The original idea for Complex Plottings is by Lawrence Crone!!

My 3D embedding of *Dunce Hat* is not a masterpiece... My general idea is to set the 3-winged edge
around a circle. I also have other better ideas; i.e. to fold the figure in two steps: first to form
a real Dunce Hat and next to glue the circular border with the cone.

The hypersphere and hypercone are my favorite pictures. From a radial parametrization of these figures
is easy to extract parallels, meridians and hypermedians. Each 4D each curve is then projected by
*Stereographic Projection* in 3D. All of these projected curves are circles or Straight Lines,
and intersect each other
at right angle (like in 4D, because Stereographic Projection is conformal).

The duocylinder (or double cylinder or 4D torus) is my favorite geometric object. While the 3D torus is intrinsically asymmetric, due to temporal significance of folding order, the 4D torus is completely symmetrical and perfect. Unfortunately draw duocylinder is not trivial: most of 3D projection of 4D torus are ... a 3D torus! The solution is an animation of rotating duocylinder, in wich the border and the hole of the torus exchanges the position. Second static image is a snapshot of animation; the hue of color is the time. The blue parallels takes the place of red meridians.

### Gallery

Hypercycle example - SVG File - CarMetal | |

Projections onto convex avg sets algorithm example - SVG File - CarMetal | |

Projections onto convex sets algorithm example - SVG File - CarMetal | |

Robbins Pentagon (V1) - SVG File | |

Robbins Pentagon (V2) - SVG File | |

Sphere Inversion - Wrml Source - C++ Source Code | |

Lonely runner conjecture (animated) | |

Complex Plot 1 - complex_plot | |

Complex Plot 2 - complex_plot | |

Complex Plots - complex_plot | |

Dunce Hat Model | |

Dunce Hat Model Animated | |

Hypersphere's coordinates - Another embedding is on top of the page. | |

Spherical Cone Coordinates | |

Hammersley's solution of Moving Sofa Problem (Animated) | |

Rotating 4D Duocylinder Ridge animated | |

Rotating 4D Duocylinder Ridge snapshots | |

Some WRL Snapshot of my graphical debugger | |

Interval Exchange - Interval_exchange.svg | |

Mobius Configuration - VRML Model 1 (mobius_conf1.wrl) - VRML Model 2 (mobius_conf2.wrl) | |

Scheme of Butterfly Lemma - Butterfly_lemma.svg | |

Centered Decagonal Number - Centered_decagonal_number.svg | |

Centered Heptagonal Number - Centered_heptagonal_number.svg | |

Centered Nonagonal Number - Centered_nonagonal_number.svg | |

Centered Octagonal Number - Centered_octagonal_number.svg | |

Reverse Perspective - Reverse_perspective.svg | |

Sacks Spiral of primes - Sacks_spiral.svg | |

Braid Theory: permutations as braid | |

Zernike Polynomials - zernike source code |

### Final Note

I have not really understood the *Zassenhaus lemma* (also named "Butterfly Lemma").
I was inspired by a real butterfly, but I do not remember his name.
The *Reverse Perspective* sample is generated by code (not drawn by hand).
I want to make a best example but I had trouble (in other words, because I am lazy).

The *Sacks Spiral*'s design is simple; The effort is to minimize the relative SVG file (see bottom
this page).
The 24 elements of a Permutation group on 4 elements as *braids* are also computer generated.
For each permutation the code search the "first" (simplest) braid that corresponds to permutation.

About the *Zernike Polynomials* nothing to say; but the Zernike function is oversampled for
antialiasing.

*Mobius configuration*: this images is a vrml snapshot. Here is the files.
Showing planes in the space is an old headache.

The last image is a set of snapshots of *WRL* objects; I use some WRL model to preview
my creations. After debugging, the finale version of obect is rendered by POVRay (or Blender).

The *Moving Sofa Problem* is a joy for the brain. The Hammersley solution is not the largest one...
I bought the Gerver's paper but the relative Sofa schema is still the upholsterer.
For a complete resuming of this problem, you must read the book *Dirk Gently* by Douglas Adams.