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Sequence-based Geometric Construction with Exponents

Sequence-based Geometric Construction with Exponents
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2020-09-17 01:58
PolygonFunctionSequenceGeometric Construction
In collections The Treehouse-Projekt - Team Architects
This GeoGebra example demonstrates the dynamic generation and arrangement of regular polygonal structures across multiple layers using sequence commands combined with arithmetic operations and exponents. Core mathematical concepts include sequences, exponentiation, polygon geometry, and algebraic addition/subtraction. Constructed primarily using the GeoGebra `Sequence` command applied to functions, parameters, and point sequences ($F_1$ to $F_n$), it automates the generation of complex geometric grids. Ideal for high school and university students learning the intersection of algebra, sequences, and dynamic geometry.

GeoGebra Commands & Parameters

  1. Sequence((((12 * cos(k)) * (a - 1.5)), (-16 * (a - 1.5)^(2)) + ((12 * sin(k)) * (a - 1.5)) + If[a > 1.5, 3]), k, -60, 240, 10) → F_1
  2. Sequence((((12 * cos(k)) * (a - 1.52)), (-16 * (a - 1.51)^(2)) + ((12 * sin(k)) * (a - 1.51)) + If[a > 1.51, 3]), k, -60, 240, 10) → F_2
  3. Sequence((((12 * cos(k)) * (a - 1.52)), (-16 * (a - 1.52)^(2)) + ((12 * sin(k)) * (a - 1.52)) + If[a > 1.52, 3]), k, -60, 240, 10) → F_3
  4. Sequence((((12 * cos(k)) * (a - 1.53)), (-16 * (a - 1.53)^(2)) + ((12 * sin(k)) * (a - 1.53)) + If[a > 1.53, 3]), k, -60, 240, 10) → F_4
  5. Sequence((((12 * cos(k)) * (a - 1.54)), (-16 * (a - 1.54)^(2)) + ((12 * sin(k)) * (a - 1.54)) + If[a > 1.54, 3]), k, -60, 240, 10) → F_5
  6. Sequence((((12 * cos(k)) * (a - 1.55)), (-16 * (a - 1.55)^(2)) + ((12 * sin(k)) * (a - 1.55)) + If[a > 1.55, 3]), k, -60, 240, 10) → F_6
  7. Sequence((((12 * cos(k)) * (a - 1.56)), (-16 * (a - 1.56)^(2)) + ((12 * sin(k)) * (a - 1.56)) + If[a > 1.56, 3]), k, -60, 240, 10) → F_7
  8. Sequence((((12 * cos(k)) * (a - 1.57)), (-16 * (a - 1.57)^(2)) + ((12 * sin(k)) * (a - 1.57)) + If[a > 1.57, 3]), k, -60, 240, 10) → F_8
  9. Sequence((((12 * cos(k)) * (a - 1.58)), (-16 * (a - 1.58)^(2)) + ((12 * sin(k)) * (a - 1.58)) + If[a > 1.58, 3]), k, -60, 240, 10) → F_9
  10. Sequence((((12 * cos(k)) * (a - 1.59)), (-16 * (a - 1.59)^(2)) + ((12 * sin(k)) * (a - 1.59)) + If[a > 1.59, 3]), k, -60, 240, 10) → F_a
  11. Sequence((((12 * cos(k)) * (a - 1.6)), (-16 * (a - 1.6)^(2)) + ((12 * sin(k)) * (a - 1.6)) + If[a > 1.6, 3]), k, -60, 240, 10) → F_b
  12. Sequence((((12 * cos(k)) * (a - 1.61)), (-16 * (a - 1.61)^(2)) + ((12 * sin(k)) * (a - 1.61)) + If[a > 1.61, 3]), k, -60, 240, 10) → F_c
  13. Sequence((((12 * cos(k)) * (a - 1.62)), (-16 * (a - 1.62)^(2)) + ((12 * sin(k)) * (a - 1.62)) + If[a > 1.62, 3]), k, -60, 240, 10) → F_d
  14. Sequence((((12 * cos(k)) * (a - 1.63)), (-16 * (a - 1.63)^(2)) + ((12 * sin(k)) * (a - 1.63)) + If[a > 1.63, 3]), k, -60, 240, 10) → F_e
  15. Sequence((((12 * cos(k)) * (a - 1.64)), (-16 * (a - 1.64)^(2)) + ((12 * sin(k)) * (a - 1.64)) + If[a > 1.64, 3]), k, -60, 240, 10) → F_f
  16. Sequence((((12 * cos(k)) * (a - 1.65)), (-16 * (a - 1.65)^(2)) + ((12 * sin(k)) * (a - 1.65)) + If[a > 1.65, 3]), k, -60, 240, 10) → F_g
  17. Sequence((((12 * cos(k)) * (a - 1.66)), (-16 * (a - 1.66)^(2)) + ((12 * sin(k)) * (a - 1.66)) + If[a > 1.66, 3]), k, -60, 240, 10) → F_h
  18. Sequence((((12 * cos(k)) * (a - 1.67)), (-16 * (a - 1.67)^(2)) + ((12 * sin(k)) * (a - 1.67)) + If[a > 1.67, 3]), k, -60, 240, 10) → F_i
  19. Sequence((((12 * cos(k)) * (a - 1.68)), (-16 * (a - 1.68)^(2)) + ((12 * sin(k)) * (a - 1.68)) + If[a > 1.68, 3]), k, -60, 240, 10) → F_j
  20. Sequence((((12 * cos(k)) * (a - 1.69)), (-16 * (a - 1.69)^(2)) + ((12 * sin(k)) * (a - 1.69)) + If[a > 1.69, 3]), k, -60, 240, 10) → F_k
  21. Sequence((((12 * cos(k)) * (a - 1.7)), (-16 * (a - 1.7)^(2)) + ((12 * sin(k)) * (a - 1.7)) + If[a > 1.7, 3]), k, -60, 240, 10) → F_l
  22. Sequence((((12 * cos(k)) * (a - 1.71)), (-16 * (a - 1.71)^(2)) + ((12 * sin(k)) * (a - 1.71)) + If[a > 1.71, 3]), k, -60, 240, 10) → F_m
  23. Sequence((((12 * cos(k)) * (a - 1.72)), (-16 * (a - 1.72)^(2)) + ((12 * sin(k)) * (a - 1.72)) + If[a > 1.72, 3]), k, -60, 240, 10) → F_n

AI Drawing Prompt Reference

Create a layered geometric figure made from regular polygon-like, radially arranged point sets. Generate points using a parameter angle that runs from -60° to 240° with a step of 10°. For each layer, scale the construction with a radius-like term of the form (a minus a layer constant), then compute x and y components using 12·cos(angle) and 12·sin(angle). Add a vertical/offset contribution of the form -16·(a minus the same constant)^2, and apply an extra height adjustment when a is greater than that layer constant. Produce multiple layers with successive constants (forming F1 through Fn) and display all layers together on the same plane to obtain a complex grid-like polygonal arrangement.
The prompt above was compiled from the original drawing commands and is for reference only.