#möbius klein

[welcome-to-dragonshead]

blood meridian / american psycho / i have no mouth and i must scream

disclaimer: these quotes are taken off-context and have been purposefully changed to fit better the context of my story. möbius is NOT like judge holden at all. and [redacted] isnt like patrick bateman either. hes still just as crazy though

[more-mathematics-mor-lam]

Finally some poetry of interest to me

[art-of-mathematics]

... if you wonder where I started to doodle the drawing of schröinger's cat in the klein bottle...

... It was on that non-linear regression visualization.

Just as I indulged in thinking about what this non-linear regression thing might depict in real life (hint: some non-linear causality stuff... so to speak.. errr... primarily conservation of momentum... ) I started to randomly draw a möbius strip.

And then I somewhat imagined a cat in that möbius strip and drew that cat body above the part of the möbius strip. (That's one of the pragmatic reasons why the cat is very dark there. The other reason is for pure aesthetical aspects... )

And then I had a cat floating in a möbius strip.

This led me to adding a klein bottle around the cat-möbius strip bundle.

This is literally recursive mindfucks...

The way this drawing of Schrödingers cat in a klein bottle emerged (process of creating it) is more art than the actual resulting artwork/drawing (the created/creation)

----- --- -- - -

This train of thought partially reminds me of the aphorism:

'The whole is more than the sum of its parts, as it contains both the parts as well as the interaction patterns between the parts.'

[strawbee-s]

she klein bottle on my möbius strip till i lemniscate

[forgottenbones]

Four Dimensional Maths: Things to See and Hear in the Fourth Dimension - with Matt Parker

[hkayakh]

Holy shit! The prophecy has been fufilled! A bug walking around on a Möbius strip can cover the whole area without having to ever cross a line!!! Now we just need to have it get onto a Klein bottle.

[lipshits-continuous]

Topological Spaces 13: Conclusions and Remaining Questions

We have almost come to the end of this series but first I'd like to give a little overview of what we've done and what more there is to topology!

I've put a few book recommendations at the end too!

Here is the link to the previous post.

In generalising the notion of continuity, we found a notion in which topological spaces are "the same" (more formally, it is possible to show that homeomorphisms define an equivalence relation on class of topological spaces). We have also found certain properties of spaces which remain invariant under homeomorphisms:

(path-)connectedness

compactness

Hausdorffness

Second Countability

These invariants allow us to tell when spaces aren't homeomorphic. Having these invariants is extremely helpful since the task of classifying topological spaces up to homeomorphism is an insurmountable one, even when restricting to "nice" spaces. These are not the only homeormophism invariants, just the common ones.

There are a couple of questions that we haven't really answered in these posts:

How do we know the spaces in post 12 are not all homeomorphic?

When is ℝⁿ≅ℝᵐ?

Although the spaces in post 12 look different, how do we show that rigorously? How do we know that S¹ isn't homeomorphic to S²? One tool is orientability. Loosely, a space is orientable if no matter how you traverse the space, your orientation is preserved. It turns out orientability is a homeomorphism invariant. Spheres and tori are orientable whereas the Möbius band, Klein bottle, and real projective spaces are non-orientable. Another invariant is something called the Euler characteristic which is a number associated to (some) topological spaces which helps us further distiguish between these spaces.

Another approach is to associate algebraic structures to topological spaces in such a way that homeomorphisms induce isomorphisms between those algebraic structures. This is the realm of algebraic topology and provides us with machinary sophisticated enough to prove a lot of things which are otherwise extremely hard to prove.

The second question may seem obvious. It must be only when n=m right? That is indeed the case, but how do we prove that? In the case where n=1, we can easily show that by removing a point from ℝ and ℝᵐ for m>1. ℝ\{0} is not connected but ℝᵐ\{0} is so they are not homeomorphic, so we can conclude that ℝ and ℝᵐ can't be homeomorphic. Technically we must show that if f:X->Y is a homeomorphism and A⊆X, then A is homeomorphic to f(A). However, doing this in general is not so easy. Sure, we could go to ℝ² and remove a line and say that ℝᵐ with a line removed is still connected but ℝ² isn't and then go to ℝ³ and repeat with a higher dimensional subspace. But this strategy of going case by case is not productive and would literally take forever. So is there a quicker way? Using something called homology groups, we find that ℝᵐ\{0} and ℝⁿ\{0} have different homology groups and therefore cannot be homeomorphic. Of course that is skipping over a tonne of theory.

Book recommendations:

Introduction to Metric and Topological Spaces by Wilson A. Sutherland. The structure of this series is heavily influenced by this book. There is an emphasis on building intuition for definitions and is great first reading of topology (in my opinion)

Topology by James Munkres. This book is full of a lot more than what we've covered here. Whilst I haven't read it cover to cover, I've found the sections I have read have been clear.

There's not anything else that I'd like to talk about in this post. If anyone has any questions about topology, don't be afraid to ask! The whole reason I started this post series is a mutual sent in an ask about topology.

[ilexdiapason]

🌻 give me your wisdom

hmmmm. youve probably heard of the möbius strip, which is a 3d shape that only has one side, but it also has two edges which means its not at peak topology. that award goes to the klein bottle, which is a 3d shape that only has one side and NO edges - whichever way you travel in one direction on the klein bottle, you'll always end up going back to where you started in the end, without ever skipping over to the opposite end of your path! and best of all these shapes can both be made in real life yippee

[knotty-et-al]

Sorry to bother you again but have you done origami? If you haven't I'd love to see some origami from you!

Hi there! Yeah, it's alright. I like receiving asks/messages, tough it often takes a lot of time for me to reply. (Sorry for that.)

I already made some Origami, and I want to share once I feel able or motivated to photograph them.

(I find photographing Origami crafts veeery difficult.)

I made some modular Origami, and I also created a Shishabo cube with 12 irregular tetrahedron modules that a friend has shown me. (My friend created those modules, and we sometimes/often do Origami together.)

- - -- ---

My recent Origami craft was this stacked flower petal thingy

(It is an art object. But it can also be used as tiny sorting box for stuff like earrings, piercing jewelry or tiny crafting utensils.)

Top view:

(I added a hook made from wire. )

Bottom view:

(with a bead and wire at the center)

Three layers: (plus I attached two tiny red leather möbius strips, a blue origami link that my friend has given me and a tiny glass klein bottle below.)

Attached to furtherly add a second layer:

(white and pretty neon colors)

These are other "flower petal boxes" I made:

Making the modules is fun. I can also do it blindly. (Helpful if my eyes are very tired but I want a calmful activity to do.)

- - -- --- -----

I want to share some other origami soon. (I will tag it as "origami". )

[welcome-to-dragonshead]

textpost goobers

[solreefs]

Klein bottle:

If you've ever heard of a mobius strip, it's kind of like that but 3d. It has no volume.

I have in fact heard of a Möbius strip! but now why is it 3D. there is no need.

for real though how does it not have volume? it’s got insides so why no volume?

[oogmonster]

He was a Möbius Strip Boy... she was a Klein Bottle Girl

[didanawisgi]

Is TIME a Möbius/Klein TORUS that BOTH 'Processes' (Forward Direction) and 'Precesses' (Backward Direction) through the Zodiac simultaneously?

Why did the ancient Egyptians and Sumerians use a 360 Day Calendar, where 1 Day = 1° of Rotation around the Sun? We don't know exactly but perhaps it is because the solar calendar is 365.24 days while the Lunar calendar is 354.8 days. The Mean Value of the Solar and Lunar Calendar is 360 Days....matching the both Egyptian and Sumerian Ancient Calendars.

'Procession' through the Zodiacal Equinox takes 360 days or 8,640 Hours (Horus) for one complete rotation. In this context its takes approximately 1 month to traverse from one zodiac into the next (for example Aquarius ♒️ into Pisces).

However, 'Precession' through the Zodiacal Equinox requires 24,000 years to complete its 'backwards' cycle. We are now in the Age of ♒️ Aquarius having just completed the Age or AEON of Pisces. Each AEON is approx 2,000 years in duration.

Time speeds up and slows down during the cycle, with the long end of the cycle being 12,960 years and the short end of the cycle being only 11,040 years. The total Cycle rotation is therefore 24,000 years. This is due to Mass-Time Dilation as a result of the increased Gravity on the short side of the cycle. 24,000 years = 8,640,000 Days...

*The Sun is approximately 864,000 miles in Diameter

*Jupiter is approximately 86,400 miles in Diameter

*There are 86,400 Seconds in One Day...

Gravitational Constant 'G' = 6.674 x 10^-11 m^3 kg^-1 s^-2

G / (√π - 1) = 8,640 (Hours in One 'Balance' Year)

{G / (√π - 1)}*10^3 = 8,640,000 (Days in One Great Year)

All of Time's 'Sexagesimal' (Base60) Calculations are derived using only the number 24 and its 'Orders of Magnitude' (2.4, 240, etc)...and 2+4 = 6.

1. The Great Year = 24,000 'Balance' Years

2. 'Balance' Days in One Year = {(2.4*2.4)+.24}*{(2.4*24)+2.4)} = 360

3. Hours in One Day = 24

4. Minutes in One Hour = (2.4*24)+2.4 = 60

5. Seconds in One Minute = (2.4*24)+2.4 = 60 (there are 86,400 seconds in One Day)

Robert Edward Grant

Illustrations by Robert Edward Grant

#robertedwardgrant #timeasakleintorus #simulationtheory

[visual-calc]

tricker treat!!!!

You get the Klein bottle! Similar to the Möbius strip, it is a one-sided, non-orientable surface— you could travel along it in a straight line and return to where you started, the only difference being that you’ve flipped upside down. Unlike the Möbius strip, a Klein bottle cannot be properly constructed in 3D space.

[thatbajablastboi]

A Klein bottle definitely has it much worse than a Möbius strip in that regard lol

A Möbius strip walks into a bar, sobbing. 

The bartender asks, “What’s wrong, buddy?” The Möbius strip replies, “Where do I even begin?”