Philosophy is written in this grand book — I mean universe — which stands continuously open to our gaze, but which cannot be understood unless one first learns to comprehend the language in which it is written. It is written in the language of mathematics, and its characters are triangles, circles and other geometric figures, without which it is humanly impossible to understand a single word of it; without these, one is wandering about in a dark labyrinth. — Galileo Galilei (1623).
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All geometric shapes below were created with basically the same plotting software that I have written in VPython.
A torus, a trivial example of a connected orientable surface of
genus one.
Trefoil knot, the simplest
example of a (non-trivial) knot.
The limpet torus.
Elliptic torus.
Double torus.
A twisted torus.
The famous Möbius strip,
perhaps the most well-known non-orientable surface.
The most well-known embedding of
Klein's bottle
in three-dimensional space.
Paul Bourke's parametrization
for the cross cap.
A sliced cross-capped disk is
homeomorphic to a self-intersecting disk.
Spherical harmonics are of the form
- the angles
$\phi \in [0, \pi]$ (latitude), and$\theta \in [0, 2\pi]$ (longitude), - the parameters
$m_0$ ,$m_1$ ,$m_2$ ,$m_3$ ,$m_4$ ,$m_5$ ,$m_6$ , and$m_7$ are all integers and$\geq 0$ , - and where
$r$ is the radius.
Dini's spiral, Dini's surface,
or twisted pseudo-sphere: characterized by a surface of constant (negative) curvature,
named after Ulisse Dini.
Nature meets mathematics: a purely mathematically generated seashell, with the parametrization
found on Paul Bourke's site.
A dented object.
Arc.
Combined ball and torus.
A surface of revolution.
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