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Coloring Difficult Shapes

Non-convex polygons with Postscript graphics

Interestingly, I happened to discover on accident that Mathematica 5.2 will properly shade a polygon even if it is not convex, and this fact makes producing pictures like this one below easier, as a series of polygons becoming less and less convex.

Categories : mathematical

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Planetary Concordance

Finding Harmony Among the Planets' Motions

Assuming that the acquired data of planet positions and velocities is accurate, the information below shows the relationships between the motions of the planets such that the angular velocities are in proportions which correspond to musical intervals.  Actually, I'm a bit unsure about the accuracy because I'm finding that the positions that I calculated aren't in agreement with the ones in the online ephemeris with JPL.

Ephemeris Report:
Date:12/25/2021 10:19:25
Julian Day: 2459574.180150463
Linear Units:Astronomical Units
Angular Units:Radians

 

Categories : mathematical, music

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Kepler's Sugar

Raytracing Images

Here is an image of an informal tea service where instead of sugar cubes there are sugar dodecahedrons. Actually, they are rhombic dodecahedrons. Should I mention that cubes have six equal sides whereas these figures have twelve equal sides? I decided to name the sugar brand as Kepler's because I first saw the geometric shape while reading about it in one of Kepler's books.

 

Categories : mathematical, computer

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