Klein Bottle by Patrick Clément

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2014-09-19 01:21

In collections surface 3D surface 3D My 3D Collection

This example demonstrates the 3D shape of a Klein bottle, a non-orientable topological object with no inside or outside. The surface of the Klein bottle is constructed using the Surface command. Suitable for high school and above students to learn topology and 3D geometry.

GeoGebra Commands & Parameters

Surface((((a * (1 + sin(u))) + ((c * (1 - cos(u) / 2)) * cos(v))) * cos(u)), ((b + ((c * (1 - cos(u) / 2)) * cos(v))) * sin(u)), ((c * (1 - cos(u) / 2)) * sin(v)) + d, u, Umin, Umax, v, Vmin, Vmax) → KleinsBottle1
Surface(((a * (1 + sin(u))) * cos(u)) - ((c * (1 - cos(u) / 2)) * cos(v)), (b * sin(u)), ((c * (1 - cos(u) / 2)) * sin(v)) - d, u, Umin + π, Umax + π, v, Vmin, Vmax) → KleinsBottle2

AI Drawing Prompt Reference

Draw a Klein bottle, a non-orientable topological object with no inside or outside. First, create a parametric surface with the formula: (((a * (1 + sin(u))) + ((c * (1 - cos(u) / 2)) * cos(v))) * cos(u), ((b + ((c * (1 - cos(u) / 2)) * cos(v))) * sin(u), ((c * (1 - cos(u) / 2)) * sin(v)) + d. Then, create another parametric surface with the formula: (((a * (1 + sin(u))) * cos(u)) - ((c * (1 - cos(u) / 2)) * cos(v)), (b * sin(u)), ((c * (1 - cos(u) / 2)) * sin(v)) - d.

The prompt above was compiled from the original drawing commands and is for reference only.