3D Projection of an 8-cell

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2017-02-20 07:06

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This example demonstrates the 3D projection of an 8-cell (16-cell), a four-dimensional polytope, onto a three-dimensional space. Key mathematical concepts include solid geometry, vector operations, and 3D coordinate transformations. The construction utilizes GeoGebra commands such as Circle, Point, Plane, OrthogonalLine, Ellipse, Mirror, and Segment. It is suitable for high school students and above studying spatial geometry, 3D transformations, and related calculations.

GeoGebra Commands & Parameters

Circle((0, -1, 1), (-2, -2, 2), (2, -2, 2)) → c
Point(c) → A
Point(c) → C
Plane(C, A, B) → a
OrthogonalLine(B, a) → f
Ellipse((3, -2, 2), (-3, -2, 2), (0, -1, 1)) → e_1
Mirror(K, xOyPlane) → K'
Mirror(N, xOyPlane) → N'
Mirror(L, xOyPlane) → L'
Mirror(M, xOyPlane) → M'
Mirror(K, p) → K'_1
Mirror(N, p) → N'_1
Mirror(L, p) → L'_1
Mirror(M, p) → M'_1
Mirror(K', p) → K''
Mirror(L', p) → L''
Mirror(M', p) → M''
Mirror(N', p) → N''
Segment(K, K') → g
Segment(N, N') → h
Segment(L, L') → i
Segment(M, M') → j
Segment(K', N') → k
Segment(N, K) → l
Segment(K, L) → m
Segment(L', K') → n
Segment(L', M') → q
Segment(M', N') → r
Segment(L, L'_1) → s
Segment(M, L) → t
Segment(L', L'') → b
Segment(M', M'') → d
Segment(N', N'') → e
Segment(K', K'') → f_1
Segment(M, N) → g_1
Segment(N, N'_1) → h_1
Segment(K, K'_1) → i_1
Segment(K'_1, K'') → j_1
Segment(N'_1, N'') → k_1
Segment(K'', N'') → l_1
Segment(N'_1, K'_1) → m_1
Segment(M, M'_1) → n_1
Segment(L'_1, M'_1) → p_1
Segment(L'_1, L'') → q_1
Segment(M'_1, M'') → r_1
Segment(M'', L'') → s_1
Segment(L'', K'') → t_1
Segment(L'_1, K'_1) → a_1
Segment(M'', N'') → b_1
Segment(M'_1, N'_1) → c_1