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Boy Surface

Boy Surface
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2015-02-04 21:19
FunctionSolid GeometryCalculusParametric Equation
In collections surface 3D /3D-Oberflächen 1 surface 3D /3D-Oberflächen 1 surface 3D surface 3D
This example demonstrates the Boy surface, a complex algebraic surface known for its unique single-sided topology. It involves core mathematical concepts such as 3D geometry, parametric equations, surface properties, and topology fundamentals. The construction utilizes GeoGebra commands like Surface and CurveCartesian to visualize this intricate shape, making it suitable for high school students, math enthusiasts, or educators exploring advanced geometric and topological ideas.

GeoGebra Commands & Parameters

  1. Surface(((2 / 3 * ((cos(u) * cos((2 * v))) + ((sqrt(2) * sin(u)) * cos(v)))) * cos(u) / (sqrt(2) - (sin((2 * u)) * sin((3 * v))))), ((sqrt(2) * cos(u)) * cos(u) / (sqrt(2) - (sin((2 * u)) * sin((3 * v))))), ((2 / 3 * ((cos(u) * sin((2 * v))) - ((sqrt(2) * sin(u)) * sin(v)))) * cos(u) / (sqrt(2) - (sin((2 * u)) * sin((3 * v))))), u, 0, 1.5708, v, 0, 1.5707963267948966) → b
  2. Surface(((2 / 3 * ((cos(u) * cos((2 * v))) + ((sqrt(2) * sin(u)) * cos(v)))) * cos(u) / (sqrt(2) - (sin((2 * u)) * sin((3 * v))))), ((sqrt(2) * cos(u)) * cos(u) / (sqrt(2) - (sin((2 * u)) * sin((3 * v))))), ((2 / 3 * ((cos(u) * sin((2 * v))) - ((sqrt(2) * sin(u)) * sin(v)))) * cos(u) / (sqrt(2) - (sin((2 * u)) * sin((3 * v))))), u, 0, 1.5708, v, 1.5708, 3.141592653589793) → d
  3. Surface(((2 / 3 * ((cos(u) * cos((2 * v))) + ((sqrt(2) * sin(u)) * cos(v)))) * cos(u) / (sqrt(2) - (sin((2 * u)) * sin((3 * v))))), ((sqrt(2) * cos(u)) * cos(u) / (sqrt(2) - (sin((2 * u)) * sin((3 * v))))), ((2 / 3 * ((cos(u) * sin((2 * v))) - ((sqrt(2) * sin(u)) * sin(v)))) * cos(u) / (sqrt(2) - (sin((2 * u)) * sin((3 * v))))), u, 1.5707963267948966, 3.141592653589793, v, 0, 1.5707963267948966) → c
  4. Surface(((2 / 3 * ((cos(u) * cos((2 * v))) + ((sqrt(2) * sin(u)) * cos(v)))) * cos(u) / (sqrt(2) - (sin((2 * u)) * sin((3 * v))))), ((sqrt(2) * cos(u)) * cos(u) / (sqrt(2) - (sin((2 * u)) * sin((3 * v))))), ((2 / 3 * ((cos(u) * sin((2 * v))) - ((sqrt(2) * sin(u)) * sin(v)))) * cos(u) / (sqrt(2) - (sin((2 * u)) * sin((3 * v))))), u, 1.5708, 3.141592653589793, v, 1.5708, 3.141592653589793) → e
  5. CurveCartesian(((2 / 3 * ((cos(u) * cos((2 * vv))) + ((sqrt(2) * sin(u)) * cos(vv)))) * cos(u) / (sqrt(2) - (sin((2 * u)) * sin((3 * vv))))), ((sqrt(2) * cos(u)) * cos(u) / (sqrt(2) - (sin((2 * u)) * sin((3 * vv))))), ((2 / 3 * ((cos(u) * sin((2 * vv))) - ((sqrt(2) * sin(u)) * sin(vv)))) * cos(u) / (sqrt(2) - (sin((2 * u)) * sin((3 * vv))))), u, 0, 3.141592653589793) → cv
  6. CurveCartesian(((2 / 3 * ((cos(uu) * cos((2 * v))) + ((sqrt(2) * sin(uu)) * cos(v)))) * cos(uu) / (sqrt(2) - (sin((2 * uu)) * sin((3 * v))))), ((sqrt(2) * cos(uu)) * cos(uu) / (sqrt(2) - (sin((2 * uu)) * sin((3 * v))))), ((2 / 3 * ((cos(uu) * sin((2 * v))) - ((sqrt(2) * sin(uu)) * sin(v)))) * cos(uu) / (sqrt(2) - (sin((2 * uu)) * sin((3 * v))))), v, 0, 3.141592653589793) → cu

AI Drawing Prompt Reference

Draw a complex Boy surface, a unique algebraic surface known for its single-sided topology. Use parametric equations to define the surface's coordinates with parameters u ranging from 0 to π/2 and v from 0 to π/2. First, plot the main body of the surface, then add two closed curves as boundaries, rotating around the u and v axes to show the overall structure and topological features.
The prompt above was compiled from the original drawing commands and is for reference only.