---
title: 'Spinning Families: Theories and Applications'
url: https://www.emergentmind.com/topics/spinning-families
type: topic
---

# Spinning Families: Theories and Applications

Searching arXiv for the primary paper and adjacent uses of “spinning families” across fields.
The expression **spinning families** appears in several mathematically and physically distinct research programs. In algebraic combinatorics it denotes generalized spinning-switch puzzles modeled by wreath products; in geometric group theory it denotes equivariant collections of subgroups satisfying large-projection “spinning” conditions on projection complexes or cone-offs; in planetary science it refers to collisional asteroid families whose members’ spin states encode Yarkovsky and YORP evolution; and in topology and contact geometry it denotes spinning constructions that pass from lower-dimensional objects to higher-dimensional foams, Legendrians, and Lagrangian cobordisms [2210.09408], [2010.10735], [2409.03419], [1411.2617]. The common vocabulary comes from rotation, twist, or spin as an organizing operation, but the technical content is discipline-specific.

## 1. Wreath products and generalized spinning-switch puzzles

In the algebraic-combinatorial setting, the basic model starts with a finite group $G$ modeling a single switch and a finite group $H$ modeling the “spinning” action on a finite set $\Omega$ of switch positions. The puzzle is modeled by the wreath product
\[
G\wr H \;=\; G^\Omega\rtimes H,
\qquad
G^\Omega = \prod_{\omega\in\Omega}G_\omega,\;G_\omega\cong G,
\]
with multiplication
\[
(k,h)\cdot(k',h')
=\bigl(k\,(h\cdot k'),\,hh'\bigr),
\quad
h\cdot(g_\omega)=(\,g_{h^{-1}\omega}\,).
\]
Here the base $G^\Omega$ records which move is applied at each corner, while the semidirect factor $H$ records the adversary’s rotation. A move is an element $(k,h)\in G\wr H$: the solver applies $k\in G^\Omega$ and then the adversary applies $h\in H$ [2210.09408].

The central notion is a **surjective strategy**. A finite sequence $(k_1,\dots,k_N)$ in the base $K=G^\Omega$ is called a surjective strategy if, for every adversarial choice $h_1,\dots,h_N\in H$, the partial products
\[
m_0=1,\quad
m_i=(k_1,h_1)\cdots(k_i,h_i)
\;\mapsto\;
p(m_i)\in K
\]
run through every element of $K$. Equivalently, no matter the hidden start-state in $K$, one of the partial products brings the system to the identity.

The main classification theorem in the abelian case states that if $G$ is finite abelian and $H$ acts faithfully on $\Omega$, then
\[
G\wr_\Omega H
\text{ admits a surjective strategy}
\quad\Longleftrightarrow\quad
G\text{ and }H\text{ are both $p$-groups (for the same prime $p$).}
\]
A second theorem states that if $G$ is a $p$-group of order $p^n$ and $H$ is any finite $p$-group acting faithfully, then $G\wr H$ admits a winning strategy. The proof uses induction on a proper normal $p$-subgroup $N\triangleleft G$, together with an interleaving construction that combines strategies for $N\wr H$ and $(G/N)\wr H$ [2210.09408].

The explicit constructions mirror the algebra. In the abelian case $G\simeq\Bbb Z_p^r$, one identifies $K=G^\Omega\cong(\Bbb F_p)^{r\cdot|\Omega|}$ and constructs a sequence of base-moves whose partial sums sweep through the entire vector space. In the general $p$-group case one fixes a chief series
\[
1=N_0\triangleleft N_1\triangleleft\cdots\triangleleft N_n=G
\]
with $N_{i+1}/N_i\cong(\Bbb Z/p)^{r_i}$, and interleaves quotient and kernel strategies. The paper also gives a nonabelian example with two interchangeable copies of the Monster group $M$: with $H=C_2$ swapping the two copies, a surjective strategy is obtained because $M$ is generated by involutions and the construction forces every pair $(x,y)\in M^2$ [2210.09408].

This line of work recasts a class of recreational puzzles as a solvability problem for wreath products. The paper closes with open questions on full classification, higher-order generators, palindromic strategies, quasigroup switches, expected turns under random play, and bounds on minimal strategy length.

## 2. Projection complexes, rotating families, and quotient geometry

In geometric group theory, a **spinning family** is defined relative to a projection complex. Let $P=P(Y,\theta,K)$ be a projection complex with vertex set $\mathrm{VP}\simeq Y$, and let $G$ act by isometries on $P$, preserving the projection data. For each vertex $c\in\mathrm{VP}$ choose a subgroup $G_c\le\mathrm{Stab}_G(c)$. Then $\{G_c\}_{c\in\mathrm{VP}}$ is an $L$-spinning family if it satisfies equivariance,
\[
\forall\,g\in G,\;c\in\mathrm{VP}:\quad g\,G_c\,g^{-1}=G_{g(c)},
\]
and the spinning condition,
\[
\forall\;c\neq c'\in\mathrm{VP},\;\forall\;1\neq h\in G_c:\quad
d_c\bigl(c',\,h(c')\bigr)\;\ge\;L.
\]
The principal theorem states that there exists $L_0=L_0(\theta,K)$ such that, when $L\ge L_0$, the normal closure
\[
N \;=\;\bigl\langle\!\bigl\langle\,\cup_{c}G_c\bigr\rangle\!\bigr\rangle_G
\]
is isomorphic to a free product
\[
N \;\cong\; \bigast_{\,c\in O}\;G_c
\]
for some subset of representatives $O\subset \mathrm{VP}$; moreover every nontrivial element of $N$ is either conjugate into some $G_c$ or acts loxodromically on $P$ [2010.10735].

A related but more elaborate framework uses **rotating families** and **composite rotating families** on several projection complexes simultaneously. Here one works with a disjoint union
\[
\mathbf Y\;=\;\mathbf Y_1\;\sqcup\;\cdots\;\sqcup\;\mathbf Y_m
\]
equipped with projection data and active sets, and with infinite subgroups
\[
T_X \;\le\;\Stab_G(X)
\]
satisfying equivariance, large-rotation conditions, and commutation for disjoint support. Dahmani’s “plate-spinning” construction organizes these data via **composite windmills** and successive unfoldings through the coordinates. The resulting subgroup
\[
\Rot\;=\;\bigl\langle\,T_X\;:\;X\in\mathbf Y\bigr\rangle
\]
admits a partially commutative presentation whose only relators are commutators
\[
[\,s,\;g\,s'\,g^{-1}\,]=1
\]
whenever the relevant supports do not interact. In the mapping-class-group application, the normal closure of a sufficiently large power of a Dehn twist has a presentation whose relators consist only of commutators between twists of disjoint support [1703.08968].

A recent extension places spinning families in the setting of hyperbolic and hierarchically hyperbolic quotients. Let $X$ be a Gromov-hyperbolic graph, let $\mathcal Y$ be a $G$-invariant collection of uniformly $K$-quasiconvex subgraphs, and let $X^1$ be the electrification obtained by coning off each $Y\in\mathcal Y$. With cone vertex $v_Y$ and extended projections $\pi_Y$, an equivariant family $\{H_Y\}_{Y\in\mathcal Y}$ is $L$-spinning if for every $Y\in\mathcal Y$, every $x\neq v_Y$ in $X^1$, and every nontrivial $h\in H_Y$,
\[
{}^\pi_Y(x,hx)\;>\;L.
\]
The quotient graph $\bar X=X^1/N$ is hyperbolic when $L$ is sufficiently large, and the same techniques show that random quotients of hierarchically hyperbolic groups are again hierarchically hyperbolic asymptotically almost surely [2507.16677].

Across these works, spinning families function as a small-cancellation-like mechanism encoded in projection data rather than in a relator set. The resulting algebraic outputs are free-product decompositions, partial-commutativity theorems, Greendlinger-type shortening lemmas, and stability of hyperbolic or hierarchically hyperbolic quotient structures.

## 3. Spin-state asymmetry in asteroid collisional families

In asteroid science, the relevant “families” are collisional families: fragments of a disrupted parent body that initially cluster in proper orbital element space and subsequently disperse under thermal forces. The key theoretical relation is the Yarkovsky drift law
\[
da/dt \propto \pm\,(1/D)\cos\gamma,
\]
with prograde rotators moving outward and retrograde rotators moving inward. In the $(1/D,a)$ plane, the fragments define a V-shaped envelope
\[
a(1/D)=a_0\pm C\cdot(1/D),
\]
where the slope encodes family age. Spin-state measurements therefore provide an independent test of family membership: one expects retrograde excess on the inward wing and prograde excess on the outward wing [2409.03419].

This pattern is observed for the X-complex Athor and Zita families in the inner main belt. For Athor, the family contains **60% of retrograde asteroids on the inward side** and **76% of prograde asteroids on the outward side**; for Zita, the inward side contains **80% of retrograde asteroids**, while the outward side has equal numbers of prograde and retrograde rotators, although kernel density estimation shows a clear prograde peak on the outward side. The analysis combines dense and sparse photometry, lightcurve construction, and convex inversion to estimate sidereal period, spin axis, and convex shape [2409.03419].

A related study of a **4 Gyr collisional family** in the inner main belt used photometric observations, literature light curves, and sparse-in-time photometry to determine spin poles for 55 candidate members. After rejecting 9 interlopers via albedo and spectral criteria, 46 confirmed family members remained, of which **31 are retrograde and 15 prograde**. Under the null hypothesis of equal prograde and retrograde probability, the one-sided binomial p-value is
\[
p \approx 0.0129 \;=\;1.29\%.
\]
The predominance of retrograde spins on the inward wing was therefore taken as corroborating evidence of common origin and long-term Yarkovsky-driven spreading [2210.02096].

Methodologically, these studies rely on convex inversion with a scattering law combining Lommel-Seeliger and Lambert terms, followed by classification of the spin sense from the ecliptic latitude $\beta$: prograde for $\beta>0$, retrograde for $\beta<0$. The family-scale inference does not require exact reconstruction of the original collision; rather, it uses the statistical persistence of the spin–orbit correlation over Gyr timescales. This suggests that spin-state analysis can validate very old and diffuse families that are difficult to isolate by orbital clustering alone.

## 4. Synthetic families, TESS rotational statistics, and normalized YORP evolution

A second asteroid-science use of the expression concerns **synthetic families** produced in impact simulations with rotating parent bodies. In SPH/N-body calculations of collisions into monolithic targets with diameters $D_{pb}=10\ \mathrm{km}$ and $100\ \mathrm{km}$, the parent-body spin modifies fragmentation, reaccumulation, and the spin distribution of the fragments. The mass ejection relation is written
\[
M_{ej}(P,\phi)=\mu_{ej}(P,\phi;Q/Q^*)\cdot M_{ej}(\infty,\phi;Q/Q^*),
\]
with $\mu_{ej}\to1$ for slow rotators and $\mu_{ej}$ reaching **up to $\approx 5$** for near-critical spins in oblique cratering events. Although individual impacts can accelerate or decelerate the target, the deceleration prevails on average, so collisions cause a **systematic spin-down of asteroid population** [1908.03248].

Observationally, TESS S1–S13 light curves were used to study **2859 family asteroids in 16 Main Belt families**. The distributions of rotation periods and amplitudes are family-specific in some groups, including the Hungaria, Maria, Juno, Eos, Eucharis, and Alauda families. Older families tend to contain a larger fraction of more spheroidal, low-amplitude asteroids. For eight families with firm ages, the log-linear regression is
\[
\mathrm{Median}\,A = 0.328 - 0.02067\cdot \mathrm{Age[Gyr]} \;\pm\;0.025\ \mathrm{mag},
\qquad p=0.015.
\]
The study also reports that very fast rotators are close to spherical, minor planets rotating slower than $\sim 11$ hour are also more spherical than asteroids in the 4--8 hour period range, and the Flora and Maria families show statistically significant core-versus-outskirts differences in period distributions [2202.04372].

A still broader synthesis is provided by a 2026 analysis of **8739 asteroids with spin period measurements and 3794 asteroids with obliquity determinations across 28 asteroid families spanning ages from 14 Myr to 3 Gyr**. The paper introduces the dimensionless time
\[
t=\frac{\tau_{\rm age}}{\tau_{\rm YORP}},
\qquad
\tau_{\rm YORP}\approx 1\;\biggl(\frac{D}{1\;\hbox{km}}\biggr)^{2}
\biggl(\frac{a_p}{2.5\;\hbox{AU}}\biggr)^{2}\quad\text{[Myr],}
\]
and studies two observables. The slow-rotator fraction $f_{\rm slow}$, defined using $P\ge 30$ h, increases steeply and saturates at
\[
f_{\rm slow}\simeq 0.25
\]
around a breakpoint
\[
t_{\rm bp}\simeq 20.
\]
The polarization fraction $f_{\rm pol}$, which measures whether the spin-pole sign matches the expected side of the family V-shape, rises to a maximum of $\simeq 0.8$ near $t\simeq 16$ and then decays toward the random limit $0.5$ for $t\gtrsim 20$ [2601.12972].

Taken together, these results show that asteroid-family “spinning” can refer both to the spin of the disrupted parent body and to the post-collisional spin evolution of family members. The first controls fragment production and angular-momentum drainage; the second controls the long-term observables used for family identification and age diagnostics.

## 5. Spinning constructions in topology, Legendrian theory, and Lagrangian cobordism

In low-dimensional and symplectic topology, spinning is a construction rather than a statistical descriptor. For **knotted trivalent graphs** (KTGs), the starting object is a spatial embedding
\[
\Gamma\colon G \hookrightarrow S^3
\]
of a trivalent graph. The relevant higher-dimensional objects are **2-dimensional foams**, spaces locally modeled on the cone on the 1-skeleton of the 3-simplex. The classical Zeeman theorem implies that the $(\pm 1)$-twist spin of any classical knot is an unknotted $2$-sphere, but the graph case is different: there exists a knotted $\theta$-curve $\Gamma\subset S^3$ and an edge $e$ such that its $(\pm1)$-twist spin
\[
t_{e,\pm1}(\Gamma)\subset S^4
\]
is a knotted foam [1411.2617].

The same paper proves a deletion criterion. If $\Gamma\setminus e$ is unknotted, then the $(\pm1)$-twist spin $t_{e,\pm1}(\Gamma)$ is an unknotted foam. Consequently, if $\Gamma$ is unknotted or almost trivial, then for each edge $e$ the $(\pm1)$-twist spin is unknotted. Kinoshita’s $\theta$-curve is almost trivial, so every $(\pm1)$-twist spin of it is unknotted; by contrast, a $\theta$-curve whose constituent knots are all granny knots has the property that every $(\pm1)$-twist spin is knotted [1411.2617].

A related but distinct spinning construction appears in the theory of **generating families** for Legendrian submanifolds. Let
\[
f:M\times\mathbb R^N\to\mathbb R
\]
be a linear-at-infinity generating family for a Legendrian in $J^1M$, with $M=H^n$ in the spinning construction. The function is called **spinnable** if it extends smoothly across the reflecting hyperplane, equivalently
\[
f(x_1,\dots,x_{n-1},-x_n;\eta)=f(x_1,\dots,x_{n-1},x_n;\eta).
\]
Introducing polar coordinates $(r,\theta)$ on the $(x_n,x_{n+1})$-plane, the spun generating family is
\[
\widetilde f(x_1,\dots,x_{n-1},r,\theta;\eta)=f(x_1,\dots,x_{n-1},r;\eta),
\]
with fiber-critical set
\[
\Sigma_{\widetilde f}= \Sigma_f\times S^1.
\]
This $\widetilde f$ generates an embedded spun Legendrian $\Lambda^s\subset J^1\mathbb R^{n+1}$. If one begins with a spinnable, gf-compatible exact Lagrangian cobordism, the same construction produces an embedded, gf-compatible exact Lagrangian cobordism between the spun Legendrians [1409.3152].

In both settings the word “spinning” is literal: it denotes a controlled geometric operation, usually an $S^1$-rotation, that creates higher-dimensional objects while preserving specific regularity or compatibility conditions.

## 6. Further specialized usages in high-energy theory, compact objects, quantum droplets, and binary black holes

A further use of spin-family language appears in a six-dimensional toy model for the spin-charge-family theory. In $d=(1+5)$, with a zweibein that curves an infinite disc into an almost $S^2$ and with two kinds of spin-connection fields $\omega_{ab\alpha}$ and $\tilde\omega_{ab\alpha}$, the theory contains
\[
2^{d/2-1}=4
\]
independent families. After the $6D\to 4D\times S^2$ split, the equations of motion separate into two decoupled pairs of families, $\{I,II\}$ and $\{III,IV\}$. Only for the fine-tuned choice
\[
0<2(F_{56}+\tilde F_{56})\le1,\quad \tilde F_{03}=0
\]
do normalizable zero-modes survive, reproducing a single-chiral 4D Weyl spinor [1212.2370].

In compact-star astrophysics, the relevant notion is not a spinning construction but **two coexisting families of compact stars**, namely neutron stars and strange stars. Fully general relativistic computations for uniformly rotating stars show that neutron-star $\to$ strange-star conversion, under baryon-number and angular-momentum conservation, causes a simultaneous spin-up and decrease in gravitational mass:
\[
\Delta M_G \sim 0.07\mbox{--}0.09\,M_\odot,
\qquad
\frac{\Delta \nu}{\nu_{\rm in}}\approx 20\mbox{--}35\%.
\]
This produces a new evolutionary channel in the $M_G$–$\nu$ plane, distinct from ordinary accretion-driven spin-up or electromagnetic spin-down [1709.02415].

In nonlinear quantum many-body theory, families of **spinning quantum droplets** arise in a two-dimensional amended Gross–Pitaevskii equation with Lee–Huang–Yang corrections and a $\mathcal{PT}$-symmetric potential. In the unbroken-$\mathcal{PT}$ region, stationary multipolar and vortex-droplet families bifurcate from linear modes; in the spinning regime, droplets become asymmetric above a critical rotation frequency $\omega_c$, but many remain stable. For the monopole family at $\mu=2$, the paper reports
\[
\omega_c\approx 1.35.
\]
The same work describes essentially elastic collisions for both spinning and nonspinning droplets under the chosen trap [2303.05210].

Finally, in gravitational-wave population inference, “spinning” and “nonspinning” **binary-black-hole populations** are disentangled by **spin sorting**. Defining
\[
\chi_A\equiv \max(\chi_1,\chi_2),\qquad \chi_B\equiv \min(\chi_1,\chi_2),
\]
one compares purely nonspinning, singly-spinning, and fully spinning synthetic populations under an IID Beta spin-magnitude model. The study concludes that current observations are inconsistent with a fully nonspinning population, but could be explained by a population with only one spinning black hole per binary or a population with up to **80% nonspinning sources** [2507.23663].

These specialized usages do not share a common formalism. What they share is a structural role for spin as a family-defining parameter: a family index in higher-dimensional fermion models, an evolutionary discriminator in compact objects and binary black holes, or a dynamical control parameter in non-Hermitian quantum droplets.

Source: https://www.emergentmind.com/topics/spinning-families