Klein Bottle by Juan Carlos Ponce Campuzano (2)
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This example demonstrates the geometric construction of a Klein bottle, a closed surface in four-dimensional space without inside or outside. It involves topological and four-dimensional geometry concepts. Using Surface, Polygon, Textfield, etc., GeoGebra commands, the shape and transparency of the bottle can be dynamically adjusted. Suitable for high school students and math enthusiasts.
GeoGebra Commands & Parameters
Surface(If[0 ≤ u < π, ((a * cos(u)) * (1 + sin(u))) + (((c * (1 - cos(u) / 2)) * cos(u)) * cos(v)), ((a * cos(u)) * (1 + sin(u))) + ((c * (1 - cos(u) / 2)) * cos(v + π))], If[0 ≤ u < π, (b * sin(u)) + (((c * (1 - cos(u) / 2)) * sin(u)) * cos(v)), (b * sin(u))], ((c * (1 - cos(u) / 2)) * sin(v)), u, 0, u1, v, 0, v1) → dPolygon(C, E, F, D) → poly1, c_1, e_1, f, d_1Textfield(u1) → InputBox1
AI Drawing Prompt Reference
Create a 3D parametric surface of a Klein bottle, formed by two segments: the first uses cosine and sine functions to generate the bottle body, while the second applies a phase shift to create self-intersection and inversion, resulting in a closed surface without inside or outside. The surface transparency is adjustable, accompanied by a quadrilateral polygon as a reference plane and an input box to control the surface extension range. This visualizes a four-dimensional topological geometry for mathematical demonstration.
The prompt above was compiled from the original drawing commands and is for reference only.