Gabriels-Horn

Generate a visualization of Gabriel's Horn in Python and Blender; also called Torricelli’s Trumpet, a geometric figure with infinite surface area and finite volume.

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Primary URL for the repository: OJB-Quantum/Gabriels-Horn

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The visualization for the rendered Gabriel's Horn is based on the formula below:

$V=\pi \int_1^a\left(\frac{1}{x}\right)^2 \mathrm{~d} x=\pi\left(1-\frac{1}{a}\right) \quad \lim _{a \rightarrow \infty} V=\lim _{a \rightarrow \infty} \pi\left(1-\frac{1}{a}\right)=\pi$

$A=2 \pi \int_1^a \frac{1}{x} \sqrt{1+\left(-\frac{1}{x^2}\right)^2} \mathrm{~d} x>2 \pi \int_1^a \frac{\mathrm{~d} x}{x}=2 \pi \ln (a) \quad \lim _{a \rightarrow \infty} A \geq \lim _{a \rightarrow \infty} 2 \pi \ln (a)=\infty$

Read more about the mathematical definition on Wiki.

Note: the rendered equation in the image shown was imported as an SVG into Blender, followed by converting it into a mesh, cleaning it up in Edit Mode, then extruding the faces into a 3D object.

Click to view Gabriel's Horn visualized in a Google Colab notebook:

Click to view the Blender Python script: