A Few of My Favorite Spaces: The Poincaré Homology Sphere
An interesting blog about spaces.
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Tau.Neutrino said:
The Hardest Mandelbrot Zoom Ever In 2014,10^198 : New record – 350 000 000 iterations
Tau.Neutrino said:
A Few of My Favorite Spaces: The Poincaré Homology Sphere
An interesting blog about spaces.
Fascinating little piece. I really helps me to together little snippets of topology that I’ve picked up over the years.
“Early topologists wanted to try to find ways of distinguishing spaces by finding invariants: numbers or other mathematical objects that could be assigned to each space. Ideally, two spaces with the same invariant would be the same space, and two spaces with different invariants would be different spaces. Poincaré came up with Betti numbers, which are informally a way to catalogue the holes of different dimensions in a space, and torsion coefficients, which sort of keep track of twistedness. In a 1900 paper, Poincaré conjectured that these Betti numbers and torsion coefficients (also known today as homology) could tell you whether or not a space was a sphere. “
“He added another invariant, known as the fundamental group, and believed that if a manifold had the same homotopy and fundamental group as a sphere, it had to be a sphere. The Poincaré conjecture was one of the most important unsolved conjectures In 2006, this conjecture was finally proved, with Russian mathematician Grigori Perelman “
“A better-known way to define it now starts with a dodecahedron”.
I’d been aware of the use of a dodecahedron in constructing odd topologies. Faces of a dodecahedron can be mapped to one another with or without twists to generate new topologies.
If you want to learn about this, start with a square rather than a dodecahedron. If the opposite edges of a square are mapped onto one another without twists then the result is a doughnut. If adjacent edges are mapped onto one another in two pairs then the result is a sphere. If opposite edges are mapped onto one another with a 180 degree twist then the result is a Klein bottle.
Tau.Neutrino said:
A Few of My Favorite Spaces: The Poincaré Homology Sphere
An interesting blog about spaces.
You’ve got to see this video of the 120-cell in 4-D.
I’ve been wanting to make up 4-D videos like this for at nearly 30 years.
From https://bakingandmath.com/2016/04/19/dodecahedral-construction-of-the-poincare-homology-sphere/
mollwollfumble said:
Tau.Neutrino said:A Few of My Favorite Spaces: The Poincaré Homology Sphere
An interesting blog about spaces.
You’ve got to see this video of the 120-cell in 4-D.
I’ve been wanting to make up 4-D videos like this for at nearly 30 years.
From https://bakingandmath.com/2016/04/19/dodecahedral-construction-of-the-poincare-homology-sphere/
If you’re interested, one day I’ll post some of the 4-D animations I’ve made of quaternion Mandelbrot/Julia sets (quandelsets).
Wocky said:
mollwollfumble said:
Tau.Neutrino said:A Few of My Favorite Spaces: The Poincaré Homology Sphere
An interesting blog about spaces.
You’ve got to see this video of the 120-cell in 4-D.
I’ve been wanting to make up 4-D videos like this for at nearly 30 years.
From https://bakingandmath.com/2016/04/19/dodecahedral-construction-of-the-poincare-homology-sphere/
If you’re interested, one day I’ll post some of the 4-D animations I’ve made of quaternion Mandelbrot/Julia sets (quandelsets).
Noted.