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8: Embedding - Mathematics


8: Embedding - Mathematics

Klein bottle

In topology, a branch of mathematics, the Klein bottle ( / ˈ k l aɪ n / ) is an example of a non-orientable surface it is a two-dimensional manifold against which a system for determining a normal vector cannot be consistently defined. Informally, it is a one-sided surface which, if traveled upon, could be followed back to the point of origin while flipping the traveler upside down. Other related non-orientable objects include the Möbius strip and the real projective plane. While a Möbius strip is a surface with boundary, a Klein bottle has no boundary. For comparison, a sphere is an orientable surface with no boundary.

The Klein bottle was first described in 1882 by the German mathematician Felix Klein.


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The Möbius strip has several curious properties. A line drawn along the edge travels in a full circle to a point opposite the starting point. If continued, the line returns to the starting point, and is double the length of the original strip: this single continuous curve traverses the entire boundary.

Cutting a Möbius strip along the center line with a pair of scissors yields one long strip with two full twists in it, rather than two separate strips the result is not a Möbius strip, but homeomorphic to a cylinder. This happens because the original strip only has one edge, twice as long as the original strip. Cutting creates a second independent edge of the same length, half on each side of the scissors. Cutting this new, longer, strip down the middle creates two strips wound around each other, each with two full twists.

If the strip is cut along about a third in from the edge, it creates two linked strips. The center third is a thinner Möbius strip, the same length as the original strip. The other is a thin strip with two full twists, a neighborhood of the edge of the original strip, with twice the length of the original strip. [2]

Other analogous strips can be obtained by similarly joining strips with two or more half-twists in them instead of one. For example, a strip with three half-twists, when divided lengthwise, becomes a twisted strip tied in a trefoil knot if this knot is unraveled, it is found to contain eight half-twists. A strip with N half-twists, when bisected, becomes a strip with N + 1 full twists. [2] Giving it extra twists and reconnecting the ends produces figures called Paradromic Rings.

One way to represent the Möbius strip embedded in three-dimensional Euclidean space is by the parameterization:

Widest isometric embedding in 3-space Edit

If the Möbius strip in three-space is only once continuously differentiable (class C 1 > ), however, then the theorem of Nash-Kuiper shows that no lower bound exists.

A method of making a Möbius strip from a rectangular strip too wide to simply twist and join (e.g., a rectangle only one unit long and one unit wide) is to first fold the wide direction back and forth using an even number of folds—an "accordion fold"—so that the folded strip becomes narrow enough that it can be twisted and joined, much as a single long-enough strip can be joined. [10] With two folds, for example, a 1 × 1 strip would become a 1 × 1 3 <3>>> folded strip whose cross section is in the shape of an 'N' and would remain an 'N' after a half-twist. This folded strip, three times as long as it is wide, would be long enough to then join at the ends. This method works in principle, but becomes impractical after sufficiently many folds, if paper is used. Using normal paper, this construction can be folded flat, with all the layers of the paper in a single plane, but mathematically, whether this is possible without stretching the surface of the rectangle is not clear. [11]

Topology Edit

A less used presentation of the Möbius strip is as the topological quotient of a torus. [12] A torus can be constructed as the square [ 0 , 1 ] × [ 0 , 1 ] with the edges identified as ( 0 , y ) ∼ ( 1 , y ) (glue left to right) and ( x , 0 ) ∼ ( x , 1 ) (glue bottom to top). If one then also identified ( x , y ) ∼ ( y , x ) , then one obtains the Möbius strip. The diagonal of the square (the points ( x , x ) where both coordinates agree) becomes the boundary of the Möbius strip, and carries an orbifold structure, which geometrically corresponds to "reflection" – geodesics (straight lines) in the Möbius strip reflect off the edge back into the strip. Notationally, this is written as T 2 / S 2 /S^<2>> – the 2-torus quotiented by the group action of the symmetric group on two letters (switching coordinates), and it can be thought of as the configuration space of two unordered points on the circle, possibly the same (the edge corresponds to the points being the same), with the torus corresponding to two ordered points on the circle.

The Möbius strip is a two-dimensional compact manifold (i.e. a surface) with boundary. It is a standard example of a surface that is not orientable. In fact, the Möbius strip is the epitome of the topological phenomenon of nonorientability. This is because two-dimensional shapes (surfaces) are the lowest-dimensional shapes for which nonorientability is possible and the Möbius strip is the only surface that is topologically a subspace of every nonorientable surface. As a result, any surface is nonorientable if and only if it contains a Möbius band as a subspace.

Computer graphics Edit

A simple construction of the Möbius strip that can be used to portray it in computer graphics or modeling packages is:

  • Take a rectangular strip. Rotate it around a fixed point not in its plane. At every step, also rotate the strip along a line in its plane (the line that divides the strip in two) and perpendicular to the main orbital radius. The surface generated on one complete revolution is the Möbius strip.
  • Take a Möbius strip and cut it along the middle of the strip. This forms a new strip, which is a rectangle joined by rotating one end a whole turn. By cutting it down the middle again, this forms two interlocking whole-turn strips.

Geometry of the open Möbius band Edit

It may be constructed as a surface of constant positive, negative, or zero (Gaussian) curvature. In the cases of negative and zero curvature, the Möbius band can be constructed as a (geodesically) complete surface, which means that all geodesics ("straight lines" on the surface) may be extended indefinitely in either direction.

Constant negative curvature: Edit

Like the plane and the open cylinder, the open Möbius band admits not only a complete metric of constant curvature 0, but also a complete metric of constant negative curvature. One way to see this is to begin with the upper half plane (Poincaré) model of the hyperbolic plane, a geometry of constant curvature whose lines are represented in the model by semicircles that meet the x -axis at right angles. Take the subset of the upper half-plane between two nested semicircles, and identify the outer semicircle with the left-right reversal of the inner semicircle. The result is topologically a complete and non-compact Möbius band with constant negative curvature.

(Constant) zero curvature: Edit

Constant positive curvature: Edit

Möbius band with round boundary Edit

The edge, or boundary, of a Möbius strip is homeomorphic (topologically equivalent) to a circle. Under the usual embeddings of the strip in Euclidean space, as above, the boundary is not a true circle. However, it is possible to embed a Möbius strip in three dimensions so that the boundary is a perfect circle lying in some plane. For example, see Figures 307, 308, and 309 of "Geometry and the imagination". [14]

A much more geometric embedding begins with a minimal Klein bottle immersed in the 3-sphere, as discovered by Blaine Lawson. We then take half of this Klein bottle to get a Möbius band embedded in the 3-sphere (the unit sphere in 4-space). The result is sometimes called the "Sudanese Möbius Band", [15] where "sudanese" refers not to the country Sudan but to the names of two topologists, Sue Goodman and Daniel Asimov. Applying stereographic projection to the Sudanese band places it in three-dimensional space, as can be seen below – a version due to George Francis can be found here.

From Lawson's minimal Klein bottle we derive an embedding of the band into the 3-sphere S 3 , regarded as a subset of C 2 , which is geometrically the same as R 4 . We map angles η, φ to complex numbers z1, z2 via

Here the parameter η runs from 0 to π and φ runs from 0 to 2π. Since | z1 | 2 + | z2 | 2 = 1 , the embedded surface lies entirely in S 3 . The boundary of the strip is given by | z2 | = 1 (corresponding to η = 0, π ), which is clearly a circle on the 3-sphere.

To obtain an embedding of the Möbius strip in R 3 one maps S 3 to R 3 via a stereographic projection. The projection point can be any point on S 3 that does not lie on the embedded Möbius strip (this rules out all the usual projection points). One possible choice is < 1 / 2 , i / 2 >>,i/> ight>> . Stereographic projections map circles to circles and preserves the circular boundary of the strip. The result is a smooth embedding of the Möbius strip into R 3 with a circular edge and no self-intersections.

The Sudanese Möbius band in the three-sphere S 3 is geometrically a fibre bundle over a great circle, whose fibres are great semicircles. The most symmetrical image of a stereographic projection of this band into R 3 is obtained by using a projection point that lies on that great circle that runs through the midpoint of each of the semicircles. Each choice of such a projection point results in an image that is congruent to any other. But because such a projection point lies on the Möbius band itself, two aspects of the image are significantly different from the case (illustrated above) where the point is not on the band: 1) the image in R 3 is not the full Möbius band, but rather the band with one point removed (from its centerline) and 2) the image is unbounded – and as it gets increasingly far from the origin of R 3 , it increasingly approximates a plane. Yet this version of the stereographic image has a group of 4 symmetries in R 3 (it is isomorphic to the Klein 4-group), as compared with the bounded version illustrated above having its group of symmetries the unique group of order 2. (If all symmetries and not just orientation-preserving isometries of R 3 are allowed, the numbers of symmetries in each case doubles.)

But the most geometrically symmetrical version of all is the original Sudanese Möbius band in the three-sphere S 3 , where its full group of symmetries is isomorphic to the Lie group O(2). Having an infinite cardinality (that of the continuum), this is far larger than the symmetry group of any possible embedding of the Möbius band in R 3 .

Projective geometry Edit

Using projective geometry, an open Möbius band can be described as the set of solutions to a polynomial equation. Adding a polynomial inequality results in a closed Möbius band. These relate Möbius bands to the geometry of line bundles and the operation of blowing up in algebraic geometry.

A realization of an open Möbius band is given by the set

where m corresponds to A / B .

There is a realization of the closed Möbius band as a similar set, but with an additional inequality to create a boundary:

A closely related 'strange' geometrical object is the Klein bottle. A Klein bottle could in theory be produced by gluing two Möbius strips together along their edges however this cannot be done in ordinary three-dimensional Euclidean space without creating self-intersections. [16]

Another closely related manifold is the real projective plane. If a circular disk is cut out of the real projective plane, what is left is a Möbius strip. [17] Going in the other direction, if one glues a disk to a Möbius strip by identifying their boundaries, the result is the projective plane. To visualize this, it is helpful to deform the Möbius strip so that its boundary is an ordinary circle (see above). The real projective plane, like the Klein bottle, cannot be embedded in three-dimensions without self-intersections.

In graph theory, the Möbius ladder is a cubic graph closely related to the Möbius strip.

In 1968, Gonzalo Vélez Jahn (UCV, Caracas, Venezuela) discovered three dimensional bodies with Möbian characteristics [18] these were later described by Martin Gardner as prismatic rings that became toroidal polyhedrons in his August 1978 Mathematical Games column in Scientific American. [19]

There have been several technical applications for the Möbius strip. Giant Möbius strips have been used as conveyor belts that last longer because the entire surface area of the belt gets the same amount of wear, and as continuous-loop recording tapes (to double the playing time). Möbius strips are common in the manufacture of fabric computer printer and typewriter ribbons, as they let the ribbon be twice as wide as the print head while using both halves evenly. [20]

A Möbius resistor is an electronic circuit element that cancels its own inductive reactance. Nikola Tesla patented similar technology in 1894: [21] "Coil for Electro Magnets" was intended for use with his system of global transmission of electricity without wires.

The Möbius strip is the configuration space of two unordered points on a circle. Consequently, in music theory, the space of all two-note chords, known as dyads, takes the shape of a Möbius strip this and generalizations to more points is a significant application of orbifolds to music theory. [22] [23]

  • A compact resonator with a resonance frequency that is half that of identically constructed linear coils [24]
  • An inductionless resistor [25] with high transition temperature [26]
  • Möbius resonator [27]
    with special characteristics (Knotane [2], Chirality)
  • Molecular engines [28]
  • Graphene volume (nano-graphite) with new electronic characteristics, like helical magnetism [29]
  • A special type of aromaticity: Möbius aromaticity
  • Charged particles caught in the magnetic field of the Earth that can move on a Möbius band
  • The cyclotide (cyclic protein) kalata B1, active substance of the plant Oldenlandia affinis, contains Möbius topology for the peptide backbone.

Arts and entertainment Edit

The Möbius strip principle has been used as a method of creating the illusion of magic. The trick, known as the Afghan bands, was very popular in the first half of the twentieth century. Many versions of this trick exist and have been performed by famous illusionists such as Harry Blackstone Sr. and Thomas Nelson Downs. [30] [31]

In creative works Edit

The universal recycling symbol (♲) design has its three arrows forming a Möbius loop. According to its designer Gary Anderson, "the figure was designed as a Mobius strip to symbolize continuity within a finite entity". [32]


8: Embedding - Mathematics

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The Feature Paper can be either an original research article, a substantial novel research study that often involves several techniques or approaches, or a comprehensive review paper with concise and precise updates on the latest progress in the field that systematically reviews the most exciting advances in scientific literature. This type of paper provides an outlook on future directions of research or possible applications.

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Interpolate_pos_encoding(x, pos_embed) doesnt return correct dimension for images that is not square (w != h) #8

I notice the generation of positional embedding in interpolate_pos_encoding method is slightly different than the one in the forward_selfattention method. The following simple modification bring both into the same page, to your interest.

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Enverfakhan commented May 2, 2021

I realized this solution is prone to floating point error, an example for such an error would be the following

This happens because w0 / math.sqrt(N) * math.sqrt(N) is not equal w0 and nn.functional.interpolate casts int on the scale_factor (I assume) and w0 / math.sqrt(N) * math.sqrt(N) is something like 60.999999999.. and it become 60 when int is casted on. Now a solution for this problem would be adding a small number (0.1) to w0 and h0 .

However this brings another question, does interpolating the positional embedings makes sense?, I mean does this operation happen during training or all the images in the training are just in the right shape and interpolation never occur? Because in that case interpolating positional embeddings (learned positional embedding) would be illegitimate right?

Mathildecaron31 commented May 2, 2021

Thanks for raising this issue. Yes I totally agree that something could be done to simplify/unify a bit the code there.

For the floating point error I've found that workaround:

if w0 != patch_pos_embed . shape [ - 2 ]:
helper = torch . zeros ( h0 )[ None , None , None , :]. repeat ( 1 , dim , w0 - patch_pos_embed . shape [ - 2 ], 1 ). to ( x . device )
patch_pos_embed = torch . cat (( patch_pos_embed , helper ), dim = - 2 )
if h0 != patch_pos_embed . shape [ - 1 ]:
helper = torch . zeros ( w0 )[ None , None , :, None ]. repeat ( 1 , dim , 1 , h0 - patch_pos_embed . shape [ - 1 ]). to ( x . device )
pos_embed = torch . cat (( patch_pos_embed , helper ), dim = - 1 )

I mean does this operation happen during training or all the images in the training are just in the right shape and interpolation never occur?

This operation actually happens during training. Indeed, with multi-crop the model is trained both with images of 224x224 and images of 96x96. So when forwarding a batch of 96^2 images we need to interpolate the encodings. As a matter of fact in my experiments I also tried having two sets of encodings. In practice that means that I was using differents encodings for the 224x224 and for the 96x96 inputs. This solution has exactly the same performance as when performing bicubic interpolation, which makes me think that the interpolation solution makes sense.

Enverfakhan commented May 2, 2021

Hi @mathildecaron31 thanks for the response and also I appreciate the insight about the interpolation vs separate pos_embed a lot. I would be curios about how would that behave in the wild with completely different sizes. I actually tried the deit_small(patch_size=8) for retrieval task on a in-house data, it seems to be working on par with a supervised vgg imagenet, however I had to set the image sizes to [224, 224] because some of the images blow the memory during the attention computation.

About the workaround for the floating point error, I feel like incrementing the w0 and h0 a small amount is more legit than zero padding the pos_embed but it is probably not a big deal especially if the image size is relatively big.

Looking forward for the Dino on large, random, uncurated dataset.

Mathildecaron31 commented May 2, 2021 •

I actually tried the deit_small(patch_size=8) for retrieval task on a in-house data, it seems to be working on par with a supervised vgg imagenet,

That's slightly disappointing :/. Have you tried the other models ? For example ViT-Base/16 should be more manageable memorywise. As a matter of fact, on copy detection datasets, I've found the base models to perform clearly better than the small ones: I get better performance with Base16x16 than with Small8x8 though Small8x8 is better at k-NN ImNet.

About the workaround for the floating point error, I feel like incrementing the w0 and h0 a small amount is more legit than zero padding the pos_embed but it is probably not a big deal especially if the image size is relatively big.

Yes your solution is definitely better ! I'll update that in the code.

Enverfakhan commented May 2, 2021

I picked the Small 8x8 because it was shown that that performs better with k-NN ImNet and because I was going to try with zero shot for retrieval task this choice mad more sense at the time. The result was indeed slightly disappointing, however I haven't experimented exhaustively and I dont have quantitative result either, I only check qualitatively which you can only do it for a handful of query, so this result is not definitive at all. But I should say the in-house data is very different than the imagenet, so I wouldn't be very surprised if I got some weird result with either model.

KeremTurgutlu commented May 4, 2021

But I should say the in-house data is very different than the imagenet, so I wouldn't be very surprised if I got some weird result with either model.

I wonder if finetuning DINO models on the in-house data you have might help? But you mentioned, that results are on par with vgg pretrained on imagenet so I am not very sure. Probably still worth trying.

Enverfakhan commented May 5, 2021 •

I actually tried the deit_small(patch_size=8) for retrieval task on a in-house data, it seems to be working on par with a supervised vgg imagenet

I guess I caused a misinformation unintentionally. The images were RGBA and I was treating them as RGB. I'm sorry if I caused any confusion. However after I accounted for that image format, the result still varies from query (image) to query. In some cases Dino outperforms a vgg_16 ImNet by far, but in some other cases they are almost on par or even worse. I haven't detected a consistent pattern for which images that Dino outperform or under-perform, but so far it seems like, anecdotally, Dino outperform in colorful images and it is on par (or worse) with vgg_16 ImNet for black and white images. By the way, I'm testing OpenAI's clip model with ViT too and Clip model seems to be the worst among the three (I was betting on the clip model that it would be the best, but couldn't be more wrong :) )

it's hard to come up with a quantitative evaluation. The images are multi--tagged and we are trying to retrieve similar images given a query. The tags are not reliable for evaluation because some similar images doesn't share any tag, or reverse is also the case, different images may share some common tag (instagram logo vs instagram app image). I'm planning to do a finetunining as multi class classification and try to get some numeric assessment out of that.

I wonder if finetuning DINO models on the in-house data you have might help?

I strongly believe it would help, but I wonder which model would perform better after finetuning with each. However the preliminary result that Dino is working better on colorful images is worth to pay attention.


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LouAnn H. Lovin is a professor of mathematics education at James Madison University (Virginia). She co-authored the first edition of the Teaching Student - Centered Mathematics Professional Development Series with John A. Van de Walle as well as Teaching Mathematics Meaningfully: Solutions for Reaching Struggling Learners, 2nd Edition with David Allsopp and Sarah Vaningen. LouAnn taught mathematics to middle and high school students before transitioning to pre-K―grade 8. For almost twenty years, she has worked in pre-K through grade 8 classrooms and engaged with teachers in professional development as they implement a studentcentered approach to teaching mathematics. She has published articles in Teaching Children Mathematics, Mathematics Teaching in the Middle School, and Teaching Exceptional Children and has served on NCTM’s Educational Materials Committee. LouAnn’s research on teachers’ mathematical knowledge for teaching has focused most recently on the developmental nature of prospective teachers’ fraction knowledge.

Karen S. Karp is at the School of Education at Johns Hopkins University-Baltimore, MD. Previously, she was a professor of mathematics education at the University of Louisville for more than twenty years. Prior to entering the field of teacher education she was an elementary school teacher in New York. She is also co-author of Elementary and Middle School Mathematics: Teaching Developmentally, Developing Essential Understanding of Addition and Subtraction for Teaching Mathematics in Pre-K―Grade 2, and numerous book chapters and articles. She is a former member of the Board of Directors of NCTM and a former president of the Association of Mathematics Teacher Educators (AMTE). She continues to work in classrooms to support teachers of students with disabilities in their mathematics instruction.

Jennifer M. Bay - Williams is a professor of mathematics education at the University of Louisville (Kentucky). Jennifer has published many articles on teaching and learning in NCTM journals. She has also coauthored numerous books, including Mathematics Coaching: Resources and Tools for Coaches and Leaders, K―12 Developing Essential Understanding of Addition and Subtraction for Teaching Mathematics in Pre-K―Grade 2 Math and Literature: Grades 6―8 Math and Nonfiction: Grades 6―8 and Navigating through Connections in Grades 6―8. Jennifer taught elementary, middle, and high school in Missouri and in Peru, and continues to work in classrooms at all levels with students and with teachers. Jennifer served as member of Board of Directors for TODOS: Equity for All, as president of AMTE, and as editor for the 2012 NCTM Yearbook.


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SIAM Journal on Algebraic Discrete Methods

We study the graph-theoretic problem of embedding a graph in a book with its vertices in a line along the spine of the book and its edges on the pages in such a way that edges residing on the same page do not cross. This problem abstracts layout problems arising in the routing of multilayer printed circuit boards and in the design of fault-tolerant processor arrays. In devising an embedding, one strives to minimize both the number of pages used and the “cutwidth” of the edges on each page. Our main results (1) present optimal embeddings of a variety of families of graphs (2) exhibit situations where one can achieve small pagenumber only at the expense of large cutwidth and (3) establish bounds on the minimum pagenumber of a graph based on various structural properties of the graph. Notable in the last category are proofs that (a) every n-vertex d-valent graph can be embedded using $O( dn^ <1/ 2>)$ pages, and (b) for every $d > 2$ and all large n, there are n-vertex d-valent graphs whose pagenumber is at least [ Omega left( frac> ight). ]


8: Embedding - Mathematics

All articles published by MDPI are made immediately available worldwide under an open access license. No special permission is required to reuse all or part of the article published by MDPI, including figures and tables. For articles published under an open access Creative Common CC BY license, any part of the article may be reused without permission provided that the original article is clearly cited.

Feature Papers represent the most advanced research with significant potential for high impact in the field. Feature Papers are submitted upon individual invitation or recommendation by the scientific editors and undergo peer review prior to publication.

The Feature Paper can be either an original research article, a substantial novel research study that often involves several techniques or approaches, or a comprehensive review paper with concise and precise updates on the latest progress in the field that systematically reviews the most exciting advances in scientific literature. This type of paper provides an outlook on future directions of research or possible applications.

Editor’s Choice articles are based on recommendations by the scientific editors of MDPI journals from around the world. Editors select a small number of articles recently published in the journal that they believe will be particularly interesting to authors, or important in this field. The aim is to provide a snapshot of some of the most exciting work published in the various research areas of the journal.


'Hard Day's Night': A Mathematical Mystery Tour

Listen Now: Hear The Interview With Math Guy Keith Devlin

"Bags of words"? What are they?

It actually goes back to the 1950s. It's used by the computer scientists who created spam filters. What you do is you take a piece of text, and you ignore the grammar, you ignore the word order, and you just regard it as a collection of words. And once you've done that, you can count the frequencies of the different words in the bag of words. To do it for music, you had to get little snippets, and the way they did that was the team analyzed, I think, about 70 songs from Lennon and McCartney, and they found there were 149 very distinct transitions between notes and chords that are present in almost all Beatles songs. And those transitions will be unique to one person or the other person.

So they'd be bags of notes and chords.

Bags of notes and chords, pairs of notes and chords. Those are the little items, and you just count them.

Part of the confusion is that Paul McCartney said he wrote the music. John Lennon said Paul McCartney wrote a section of music. So what did this trio of mathematicians detect?

Religion

Inspired By The Beatles' Love Gospel, 'Submarine Churches' Bucked Tradition

Cutting to the chase, it turns out Lennon wrote the whole thing. When you do the math by counting the little bits that are unique to the people, the probability that McCartney wrote it was .018 — that's essentially zero. In other words, this is pretty well definitive. Lennon wrote the music. And in situations like this, you'd better believe the math because it's much more reliable than people's recollections, especially given they collaborated writing it in the '60s with an incredibly altered mental state due to all the stuff they were ingesting.

I know what you are saying.

I would go with mathematics.

Keith, alright, I ask you — what about the artistic process of collaboration? Isn't it possible they were such close and accomplished collaborators that they inhaled a little bit of each other's technique and Lennon could write like McCartney and McCartney like John Lennon?

For sure. And that's why it's hard for the human ear to tell the thing apart. It's also hard for them to realize who did it and this is why actually the only reliable answer is the mathematics because no matter how much people collaborate, they're still the same people, and they have their preferences without realizing it. Lennon would use certain kinds of things over and over again. So would McCartney. It was the collaboration. Those two things come together that works, but they were still separate little bits. The mathematics isolates those little bits that are unique to the two people.


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