Hopf Fibration Explained Better than Eric Weinstein on Joe Rogan

[Music] by the end of this video you'll understand this a hopf vibration you may have heard mathematician eric weinstein comment on joe rogan pop and this is going to be hopf hopf you are looking at the most important object in the universe what discovered by heinz hoff in 1931 huff fiber bundles pop up in at least eight different physics situations so what you are seeing is as legendary physicist roger penrose put it an element of the architecture of our world will build to an understanding of the hop vibration and the fewest steps necessary if you're not yet familiar with higher dimensional shapes you might want to first watch my video explaining a 4d hypercube known as the tesseract linked above and below in my opinion hypercubes are easier to understand than hyperspheres and that video more gradually builds from the zeroth to the fourth dimension so what's the simplest definition of a hopf vibration it's just a map from a hypersphere in 4d onto a sphere in 3d how is it mapped a sphere is fully covered by points each point is mapped to a circle from the hypersphere these circles are the fibers that compose the hypersphere we'll unpack this in a bit but first we need to briefly cover the concept of stereographic projection which is the process of mapping a sphere onto a plane first imagine a light source at the north pole of this circle in 2d as this light beam passes through the circumference you can see there is projection of a purple point on the x-axis in one d now here we have the light source at the north pole of this sphere in 3d as the light passes through the boundary you can see a circle is projected onto the 2d plane note how the circle is larger closer to the north pole next we inflate a polyhedron with four colored faces we project in all directions from a fixed point on the north pole you can see how the faces on the sphere are projected onto the two-dimensional plane below we are demonstrating how a circle or sphere appears in lower dimensions so we can build intuition of how a hypersphere projects into one lower dimension the third dimension the top row is the view of a circle or sphere in what you might call their traditional space 2d for the circle and 3d for the sphere the bottom row is how that is projected onto a lower dimensional plane before we explain the hop vibration it's important to note i'm skimming over imaginary numbers quaternions and the complex plane to keep this as straightforward as possible but if you wish to dive into the nitty-gritty math there are links in the description [Music] in our first example we are stereographically projecting from the north pole of a sphere as we rotate around the circle you can see how this one point corresponds to one circle if we view this continuously all of these circles fill up the hypersphere [Music] let's track just one point as it moves from north to south we'll start at the equator the red point on the sphere relates to the red circle within the hypersphere the white torus or donut corresponds to connected points around an axis as the point spins southward you can see the circle getting smaller and tighter around the center until it's a non-skewed circle through the south pole as the point moves north the corresponding circle gets much larger until when it goes to true north it appears as a straight line it's hard to imagine but this is actually a circle through infinity this is how the circles would appear around a single axis say the equator it's easier to see this interaction if we look at just two complex points and their complementary circles as you view them move around an axis you can see how they are linked like a chain a third linked circle joins the dance [Music] and we can keep on adding circles to fit the entire hypersphere algebraic topology professor niles johnson at ohio state university created this excellent visualization of the hop vibration with its corresponding points on a sphere some hopf fibration facts none of the circles intersect each circle links to every other circle exactly once true south in 3d is a circle at the core of the hop vibration visualize this as the tightest circle in the center of the torus or donut true north in 3d is that circle through infinity remember this is the stereographic projection of the hypersphere in our familiar three dimensions while this is running two plugs first if you're enjoying this video please subscribe to my channel and second i highly recommend a similar visualizer and interactive tool created by nico belmonte it allows you to draw your own circles within the hop vibration and rotate the structure as well this awesome learning tool is linked below in the description and now we stereographically project these velarco circles on a few latitudes on the sphere as this runs i need to recommend the excellent resource dimensions dimensions-math.org and the corresponding joe slays youtube channel jos and a team of brilliant minds created an amazing nine-part series on dimensions which you should check out many of the visuals in this video come from that series which they so graciously distributed under a creative commons license everything is linked below in the description we then rotate around 4d space [Music] i also want to share some still images of the hop vibration from another angle which i think better illustrates the fiber bundledness of the structure while we're here why not take a ride inside a hop vibration you can see all the circles which look like lines are made up of zero dimensional points which kind of look like spheres [Music] as we mentioned earlier the hopf vibration is a fundamental element across a range of physics applications even if you'll never understand these advanced physics concepts i hope you gained an intuitive understanding of an essential feature of our universe the hopf vibration [Music] thank you for watching and please like and subscribe [Music] you

Channel: Carlos Farias

Views: 2,370,691

Keywords: Hopf, hopf fibration, hopf fiber bundle, hopf map, topology, topology mathematics lecture, eric weinstein, joe rogan, lex fridman, geometric unity, sacred geometry, jordan peterson, dimensions, 4th dimension, 4th spatial dimension, hypersphere, math, hopfion, heinz hopf, physics, geometric physics, planet hopf, hopf explained, n-spheres, stereographic projection, niles johnson, jre, joe rogan podcast, joe rogan experience, stephen wolfram, garrett lisi

Id: PYR9worLEGo

Length: 9min 42sec (582 seconds)

Published: Thu Jul 22 2021