#constant negative curvature

[art-of-mathematics]

I know I have drawn this shape so often already, but the process of drawing it is so soothing.

And for that, I have drawn a kind of step-by-step guide how to draw that shape in the top of this drawing:

(from left to right: ) [sorry in advance if I make it sound more complicated than it actually is. If you want to draw it, I would advice you to focus more on these illustrations rather than on my gibberish-text.]

1. draw a 2-dimensional Cartesian plane - or, in other words: just draw a cross like depicted

1.1. mark 2 points on the y-axis/vertical line with same distance to the coordinate origin, then mark 2 points on the x-axis/horizontal line with the same distance to the coordinate origin. (The markings on the y-axis need to be farther away from the origin than the markings of the x-axis)

2. connect the 4 marked points like depicted above. This is a function plot of a tractrix. (it has two mirror symmetry axes. )

3. draw an ellipse and connect the two markings on the x-axis. This becomes a kind of "belt" for the pseudosphere (4th picture)

4. part the ellipse into whatever-amount-you-want of partings (like you would cut a cake) and slightly mark these.

5. now imagine you cut that shape horizontally on the outer surface. (In the 5th picture I depicted that with red-ish pen across the pseudosphere. ) -

6. then the cut shape needs to be "(shape) shifted". For that we use a set of marked points we did in step 4). Furtherly, we "cut" the ellipse open, and push one end of it to the top, and the other end to the bottom. (depicted in picture 6 )

7. Then we connect the rest to get that shape:

[mysticstronomy]

IS TIME TRAVEL IN THE PAST POSSIBLE??

Blog#315

Wednesday, July 19th, 2023

Welcome back,

We have all watched movies that deal with the subject of time travel. Most of the time, such movies use an exotic element of physics, such as wormholes, faster than light speed (warp drives) and other similar mechanics. There are a huge number of movies on the subject with back to the future, really standing out as the perfect flick. But our current understanding of physics does not support any of these elements as the viable methods of time travel.

However, there is one theory that has not been disproven by science and it is probably only way this science fiction can exist, albeit it would limit us to only time travel to the past. And that is the theory of a rotating universe, given by the mathematician Kurt Godel in 1949. Lets take a closer look at how time travel would work as per this theory.

Kurt Godel was a logician, mathematician and philosopher. And one of the notable works of his life was the formulation of a rotating universe. To describe this universe, he used the language Albert Einstein's theory of general relativity. The main purpose of this theory was to highlight that general relativity was not complete in itself and there were gaps that this parallel theory could fill in.

Time travel and Rotating Universe

Let’s take a closer look at this theory. The theory spoke of a universe which was rotating, much like our planets rotate along its axis. It deviates from our current understanding of the universe where we believe that it is expanding and planets, solar systems and even galaxies are constantly in motion. However, in Godel’s theorem, the universe, instead of moving, keeps rotating. He also imagines a negative constant that resists the centrifugal force to keep the universe static. Now, in this world, if you were standing still at one point, you could theoretically see galaxies and stars moving in the sky.

In fact, you could leave the Earth and be in space and you would still see everything rotating around you. But this would also mean, the farther you were from a celestial object, the faster it would appear to spin. And this is a rotation of space-time itself. So, light itself would be moving in curves to reach an observer. This curvature of light would also mean that it will take longer for information to come from larger distances compared to the closer ones. Extending the logic would also mean that at some point, light would completely curve and go back to the observer if they tried looking beyond a certain point.

In theory, it means, an observer could take a telescope and see reflective images of our world in it, just with things in the past. And if you applied the same logic to an object and sent them far away in space, they too would get caught up in the rotation and return to their starting point. But they would find that they returned to a point earlier than they left. So, in short, they time traveled to the past. Interestingly, while there is no proof of an universe which displays such characteristics, in our knowledge of physics, nothing stops such a universe from existing either. It’s just that we have not found such a universe so far. So, in reality, it is still, just a theory.

Originally published on tech.hindustantimes.com

COMING UP!!

(Saturday, July 22nd, 2023)

"ASTRONOMERS HAVE DETECTED A NEW GRAVITATIONAL-WAVE??"

[oliviabutsmart]

Physics Friday #6: WTF is Dark Energy/Dark Matter? (Part 3)

Preamble

Education level: High School (Y9/10)

Topic: Cosmology (Astronomy)

This is the last part of the Dark Energy vs. Dark Matter post. In this part, I'm going to cover the possible ideas behind what dark energy is. And more importantly, what exactly it does.

From the last two parts, what you need to know is that:

The "cosmological constant" was created by Einstein in order keep general relativity in a flat universe. He later thought of this as a really bad mistake

We later found out that we do need this constant, so that we can explain the accelerating expansion of the universe

In the Friedmann equations, the cosmological constant appears as a sort-of 'negative' energy density

Introduction: So what does this whole "Dark Energy" thing mean?

The Equation with a bunch of Ω

Dark energy is great and all, but what does this substance really do? What actually happens because of an accelerating universe?

Well, let's take a look at the Freidman equations again, however this time, I'm going to rephrase it into something else:

Image source: Wikipedia

Now what do all these terms mean?

The H's are the Hubble constant (H0) and the Hubble parameter (H), are just expressions of how quickly the universe is expanding. The larger the value H the larger the expansion.

The a variable is the scale factor. It basically says how big the universe is at a particular point in time. To make things easy, we always set a = 1 at our present time. And at the big bang, a = 0 (the universe has no size).

These omegas are the interesting parts. They represent the density of particular substances in the universe. The omega term with the R in it, is the density of radiation (like light). The omega term with the M in it is the matter density. Lastly, the Λ term is the dark energy density.

The omega with the k in it is a sort of 'fake density' - it actually refers to the innate curvature of spacetime, however we can conveniently express it as a sort of 'density' of space itself.

Each of these omega terms are actually unitless. As they are the ratio between the actual density and the so-called 'critical density' - the density at which we achieve a flat spacetime.

How we get Expansion

So notice, how, the matter and radiation terms shrink very quickly with increasing scale factor? While because of E = mc^2 we know that these two are equivalent, radiation shrinks faster because it undergoes redshift whereas matter just becomes less dense.

Unlike this, dark energy always remains constant. This means that dark matter will always try to expand the universe at an accelerating rate, as if it's there, and it's constant, it will always be there.

Now there are multiple ways we can play around with end-of the universe scenarios, but we'll leave that for later.

A "big crunch" scenario occurs when there is no dark energy at all to counteract the pull of gravity. All of the universe's stuff starts falling into itself suddenly.

If the dark energy density is large enough, but the curvature of the universe is large enough to counteract this, we get a "big bounce" where the universe will start shrinking and then suddenly jump up into an expanding universe.

The "big chill", is simply where we have just the perfect mixture of everything that the universe continues to expand together.

But what if dark matter was not constant? What if more continued to get added, more and more?

What we'd get is the "big rip" where the density of dark matter overcomes the strongest of gravitational interactions.

Image source: my lecture notes

So What is it? What is it made of?

Just like dark matter, we need to find a source. Unlike dark matter, we're less concerned with what it's made of because we already kinda know what it does and how it behaves, and we know it's likely it's own thing.

But let's assess the properties of dark energy:

It is everywhere

It is of constant density; expansion and curvature does not affect the amount of it

It either is intrinsically tied to space itself or more of it keeps getting created as space expands

It behaves as a negative energy density, pushing space apart instead of pulling it together

Quintessence and Quantum Mechanics

So what this dark energy could be is some sort of quantum field, as fields permeate the entire universe, and aren't affected suddenly by a change in spacetime. In fact, they are special because they are locally invariant under changes in things like the speed of an observer.

Perhaps the vacuum itself has a constant negative energy density? You may object to that because it violates the conservation of energy. But conservation doesn't mean it all sums up to zero, it just means it sums up to the same number.

Under the quintessence model, there exists a new field, which has a non-zero potential energy. But unlike this "vacuum energy field", this new field, can change it's strength and can interact with gravity easier.

This changing strength is where we get models like the big rip. It can even explain the cosmological crisis, which is another future post for sure.

Related to Dark Matter?

As the title says, maybe dark energy and dark matter are very similar? Current theories of this form state that dark matter actually decays into dark energy.

And additionally, the DE/DM relation is related to by the mass-energy equivalence E = mc^2.

This type of dark energy would interact with itself via the method of dark matter.

Black Holes! Probably ...

When we do calculations in the universe we usually do them in isolation.

When we formulated the physics of black holes, we formulated them in isolation, and assumed that normal newton's laws just applied far away from the event horizon.

But what if we instead assumed that far out the space would follow the Friedmann solutions?

What we get is that black holes should gain mass as the universe expands. In fact they expand in such a way that the black hole energy density remains constant.

Perhaps this imitation of the effects of dark energy is the real dark energy.

Maybe we're just Wrong?

Perhaps the universe is not as homogenous as we think, and that the density of what we see changes as we look out into the universe. Maybe this explains the affects of redshift and distance models, and because of this, we see that the universe is expanding, when it's actually just a visual illusion caused by matter distributions.

There are many alternative solutions that try to explain away dark energy by simply showing that one of the other assumptions we make when doing cosmological calculations are wrong.

The problem with this is that it creates more questions than answers. Why would the universe not be homogenous? Why should our view of the universe suddenly change?

Conclusion

Well, it's clear that dark energy still remains pretty elusive. That we don't really know what it is. But I'd certainly say that the idea of it being a quantum field or effect of general relativity may be the right direction. The problem, of course, is that we might require a theory of quantum gravity to answer this sort of stuff.

These posts were a doozy, covering the discovery of, and the search for dark energy and dark matter. The topics obviously go much much much deeper, however I certainly hope I got into the weeds enough.

Both mysteries are really important to our modern-day understanding of cosmology and physics. They offer new ideas to fields, particles, or phenomena that might affect our perceptions of gravity itself.

Clearly gravity is part of the problem here, because both of these theories are related to gravity in some way, one on the local scale, and one on the global scale.

As always follow if you would like to see more of these posts! And please, feedback is very welcome.

Next week I was considering talking about thermodynamics, but I might want to do one on E = mc^2. I might actually want to poll this one. I know not a lot of people will vote on it but if I were to do anything else I would just flip a coin.

Edit: the poll is on this post

[celestialmazer]

Coral reefs inspired the crochet exhibition “Austrian Satellite Reef,” by Margaret and Christine Wertheim. It is on view at the Schlossmuseum Linz in Austria.Credit...David Payr for The New York Times

The Crochet Coral Reef Keeps Spawning, Hyperbolically

The long-running project, sometimes described as the environmental version of the AIDS quilt, thrives on convoluted math and a sea of volunteers.

By Siobhan Roberts

Published Jan. 15, 2024 Updated Jan. 16, 2024

Every year after the full moons in late October and November, Australia’s Great Barrier Reef begins its annual spawning — first the coral species inshore, where waters are warmer, then the offshore corals, the main event. Last year, this natural spectacle coincided with the woolly propagation of two new colonies of the Crochet Coral Reef, a long-running craft-science collaborative artwork now inhabiting the Schlossmuseum in Linz, Austria, and the Carnegie Museum of Art in Pittsburgh.

To date, nearly 25,000 crocheters (“reefers”) have created a worldwide archipelago of more than 50 reefs — both a paean to and a plea for these ecosystems, rainforests of the sea, which are threatened by climate change. The project also explores mathematical themes, since many living reef organisms biologically approximate the quirky curvature of hyperbolic geometry.

Within the realm of two dimensions, geometry deals with properties of points, lines, figures, surfaces: The Euclidean plane is flat and therefore displays zero curvature. By contrast, the surface of a sphere displays constant positive curvature; at all points, the surface bends inward toward itself. And a hyperbolic plane exhibits constant negative curvature; at all points, the surface curves away from itself. Reef life thrives on hyperbolism, so to speak; the curvy surface structure of coral maximizes nutrient intake, and nudibranchs propel through water with frilly flanges.

Christine Wertheim, a creator of the project and the artistic force behind it. Credit...David Payr for The New York Times

Margaret Wertheim, a science writer and twin sister to Christine, is in charge of the project’s scientific and mathematical components and its management. Credit...David Payr for The New York Times

In the artworks, marine morphologies are modeled — crocheted — with loopy verisimilitude. A bit like Monet’s water lilies, the crochet corals are abstract representations of nature, said Christine Wertheim, an artist and writer now retired from the California Institute of the Arts. Dr. Wertheim is the driving artistic force behind the project, which she created with Margaret Wertheim, her twin sister, a science writer who is in charge of scientific and mathematical components as well as management. The Wertheims, Australians who live together in Los Angeles, spun out the mother reef from their living room many moons ago, in 2005.

Crochet Coral Reef exhibitions typically have two main components: The Wertheims provide an anchor, of sorts, with works from their collection that they have crocheted over the years. They also incorporate pieces by select skilled international contributors. One is a “bleached reef,” evoking corals stressed by increases in ocean temperature; another, a “coral forest” made from yarn and plastic, laments the debris that pollutes reef systems.

Then in response to an open call, volunteers far and wide crochet a pageant of individual specimens that agglomerate in a “satellite reef,” staged by a local curatorial team with guidance from the Wertheims. The Wertheims liken this hive mind to a friendly iteration of the Borg from “Star Trek: Next Generation.” All contributors are credited.

The Linz satellite reef takes colorful inspiration from traditional Austrian craftswomanship, such as blue-and-white fabric (Blaudruck) or gold-and-black hats (Goldhauben). Credit...David Payr for The New York Times

The largest satellite reef thus far coalesced in 2022 at the Museum Frieder Burda in Baden-Baden, Germany, with some 40,000 coral pieces by about 4,000 contributors. The Wertheims call this the Sistine Chapel of crochet reefs (documented in a splashy exhibition catalog). But the show at the Linz Schlossmuseum, which is dedicated to natural science as well as art and culture, is reminiscent of the work of the painter Giuseppe Arcimboldo, whose collage portraits from depictions of fruits, vegetables and flowers are “fantastically heterogeneous, also very funny and clever,” Ms. Wertheim said.

The Linz satellite reef unites some 30,000 pieces by 2,000 crocheters. The disparate parts take colorful inspiration from traditional Austrian “craftswomanship,” as the exhibit text puts it, and there is a vast, glittery coral wall that gives a nod to the artist Gustav Klimt. In the Wertheims’ view, however, the crochet coral project is proof that it is not always lone geniuses who create great art, but also communities. In the art world, that is a radical idea, they noted, yet in science big collaborative projects and papers with thousands of authors are not unprecedented.

Primordial bamboozlement

Scientifically, the Linz exhibition holds special symbolism since, as the narrative explains, the region was previously occupied by an “ancient primordial sea, filled with corals whose remains can still be found in the basins and Alps of Upper Austria.”

The mathematical dimension of the story intersects (from afar) with research by the applied mathematician Shankar Venkataramani and his students at the University of Arizona. They use idealized models to study hyperbolic surfaces in nature. “It’s all around us,” Dr. Venkataramani said — consider the ubiquity of curly kale. “The question is, Why is it all around us?” The textbook evolutionary benefit, he said, is that it helps optimize processes like circulation and nutrient absorption. His research group’s studies show additional advantages, such as affording a structural “sweet spot,” making organisms neither too rigid nor too flexible and allowing them “to move and change shape with a small energy budget.”

The Wertheim sisters share an appreciation for Dr. Seuss’ playful absurdity, which is a touchstone for the project. Credit...David Payr for The New York Times

When Margaret Wertheim, who studied math, physics and computer science at university, learned hyperbolic geometry, she found it “a bit bamboozling.” She took the principles more on faith than understanding. Yet through crocheting models, she said, “you really do learn in a very deep way what a hyperbolic structure is, and in a way that I think is very powerfully pedagogical.”

Feeling the frills

That the hyperbolic plane could be looped into existence with a crochet hook became apparent only a quarter-century ago. Daina Taimina, a mathematician now retired from Cornell University, made this discovery while preparing a geometry course. “I needed to feel it,” Dr. Taimina said. Investigations with the Wertheims in the early to mid-2000s planted a seed for their coral-reef project (and a chap book, “A Field Guide to Hyperbolic Space”) and for Dr. Taimina’s outreach workshops and shows (and a book of her own, “Crocheting Adventures With Hyperbolic Planes”).

Further back, in 1868, the Italian mathematician Eugenio Beltrami constructed a parchment model of the hyperbolic plane — and he rolled it into a negatively curved surface called a pseudosphere (as one does). A century later the mathematician William Thurston independently had a similar idea, using paper and tape.

Dr. Taimina encountered a crumbling paper version in 1997 at a workshop by David Henderson, a Cornell mathematician and her partner. Dr. Henderson had learned the model-making technique from Dr. Thurston. On the spot, Dr. Taimina set out to construct something more pliable and durable for her course. When she tried knitting, the result was too floppy, unwieldy. Crochet proved the perfect medium. Dr. Taimina devised a simple algorithm: Increase the number of stitches in the constant ratio N+1. For instance, say N=6: crochet six stitches, and on the seventh stitch, increase by crocheting two stitches into one; repeat, row upon row.

Eugenio Beltrami’s 19th-century parchment model of the hyperbolic plane, displaying constant negative curvature. Credit...Department of Mathematics, University of Pavia

Daina Taimina’s largest hyperbolic crochet model, using 4.8 miles of yarn and stretching more than two feet wide, earned a 2013 Guinness World Record. Credit...Daina Taimina

“You can experiment with different ratios, but not in the same model,” she cautioned in an article for “The Mathematical Intelligencer” that she wrote with Dr. Henderson. “You will get a hyperbolic plane only if you increase the number of stitches in the same ratio all the time.”

Dr. Taimina also joined Dr. Henderson, who died in 2018, as a co-author for revised editions of his book “Experiencing Geometry,” wherein he described his belief “that mathematics is a natural and deep part of human experience and that experiences of meaning in mathematics are accessible to everyone.”

The Wertheims adopted a similar vision with their Institute for Figuring, a nonprofit where projects are motivated by the belief that people can play with and aesthetically appreciate — and thereby acquire an understanding of — mathematical ideas.

With her science training, Margaret’s instinct had been to follow Dr. Taimina’s algorithm to the stitch. But Christine’s artistic sensibility was to break the rules and go wild. For instance, crochet a few rows, increasing every third stitch, and then switch to every fifth stitch, and then to every second stitch — the result is not perfectly hyperbolic, because the piece does not have regular curvature.

For the Wertheims, embracing that irregular frilliness was the moment their crochet reef project was born: The erratic algorithms begot a riotous taxonomy, a woological seascape of creatures that all the more closely emulated the geometrically aberrant curvatures of their real-life biological counterparts.

Patterns of change

Christine Wertheim noted that while crocheting, she could tell when a structure wanted to form another ruffle. “It’s interesting to feel that in your hands,” she said. Credit...David Payr for The New York Times

Another crochet-coral incarnation recently emerged from a pond of creativity organized by the Carnegie Museum of Art in Pittsburgh, a city known for its three rivers: The Allegheny and the Monongahela Rivers converge to form the Ohio, which empties into the Mississippi, which empties into the Gulf of Mexico, where coral spawns after July and August full moons. This show, organized by Alyssa Velazquez, a curatorial assistant of decorative arts and design, features only a satellite reef made by 281 community crocheters.

Ms. Velazquez noted that the Wertheims’ project takes inspiration from the fiber-art movement — advanced by mostly women, among them Sheila Hicks, Tau Lewis and Marie Watt — and then democratizes it. As (mostly) women gathered and interlocked loops of yarn, Ms. Velazquez observed the lines of conversations: memories of time spent at local waterways, recycling habits, the chance to crochet something other than baby bootees. At that, the enterprise represents “the creative potential for environmental dialogue and new ecological behaviors,” she said — invoking imaginative yet concrete patterns of change.

A version of this article appears in print on Jan. 16, 2024, Section D, Page 1 of the New York edition with the headline: Crochet Coral Reef Grows Hyperbolically.

More at This Is Colossal

https://www.thisiscolossal.com/2022/05/crochet-coral-reef-wertheim/

[harbourfronttechnologies]

Volatility Smile in the Commodity Market

The volatility smile is a phenomenon observed in the options market where implied volatility tends to be higher for out-of-the-money (OTM) options compared to at-the-money (ATM) options. It refers to the graphical shape of the volatility curve, resembling a smile when plotted against the strike prices of options. The volatility smile suggests that options traders are willing to pay higher premiums for OTM options, indicating a higher perceived risk of extreme price movements in the underlying asset. This departure from a flat volatility curve challenges the assumptions made by the Black-Scholes option pricing model, which assumes constant and symmetric volatility across different strike prices.

Reference [1] proposed to use the new method developed by Carr and Wu in order to study the volatility smile of commodities. Specifically, the authors examined the volatility smile of the United States Oil ETF, USO.  They pointed out,

In this paper, we study the information extracted from the no-arbitrage Carr and Wu (2020) formula based on a new option pricing framework in the USO option market and investigate the predictability of the information in forecasting the future USO returns. The risk-neutral variance and covariance estimates can be obtained from the no-arbitrage formula under the new framework. We document the term structure and dynamics of the risk-neutral estimates which lead to a “U”-shaped implied volatility smile with a positive curvature.

The figure below shows the volatility term structure for the United States Oil ETF,

[caption id="attachment_9704" align="aligncenter" width="427"] Volatility smile of the United States Oil ETF, USO[/caption]

Usually, an equity index such S&P 500 exhibits a downward-sloping implied volatility pattern, i.e. a negative implied volatility skew. Oil, on the other hand, possesses a different volatility smile. This is because while equities are typically associated with crash risks, oil prices exhibit both sharp spikes and crashes, leading to a different implied volatility pattern. This highlights the importance of considering the specific characteristics and dynamics of different asset classes when analyzing and interpreting implied volatility patterns.

Let us know what you think in the comments below or in the discussion forum.

References

[1] Xiaolan Jia, Xinfeng Ruan, Jin E. Zhang, Carr and Wu’s (2020) framework in the oil ETF option market, Journal of Commodity Markets, Volume 31, September 2023, 100334

Originally Published Here: Volatility Smile in the Commodity Market

from Harbourfront Technologies - Feed https://ift.tt/dBFR1Yf

[semcoinfratechworld]

Stacked vs. Wound Batteries: The Shocking Showdown!

Prism, pouch, and cylindrical batteries are the different classifications for lithium-ion batteries based on how they are packaged. Prism and pouch batteries offer flexibility in packaging, allowing stacked or even winding from the inside out. Cylindrical batteries, like 18650, 21700, and 4680, are limited to winding due to their inherent design with curved walls.

The winding process has a relatively long development time, low cost, and a high yield rate. In contrast to winding and tandem batteries, the tandem process has gained popularity as electric vehicles have become more commonplace. This is because of its advantages, which include long cycle life, small internal resistance, stable structure, and high-volume utilization.

Production Process of Stacked & Wound Batteries

The two main components of the assembly process for intermediate lithium battery cells are winding and lamination. Winding transforms individual components into rolled products with precise dimensions, integrating cores, diaphragms, belts, and more. Laminated cells skip the winding, stacking pole pieces and separators into layered cores.

Based on the battery's shape, pouch cells are limited to using stacking technology. Prismatic batteries can be wound or layered, with wind being the more popular method at the moment. The only way to wind cylindrical batteries is through stacked battery technology.

Advantages of Stacked Batteries Over Wound Batteries

Higher Battery Energy Density The winding's corner has a curvature, and its space utilization rate is lower than stacking's. Stacking squeezes more energy in by filling corners, boosting density. The energy density in stacked and wound batteries can be raised by roughly 5% in comparison to wound batteries.

The Internal Structure is More Stable The uneven stress in the corners of stacked batteries is not an issue, in contrast to wound batteries. Because each layer's expansion force during repeated battery usage is comparable, stacking the batteries helps maintain the interface's flatness. The positive and negative electrodes of the winding battery will expand when lithium ions are embedded in them during operation. This will cause uneven internal stress in the inner and outer layers of the winding corner, which will result in wavy deformation. This deformation will speed up the instability of the battery's internal structure, cause an uneven current distribution, and deteriorate the battery interface.

Increased Security The stacked batteries are evenly stressed, and there is no bending problem at both ends, making the battery safer. In contrast to stacked batteries, the coating material will be significantly bent and deformed after the pole pieces at both ends of the winding are bent, and the "C corner" bend is more likely to experience burr issues and powder loss. In extreme circumstances, it will result in a battery short circuit and thermal runaway.

Longer Cycle Life The number of tabs in stacked batteries is twice that of wound batteries, and the more tabs there are, the lower the resistance and the shorter the electron transport distance. As is common knowledge, heat generation decreases with increasing resistance and increases with decreasing resistance when voltage and time remain constant. For this reason, stacked batteries have a longer service life than wound batteries. Wound vs. stacked battery lifecycle.

Disadvantages of Stacked Batteries over Wound Batteries

The equipment efficiency is low, and the area occupied by the stacker crane is larger than that of the winding machine.

High Investment in Equipment Ten winding machines are required for a single production line. A winding machine on a production line costs roughly 30-35 yuan, based on a price of 3-3.5 million yuan per unit. The number of cells on a production line determines how many stacker cranes are needed.

The Yield is Low The wound battery has a high pass rate and is simple to cut. It is easier on each cell because it only needs to cut the positive and negative poles once. However, when stacking and cutting, each cell has dozens of small pieces, and each small piece has four cutting surfaces, making it easier to produce defective products than with wound or stacked batteries.

Difficult to Control The battery can be wound with just two pole pieces, and spot welding is simple and requires only two spot welds per battery, making control over the process easy. Nevertheless, there are a lot of stacked pole pieces, and virtual welding is more likely to happen than with stacked vs. wound battery performance. It is challenging to operate when all of the pole pieces need to be spot-welded to a single solder joint.

After slitting, the diaphragm, positive and negative electrodes, and pole piece are wound together by adjusting the pole piece's speed, tension, relative position, and other factors. This process is known as winding. Only lithium batteries with a regular shape can be prepared due to the process's characteristics.

In contrast to wound batteries and laminated cells, laminated cells are created by alternatingly stacking the positive and negative electrodes and separators via a sheet feeding mechanism. This process produces more flexible, regular, or specially-shaped lithium batteries.

Conclusion

The wound battery has a high internal resistance, which can be significantly decreased by improving the structure. An all-tab structure, for instance, can accomplish a comparable degree of internal resistance to a stacked structure, but it needs more sophisticated machinery and tighter quality control. Stackable batteries have a flat structure, low internal resistance, and high space utilization compared to wound and stacked batteries, and excel in the field of energy-storage batteries.

[chiliahedron]

A lot of traditional origami figures start with making a ‘base’, which has the same number of flaps as the figure will have appendages. Usually the base will have four flaps, since you’re starting from a square and if you fold its diagonals you have four sections which are congruent isosceles right triangles. The traditional crane has four appendages: two wings, a head, and a tail, each one in some sense formed from one of those triangles.

If you had five isosceles right triangles stuck together in the same way, it wouldn’t lay flat, because you can’t make a regular pentagon with 90° angles in the Euclidean plane. But you can fold a square sheet of paper into five slightly smaller isosceles right triangles by layering some parts over each other, into a kind of inverted cone shape (specifically, “Montroll’s five sided square”), and then do the same thing to those triangles as you would the four you’d get from a square sheet of paper to get a flat base with five flaps instead of four. This lets you add an extra appendage to many traditional origami figures, like this two-headed crane!

If you have a large enough sheet of paper with constant negative curvature (hyperbolic paper), you can make a regular n-gon out of 90° angles for any n. This makes folding the two-headed crane easier, if you already have enough hyperbolic paper and a hyperbolic “desk” to fold it against, like Alperin, Hayes, and Lang.

From step five onwards, the instructions at the end of the linked paper for folding their hyperbolic crane also work if you are starting from Montroll’s five sided square, which is much easier to make/obtain than hyperbolic paper.

[timetravelandgravityengineering]

Looks amazing

From what I understand this isn't a de sitter space. Though it incorporates those spaces in a progressive form. It seems be the creation of time space time and thus a new time/space in of itself in a dynamic model.

Requires more study

Using a wave collapse from a string through uncertainty to milwoski space and thus Einstein from a holographic born from a collapsed ratio and probalistic amplitudes.

Anti de sitter - 2 negative space curvatures

Minkowski space - 0 space

De sitter space - one hyperbolic sheet

This is 3 (+-ņ0 ţ) time in a 1d space Sphere arising from string dimensions of order 4 expanded resulting in Minkoski space resulting in a 2x2d surface and a creation of an infrance pattern in a 4d manifold.

This is a string time space geometry into atomic scales of the standard and space time which relates to inflation but also allows a cyclical pattern in fractal scales from a string quantum cascade resolving the DeM ratios

It would seem to support a many worlds but a local version of it being that fundamental time is a 2d non localised constant but realtive time is a 4d localised event. Now being 4.8L~ sec of probability amplitudes

Which may help description of models.

Being a neg and positive resolution into a ⁰⁰⁰ minkoski space with 1 time dimension

However this is compatible with these spaces and a time space and metric in its own right.

It could be described as anti minkoski space being 3 of time and 1 of space. Resolves string dimensions with DeM and uncertainty. However the surprise is the wave collapse and wave particle duality as a many positioned particle in time/space.

Though I may be gave a different name 😎

String geometry from ratio potential

Yeah it essentially states that on a infinitesimal scale all points are the same point their proximity close and that all energy is the same energy. That every atom and quanta is bound up in a potential. Inculding time and space and especially electrons.

[rhodesmusic]

Listen man constant negative curvature surfaces can get you through some tough nights

It's his personal pseudosphere for personal use.

[art-of-mathematics]

Pseudosphere transforming into Dini surface (WIP)

Here is a gif to what I try to depict:

Great website:

[slack-wise]

Hiroshi Sugimoto. Surface of revolution with constant negative curvature

[mornyavie]

@mornyavie too true lol. PLEASE TELL ME MORE ABOUT THE EXPANDING AND THE DARK ENERGY?! 

ok hello!

I am only one person so others feel free to chime in if I make a mistake! Also if I say anything too confusing feel free to ask for more explanation.

To start: in physics we talk about space having a structure. Even if there is no stuff there, the empty space can have a structure to it, and that has important effects on questions like “how far is it from point a to point b.”

Fortunately there is a good analogy: the surface of the earth is a sphere. This is a type of structure of space. If you try to represent this sphere on a flat piece of paper like a map, you have to make compromises, because the structure of these two spaces is different.

[image from wikipedia; this is a mercator projection]

The main important thing to talk about here is curvature. A sphere has positive curvature, because it curves inwards until it meets itself. A flat piece of paper is, well, flat, and has zero curvature. The other type of possibility would be an object called a hyperbola, which has negative curvature and bends outwards to infinity.

Actual space, of course, is not 2-d like a sheet of paper; we have three spatial dimensions (up/down, left/right, forwards/backwards) as well as time. (there’s proposals about there maybe being more dimensions but that’s not important right now). This is sometimes called “3+1-d space.” It’s harder to visually understand the difference between flat and curved spaces in higher dimensions, but mathematically it’s the same as the sphere vs. sheet of paper thing.

.

The other useful thing to talk about is the radius of curvature. You can have a really big sphere (big radius), or you can have a tiny sphere, or anything in between. This is often compared to blowing up a balloon; as the balloon gets bigger, the distance between two dots on the balloon will increase even though the dots aren’t actually moving.

[hilarious picture obtained from stack exchange]

As far as we can tell, our universe is flat probably, not curved, but it is getting bigger. You can tell this by measuring the light from stars. If the star is moving away from us, the light will look more red, while if it’s moving towards us the light will look more blue. Pretty much uniformly all stars are moving away from us, which means either we’re the center of the universe or the universe is getting bigger, and it would be pretty wierd if Earth Specifically was the center of the universe.

This is what happened in the Big Bang. We had a very small universe that suddenly became very big, and the expansion has been slowing down ever since.... we thought?

[wikipedia again]

Gravity pulls stuff closer together, so all of the stars and planets and whatnot in our universe should be causing that expansion to slow down. We assumed that was true for a while. But lately, by doing that “is the light more red than normal” measurement more carefully, we’ve found that actually the expansion seems to be getting faster again.

It’s called “dark energy” because in order for the universe to get bigger it needs an energy source “pushing” it, but other that that we have basically no solid understanding of why this is happening.

Maybe we just don’t understand gravity properly and once we find a better theory “dark energy” will be part of it.

Maybe Space just has energy associated with it existing; this is called a cosmological constant because 100 years ago Einstein made up a cosmological constant, realized it didn’t make sense, and called it his biggest blunder, and then later we found out that he may have been right but for the wrong reasons (he put in a cosmological constant because he thought the universe could not be expanding or contracting, and used his version of the constant to make a static universe).

Maybe there’s extra energy hanging out in the form of certain types of matter that we don’t know about yet, or something like that.

Maybe the measurements were bad and this is just wrong.

Who knows!

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One more thing: if we go back to the whole curvature thing, I said that the universe seems to be flat. However, we can also calculate whether the universe should be flat, and if the only stuff in the universe was the stuff we could see, it would actually have positive curvature like a sphere. Some kind of extra energy source has to be there for it to be flat. This is where you get numbers like “70% of the universe is dark energy;” that’s based on trying to calculate the curvature of the universe.

[wikipedia again]

Maybe we’re wrong about how this works or about how the energy works? maybe dark energy exists? who knows!

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(Side note: dark matter and dark energy, despite the names, are unrelated. Dark matter is the fact that we can measure the weight of galaxies and they’re heavier than they look, so there must be something there we can’t see. Dark energy is this whole expanding-universe thing)

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Ok hopefully that explains a bit of why physicists talk about dark energy. Now the “end of the universe” stuff; this is where it might get a bit existential for those who don’t like contemplating eternity, so feel free to tap out.

If the universe stops getting bigger and in fact starts getting smaller again, that may lead to something jokingly called the “Big Crunch,” where everything gets closer together until it all becomes one enormous fireball and then vanishes. Whether this would actually happen or a difference process would kick the universe bigger again, who knows; in fact, some people think the universe may be a sort of cycle, where the Big Crunch reaches a critical point and the universe gets thrown back outwards as a new Big Bang. But this is basically all talking; we don’t have any real evidence about it.

The other option is called Heat Death. For this one it doesn’t actually matter whether the universe is staying the same size or getting bigger; what matters is that it doesn’t collapse and so it exists forever.

Forever is a long time. You know how, if you put ice in water, eventually it melts and you have one big tub of slightly colder water? That’s happening, slowly slowly slowly, on a cosmological scale. Heat in the universe is very much not evenly distributed; most of it is in the stars burning, or some dust and rocks and planets, and the void of space in between is very cold. The laws of thermodynamics show that this won’t be true forever. Entropy increases; heat flows from hot areas to cold. A star explodes, and the new stars that form don’t use all of its energy, some of that energy escapes and mixes into the void. Eventually the universe will be more or less uniformly mixed together, all at the same temperature and roughly the same everywhere, and so stuff won’t happen because sort of by definition “stuff” is “a place where things are different from the things around them.” Stars stop existing, planets fall apart, etc.

When I say “eventually” I mean it. You know those pictures that try to explain the difference between a million and a billion?

[image from The Daily Dot]

A million is 10^6 (1,000,000; one followed by six zeros). A billion is 10^9. Our universe has existed for about 10 billion years. At a rough estimate, this kind of “heat death” couldn’t happen for at least 10^100 years.

[this is just a screenshot of a program called Mathematica because I didn’t feel like typing 100 zeros]

Take that little pile of rice representing a billion. Now fill up the entire milky way galaxy with rice, and you’ve gotten to about 10^50 if my estimate is correct. But don’t be fooled; you aren’t halfway there. It’s going to take 10^50 more galaxies to actually reach 10^100. 10^100 would fill up the entire known universe with rice, and you still wouldn’t have enough space to fit it all.

So the heat death of the universe is gonna take a while.

Don't talk about it if you don't want to, but DO you have any idea how you might try to help someone experiencing The Existentialism? This might sound silly but I'm a physicist so I've been around a couple people when they sort of process The Concept Of Space for the first time, and I never know how to help them because it's just not something I experience at all. And I can definitely see it happening again in the future so it'd be nice to have a better perspective on how to help.

Hmmmmmm I'm not sure tbh. I think probably the most helpful thing is. Like. Grounding in everyday Animal Reality? So maybe just go "yeah it's pretty wild" and give them a lollipop or something 😂😂😂

[wumblr]

what... does radiation look like in positive curvature. i mean not like nuclear radiation but the literal, underlying concept of radiating (i.e. water sprinklers radiate water)

i mean that's a silly hair to split because it's not exactly relevant (i could talk about the same things in the context of nuclear radiation specifically) but, in a radiating physical event, particles (or whatever) move outwards from the epicenter and in lines that grow progressively further apart from each other

in negative curvature, parallel lines eventually intersect

does positive curvature cancel out the radiating effects of radiating? is this why positive value for lambda is said to cause "expansion" (a theory i don't believe is correct)?

because it seems to me like positive curvature is likelier to increase collision and accretion events. that seems like way more of a direct, mechanical explanation for the reason the distances to other galaxies appear to grow vaster

(or, in my interpretation, it requires more energy to cross the distance, while the objects stay in approximately the same place, because accretion events have increased the pull of the local supermassive, over time, which changes the escape velocity for a galaxy -- and, crucially, for every object and interacting object pair within it. this of course would also mean the strength of gravity is not a constant. i think the problem there, and the reason expansion has been more commonly accepted for a century, is because you can't really measure a change in the strength of gravity. because every other physical quantity would change with it, wouldn't they?)

again, it seems like a binary supermassive merger is the relevant observational test arena for this idea. maybe the standard candle discrepancy but it seems like trying to start by sorting that out would drive you mad. it seems easier to forge the theory in a different fire, and then test the results on the standard candle discrepancy to see what new distance ladder it constructs

[virtualgeometry]

HIROSHI SUGIMOTO

Surface of Revolution with Constant Negative Curvature, 2006

https://www.sugimotohiroshi.com/mathematical-models

[seeuonadarknite]

one of the boys — yandere overhaul x f. reader

apparently woman are looked down upon in the yakuza,, shame on y'all ;( forgive me for my poor smut writing skills, i might write longer smut in the future if y’all want it 😳

warnings: dubcon, stalking, gore (overhaul's quirk), explicit smut.

All it took was a dramatic wardrobe change, a short haircut, extensive vocal practice, and the removal of all of your files to worm your way into the Shie Hassaikai.

That's right, you were a woman that was a part of the Yakuza. It wasn't a simple task on your end; you had to let go of everything you knew in order to live this dangerous life.

You believed it was unfortunate how women were looked down upon in the underground world. It took months of preparations to prepare yourself for this role, not even including your extensive quirk training.

Finally, after months of trying to remove all personal and medical files on yourself, you had successfully managed to fake your identity and wow the Shie Hassaikai with your unique quirk.

You had an informational quirk that allowed you to solve any case within seconds. The only downside to this quirk was how you needed the names of the people involved in the case. Your quirk also didn't give you any sort of super intuition, so you still needed to be on guard most of the time.

This quirk was honestly quite useful when it came down to it, so why did you decide use it in the way that you did? Why not become a hero, or an assistant, or even a villain for that matter? Why give your life away and allow yourself to become a slave to the underground world?

It was simple, really. It was all because of the boss of the Shie Hassaikai; Chisaki Kai. Also known as Overhaul.

This man ridded you of the only parental figure you had in your life. Your father. It was only one year ago when you arrived to an empty house and a note on the counter left by your father. It was a warning, telling you to stay away from the mob boss that went by Chisaki Kai.

Just his name was enough information to use your quirk, allowing you to crack the case in a matter of 5 seconds. He had blown your father up. All because he was beginning to figure out the secrets on who he really was.

This was enough to encourage you to avenge your father, and finish the job. It was a miracle that you were recruited, and no one had further questioned your mysterious backstory.

Well, almost no one.

The mob boss himself had questioned your presence the second you become apart of Shie Hassaikai. Why were your files almost completely nonexistent? Why did you seem so on edge around the other members? Why did you seem so interested in the name of the past leader?

Oh. Oh.

Now he got it. He pieced your suspiciousness and your quirk together. You were trying to get the dirt on him, weren't you? Why else would you willingly join the Yakuza with your useful quirk with open arms?

Two could play at that game, then. Kai began discreetly following you around, keeping tabs on important things like who you talked to, what you did on your phone or laptop, what times you'd leave the vicinity, and other things of the sort. Perhaps he could admit it was getting a little bit obsessive.

But he swore to himself that he was just trying to protect his status and not allow any traitors to get away with anything. That's why he took it upon himself to follow you home one night. He needed to make sure that you weren't spreading information or doing anything suspicious.

But boy, did he not expect to see what he saw through your bedroom window. You had taken your uniform off, along with layers of binding clothing to hide any type of curvature. What really surprised him, was the fact that you had breasts and what looked to be laced panties covering your womanhood.

One would assume that this would infuriate the mob boss, knowing that he had been betrayed and lied to by one of his men. But the tent growing in his pants proved otherwise.

Oh, he would have fun with this one.

—

You had been summoned to the boss's private office. Odd, on normal circumstances if the boss wanted to meet with you, it'd be planned out and in a meeting room. But you had received the order on one of your night shifts by a tall, hooded, masked man.

Shaking off any paranoid thoughts, you gently knocked on the large mahogany door, almost hoping you wouldn't receive an answer. To your dismay, you heard the monotone voice of your boss. "Come in."

As soon as you stepped foot into the office, you almost gasped at the sight before you. His office was huge. It was probably the size of the first floor of your house. Before you could gawk at the nicely furnished office any longer, Kai cleared his throat, successfully gaining your undivided attention.

"Boss! Sorry, it's just that your office is so nice, I've never seen anything l—" Your ramblings were cut short by the masked man before you.

"Quiet. I want you to come here." He curled his gloved finger a few times, indicating that he wanted you to meet him behind his desk.

Okay, now things were getting really weird. You didn't question his orders in fear of getting your body blown into nothing. Reluctantly, you made your way over behind the large, polished desk. As soon as you were standing next to Kai's sitting form, he shot up from his seat and slammed you onto the desk, causing a few items to fall onto the floor.

"Did you really think you could get away with deceiving me like this? Pretending to be a man just to get the dirt on me? Pathetic." Your eyes were as wide as saucers as you trembled in fear. This only caused Kai's pants to tighten around his cock, feeling an immense power trip.

You should've looked into this meeting before you carelessly walked into the clutches of the devil himself. Here you were, getting your uniform quite literally ripped off of your body.

Kai clicked his tongue at the bindings that covered your chest. "Your body must be in constant agony because of that poor binding job." He slipped his gloves off with ease, allowing his quirk to destroy the bindings and free your breasts.

What you didn't notice, was the shocked expression that covered Kai's features. As he touched you, he didn't receive a single hive. He wanted to savor the feeling of your soft skin, knowing it wouldn't have any negative effects on his.

Before you could retaliate, you felt smooth, large hands begin to roughly palm at your breasts. Kai wasted no time and leaned his head down to swirl his tongue around the hardened bud on one of your breasts, while the other one was being fondled and pinched by Kai's hand. You whimpered as his teeth began leaving bite marks all over your soft mounds.

As much as you wanted to scream and shout, you couldn't help but feel a pool begin to form between your legs. He was just so good and seemed to know exactly what he was doing, even if he had never touched a female in such a way.

"Aren't you a little slut? I've barely even touched you and you're already soaking wet." The smug man pulled away from your breasts, opting to slide his hand under your panties. You couldn't help the moans that escaped your lips as he practically shoved a finger inside of you, feeling your tightened cunt squeeze around it.

He grinned, gradually slipping in two more fingers while rubbing your clit with the palm of his hand. It was quite the show for him, seeing you shamelessly grind yourself onto his hand. He could only imagine how you felt around his swelling cock.

And before you knew it, you had been clenching around his fingers and twitching in place, allowing yourself to cum all over his hand.

On any normal circumstances, Kai would find this act utterly repulsive. He would feel the need to scrub his hands until they bled. But for some ungodly reason, he felt the urge to fuck you until you couldn't form coherent words.

Hastily pushing your panties to the side, Kai wasted no time in pulling his lengthy cock out. Before you had the chance to even ask about protection, you felt your insides being stretched by Kai's massive cock.

He didn't even give you a chance to adjust to his length. "O-Overhaul, wait! You're gonna rip me apart at this rate..!" You stuttered, silently cursing yourself out for appearing so pathetic to your superior. Your pitiful attempt at retaliation only caused Kai's thrusts to pick up in speed, rivaling the movements of a wild animal.

You wanted to hate it, you really did. You were allowing yourself to get ravaged by the man that killed your father. But he pleasured you in a way that your mere left hand would never be able to accomplish. It almost felt like his cock was meant to be inside of you.

"Good. I want your body to become accustomed to mine. Only I'll be allowed to fuck you like this." You hadn't even realized he had taken his mask off before he leaned in and attacked your lips with his. Your mouth had moved on its own, as you allowed his tongue to ravage your oral cavity.

After pulling away with a light string of saliva separating the two of you, Kai forcefully folded your legs into your chest. This action allowed him to bury his cock deep inside of you; deep enough to hit your cervix.

Now this had you going. You couldn't even contain yourself as you clenched around his cock, feeling you near your climax. Kai soon interrupted the sound of grunts, moans, and slapping by leaning forward and whispering in your ear.

"I want you to pledge that you'll keep your dirty secret, that you won't let the others know that you're not a sophisticated detective, but dirty little cock slut. Only I'm allowed to fuck you like this. Only I'm allowed to touch you. Come on. Say it!" He grunted, raising his voice towards the end of his command.

"I-I promise not to tell a soul..! You're the only one who knows my secret and the only one allowed to relish in it!" You couldn't believe the words coming out of your mouth. You were basically allowing the man who ruined your life to be the only person in your life that knew your true identity. To be the only person allowed to touch you.

It was wrong in so many ways, but you couldn't help but submit to him. The way his lengthy cock slammed against your cervix was enough to make you fall for his spell.

"Good girl. Good.." He grunted, feeling you clench around him and practically milk the semen right out of him as you also reached your limit. Your moans rivaled with ones of a pornstar's as you felt him release his warm, sticky seed into your womb.

He found himself reluctantly pulling his dick out of the warmth of your hole, almost cringing at the feeling of the cool air as soon as he pulled out. Meanwhile, you were finally beginning to come to your senses and realize how fucked up all of this was. The now masked man simply just sighed, walking away from the desk that you laid atop.

"I'm going off to take a shower. If I come back and you're not changed and cleaned up, there will be further punishments."

This was not how you expected this to play out.

[art-of-mathematics]

Some of my yesterday's art.

The first one is some zentangle stuff.

Some years ago I glued a piece of paper on the cover of this sketchbook, then I removed the paper, but some rests remained. Yesterday I drew these zentangle-like tentakles on top of the paper's remains...

The second drawing was inspired by the Dini surface - a surface with constant negative curvature. You get the Dini surface by twisting a pseudosphere, so to speak.

Knotty's art of today. [2023/07/26]