We Were Wrong About The Shape Of The Universe
Added 2024-09-30 15:34:11 +0000 UTCEnjoy the latest video!
Comments
What has all this got to do with Physics though? Well, space-time itself is a partial order on events, so this gives an empirical basis for the kind of physics I like, which I described in that video I made for you about the Norton's Dome script you sent me. I really enjoyed thinking about that stuff, even if some of the philosophical rambling was a little hard to digest! https://youtu.be/8pxJ6_2J7Fg
Ian Grant
2024-10-09 12:07:26 +0000 UTCThat graph deformation animation around 1:13 really made lights flash in my head when I saw it. It's awesome! I have been reading Eugenia Cheng's really superb book "The Joy of Abstraction" and I have finally read Chapter 10 on Order Theory. It starts describing Total Orders, which are relations where everything is related to everything. For example "less than or equal" is a relation that connects every natural number. So whenever you have two numbers a and b, say, either a is less than or equal to b or b is less than or equal to a. These relations can be described as a total ordering on the elements of the set, so you get a canonical way to write the set in ascending order, say. Then it goes on to describe Partial Orders, where this isn't the case. For example, if a and b are subsets of some set X then the relation "is a subset of" is not a total order, because you can have two disjoint sets like {1,2,3} and {4,5,6} and neither is a subset of the other. These relations can not be put into a canonical order. Then there are Pre-orders, which are relations where sometimes there are loops, like the ones in the graph of the Konigsberg Bridges. But pre-orders can be turned into partial orders by taking the quotient of an equivalence relation. That just means that wherever there is a loop in the graph, you consider all the vertices in that loop as being identical. Then the resulting relation is a partial order. So what is the connection with the topology? There is an idea which seems to have originated with Poincare, that if you consider all the functions h which map paths to paths in such a way that there is a real number parameter t, say, in the interval [0,1] and a function h(x,t)=y which takes values x on one path X to values y on the other path Y in such a way that if t=0 they remain on the path X and if t=1 then they are entirely on the path Y, and for other values of t between 0 and 1 they lie on paths that are linear interpolations of the paths X and Y, then two paths are said to be homotopic and h is called a homotopy. Then when you think of all the maps of the underlying space which preserve homotopy there is an equivalence relation: from Wikipedia, "A map f is called a homotopy equivalence if there is another map g such that f ∘ g and g ∘ f are both homotopic to the identity maps in their respective spaces. Two spaces are said to be homotopy equivalent if there is a homotopy equivalence between them." So homotopy equivalence is another way to identify circular paths in a preorder and make it a Partial Order. And there is theorem by Vladimir Voevodsky which connects this with intuitionistic logic. This field is called Homotopy Type Theory or HoTT. But there is a much more direct connection with Partial Orders via ideals, which are a kind of analogue of prime numbers for partial orders. This gives rise to a very abstract set of partial orders which has connections to all sorts of wildly different areas of mathematics and physics and Norman Wildberger has made a short introductory video about it here https://youtu.be/UieK7D7QLyA It seems these objects are in some sense the building blocks of everything we can reasonably say about any partial order.
Ian Grant
2024-10-09 11:46:53 +0000 UTCCliff Stoll! Even after almost 40 years, “Cuckoo’s Egg” is still one of my favorite non-fiction books of all time.
Gary Wilmot
2024-10-01 16:53:29 +0000 UTCI had seen a Numberphile video in which Cliff Stoll explained Gaussian curvature of 3D shapes by the sum of angles in triangles but I'd bever thought their effect on the nearby and entire universe to be so enormous. Thanks explaining these facts in detail yet still easily comprehensable.
Armin Quast
2024-10-01 12:13:10 +0000 UTCJade, great video. I was hoping something new would come out soon. It was a nice surprise to wake up and find this. That background radiation map is fascinating. I've never seen it before.
Jonathan Rayback
2024-09-30 18:56:00 +0000 UTC
