265625), (1.921875, 1.28125)}, {(1.9375, 1.359375), (1.953125, 1.328125), (1.96875, 1.3125), (1.984375, 1.328125), (1.984375, 1.34375)}, {(1.9375, 1.421875), (1.953125, 1.390625), (1.96875, 1.375), (1.984375, 1.390625), (1.984375, 1.40625)}}, {{(1.75, 1.734375), (1.765625, 1.703125), (1.78125, 1.6875), (1.796875, 1.703125), (1.796875, 1.71875)}, {(1.8125, 1.609375), (1.828125, 1.578125), (1.84375, 1.5625), (1.859375, 1.578125), (1.859375, 1.59375)}, {(1.875, 1.546875), (1.890625, 1.515625), (1.90625, 1.5), (1.921875, 1.515625), (1.921875, 1.53125)}, {(1.9375, 1.609375), (1.953125, 1.578125), (1.96875, 1.5625), (1.984375, 1.578125), (1.984375, 1.59375)}, {(1.9375, 1.671875), (1.953125, 1.640625), (1.96875, 1.625), (1.984375, 1.640625), (1.984375, 1.65625)}}}, {{{(2, 0.984375), (2.015625, 0.953125), (2.03125, 0.9375), (2.046875, 0.953125), (2.046875, 0.96875)}, {(2.0625, 0.859375), (2.078125, 0.828125), (2.09375, 0.8125), (2.109375, 0.828125), (2.109375, 0.84375)}, {(2.125, 0.796875), (2.140625, 0.765625), (2.15625, 0.75), (2.171875, 0.765625), (2.171875, 0.78125)}, {(2.1875, 0.859375), (2.203125, 0.828125), (2.21875, 0.8125), (2.234375, 0.828125), (2.234375, 0.84375)}, {(2.1875, 0.921875), (2.203125, 0.890625), (2.21875, 0.875), (2.234375, 0.890625), (2.234375, 0.90625)}}, {{(2.25, 0.484375), (2.265625, 0.453125), (2.28125, 0.4375), (2.296875, 0.453125), (2.296875, 0.46875)}, {(2.3125, 0.359375), (2.328125, 0.328125), (2.34375, 0.3125), (2.359375, 0.328125), (2.359375, 0.34375)}, {(2.375, 0.296875), (2.390625, 0.265625), (2.40625, 0.25), (2.421875, 0.265625), (2.421875, 0.28125)}, {(2.4375, 0.359375), (2.453125, 0.328125), (2.46875, 0.3125), (2.484375, 0.328125), (2.484375, 0.34375)}, {(2.4375, 0.421875), (2.453125, 0.390625), (2.46875, 0.375), (2.484375, 0.390625), (2.484375, 0.40625)}}, {{(2.5, 0.234375), (2.515625, 0.203125), (2.53125, 0.1875), (2.546875, 0.203125), (2.546875, 0.21875)}, {(2.5625, 0.109375), (2.578125, 0.078125), (2.59375, 0.0625), (2.609375, 0.078125), (2.609375, 0.09375)}, {(2.625, 0.046875), (2.640625, 0.015625), (2.65625, 0), (2.671875, 0.015625), (2.671875, 0.03125)}, {(2.6875, 0.109375), (2.703125, 0.078125), (2.71875, 0.0625), (2.734375, 0.078125), (2.734375, 0.09375)}, {(2.6875, 0.171875), (2.703125, 0.140625), (2.71875, 0.125), (2.734375, 0.140625), (2.734375, 0.15625)}}, {{(2.75, 0.484375), (2.765625, 0.453125), (2.78125, 0.4375), (2.796875, 0.453125), (2.796875, 0.46875)}, {(2.8125, 0.359375), (2.828125, 0.328125), (2.84375, 0.3125), (2.859375, 0.328125), (2.859375, 0.34375)}, {(2.875, 0.296875), (2.890625, 0.265625), (2.90625, 0.25), (2.921875, 0.265625), (2.921875, 0.28125)}, {(2.9375, 0.359375), (2.953125, 0.328125), (2.96875, 0.3125), (2.984375, 0.328125), (2.984375, 0.34375)}, {(2.9375, 0.421875), (2.953125, 0.390625), (2.96875, 0.375), (2.984375, 0.390625), (2.984375, 0.40625)}}, {{(2.75, 0.734375), (2.765625, 0.703125), (2.78125, 0.6875), (2.796875, 0.703125), (2.796875, 0.71875)}, {(2.8125, 0.609375), (2.828125, 0.578125), (2.84375, 0.5625), (2.859375, 0.578125), (2.859375, 0.59375)}, {(2.875, 0.546875), (2.890625, 0.515625), (2.90625, 0.5), (2.921875, 0.515625), (2.921875, 0.53125)}, {(2.9375, 0.609375), (2.953125, 0.578125), (2.96875, 0.5625), (2.984375, 0.578125), (2.984375, 0.59375)}, {(2.9375, 0.671875), (2.953125, 0.640625), (2.96875, 0.625), (2.984375, 0.640625), (2.984375, 0.65625)}}}, {{{(3, 1.984375), (3.015625, 1.953125), (3.03125, 1.9375), (3.046875, 1.953125), (3.046875, 1.96875)}, {(3.0625, 1.859375), (3.078125, 1.828125), (3.09375, 1.8125), (3.109375, 1.828125), (3.109375, 1.84375)}, {(3.125, 1.796875), (3.140625, 1.765625), (3.15625, 1.75), (3.171875, 1.765625), (3.171875, 1.78125)}, {(3.1875, 1.859375), (3.203125, 1.828125), (3.21875, 1.8125), (3.234375, 1.828125), (3.234375, 1.84375)}, {(3.1875, 1.921875), (3.203125, 1.890625), (3.21875, 1.875), (3.234375, 1.890625), (3.234375, 1.90625)}}, {{(3.25, 1.484375), (3.265625, 1.453125), (3.28125, 1.4375), (3.296875, 1.453125), (3.296875, 1.46875)}, {(3.3125, 1.359375), (3.328125, 1.328125), (3.34375, 1.3125), (3.359375, 1.328125), (3.359375, 1.34375)}, {(3.375, 1.296875), (3.390625, 1.265625), (3.40625, 1.25), (3.421875, 1.265625), (3.421875, 1.28125)}, {(3.4375, 1.359375), (3.453125, 1.328125), (3.46875, 1.3125), (3.484375, 1.328125), (3.484375, 1.34375)}, {(3.4375, 1.421875), (3.453125, 1.390625), (3.46875, 1.375), (3.484375, 1.390625), (3.484375, 1.40625)}}, {{(3.5, 1.234375), (3.515625, 1.203125), (3.53125, 1.1875), (3.546875, 1.203125), (3.546875, 1.21875)}, {(3.5625, 1.109375), (3.578125, 1.078125), (3.59375, 1.0625), (3.609375, 1.078125), (3.609375, 1.09375)}, {(3.625, 1.046875), (3.640625, 1.015625), (3.65625, 1), (3.671875, 1.015625), (3.671875, 1.03125)}, {(3.6875, 1.109375), (3.703125, 1.078125), (3.71875, 1.0625), (3.734375, 1.078125), (3.734375, 1.09375)}, {(3.6875, 1.171875), (3.703125, 1.140625), (3.71875, 1.125), (3.734375, 1.140625), (3.734375, 1.15625)}}, {{(3.75, 1.484375), (3.765625, 1.453125), (3.78125, 1.4375), (3.796875, 1.453125), (3.796875, 1.46875)}, {(3.8125, 1.359375), (3.828125, 1.328125), (3.84375, 1.3125), (3.859375, 1.328125), (3.859375, 1.34375)}, {(3.875, 1.296875), (3.890625, 1.265625), (3.90625, 1.25), (3.921875, 1.265625), (3.921875, 1.28125)}, {(3.9375, 1.359375), (3.953125, 1.328125), (3.96875, 1.3125), (3.984375, 1.328125), (3.984375, 1.34375)}, {(3.9375, 1.421875), (3.953125, 1.390625), (3.96875, 1.375), (3.984375, 1.390625), (3.984375, 1.40625)}}, {{(3.75, 1.734375), (3.765625, 1.703125), (3.78125, 1.6875), (3.796875, 1.703125), (3.796875, 1.71875)}, {(3.8125, 1.609375), (3.828125, 1.578125), (3.84375, 1.5625), (3.859375, 1.578125), (3.859375, 1.59375)}, {(3.875, 1.546875), (3.890625, 1.515625), (3.90625, 1.5), (3.921875, 1.515625), (3.921875, 1.53125)}, {(3.9375, 1.609375), (3.953125, 1.578125), (3.96875, 1.5625), (3.984375, 1.578125), (3.984375, 1.59375)}, {(3.9375, 1.671875), (3.953125, 1.640625), (3.96875, 1.625), (3.984375, 1.640625), (3.984375, 1.65625)}}}, {{{(3, 2.984375), (3.015625, 2.953125), (3.03125, 2.9375), (3.046875, 2.953125), (3.046875, 2.96875)}, {(3.0625, 2.859375), (3.078125, 2.828125), (3.09375, 2.8125), (3.109375, 2.828125), (3.109375, 2.84375)}, {(3.125, 2.796875), (3.140625, 2.765625), (3.15625, 2.75), (3.171875, 2.765625), (3.171875, 2.78125)}, {(3.1875, 2.859375), (3.203125, 2.828125), (3.21875, 2.8125), (3.234375, 2.828125), (3.234375, 2.84375)}, {(3.1875, 2.921875), (3.203125, 2.890625), (3.21875, 2.875), (3.234375, 2.890625), (3.234375, 2.90625)}}, {{(3.25, 2.484375), (3.265625, 2.453125), (3.28125, 2.4375), (3.296875, 2.453125), (3.296875, 2.46875)}, {(3.3125, 2.359375), (3.328125, 2.328125), (3.34375, 2.3125), (3.359375, 2.328125), (3.359375, 2.34375)}, {(3.375, 2.296875), (3.390625, 2.265625), (3.40625, 2.25), (3.421875, 2.265625), (3.421875, 2.28125)}, {(3.4375, 2.359375), (3.453125, 2.328125), (3.46875, 2.3125), (3.484375, 2.328125), (3.484375, 2.34375)}, {(3.4375, 2.421875), (3.453125, 2.390625), (3.46875, 2.375), (3.484375, 2.390625), (3.484375, 2.40625)}}, {{(3.5, 2.234375), (3.515625, 2.203125), (3.53125, 2.1875), (3.546875, 2.203125), (3.546875, 2.21875)}, {(3.5625, 2.109375), (3.578125, 2.078125), (3.59375, 2.0625), (3.609375, 2.078125), (3.609375, 2.09375)}, {(3.625, 2.046875), (3.640625, 2.015625), (3.65625, 2), (3.671875, 2.015625), (3.671875, 2.03125)}, {(3.6875, 2.109375), (3.703125, 2.078125), (3.71875, 2.0625), (3.734375, 2.078125), (3.734375, 2.09375)}, {(3.6875, 2.171875), (3.703125, 2.140625), (3.71875, 2.125), (3.734375, 2.140625), (3.734375, 2.15625)}}, {{(3.75, 2.484375), (3.765625, 2.453125), (3.78125, 2.4375), (3.796875, 2.453125), (3.796875, 2.46875)}, {(3.8125, 2.359375), (3.828125, 2.328125), (3.84375, 2.3125), (3.859375, 2.328125), (3.859375, 2.34375)}, {(3.875, 2.296875), (3.890625, 2.265625), (3.90625, 2.25), (3.921875, 2.265625), (3.921875, 2.28125)}, {(3.9375, 2.359375), (3.953125, 2.328125), (3.96875, 2.3125), (3.984375, 2.328125), (3.984375, 2.34375)}, {(3.9375, 2.421875), (3.953125, 2.390625), (3.96875, 2.375), (3.984375, 2.390625), (3.984375, 2.40625)}}, {{(3.75, 2.734375), (3.765625, 2.703125), (3.78125, 2.6875), (3.796875, 2.703125), (3.796875, 2.71875)}, {(3.8125, 2.609375), (3.828125, 2.578125), (3.84375, 2.5625), (3.859375, 2.578125), (3.859375, 2.59375)}, {(3.875, 2.546875), (3.890625, 2.515625), (3.90625, 2.5), (3.921875, 2.515625), (3.921875, 2.53125)}, {(3.9375, 2.609375), (3.953125, 2.578125), (3.96875, 2.5625), (3.984375, 2.578125), (3.984375, 2.59375)}, {(3.9375, 2.671875), (3.953125, 2.640625), (3.96875, 2.625), (3.984375, 2.640625), (3.984375, 2.65625)}}}}
  6. Sequence(Translate[Dilate[l3, 1 / 4], Element[l0, u]], u, 1, 5) → l4, {{{{{(0, 3.99609375), (0.00390625, 3.98828125), (0.0078125, 3.984375), (0.01171875, 3.98828125), (0.01171875, 3.9921875)}, {(0.015625, 3.96484375), (0.01953125, 3.95703125), (0.0234375, 3.953125), (0.02734375, 3.95703125), (0.02734375, 3.9609375)}, {(0.03125, 3.94921875), (0.03515625, 3.94140625), (0.0390625, 3.9375), (0.04296875, 3.94140625), (0.04296875, 3.9453125)}, {(0.046875, 3.96484375), (0.05078125, 3.95703125), (0.0546875, 3.953125), (0.05859375, 3.95703125), (0.05859375, 3.9609375)}, {(0.046875, 3.98046875), (0.05078125, 3.97265625), (0.0546875, 3.96875), (0.05859375, 3.97265625), (0.05859375, 3.9765625)}}, {{(0.0625, 3.87109375), (0.06640625, 3.86328125), (0.0703125, 3.859375), (0.07421875, 3.86328125), (0.07421875, 3.8671875)}, {(0.078125, 3.83984375), (0.08203125, 3.83203125), (0.0859375, 3.828125), (0.08984375, 3.83203125), (0.08984375, 3.8359375)}, {(0.09375, 3.82421875), (0.09765625, 3.81640625), (0.1015625, 3.8125), (0.10546875, 3.81640625), (0.10546875, 3.8203125)}, {(0.109375, 3.83984375), (0.11328125, 3.83203125), (0.1171875, 3.828125), (0.12109375, 3.83203125), (0.12109375, 3.8359375)}, {(0.109375, 3.85546875), (0.11328125, 3.84765625), (0.1171875, 3.84375), (0.12109375, 3.84765625), (0.12109375, 3.8515625)}}, {{(0.125, 3.80859375), (0.12890625, 3.80078125), (0.1328125, 3.796875), (0.13671875, 3.80078125), (0.13671875, 3.8046875)}, {(0.140625, 3.77734375), (0.14453125, 3.76953125), (0.1484375, 3.765625), (0.15234375, 3.76953125), (0.15234375, 3.7734375)}, {(0.15625, 3.76171875), (0.16015625, 3.75390625), (0.1640625, 3.75), (0.16796875, 3.75390625), (0.16796875, 3.7578125)}, {(0.171875, 3.77734375), (0.17578125, 3.76953125), (0.1796875, 3.765625), (0.18359375, 3.76953125), (0.18359375, 3.7734375)}, {(0.171875, 3.79296875), (0.17578125, 3.78515625), (0.1796875, 3.78125), (0.18359375, 3.78515625), (0.18359375, 3.7890625)}}, {{(0.1875, 3.87109375), (0.19140625, 3.86328125), (0.1953125, 3.859375), (0.19921875, 3.86328125), (0.19921875, 3.8671875)}, {(0.203125, 3.83984375), (0.20703125, 3.83203125), (0.2109375, 3.828125), (0.21484375, 3.83203125), (0.21484375, 3.8359375)}, {(0.21875, 3.82421875), (0.22265625, 3.81640625), (0.2265625, 3.8125), (0.23046875, 3.81640625), (0.23046875, 3.8203125)}, {(0.234375, 3.83984375), (0.23828125, 3.83203125), (0.2421875, 3.828125), (0.24609375, 3.83203125), (0.24609375, 3.8359375)}, {(0.234375, 3.85546875), (0.23828125, 3.84765625), (0.2421875, 3.84375), (0.24609375, 3.84765625), (0.24609375, 3.8515625)}}, {{(0.1875, 3.93359375), (0.19140625, 3.92578125), (0.1953125, 3.921875), (0.19921875, 3.92578125), (0.19921875, 3.9296875)}, {(0.203125, 3.90234375), (0.20703125, 3.89453125), (0.2109375, 3.890625), (0.21484375, 3.89453125), (0.21484375, 3.8984375)}, {(0.21875, 3.88671875), (0.22265625, 3.87890625), (0.2265625, 3.875), (0.23046875, 3.87890625), (0.23046875, 3.8828125)}, {(0.234375, 3.90234375), (0.23828125, 3.89453125), (0.2421875, 3.890625), (0.24609375, 3.89453125), (0.24609375, 3.8984375)}, {(0.234375, 3.91796875), (0.23828125, 3.91015625), (0.2421875, 3.90625), (0.24609375, 3.91015625), (0.24609375, 3.9140625)}}}, {{{(0.25, 3.49609375), (0.25390625, 3.48828125), (0.2578125, 3.484375), (0.26171875, 3.48828125), (0.26171875, 3.4921875)}, {(0.265625, 3.46484375), (0.26953125, 3.45703125), (0.2734375, 3.453125), (0.27734375, 3.45703125), (0.27734375, 3.4609375)}, {(0.28125, 3.44921875), (0.28515625, 3.44140625), (0.2890625, 3.4375), (0.29296875, 3.44140625), (0.29296875, 3.4453125)}, {(0.296875, 3.46484375), (0.30078125, 3.45703125), (0.3046875, 3.453125), (0.30859375, 3.45703125), (0.30859375, 3.4609375)}, {(0.296875, 3.48046875), (0.30078125, 3.47265625), (0.3046875, 3.46875), (0.30859375, 3.47265625), (0.30859375, 3.4765625)}}, {{(0.3125, 3.37109375), (0.31640625, 3.36328125), (0.3203125, 3.359375), (0.32421875, 3.36328125), (0.32421875, 3.3671875)}, {(0.328125, 3.33984375), (0.33203125, 3.33203125), (0.3359375, 3.328125), (0.33984375, 3.33203125), (0.33984375, 3.3359375)}, {(0.34375, 3.32421875), (0.34765625, 3.31640625), (0.3515625, 3.3125), (0.35546875, 3.31640625), (0.35546875, 3.3203125)}, {(0.359375, 3.33984375), (0.36328125, 3.33203125), (0.3671875, 3.328125), (0.37109375, 3.33203125), (0.37109375, 3.3359375)}, {(0.359375, 3.35546875), (0.36328125, 3.34765625), (0.3671875, 3.34375), (0.37109375, 3.34765625), (0.37109375, 3.3515625)}}, {{(0.375, 3.30859375), (0.37890625, 3.30078125), (0.3828125, 3.296875), (0.38671875, 3.30078125), (0.38671875, 3.3046875)}, {(0.390625, 3.27734375), (0.39453125, 3.26953125), (0.3984375, 3.265625), (0.40234375, 3.26953125), (0.40234375, 3.2734375)}, {(0.40625, 3.26171875), (0.41015625, 3.25390625), (0.4140625, 3.25), (0.41796875, 3.25390625), (0.41796875, 3.2578125)}, {(0.421875, 3.27734375), (0.42578125, 3.26953125), (0.4296875, 3.265625), (0.43359375, 3.26953125), (0.43359375, 3.2734375)}, {(0.421875, 3.29296875), (0.42578125, 3.28515625), (0.4296875, 3.28125), (0.43359375, 3.28515625), (0.43359375, 3.2890625)}}, {{(0.4375, 3.37109375), (0.44140625, 3.36328125), (0.4453125, 3.359375), (0.44921875, 3.36328125), (0.44921875, 3.3671875)}, {(0.453125, 3.33984375), (0.45703125, 3.33203125), (0.4609375, 3.328125), (0.46484375, 3.33203125), (0.46484375, 3.3359375)}, {(0.46875, 3.32421875), (0.47265625, 3.31640625), (0.4765625, 3.3125), (0.48046875, 3.31640625), (0.48046875, 3.3203125)}, {(0.484375, 3.33984375), (0.48828125, 3.33203125), (0.4921875, 3.328125), (0.49609375, 3.33203125), (0.49609375, 3.3359375)}, {(0.484375, 3.35546875), (0.48828125, 3.34765625), (0.4921875, 3.34375), (0.49609375, 3.34765625), (0.49609375, 3.3515625)}}, {{(0.4375, 3.43359375), (0.44140625, 3.42578125), (0.4453125, 3.421875), (0.44921875, 3.42578125), (0.44921875, 3.4296875)}, {(0.453125, 3.40234375), (0.45703125, 3.39453125), (0.4609375, 3.390625), (0.46484375, 3.39453125), (0.46484375, 3.3984375)}, {(0.46875, 3.38671875), (0.47265625, 3.37890625), (0.4765625, 3.375), (0.48046875, 3.37890625), (0.48046875, 3.3828125)}, {(0.484375, 3.40234375), (0.48828125, 3.39453125), (0.4921875, 3.390625), (0.49609375, 3.39453125), (0.49609375, 3.3984375)}, {(0.484375, 3.41796875), (0.48828125, 3.41015625), (0.4921875, 3.40625), (0.49609375, 3.41015625), (0.49609375, 3.4140625)}}}, {{{(0.5, 3.24609375), (0.50390625, 3.23828125), (0.5078125, 3.234375), (0.51171875, 3.23828125), (0.51171875, 3.2421875)}, {(0.515625, 3.21484375), (0.51953125, 3.20703125), (0.5234375, 3.203125), (0.52734375, 3.20703125), (0.52734375, 3.2109375)}, {(0.53125, 3.19921875), (0.53515625, 3.19140625), (0.5390625, 3.1875), (0.54296875, 3.19140625), (0.54296875, 3.1953125)}, {(0.546875, 3.21484375), (0.55078125, 3.20703125), (0.5546875, 3.203125), (0.55859375, 3.20703125), (0.55859375, 3.2109375)}, {(0.546875, 3.23046875), (0.55078125, 3.22265625), (0.5546875, 3.21875), (0.55859375, 3.22265625), (0.55859375, 3.2265625)}}, {{(0.5625, 3.12109375), (0.56640625, 3.11328125), (0.5703125, 3.109375), (0.57421875, 3.11328125), (0.57421875, 3.1171875)}, {(0.578125, 3.08984375), (0.58203125, 3.08203125), (0.5859375, 3.078125), (0.58984375, 3.08203125), (0.58984375, 3.0859375)}, {(0.59375, 3.07421875), (0.59765625, 3.06640625), (0.6015625, 3.0625), (0.60546875, 3.06640625), (0.60546875, 3.0703125)}, {(0.609375, 3.08984375), (0.61328125, 3.08203125), (0.6171875, 3.078125), (0.62109375, 3.08203125), (0.62109375, 3.0859375)}, {(0.609375, 3.10546875), (0.61328125, 3.09765625), (0.6171875, 3.09375), (0.62109375, 3.09765625), (0.62109375, 3.1015625)}}, {{(0.625, 3.05859375), (0.62890625, 3.05078125), (0.6328125, 3.046875), (0.63671875, 3.05078125), (0.63671875, 3.0546875)}, {(0.640625, 3.02734375), (0.64453125, 3.01953125), (0.6484375, 3.015625), (0.65234375, 3.01953125), (0.65234375, 3.0234375)}, {(0.65625, 3.01171875), (0.66015625, 3.00390625), (0.6640625, 3), (0.66796875, 3.00390625), (0.66796875, 3.0078125)}, {(0.671875, 3.02734375), (0.67578125, 3.01953125), (0.6796875, 3.015625), (0.68359375, 3.01953125), (0.68359375, 3.0234375)}, {(0.671875, 3.04296875), (0.67578125, 3.03515625), (0.6796875, 3.03125), (0.68359375, 3.03515625), (0.68359375, 3.0390625)}}, {{(0.6875, 3.12109375), (0.69140625, 3.11328125), (0.6953125, 3.109375), (0.69921875, 3.11328125), (0.69921875, 3.1171875)}, {(0.703125, 3.08984375), (0.70703125, 3.08203125), (0.7109375, 3.078125), (0.71484375, 3.08203125), (0.71484375, 3.0859375)}, {(0.71875, 3.07421875), (0.72265625, 3.06640625), (0.7265625, 3.0625), (0.73046875, 3.06640625), (0.73046875, 3.0703125)}, {(0.734375, 3.08984375), (0.73828125, 3.08203125), (0.7421875, 3.078125), (0.74609375, 3.08203125), (0.74609375, 3.0859375)}, {(0.734375, 3.10546875), (0.73828125, 3.09765625), (0.7421875, 3.09375), (0.74609375, 3.09765625), (0.74609375, 3.1015625)}}, {{(0.6875, 3.18359375), (0.69140625, 3.17578125), (0.6953125, 3.171875), (0.69921875, 3.17578125), (0.69921875, 3.1796875)}, {(0.703125, 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(3.9609375, 2.640625), (3.96484375, 2.64453125), (3.96484375, 2.6484375)}, {(3.96875, 2.63671875), (3.97265625, 2.62890625), (3.9765625, 2.625), (3.98046875, 2.62890625), (3.98046875, 2.6328125)}, {(3.984375, 2.65234375), (3.98828125, 2.64453125), (3.9921875, 2.640625), (3.99609375, 2.64453125), (3.99609375, 2.6484375)}, {(3.984375, 2.66796875), (3.98828125, 2.66015625), (3.9921875, 2.65625), (3.99609375, 2.66015625), (3.99609375, 2.6640625)}}}}}

AI Drawing Prompt Reference

Draw a “5-point substitution fractal” geometry. Start by creating a set of grid points on a 4×4 integer lattice (all points with integer coordinates from 0 to 3), and organize them into indexed positions for 15 steps. Randomly choose 5 starting small geometric units. For the first layer, scale each unit to 1/4 of its size and translate it to selected lattice locations to form a collection of five small shapes. Then repeat the same replacement rule: take each existing small shape, shrink it by 1/4, and translate it to lattice positions according to the indexing. Continue this iterative substitution for several layers so the result becomes a nested, self-similar cluster of tiny five-point patterns, with dense refinement toward finer scales.
The prompt above was compiled from the original drawing commands and is for reference only.