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Geometric construction from cantilever to designed muqarnas

Geometric construction from cantilever to designed muqarnas
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2020-08-17 14:50
PolygonTrigonometrySolid GeometryGeometric Construction
In collections Fatin Uluengin - muqarnas
This example demonstrates how a cantilever-style geometric organization can be transformed into a designed muqarnas (stalactite vault) pattern with a clear rule-based structure. It focuses on polygonal decomposition and the geometric relationships underlying layered ornamentation. Key math concepts include parallelogram-based geometric reasoning, angle-related structure, and trigonometric functions that guide the shape splitting and placement. Construction in GeoGebra uses Point to define control vertices, Polygon to create the partitioned faces for different layers, Prism to form 3D/thickness effects of the structured blocks, and If to apply conditional logic so that different regions follow different construction branches. The model involves multiple polygons (veelhoek1–veelhoek4) and a set of parameter-like objects (e.g., h_1, h_2, g_1, g_2) that govern the resulting geometry. Suitable for high school through early university-level geometry construction, graph/design-oriented modeling, and lessons connecting functions/trigonometry with architectural geometric patterns.

GeoGebra Commands & Parameters

  1. Point(xAxis) → A
  2. Point(xAxis) → B
  3. Polygon(A, B, 4) → veelhoek1, f, g, h, i, C, D
  4. Point(xAxis) → E
  5. Point(xAxis) → H
  6. Polygon(H, A, D, C, B, E, F, G) → veelhoek2, h_1, a, d, c, b, e, f_1, g_1
  7. Point(xAxis) → I
  8. Point(xAxis) → L
  9. Polygon(E, I, J, K, L, H, G, F) → veelhoek3, e_1, i_1, j, k, l, h_2, g_2, f_2
  10. Point(xAxis) → M
  11. Point(xAxis) → P
  12. Polygon(I, M, N, O, P, L, K, J) → veelhoek4, i_2, m, n, o, p, l_1, k_1, j_1
  13. Point(xAxis) → Q
  14. Point(xAxis) → T
  15. Polygon(M, Q, R, S, T, P, O, N) → veelhoek5, m_1, q, r, s, t, p_1, o_1, n_1
  16. Prism(veelhoek1, 1) → p1, U, RibbeAU, RibbeBV, RibbeCW, RibbeDZ, RibbeUV, RibbeVW, RibbeWZ, RibbeUZ, V, W, Z, zijvlakABVU, zijvlakBCWV, zijvlakCDZW, zijvlakADZU, zijvlakUVWZ
  17. Prism(veelhoek2, 1) → p2, A_1, zijvlak1, zijvlak5, zijvlak3, zijvlak4, zijvlak6, zijvlak7, zijvlak8, zijvlak9, Ribbe8, Ribbe1, B_1, Ribbe4, Ribbe3, Ribbe2, Ribbe5, Ribbe6, Ribbe7, Ribbe9, Ribbe11, Ribbe12, Ribbe13, C_1, Ribbe14, Ribbe15, Ribbe16, Ribbe10, D_1, E_1, F_1, G_1, H_1, zijvlak2
  18. Prism(veelhoek3, 1) → p3, I_1, zijvlak15, zijvlak16, zijvlak17, zijvlak14, zijvlak13, zijvlak12, zijvlak10, zijvlak18, Ribbe17, Ribbe21, J_1, Ribbe22, Ribbe23, Ribbe24, Ribbe20, Ribbe19, Ribbe18, Ribbe25, Ribbe27, Ribbe28, Ribbe29, K_1, Ribbe30, Ribbe31, Ribbe32, Ribbe26, L_1, M_1, N_1, O_1, P_1, zijvlak11
  19. Prism(veelhoek4, 1) → p4, Q_1, zijvlak24, zijvlak25, zijvlak26, zijvlak23, zijvlak22, zijvlak21, zijvlak19, zijvlak27, Ribbe33, Ribbe37, R_1, Ribbe38, Ribbe39, Ribbe40, Ribbe36, Ribbe35, Ribbe34, Ribbe41, Ribbe43, Ribbe44, Ribbe45, S_1, Ribbe46, Ribbe47, Ribbe48, Ribbe42, T_1, U_1, V_1, W_1, Z_1, zijvlak20
  20. Prism(veelhoek5, 1) → p5, A_2, zijvlak33, zijvlak34, zijvlak35, zijvlak32, zijvlak31, zijvlak30, zijvlak28, zijvlak36, Ribbe49, Ribbe53, B_2, Ribbe54, Ribbe55, Ribbe56, Ribbe52, Ribbe51, Ribbe50, Ribbe57, Ribbe59, Ribbe60, Ribbe61, C_2, Ribbe62, Ribbe63, Ribbe64, Ribbe58, D_2, E_2, F_2, G_2, H_2, zijvlak29
  21. If(ss < 1, Translate[p5, Vector[(0, 0, (2 * ss))]], Translate[p5, Vector[(0, 0, 2)]]) → pp5
  22. If(1 < ss < 2, Translate[p4, Vector[(0, 0, (3 * (ss - 1)))]], Translate[p4, Vector[(0, 0, 3)]]) → pp4
  23. If(2 < ss < 3, Translate[p3, Vector[(0, 0, (4 * (ss - 2)))]], Translate[p3, Vector[(0, 0, 4)]]) → pp3
  24. If(3 < ss < 4, Translate[p2, Vector[(0, 0, (5 * (ss - 3)))]], Translate[p2, Vector[(0, 0, 5)]]) → pp2
  25. If(4 < ss < 5, Translate[p1, Vector[(0, 0, (6 * (ss - 4)))]], Translate[p1, Vector[(0, 0, 6)]]) → pp1

AI Drawing Prompt Reference

Create a 3D geometric construction of a cantilever-style muqarnas vault with layered, rule-based ornamentation. Start by placing a series of points along a horizontal axis)Skip, then connect them to form multiple nested polygons, each representing a distinct decorative layer. Extrude each polygon vertically into a prism to give thickness and depth. Use conditional logic to translate each prism along the vertical axis within specific height intervals, creating a stepped, cascading effect that mimics stalactite vaults. The design emphasizes polygonal decomposition, angular relationships, and trigonometric functions, resulting in a structured, architectural pattern suitable for studying geometric construction and layered ornamentation.
The prompt above was compiled from the original drawing commands and is for reference only.