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## Gianluca Gorni

# Graphics Gallery

A QuickTime movie
(700K) showing an approximation to Peano's continuous, square-filling
curve.

A continuous but not differentiable function
of two variables:

A text written on a Moebius strip:

See also other related
files.

An animation, intended to illustrate Euler's famous formula combining the numbers e, pi, i, 1, 0.
Also, a flash version, if you can display it.

### Schwarz's accordions.

For viewing the following interactive pictures you will need the free Wolfram CDF Player plugin:

Until the late 19th century, mathematicians did not suspect that the concept of *"area of a curved surface"* could be quite very complicated to define precisely. In 1868 J.A. Serret for example made a naive description of how to approximate the area of a curved surface with the areas of finer and finer triangulations.

Actually, Serret's approach was wrong, as Hermann Amandus Schwarz in 1881 showed with a two-page article: even a surface as simple as a cilynder could be triangulated in such a way as to make the total area of the triangles as large as desired. Here above you can see two triangulations made according to Schwarz's recipe. From their look, objects like these are variously known as *Schwarz's accordion*, or pineapple, or boot, or lantern.

A 3D graphic version of the Moebius strip.

### Visualization of periodic functions.

The following is a graph of an a period of the sound wave of a
piano chord, seen as a function on the circle. See here for more
information.

The following picture is a result of playing around with POV-ray.
It shows a toothsaw wave (green) and its approximation by a truncated
Fourier series (black & white), seen as functions on the unit
circle, not as periodic functions of time. The raw objects were
calculated with *Mathematica*,
rendered (after some editing) with POV-ray,
and JPEG-compressed with GraphicConverter.

###

### Energy surface of the simple pendulum

Here is a surface graph of the energy of the simple pendulum as a
function of the angle and of the angular velocity, with the level
sets highlighted. The green, red and blue balls are stroboscopic
images of pendulums with different amplitudes (a whole circle for the
blue ball). The radius is not shown The mathematical description of
the objects was done with *Mathematica*
and the rendering with POV-ray.

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[Dept.of
Math&Info][Faculty
of Sciences][UdineUniv.]