Sunday, October 12, 2025

She Loves You

Republished from here.



The revolution was quick and complete. Even as a child I could see that something new and magical had come into the world.

On February 9, 1964, the Beatles were launched as a worldwide phenomenon from the stage of the Ed Sullivan Show. *As Sullivan’s introduction faded, John, Paul, George and Ringo broke into All My Loving and established their pop-rock credentials. They followed with Til There Was You from the 1957 Broadway hit “The Music Man”, lulling unsuspecting parents as they displayed their innate musicality. Bizarre as these young men looked, Paul’s sweet rendering of this beautiful Broadway tune was simply disarming. The third song they sang was She Loves You.

I remember the world changing before my eyes.

Generations have passed since that night. The cultural revolution that started then is with us still, and much of it can trace its origin, in greater or lesser degree, to the music of Lennon/McCartney. This would not be true had the Beatles not been accepted as the leaders of their generation. They maintained their position through 17 Number One singles and 300 million total records sold worldwide in the ‘60s, but their worldwide influence began with, and was cemented by, their initial performance on Ed Sullivan. She Loves You was the essential catalyst of the phenomenon.

That begs the question: what gave She Loves You its power?

Of the scores of books written on the Beatles, and the few specifically on their music, one would think that this question would have been answered by now. Wilfred Mellers, the eminent musicologist whose book “Twilight of the Gods: the Music of the Beatles” regarded Yesterday as the first Beatles masterpiece, dismissed She Loves You as “simply an affirmation”. Alan W. Pollack delves deeper, but only notes the key musical aspects of the song without explaining either how or why they work to produce the power of the song. And Jonathan Gould, in whose excellent book “Can’t Buy Me Love: The Beatles, Britain and America”, seems so dazzled by the song that he makes the following mistake: “John and Paul sing most of ‘She Loves You’ in unison, not harmony…” The intro is sung in unison until the final note. Including the grace note at the beginning of the first verse, John and Paul are in unison for the first five notes. On the word “love” they break into harmony and proceed to smash every expectation of what a “pop” song can deliver.

The place to begin the analysis of She Loves You is to put it in context. It is easy to consider it as one of the Beatles’ “early efforts”, as it was the fourth song the Beatles released as a single. But Lennon and McCartney had been collaborative songwriters since 1957; this song was the result of a process years in the making.

Secondly, this was a true collaboration. According to accounts from both John and Paul, they produced the song in a few hours, sitting face to face in a hotel room. Both were integrally involved in the creative process, and it is one of the rare Lennon/McCartney compositions in which neither collaborator would predominate – it was truly and completely co-written by two songwriters with years of songwriting experience informed by years of perfecting the playing of hundreds of songs of varying styles – with the emphasis on rock & roll. Every lesson learned through untold hours of playing and writing was compressed into a revelatory 2 minutes and 19 seconds. Appropriately, this was the first composition credited to “Lennon/McCartney”: previous published songs are credited to “McCartney/Lennon”, marking She Loves You as the first actual “Lennon/McCartney” song.

The following analysis takes as its starting point the belief that it was the vocals that gave the song its power. Each vocal line was isolated to examine the musical relationship between the voices, treating the combined voices as a separate musical instrument, producing unison notes and two-note chords that informed an understanding of the underlying music. It was only through this method that the complete structure of the song revealed itself.

For those readers with little or no music theory, the following analysis may be daunting. It must be pointed out to beginners that there are seven notes in a scale, with the octave of the first note being the start of a repeat of the same scale, one octave higher than the first:

I

II

III

IV

V

VI

VII

VIII

G

A

B

C

D

E

F#

G

Therefore, the fifth tone in a G scale (V) is D.

This article examines the effects of inverted chords. As we deal with the two-note chords of John and Paul’s vocals, as well as the instrumental chords in one crucial section, I will only point out that two voices, singing the notes G and D, are either voicing a G major chord when they sing tonic G with a D above or a D suspended chord when singing D with an octave G above. Singing the former while playing a G major chord simply reinforces the harmonic simplicity. Singing the latter maintains the same relative harmonic relationship but adds a totally different aural texture.

As the song begins, Ringo’s drums tumble down as though a person is arriving, breathless, with the message:

“She loves you…”

The opening phrase is sung three times, first to the opening chord of E minor – John’s tonal center – implying that the news is not necessarily welcome. The chord changes to an A7 (an unresolved relative fourth to the original E minor), then up a minor third to a C major chord for the final repetition of the phrase. Throughout this, the voices are in unison. The ambivalence is generated by the chord structure that is striving to find its key signature, and the tension is supplied by the drums which have maintained the sense of breathlessness, refusing to establish a steady beat.

The key of the song is G major, which enters the song as the final “Yeah!” is sung and the drums settle into a steady backbeat. But just as the drums and resolving chord should give a sense of footing to the song, John and Paul abandon their unison singing, John supplying the fifth to Paul’s octave (the effect being a D suspended chord over the underlying G major), and Ringo opens his hi-hat cymbals to add a truly jarring effect – the sound of raw nerves. There is no true resolution; the battle of conflicting emotions has begun.

The next measure contains George Harrison’s first inspired recorded guitar riff. The changes come at eighth-note speed, hitting the G major, slipping to an F# major and bouncing immediately back to G major, as though the strong G major feel that John and Paul established is barely held in control. The repeated G major chord that George then plays is arpeggiated in such a way that it sounds like it is exploding, followed by a repetition of the G-F#-G bounce. This arrangement not only avoids a momentary lull; it blasts the song into orbit before the first verse begins.

The first verse starts with John and Paul singing unison over a G major chord, breaking into harmony as the chord changes to E minor on the word “love”.* The harmony here is of interest on a few levels. Paul is singing a D note, which would be a strong fifth in a G major construction if that were the chord still being played, but becomes an unresolving relative seventh within the reigning E minor chord. John, on the other hand, is singing a B note –the strong fifth in the underlying chord. The battle between the two vocals is engaged, and John’s minor-key position takes the dominant position.

The chord that John and Paul’s voices form (B and D) is a B minor. This song is being written by the two voices, and the music is pulled along – the next chord becomes B minor. This cannot be overstated; She Loves You is the result of the creative battle of two emotional states singing about the same situation from starkly different positions, Paul’s joyous optimism and John’s wary ambivalence. The voices literally lead the way.

During this measure, the third of the verse, the voices form a D major chord over the underlying music’s B minor chord on the word “saw”. The voices continue to lead the song, as the music goes to a D7 in the next measure.

Again, things get interesting. The D7 surely bows to the key of G major. Paul is singing the D note, the tonic of the chord being played and the fifth of the key signature – very strong ground.

What is John singing? A “B” note – nowhere to be found in the D7 chord that is being played, but the root of the relative minor chord (B minor) of the current D7 chord. *This time, it is John who is unsupported by the underlying music. The effect is almost sinister, the voices forming a B minor triad over the D7 chord on the word “yesterday”. Furthermore, both singers join in a glissando – Paul shifting up a third to an F# and John sliding up a minor third to the D before shifting back to their prior positions at B and D in the space of two eighth note beats. Thus, over an underlying D7 chord the voices join to sing [B minor – D – B minor] , framing Paul’s world view within John’s, trapping the joy within the ambivalence.

It is worth observing that the notes being sung not only form a B minor triad, they are also the third and fifth of the G chord that the music then goes to – an implied G triad without its tonal center. The voices continue to lead the way, and the music follows.

The harmonic relationships between the voices in the verse sections are as follows: unison, minor, major, minor, and unison again at the chorus. We see within the larger structure the same enclosing of Paul’s world view within John’s – there is joy, surrounded by and at war with ambivalence.

The verses showcase the clash of emotion via the competing tonal centers of G major and E minor. As the song transitions into its chorus, there is a momentary sense that joy has won:

“She says she loves you!”

…is sung in unison over a G major, and the delirium is palpable. But here is where the collaboration becomes delicious. Paul, who up to this point has maintained the tonic center with I-V-I bounces on the bass guitar, plays a B note instead, a third to the underlying G major chord but a fifth to the E minor tonal center that John has been forced, momentarily, to abandon. The entire band hits a syncopated accent with Paul on the B note, and the combined effect is the aural equivalent of a kick in a suddenly queasy stomach – the message is not received with unalloyed joy after all. The sound effects that will, years later, be supplied by George Martin to add texture to Beatles songs are rendered here by the arrangement itself.

The importance of this musical moment cannot be overstated. Were this a typical composition, there would be no syncopation and the bass would alternate between G and D. Paul’s playing a B note on his bass establishes that he is indeed aware there are two tonal centers to the song. Even more importantly, he realizes that the demands of the song required John to abandon his tonal center (emotional state) to adopt Paul’s. Paul therefore adopts John’s tonal center (emotional state) for that split second. This highlights the effect of a world turning upside down. I’m not aware of another moment like this in the canon of the Beatles. A lesser collaborator would have taken the momentary tonal victory and moved on. Such moments of awareness and grace define the collaborative genius of Lennon/McCartney.

The playing of a B note in the bass of a G chord (G/B bass) sets up the rest of this chorus. In a lesser song, the chorus would be sung over a predictable I-VI-IV-V chord pattern, in this case G-Em-C-D7. Instead we find G-Em-Cm6-D7! Astonishingly, not one commentator to my knowledge has noted that a Cm6 chord does not belong in the key of G major, or has attempted an explanation of how this chord is included in what is supposed to be a simple tune of affirmation. The anomalous Cm6 chord is the key to this entire composition.

Coming off the unison singing on the G major, John sings the tonic on the following E minor while Paul sings the fifth, a B note. Paul then slides an easy half-step up to hit the apparent tonic of the C chord his cheery world view demands; indeed, his bass establishes a bouncy I-V-I between C and G notes.

John has other plans. Listen to the chord that the two voices form here. John is no longer content to let the voices shade the chord structure; his voice bends the entire chord structure to his will.

As Paul slides up to the C note with his voice, John counters with an A note. The relative minor reestablishes itself, but that’s just the beginning. The clash of emotions turns into a war! Paul’s C major chord is breached by the implied Am7 that John counters with, and the most glorious moment of the song is reached: Paul hears the minor chord that John is playing and singing, and adjusts his C major chord to accommodate what he hears, reflexively adjusting his chord from major (C-E-G) to the C-Eb-G of a C minor with the added sixth – John’s A note. But this forces John’s interpretive chord inversion Am7 (A-C-E-G) into an Am7b5 (A-C-Eb-G)! The best designation for this chord, as it is in the key of G, is not Cm6; in the key of G major, this can only be seen as Am7b5 with a C bass. This is discordance, tension and irresolution on a massive scale. Had aficionados of free jazz masters such as Thelonious Monk or Ornette Coleman been paying attention, they would have heard harmonic structures they could relate to. In the attempt to reconcile the irreconcilable differences of John and Paul’s emotional states, the Beatles introduced jazz voicings to the rock ‘n roll universe and transcended the limits of the genre.

“[The Beatles] were doing things that nobody was doing. Their chords were outrageous, just outrageous, and their harmonies made it all valid.” Bob Dylan

Dylan was canny enough to comprehend the magic of the Beatles, but did not make the connection to the source: their harmonies did not make their “outrageous” chords valid; their harmonies literally made the chords.

Again, this must be seen in context. This chord is clearly beyond the capabilities of either of the songwriters alone at that stage in their musical development. It could only have come from a true collaboration, not what the relationship turned into in later years where one would hand unfinished works to the other, to have the different sensibilities show up in different parts of the song (“Michelle” and “We Can Work it Out” being stark examples); She Loves You was forged in the crucible of two world views at once clashing and supporting the other, face to face in the moment of creation. It could not have been written any other way. This is the defining moment of the Lennon/McCartney songwriting partnership.

The next section of the song guaranteed that the musical cognoscenti would not pay attention to the extraordinary musical moment just achieved.

Having transitioned from the Cm6 (Am7b5 /C bass) to a D7 chord on the word “glad”, the song structure requires the D7 to be played for an extra measure. At this point, John and Paul shift their voices from a D tonic-F# (D major) chord to a falsetto F#-D octave (F#augmented) chord with a Little Richard style “Ooooooo!” The effect is of an F# augmented chord superimposed on a D7 chord, underpinned by a stuttering drum fill that threatens to fall apart through the measure. This measure takes the unbearable musical tension that precedes it and releases it in a manner utterly nonsensical to an academic, but one that entered into, and took up residence in, the psyche of a generation. No pop song in history built up so much musical pressure; its release was a psychic explosion.

Parenthetically, this singular moment in pop history included performance art as part and parcel of the experience, giving any legitimate music critic reason enough to dismiss not only this song but the entire Beatles phenomenon. As Paul and George approached their shared microphone, the sung “Ooooo!” was accompanied by a ferocious shaking of their heads, sending their culturally anarchic “mop top” hair flying wildly in a visual metaphor for the musical moment – it was electrifying! In that moment of maximum tension, maximum compression of conflicting emotions and the accompanying falsetto scream of release, rock music was born and the Beatles became the avatars of their era.

Combining jazz harmonic structures with signature Little Richard vocals and flying hair proved enough to knock fans and detractors alike from reflecting on all that had come before. Most commentary has tended to focus on the next line:

“She loves you, YEAH, YEAH, YEAH…”

The use of slang accompanied by the unison singing in G major has allowed many people to dismiss the entire song as a simple, happy ditty. But the first time the line is sung, the underlying chord is E Minor – John’s tonal center. The first repetition of the line is over an A7 chord, refusing resolution.

Put in context, romantic involvement (“She loves you…”) is in conflict with the desire for sexual gratification (“Yeah, yeah, yeah…”), and the story line becomes transparent:

“She said you hurt her so

She almost lost her mind…

…Pride can hurt you too

Apologize to her…”

Listen to John’s singing of “apologize” (C-D-B-B, with a canny slide up to the D). As in every verse, the chords voiced by the singers at this point are a D major constrained by B minor voicings on both sides – Paul’s joy is literally corralled by John, who goes out of his way to sound sickened by the idea. It is plain that the concept of apologizing is foreign to John’s world view. Paul, on the other hand, treats the suggestion as self-evident.

John and Paul are hardly finished as they head down the stretch. As they sing, “With a love like that”, Paul is firmly entrenched in the key of G major, sliding from a B note to C. John, abandoning the unison singing once more, slides from G to A, apparently staying in the key of G major but necessitating the tension-filled Am7b5/C bass chord to be repeated, right on the words “love like that”.

Then the music stops.

The music suspends for an entire quarter note rest, and the effect of the implied Am7b5 tension stopping the song dead in its tracks brings home the point; is the listener ready to commit to a “love like that”?

To those listening to Paul, the answer is a resounding YES! Paul’s melody is the essence of pop happiness, sliding down the key of G major, starting on the D note as the D7 sets up its resolution to G major and ending on the major-triad B.

John has other plans. He stays in the key of G, bowing to Paul and necessity, but slides up not to the major-third B note, but to a Bb! This means that John transitions to a G minor scale as Paul sings in the G major scale. Again, John twists the underlying chord to his will, turning the D7 into a D7+5 and forcing the vocal harmonies from the expected and tuneful Bm-Am-G to the distressing Bb-Am-G. Perhaps “you know you should be glad”, but the vocals certainly imply that this is not the case.

The final “Be glad!” starts on the E minor chord, just as the song itself starts, and the final “YEAH!” refuses to fully resolve in the end; the song ends on a G6 chord.

The choice of a G6 to end the song is the final flourish. As during the introduction, John sings D under Paul’s G, forming a D suspended chord with the voices – lack of resolution. The chord inversion of the underlying music that fits John’s tonal center is an Em7, with John singing the seventh within his tonal center – a D within the implied Em7 chord – an extreme lack of resolution on the final note. Paul, on the other hand, plays a G bass and sings the G octave, enclosing John’s minor key feelings within a major key frame that utilizes John’s tonal center E (minor) into a sixth note which acts as a cloud on Paul’s sunny ending.

We return to the initial question: what gave She Loves You its power?

The key aspect of this composition is that John and Paul, singing the same lyrics but with competing tonal centers, are not representing the emotional point of view of the friend who comes with the news, but the person who is hearing the news – the Beatles are representing the mix of emotions in the listener. She Loves You is not a description of or commentary on an emotional state; it is the emotional state itself. *The listener enters the psychodrama as both the protagonist and the antagonist, whose conflicting emotions wage war throughout the song.* The song revealed, from the inside out, the power and complexity of its audience’s adolescent reality; it acted as a validation of their vulnerabilities and swirling emotions, resonating with a young audience so intensely that the Beatles won over a generation.

A masterpiece presents an artist’s vision of the world, exposing truths about the artists themselves. In She Loves You, the world was presented with John’s wary ambivalence and Paul’s sunny optimism. These polarized world views wove through numerous Lennon/McCartney tunes, and each found their individual masterpieces.

John brought his essential world view to its greatest expression in Strawberry Fields Forever, an ode to ambivalence so strong that the recording itself *is an amalgamation of two separate recordings, performed in different keys and with different tempos; John didn’t even choose between two different arrangements of the song. The lyric perfectly complements the music; as opposed to the tension generated in She Loves You, the ambivalence of Strawberry Fields creates a feeling of disconnectedness, even from one’s own beliefs:

“I think ah no, I mean ah yes, but it’s all wrong

That is, I think I disagree”

Paul would enjoy one of his greatest successes by playing against type, confronting pain and loss head-on in Yesterday. But his individual masterpiece is Hey Jude. Here, as in so many of his songs, Paul looks at the misunderstandings, suffering and pain of life and has the response of the eternal optimist, made plain with a lyric which turned it from a simple song of commiseration into the Beatles’ greatest hit, as well as the anthem, and theme, of his life:

“Don’t make it bad

Take a sad song and make it better”

Before Hey Jude, before Getting Better or We Can Work it Out, the Beatles created a musical tale that took the immovable wall of John’s loss, pain and hurt pride and applied to it the irresistible force of Paul’s optimism and joy, and the first of many sad songs was transformed for the better. She Loves You stands as the apotheosis of the Lennon/McCartney collaborative partnership; among all of their compositions, no other song simultaneously distilled the essence of both John and Paul as singers and songwriters. This combined roar of two musical lions beguiled the world and launched them as the musical icons of their era.


quoted from “Bob Dylan: An Intimate Biography” by Anthony Scaduto, Grosset & Dunlap, ISBN 0-448-02034-3, ©1971 page 175

Thursday, June 19, 2025

self-similar cross sections by class

What are the classes of cross sections through self-similar objects, according to my self-similar shape classification? Let's look at the 2D self-similar structures first:

2D

These are approximate, I will try to update these tables when I get more time and find mistakes. They are the 'almost surely' cross section structures. For instance a cross section through a 2D void-cluster almost surely will not hit any points in 2D so will be a completely empty void-void structure. Likewise, a cross-section through a cluster-cluster will almost surely cut through infinitely many of the blobs (over all cluster-clusters, not just a geometrically aligned one), making it also a cluster-cluster in 1D.

3D

In 3D things get more difficult so expect errors in this table. The main difficulty is that we get shapes that are combinations of more than two structural types. For instance a cross section through a sponge-sponge will definitely have separated objects (a cluster) but the objects will also have loops (sponge) when the cross section aligns with the loop, and hierarchical protrusions (tree) when we capture just the start of these loops in the cross-section. So it has the characteristics of all three. 

Typically this is written from the denser to the sparser type, in this case a sponge-tree-cluster. This is either interpreted as sponge-tree cluster: a cluster of sponge-trees (trees with holes in them in 2D); or as a sponge tree-cluster: a cluster of trees, but made of sponge (holes in it). These are the same thing, suggesting that A-B-C classes are associative. i.e. (A-B)-C = A-(B-C).


There is less obvious a pattern than in the 2D case, but it does have some consistency. The first word (the qualifier) is void, cluster, tree, sponge, sponge, sponge, solid from left to right and the last word (the noun) is void, cluster, cluster, cluster, tree, sponge, solid from top to bottom. 

These have both dealt with the two missing classes (shell & foam) at different ends. The first by repeating the second-last type (sponge) and the second by repeating the second type (cluster).

In order to accomodate the middle terms, there is a simpler way to work this out. Firstly, we look at the A-A diagonals. This has a pattern to it. 

Then all of the other cells of the form A-B are just the cross section result for A-A followed by the corss section result for B-B. There is some reduction going on. A tree-cluster-cluster will reduce to a tree-cluster and a tree-cluster-tree-cluster will reduce to a tree-cluster. Otherwise the cross section of the non-diagonal cells are just the piecewise combination of the cross section of the corresponding diagonal cells.

As for the A-A diagonals, they follow this pattern:

Numerically (with void-solid being 1-5) it is: 1-1, 2-2, 3-2, 4-3-2, 4-3, 4-4, 5-5. In 2D the pattern is 1-1, 2-2, 2-2, 2-2, 3-3. In general the n-dimensional cross section of the n+1 dimensional structure is:


with repeated terms removed in the triplet. Then X(A,B) is just X(A)-X(B) with appropriate reduction of repeated terms. '-' being a hyphen here.

The use of these tables is that you can guess the class of a higher dimensional structure just from its cross section. The most common case of a cross section is when the structure is n+1D so is a moving structure (a behaviour), and you have access to still images of it. So the above table can suggest the 2+1D structure from its 2D structure at any one time.

To guess a 3+1D structure we would need a 4D table.

My guess as to the A-A pattern would be: 1-1, 2-2, 3-2, 4-3-2, 5-4-3, 6-5-4, 6-5, 6-6, 7-7.

So the cross section of a shell-shell (number 5-5) would be: 5-4-3 which is a shell-sponge-tree. 

Is turbulence a shell?

What shape is turbulence?

It sounds odd to give the chaotic motion of countless fluid molecules a shape. But we might be able to answer by being a bit broader with our use of the word shape.

We know that turbulence is statistically self-similar so it may fit into one of my structural classes of self-similar shapes. But turbulence is also moving, so we have to look at a 3+1D version of these structural classes. It turns out that mixing temporal and spatial dimensions doesn't have any significant effect on the set of classes, so the 3+1D 'behaviours' are the same classes as the 4D spatial structures. There is a 9x9 table of these.

A solid body of fluid still doesn't have a shape or structure in the conventional sense of a subset of 3+1D. But there are various features in turbulent fluid that we can use to define a shape which is quite consistent over time. The main ones are present in 2D also:

Left is a hyperbolic field, defined by its centre point, magnitude and its two 'invariant' axes. In 3D it is defined by the central line (or curve) and the two intersecting surfaces (a type of 'coherent turbulent structure'). 

Right is a vortex, defined by its centre point and magnitude, in 3D it is defined by the central vortex line (or curve) and a magnitude.

In 2D all fluid flow is composed of these two features, for example we define the vortex in red and the hyperbolic structure in green here:
The infinitesimal transformation at the vortex is defined by an anti-symmetric matrix (a rotation) and at the hyperbolic point by a symmetric matrix (a nonuniform scale), both are unitary (determinant=1) for incompressible fluids. Halfway between these two matrices are the lower or upper triangular matrices, which represent a shearing action rather than rotation or 'squashing'. This can be our boundary between points inside the 3+1D set and points outside.

In other words, the shape that we are classifying is the set of points in 3+1D whose infinitesimal transform according to the vector field is more anti-symmetric than symmetric. It is the set of points that are in the vortex and not those between vortices.

In 3D vortices look like long limbs, which tend to stretch out over time. Here is a nice image where the vortex curves in a turbulent flow are separated by approximate scale:
This shows the scale symmetry, with many vortices at the small (blue) scale and a few at the large (red) scale.

Even though they look to be distinct curve segments, this is just a threshold on the vorticity, each vortex should extend until it meets another vortex or drops to exactly zero vorticity. So I *think* that most of these curves would connect either to themselves or to the boundary. 

That means that the central vortex lines would make something like a void-sponge structure in 3D and therefore either a void-shell or void-foam structure in 3+1D. If we are talking about the solid shape I described above with the thicker vortex limbs, then it would be a sponge in 3D and either a shell or foam in 3+1D.

To further investigate its 3+1D behaviour we need to look at how it behaves. Much of the time these vortices move around but do not interact. They seem to somewhat repel each other. However, there are interactions that occur, such as this:


In 3+1D this is just a single surface in a saddle-like shape. Likewise, it is possible for a ring vortex to appear from nothing, e.g. from a pushed limb of water such as out of a pipe. Again, in 3+1D this looks like a (paraboidal) surface.

So is the structure in 3+1D just a collection of wobbly surfaces? Well, I think that it also has branching, just not very commonly. Firstly, I think there may be branching at the points where the vorticity goes to 0, such as when 4 vortex lines meet in a plane, or where 8 meet as though from the vertices of a cube to the centre. Secondly, we can see examples of a thick vortex limb flattening out and splitting into two thinner limbs:


This is real data, and even shows a second splitting to even smaller vortex limbs:

So the structure can be branching limbs in 3D and therefore branching surfaces in 3+1D. This makes it a shell or a foam.

In this paper we also see branching happening in the last images, and at a smaller scale. For anti-parallel vortex streams:
and for orthogonal vortex streams:



The difference with a 3+1D foam is that these loops always reconnect back into thicker limbs. With a shell they may branch again and again until they disappear, in terms of turbulence that is descending the energy cascade until the movement becomes thermal energy.

So, all of this considered, I think that there is a good chance that turbulence is a 3+1D shell (specifically a shell-shell), if we are describing the vortex regions. If we describe just the non-vortex regions we will get the set complement of the shell-shell in 3+1D, which is in fact also a shell-shell. 

And none of this is affected by how thick we make the vortices, so we needn't necessarily stop at the 'shear surface'. What we can't do is use the 'vortex surface' used in many visualisations because this is based on a fixed vorticity value, so it disconnected and therefore looks more like blobs than a connected sponge-like structure in 3D. 

Incidentally, there is also a lot to be said about knots in turbulent vortices, but it doesn't matter for the 3+1D structural class because a knotted string in 3D is not a knotted surface in 3+1D. 


Thursday, May 22, 2025

Inversive cluster-shells

Here is a cluster-shell:

to tell that is is made from a cluster (of balls) we can take a cross section through it:


here it is from a more top-down angle:

This one is a bit of a complicated shape to engineer, it has five steps. Step 1, make an airtight strip of spheres:

Then add a strip at half that scale below it, and half again below that etc, making the resulting strip twice as deep:

Now do the same stacking but on a 2D grid of these strips instead, with the big spheres at the grid vertices:

It is now already a cluster-shell, but it has smooth areas and it is unbounded, so we can do better.
Next stereographically project this onto a sphere:

Lastly replace each sphere in the above with this shape, as shown in the first image. Here's a last pic from a more side on angle:


The inversive tree-foam is just the set complement of this. Here I have applied a sphere inversion around the largest inner ball to make it bounded in a sphere, and I'm rendering it backwards (with transparent and solid swapped) so you can see the inner structure:

------------ OLD --------------------

The below almost works but doesn't quite work, but I'm leaving in here for reference:

Cluster-shells are another structural class that are hard to generate with inversive geometry. They therefore are one of the five unspecified classes in this description of all 49 classes of the 7x7 structural classification of self-similar shapes in 3D.

One way that I *think* counts as a cluster-shell is shown here:

It is built in a similar way to the cluster-tree. It starts with a Ford sphere packing, using the triangular lattice version, here the spheres are shown at half size:


At full size they all are packed in so no sphere can move:

Between each touching pair of spheres we can make a watertight concave triangle bridging them. This is true for the underneath spheres too:

This is a cluster-foam as the lower 'triangular mesh' of spheres butt up against the big spheres. We can now stereographically project this infinite mesh of spheres onto the (Riemann) sphere:


Much like the sphere tree (cluster-tree) structure, we can then recursively replace each sphere with this above shape (ignore the thin gaps):


The problem with this is that it remains a cluster-foam as the recursed sphere is aligned with the small underneath sphere mesh, so it has millions of air pockets.

Instead we can turn the sphere shape upside down every time we recurse each sphere. Giving the resulting shape, the (probable) cluster-shell:


A slice-through shows the solid spheres from which is it made:

Unforunately this isn't a true cluster shell as some of the intermediate spheres don't match together properly.




Tuesday, April 29, 2025

Inversive void-shells

The void-shell is one of the trickiest structural classes to generate using sphere inversion, being one of only five that are missing in the 7x7 classification of self-similar geometry here.

A void-shell is the boundary surface of a space-filling tree, so is a fractal with no volume. There are two simple ways to make this using linear transformations, the Menger-shell:

and the Viscek-shell:

(both my own naming convention as I don't know of any better names).

My first successful attempt at this class with sphere inversions is based around the fairly well-known square-symmetric Apollonian gasket:

We use this pattern on each each principle axis. The large white circles are mapped to the full disk such that the two inner contact points become vertical or horizontal opposites. This creates a sort of cocoon with the triangular hole seen in the centre of the below images. 

The four white circles remain as 3-plane shapes after being transformed to the full disk. The remaining asymmetrical concave quadrilateral black areas are Mobius transformed into the inner concave quadrilateral, and this is then treated as a single-plane structure like the one above.



A tree-solid is simply the set inversion of the above. However, inkeeping with having the structure within a sphere, we can do this by:
1. orthographically transform the unit sphere to an infinite plane
2. repeatedly scale down any points larger than the above 'tri-disk's radius.
3. using the single-plane structure, but one-sided at the start, so features are only on the inside of the sphere

The result is an inversive tree-solid:



Friday, April 11, 2025

Rules for Inversive Classes

This classification of recursive shapes can be applied to inversive geometry in 3D, which is geometry that is symmetric to inversions around various 'generating spheres'. It is a 7x7 table that looks like this:



How we go from generating spheres to the class above is what I describe in my new paper preprint. It is a connection through about 4 layers of abstraction. 

Firstly, we have the generating spheres. The inversive limit sets are generated by repeatedly inverting points within these spheres such that the sphere surface remains unchanged. 

Secondly we have the connectivity graph, which represents many sets of generating spheres that all have the same connectivity. The nodes are the spheres and the edge numbers represent the 'branch order' n of the two intersecting surfaces. The dihedral angle of intersection is pi/n.

Thirdly we have the hypergraph, which represents many connectivity graphs with the same broad limit-set connectivity. In this diagram the branch orders are not important, just the presence of not of pairwise sphere intersections (an edge), triplet-wise sphere intersections (a semi-translucent face) and quad-wise sphere intersections (a solid volume). 

Fourthly and lastly we have an abstract hypergraph, which represents many types of hypergraph that all map to the same structural class in the 7x7 table. This structure is a schema for generating hypergraphs. The key component is the polyhedral sphere, which represents any polyhedron in the hypergraph that is intersected by a single orthogonal sphere. 


In the above abstract hypergraph, the red polyhedral sphere is the parent and the single curved edge represents one or many child edges in the hypergraph. In the hypergraph above it, there are four such edges and the spherical polyhedron is an irregular tetrahedron.

This particular abstract hypergraphs generates cluster-sponges. These are sponges that would be clusters if boundary points were removed. 

You can see this limit set in the table at the top of the page. These can be viewed in realtime here.

So this new paper is an extension of the previous paper but rather than creating a single family of limit sets for the five main diagonal classes, it is a method for generating a broad family of limit sets for all 49 classes.

Well... nearly, you can see in the top table that five limit sets are missing. These are easy to generate with linear limit-sets (see them rendered here) but hard to find for inversive limit sets. It is not certain whether they even can be generated, at least with simple sphere inversions. Working out whether these last 5 classes is possible with sphere inversions is the last major open question for this structural classification.