Roger Penrose and the Impossible Triangle #shorts

[charlesthompson]

Our perception can bascially form things outside of reality

[kdog]

Topology is an area of maths i'd never even heard of, until recently. Fascinating AF.

[alessandrogiardino]

It is exactly the triangle on the cover of a Pulitzer book about Bach, Godel and Escher 😊

[scottychen]

That’s an excellent example in the study of orientation:


Namely, there’s a proof in Nigel Hitchin’s geometry of surfaces notes that suggests the orientability of all 2-manifolds that admit a smooth complex analysis (through atlas, as it were):


If youre claiming that the physical example of this (without using computer illusions to demonstrate implications of unorientable, or the modeling of platonic surface on a substantive trivial-geometrically subspaced 3-fold, e.g. the actual version of this which could be considered to not be existing in time as a platonic structure…..



then this would be an excellent way of exploring the proof by contradiction which uses a sneaky contradiction, and therefore axiom by choice:


Some may not consider this contradiction: that the hypothesis of contradiction is contradicted by construction of atlas which preserves a geometrically 2-fold orientation:

This invites the difficulties in 1-manifolds as things that are more difficult than 2-manifolds:

and so making the concept of ‘
unorientable’ inappropriate for the geometric 2-fold with atlas of smooth complex analysis.


Now, this structure, if it exists in life (no computer hacking is referred to by me, the observer) doesn’t change its orientation depending on whether or not you want there being a smooth complex analysis or just a smooth homeomorphic atlas, with no requirement for holomorphism.


So this object could be used to explore the construction involving a 1 fold that is removed on a choice of - perhaps, finite - covering of holomorphic transitions.


Gaussian curvature can be understood in many ways here, but may be considered to very irrelevant in this particular analysis.



That is, this real world object is portraying a definitite presentation of unorientability, and is therefore a stucture that is not a reimann surface.


Feynman’s brick can suggest otherwise as the unknowability of how to model this object still exists - alas; that is not the point of this demonstration.


The problem of orientation is
(1) metaphysically different conscerning a 1-manifold as opposed to a 2- manifold


(2) unorientable for a surface means you can’t just pick an INSIDE and an OUTSIDE as per the exactly 2-element image of the gauss map. Namely a normal vector to the surface is the result of having mapped that from the vector identifying the point of the surface. This is a discussion of exact surfaces only. - ref. Feynman’s brick.


(3) therefore the construction in Nigel’s notes finds a single orientation to the surface and traces that all the way around a covering of atlas codomains. Only after removal of the parameterizing curve is the unorientability something that’s constructed:

(4) apparently paradoxically, by a construction of exactly 1 orientation for the entire surface.

(5) that’s the contradiction: a contradiction by construction of an orientation, and therefore rendering the surface with smooth complex analysis unorientable. Ref. Feynman’s mirror


(6) and ref. To the application of the divergence theorem regarding fluid compressibility at the surface: consider the need for exactly 2 distinct domains for the image of the gauss map - inside and out.


There’s a great sensitivity here to the well-identifiability of exact 2-fold in the name of truth regarding the theorem that makes it ‘implausible’ to a real life structure.


Alas; that is not the point of the construction.



Nigel’s construction explicitly is that of a Mobius band: that’s the nature of these ‘constructed’ propertied of abstract nature.




This is a message to Kevin Mcgerty who gave me a failing mark for expressing the need for this shit about ‘surfaces of revolution’ as a structure that is more importantly ought to be applicable to a saxophone in the canonically observed sense of genus: more importantly in the sense that one should stop right at its conception and take care of any results that involve a study of unorientable manifolds found in the condensed matter of the object sitting in 3 -space.


Then the problem of genus can be dealt with experimentally.


So no GoS for that 17-holed 2 fold.


The results about concatonation and hitchin surgery on plane formations is the lesson that:

All exact unorientability is of the nature of being ‘contained’ in just a mobius band : in reference to a concatenation with something orientable and therefore identified by the number of holes

[senatorjosephmccarthy]

Is everyt🐕hing possible?
If everythin🐄g is possible,
is it possible something is impossible?
If something isn't impossible,
ev🎉erything isn't possible.
If something is impossible,
every🕊️thing isn't possible.

[Onelove-Oneheart-hc]

Are you gonna believe me or your lying eyes ? 😅

[SandeepKumar-nnpr]

Only in the video it appears like this.

[janetones]

Whaaat??? Green screen or what kind of illusion?⚠️🔺️🔻???