4D Frame Soccer Ball: Symmetry and Transformation in Polyhedra

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2016-05-11 22:30

① This GeoGebra example constructs a soccer-ball-like polyhedral framework, demonstrating spatial symmetry, rotation, and reflection operations, with a conceptual nod to 4D visualization. ② Core mathematical concepts include regular polyhedra, symmetry groups, rotational transformations, mirror reflections, and polygon mesh construction. ③ Construction highlights: commands such as Polygon, Rotate, Mirror, Point, Intersect, Line, Segment, Midpoint, and Angle are used to create multiple polygons (poly1, poly2, etc.) and their rotated/mirrored copies, forming an intricate 3D structure through interactive points, lines, and faces. ④ Suitable for high school to university levels, ideal for teaching solid geometry, transformation geometry, or as a resource for mathematical modeling and artistic mathematics.

GeoGebra Commands & Parameters

Polygon(A_1, A_2, A_3, A_4) → poly1, a_1, a_2, a_4, a_3
Rotate(poly1, 90°, zAxis) → poly1'
Rotate(poly1', 90°, xAxis) → poly1''
Rotate(poly1', -90°, yAxis) → poly1''_1
PointIn(poly1'') → A
Polygon(A_1, A, B) → poly2, b, a_5, a
Polygon(A_1, B, A_2) → poly3, a_6, a_7, b_1
Polygon(B, A_2, C) → poly4, c, b_2, a_8
Polygon(B, C, D) → poly5, d, b_3, c_1
Polygon(D, A, B) → poly6, b_4, d_1, a_9
Polygon(C, D, A_3) → poly7, a_{10}, c_2, d_2
Polygon(A_3, D, A_4) → poly8, a_{11}, a_{12}, d_3
Polygon(D, A, A_4) → poly9, a_{13}, d_4, a_{14}
Plane(A_1, A_2, A_3) → e
Mirror(B, e) → B'
Mirror(C, e) → C'
Mirror(D, e) → D'
Mirror(A, e) → A'
Polygon(A_4, A, A') → poly10, a', a_{15}, a_{16}
Polygon(A, A_1, A') → poly11, a'_1, a_{17}, a_{18}
Polygon(A', A_1, B') → poly12, b', a'_2, a_{19}
Polygon(A_2, A_1, B') → poly13, b'_1, a_{20}, a_{21}
Polygon(A_2, C, C') → poly14, c', a_{22}, c_3
Polygon(C, A_3, C') → poly15, c'_1, c_4, a_{23}
Polygon(A_3, D', A_4) → poly16, a_{24}, a_{25}, d'
Polygon(A_3, D', C') → poly17, c'_2, a_{26}, d'_1
Polygon(A_4, A', D') → poly18, d'_2, a_{27}, a'_3
Polygon(A', D', B') → poly19, b'_2, a'_4, d'_3
Polygon(A_2, C', B') → poly20, b'_3, a_{28}, c'_3
Polygon(C', B', D') → poly21, d'_4, c'_4, b'_4
Point(a_4) → E
Plane(D, A_4, D') → k
Plane(E, k) → l
Intersect(a_{11}, l) → F
Intersect(d_2, l) → G
Intersect(d'_1, l) → H
Intersect(c'_2, l) → I
Intersect(a_4, l) → J
Polygon(F, J, I, H, G) → poly22, f, j, i, h, g
Mirror(poly22, zAxis) → poly22'
Mirror(poly22, xAxis) → poly22'_1
Mirror(poly22', xAxis) → poly22''
Line(O, K) → m
Rotate(poly22, 120°, m) → poly22'_2
Rotate(poly22, -120°, m) → poly22'_3
Mirror(poly22'_3, zAxis) → poly23
Mirror(poly22'_2, zAxis) → poly24
Mirror(poly22'_3, xAxis) → poly25
Plane(A_1, A_2, A_3) → n
Mirror(poly22, n) → poly22'_4
Mirror(poly22'_2, n) → poly26
Mirror(poly22'_3, n) → poly27
Mirror(poly22'_4, n) → poly28
Mirror(poly23, n) → poly23'
Mirror(poly24, n) → poly24'
Plane(H, G, F) → o
Mirror(F, xAxis) → F_{22_1}'
Mirror(G, zAxis) → G_{22}'
Mirror(H, zAxis) → H_{22}'
Mirror(I, zAxis) → I_{22}'
Mirror(J, zAxis) → J_{22}'
Mirror(G_{22}', xAxis) → G_{22}''
Mirror(H_{22}', xAxis) → H_{22}''
Mirror(I_{22}', xAxis) → I_{22}''
Mirror(J_{22}', xAxis) → J_{22}''
Segment(J, J_{22}'') → q
Mirror(F, zAxis) → F_{22}'
Mirror(F_{22}', xAxis) → F_{22}''
Mirror(G, xAxis) → G_{22_1}'
Mirror(H, xAxis) → H_{22_1}'
Mirror(I, xAxis) → I_{22_1}'
Mirror(J, xAxis) → J_{22_1}'
Segment(J_{22}', J_{22_1}') → r
Rotate(F, 120°, m) → F_{22_2}'
Rotate(G, 120°, m) → G_{22_2}'
Rotate(H, 120°, m) → H_{22_2}'
Rotate(I, 120°, m) → I_{22_2}'
Rotate(J, 120°, m) → J_{22_2}'
Segment(H_{22_2}', I_{22}'') → s
Segment(G_{22_2}', F) → t
Rotate(F, -120°, m) → F_{22_3}'
Rotate(G, -120°, m) → G_{22_3}'
Rotate(H, -120°, m) → H_{22_3}'
Rotate(I, -120°, m) → I_{22_3}'
Rotate(J, -120°, m) → J_{22_3}'
Segment(G_{22_3}', F_{22_2}') → e_1
Segment(F_{22_3}', G) → f_1
Segment(H_{22_1}', I_{22_3}') → g_1
Mirror(F_{22_3}', zAxis) → F_{23}
Mirror(G_{22_3}', zAxis) → G_{23}
Mirror(H_{22_3}', zAxis) → H_{23}
Mirror(I_{22_3}', zAxis) → I_{23}
Mirror(J_{22_3}', zAxis) → J_{23}
Segment(I_{22_2}', H_{23}) → h_1
Segment(H_{22}'', I_{23}) → i_1
Mirror(F_{23}, n) → F_{23}'
Mirror(G_{23}, n) → G_{23}'
Mirror(H_{23}, n) → H_{23}'
Mirror(I_{23}, n) → I_{23}'
Mirror(J_{23}, n) → J_{23}'
Segment(J_{23}, J_{23}') → j_1
Segment(G_{22}'', I_{23}') → k_1
Segment(F_{23}, G_{22}') → l_1
Segment(F_{23}', H_{22}') → m_1
Mirror(F_{22_2}', zAxis) → F_{24}
Mirror(G_{22_2}', zAxis) → G_{24}
Mirror(H_{22_2}', zAxis) → H_{24}
Mirror(I_{22_2}', zAxis) → I_{24}
Mirror(J_{22_2}', zAxis) → J_{24}
Segment(G_{23}, F_{24}) → n_1
Segment(J_{22_2}', J_{24}) → p_1
Segment(G_{24}, F_{22}') → q_1
Segment(H_{24}, I_{22_1}') → r_1
Segment(I_{24}, H_{22_3}') → s_1
Mirror(F_{24}, n) → F_{24}'
Mirror(G_{24}, n) → G_{24}'
Mirror(H_{24}, n) → H_{24}'
Mirror(I_{24}, n) → I_{24}'
Mirror(J_{24}, n) → J_{24}'
Segment(I_{22}', G_{24}') → t_1
Segment(F_{22_1}', H_{24}') → e_2
Segment(G_{23}', F_{24}') → f_2
Mirror(F_{22_2}', n) → F_{26}
Mirror(G_{22_2}', n) → G_{26}
Mirror(H_{22_2}', n) → H_{26}
Mirror(I_{22_2}', n) → I_{26}
Mirror(J_{22_2}', n) → J_{26}
Segment(I, G_{26}) → g_2
Segment(F_{22}'', H_{26}) → h_2
Segment(I_{26}, H_{23}') → i_2
Segment(J_{26}, J_{24}') → j_2
Mirror(F_{22_3}', n) → F_{27}
Mirror(G_{22_3}', n) → G_{27}
Mirror(H_{22_3}', n) → H_{27}
Mirror(I_{22_3}', n) → I_{27}
Mirror(J_{22_3}', n) → J_{27}
Segment(J_{22_3}', J_{27}) → k_2
Segment(G_{22_1}', I_{27}) → l_2
Segment(F_{27}, H) → m_2
Segment(G_{27}, F_{26}) → n_2
Segment(I_{24}', H_{27}) → p_2
Plane(H, G, F_{22_3}') → u
Segment(G_{26}, F_{26}) → q_2
Segment(G_{22_2}', F_{22_2}') → r_2
Segment(G_{22_1}', H_{22_1}') → s_2
Polygon(J_{22_3}', J_{27}, 6, u) → poly29, t_2, P, e_3, f_3, g_3, h_3, i_3, L, M, N
Midpoint(g_3) → Q
Midpoint(k_2) → R
Line(Q, R) → j_3
Midpoint(s_2) → S
Line(R, S) → k_3
Point(j_3) → T
Plane(T, R, S) → o_1
Line(R, Q) → l_3
Line(S, R) → m_3
Angle(S, R, T) → α
Segment(B_1, C_1) → n_3
Point(n_3) → D_1