Welcome to the
Rocchini's Math Others

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.


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.

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