## Abstract

Our Local Group of galaxies is moving with respect to the cosmic microwave background (CMB) with a velocity^{
1
} of *V*
_{CMB} = 631 ± 20 km s^{−1} and participates in a bulk flow that extends out to distances of ~20,000 km s^{−1} or more^{
2,3,4
}. There has been an implicit assumption that overabundances of galaxies induce the Local Group motion^{
5,6,7
}. Yet underdense regions push as much as overdensities attract^{
8
}, but they are deficient in light and consequently difficult to chart. It was suggested a decade ago that an underdensity in the northern hemisphere roughly 15,000 km s^{−1} away contributes significantly to the observed flow^{
9
}. We show here that repulsion from an underdensity is important and that the dominant influences causing the observed flow are a single attractor — associated with the Shapley concentration — and a single previously unidentified repeller, which contribute roughly equally to the CMB dipole. The bulk flow is closely anti-aligned with the repeller out to 16,000 ± 4,500 km s^{−1}. This ‘dipole repeller’ is predicted to be associated with a void in the distribution of galaxies.

The large-scale structure of the Universe is encoded in the flow field of galaxies. A detailed analysis of the flow uncovers the rich structure manifested by the distribution of galaxies, such as the prominent nearby clusters^{
10,11,12,13
}, the Laniakea supercluster^{
14
} and the Arrowhead mini-supercluster^{
15
}. A close correspondence between the observed density field, derived from redshift surveys, and the reconstructed three-dimensional (3D) flow field has been established out to beyond 100 megaparsecs and down to a resolution of a few megaparsecs^{
16
}. Yet the flow contains more information on distant structures, as found from tides and from continuity across the zone obscured by the Galactic disk, the ‘zone of avoidance’^{
11
}. The Cosmicflows-2 dataset of galaxy distances^{
16
} provides reasonably dense coverage to *R* ≈ 10,000 km s^{−1} (distances are expressed in terms of their equivalent Hubble velocity).

The linear 3D velocity field is reconstructed here from the Cosmicflows-2 data by the Bayesian methodology of the Wiener filter and constrained realizations (see Methods). The Wiener filter is a Bayesian estimator that assumes a prior model: here it is the ΛCDM model. It is a conservative estimator which balances between the data and its errors and the assumed prior model. Where the data are weak, the Wiener-filter estimation tends to the null hypothesis of a homogenous universe: that is, a vanishing peculiar velocity. Yet it uncovers structures out to ~16,000 km s^{−1} at the extremity of the Cosmicflows-2 coverage. The variance around the mean Wiener-filter estimator is sampled by the constrained realizations.

The Wiener-filter flow field is used to construct the cosmic web, defined here by means of the velocity field^{
17
} (V-web; see Methods) This is done by evaluating the velocity shear tensor on a grid and counting the number of its eigenvalues above a threshold value: 3 corresponds to a knot, 2 to a filament, 1 to a sheet and 0 to a void. In the linear regime, the flow is proportional to the negative of the gravitational field; hence it constitutes a gradient of a scalar potential. Figure 1 shows the large-scale structure out to a distance of 16,000 km s^{−1} in a plane that contains the Local Group, the Shapley attractor and the dipole repeller. Three different aspects of the flow are depicted: streamlines, the V-web and the velocity potential.

When describing the gravitational dynamics in co-moving coordinates, by which the expansion of the Universe is factored out, underdensities repel and overdensities attract. The velocity field is represented here by means of streamlines (see Methods), the sources and sinks of which are the attractors and repellers of the large-scale structure (see Fig. 1 and the Supplementary Video). Figure 2 shows a 3D visualization of the streamlines in a box of length 40,000 km s^{−1} centred on the Local Group. The streamlines of the left plot of Fig. 2 either converge onto an attractor located roughly at [−12,300, 7,400, −300] km s^{−1} or cross out of the box. The plot uncovers the existence of a repeller at the upper right-hand side of the box — a region from which streamlines diverge. Repellers are best manifested by the anti-flow, namely the negative of the velocity field. The right plot of Fig. 2 depicts the convergence of the anti-flow streamlines onto a repeller at supergalactic coordinates [11,000, −6,000, 10,000] km s^{−1}. The Wiener-filter reconstruction detects a single attractor and a single repeller, the Shapley attractor and the dipole repeller. (The accompanying Supplementary Video provides a further visualization.)

Multipole expansion provides a different insight into the nature of the observed flow. The dipole (bulk flow) and quadrupole (shear tensor) moments are evaluated here by spherical top-hat window functions^{
11,18,19,20
} centred on the Local Group with variable radii, *R.* The telltale signature of a single dominant attractor or repeller is the close alignment of the dipole with the expansion eigenvector of the shear tensor and a degeneracy of the other two eigenvalues/eigenvectors. The observed flow is clearly not dominated by either a single attractor or repeller. In the following, we emphasize the directional aspects of the dipole and shear eigenvectors, which are robustly recovered by the Wiener filter. Figure 3 presents an Aitoff projection of the following directions: (1) the dipole repeller; (2) the Shapley attractor; (3) the CMB dipole and its anti-apex; (4) the bulk velocity of top-hat spheres of *R* = (2,000, 3,000,..., 15,000) km s^{−1}, **V**
_{bulk}(*R*), of the Wiener-filter reconstructed flow field; (5) the three eigenvectors of the shear tensor (
${\stackrel{\mathbf{\u02c6}}{\mathbf{e}}}_{i}$
, *i* = 1,2,3) of the Wiener filter field. The figure shows the strong anti-alignment of the bulk flow out to 15,000 km s^{−1} with the dipole repeller. Beyond that radius, the bulk flow loses its coherence, as the scatter in direction steadily increases. The eigenvector of the shear tensor that reflects the direction of maximal expansion (
${\stackrel{\mathbf{\u02c6}}{\mathbf{e}}}_{3}$
) is aligned with the direction of the Shapley attractor out to *R* = 7,000 km s^{−1}. Supplementary Fig. 2 further presents the mean and the scatter around the cosine of the angles formed between the bulk velocity and the dipole repeller,
${\mu}_{\mathrm{bulk}}(R)=\phantom{\rule{.25em}{0ex}}\mathrm{cos}({V}_{\mathrm{bulk}},{R}_{\mathrm{GR}})$
, and between **e**
_{3} and the Shapley attractor,
${\mu}_{{\mathbf{e}}_{3}}(R)=\phantom{\rule{.25em}{0ex}}\mathrm{cos}({\stackrel{\mathbf{\u02c6}}{\mathbf{e}}}_{3}(R),\phantom{\rule{.25em}{0ex}}{\mathbf{R}}_{\mathrm{Shapley}})$
.

Two aspects of the bulk velocity — the anti-alignment itself (*μ*
_{bulk}(*R*) = −0.96 ± 0.04) and its distance scale (*R* = 16,000 km s^{−1}) — strongly corroborate the Wiener-filter finding of the dipole repeller and its dominant role in dictating the observed flow. The direction and distance coincide with the position of the dipole repeller — a non-trivial occurrence. It is interesting to study the shear tensor. The expansion eigenvector is closely aligned with the direction to the Shapley attractor out to *R* ≈ 7,000 km s^{−1}, the distance of the foreground (Norma–Centaurus–Hydra) Great Attractor, located at the bottom of the Laniakea basin of attraction^{
14
} at supergalactic coordinates (−4,700, 1,300, 500) km s^{−1}. It is the combined mass distribution within the Laniakea and Shapley superclusters that dominates the tidal field, with the inverse cubic distance dependence of the tidal interaction tipping the balance at our location towards the Laniakea/Great Attractor.

Our main findings are tested against statistical and systematic uncertainties. There is no doubt about the existence of the Shapley concentration, and therefore we focus our attention mostly on the dipole repeller. The strong support for the existence of the dipole repeller comes not only from its close alignment of the bulk velocity but also from the small scatter around the mean Wiener-filter value, *μ*
_{bulk}(*R*) = −0.96 ± 0.04 for *R* ≈ 16,000 km s^{−1} (Supplementary Fig. 2). Assuming that the dipole repeller is the dominant structure that determines the direction of the bulk flow, the scatter in *μ*
_{bulk}(*R*) can be translated to uncertainty in the position of the repeller, Δ*R*
_{DR} ≈ 4,500 km s^{−1} (see Methods). The basins of repulsion and attraction around the dipole repeller and Shapley attractor, out to a distance of ~8,000 km s^{−1}, contribute fairly evenly to the velocity of the Local Group. At 8,000 km s^{−1}, the repeller and the attractor contribute 59 ± 26 and 67 ± 27 km s^{−1}, respectively, to the CMB dipole (see Methods). Next, the robustness of the dipole repeller is tested against subsampling of the data. These consist of cuts either by distance (6,000, 8,000 and 10,000 km s^{−1}), or by galaxy and data type (see Methods), corresponding to a degradation in the quality of the data by volume coverage, number of data points and magnitude of errors (see Methods). All subsamples considered here locate the dipole repeller in an underdense region and recover a basin of repulsion that pushes the Local Group in the direction of the CMB dipole.

The general picture that emerges here is of a complex flow that cannot be explained by a simple toy model, yet the main structures that shape the observed flow can be identified. The Wiener filter recovers a flow dominated by a single attractor and a single repeller, which roughly equally contribute to the CMB dipole. The role played by the Shapley attractor is not surprising; the earlier findings on influences beyond the Great Attractor^{
6,7,9,21,22
} suggested it. The existence of the dipole repeller was only vaguely hinted at before. A study of the all-sky distribution of X-ray-selected clusters uncovered a significant underdensity of clusters in the northern hemisphere roughly 15,000 km s^{−1} away^{
9
}. It suggested that this underdensity may be as significant as the overdensity of clusters in the southern hemisphere in inducing the local flow. Earlier examinations of galaxy peculiar velocities found a north–south anisotropy in (galactic) *y*-component of the velocities^{
3
} and found^{
20
} that the sources responsible for the bulk flow are at an effective distance >30,000 km s^{−1}. Here, the source of the repulsion is identified for the first time. The dual dominance of the dipole repeller and the Shapley attractor is the main new finding of this study. The strong anti-alignment of the CMB dipole with the dipole repeller out to a distance of 16,000 ± 4,500 km s^{−1} suggests the possible dominance of the repeller over the attractor. The predicted position of the dipole repeller is in a region that is as yet poorly covered by existing redshift surveys. We predict the dipole repeller to be associated with a void in the distribution of galaxies.

In the linear regime of gravitational instability, repellers are as abundant and dominant as attractors. Yet, observationally, repellers are much harder to identify than attractors. The association of repellers with underdensities renders them strongly deficient in galaxies, in general, and clusters of galaxies, in particular, and thereby makes their direct detection challenging. Our use of peculiar velocities as tracers of the large-scale structure overcomes that observational hindrance and unveils the existence of the new extended structure located at *α* = 22 h 25 min, *δ* = +37° (galactic longitude *l* = 93°, galactic latitude *b* = −17°; supergalactic latitude and longitude SGL = 332°, SGB = 39°) that we call the ‘dipole repeller’.

## Methods

## Cosmicflows-2 dataset

The present study is based on the second release catalogue^{
13
} of galaxy distances and peculiar velocities, Cosmicflows-2, which extends sparsely to recession velocities of 30,000 km s^{−1} (redshift *z* ≈ 0.1). It consists of 8,161 entries with high density of coverage inside 10,000 km s^{−1}. Here we used a grouped version of the Cosmicflows-2 data, in which all galaxies forming a group (of two or more) are merged to one data entry. The grouped Cosmicflows-2 data consists of 4,885 entries. Six methodologies are used for distance estimation: Cepheid star pulsations; the luminosity terminus of stars at the tip of the red giant branch; surface brightness fluctuations of the ensemble of stars in elliptical galaxies; type Ia supernovae; the fundamental plane in luminosity, radius and velocity dispersion of elliptical galaxies; and the Tully–Fisher (TF) correlation between the luminosities and rotation rates of spiral galaxies.

## Wiener filter and constrained realizations

In the standard model of cosmology, the linear velocity field constitutes a Gaussian random vector field^{
23
}. The Cosmicflows-2 dataset, as all other available velocity surveys, provides a sparse, incomplete, inhomogeneous and very noisy sampling of the observed flow. The Bayesian formalism of the Wiener filter and constrained realizations provides the optimal methodology for the reconstruction (estimation) of the underlying velocity field and the associated uncertainties in the linear regime^{
11,24,25,26
}. The Wiener-filter/constrained-realizations reconstruction is based on an assumed prior cosmological model — the ΛCDM model with cosmological parameters inferred from the Wilkinson Microwave Anisotropy Probe (WMAP). The current Wiener-filter and constrained-realization fields are the ones reported in our bulk velocity article^{
4
}. The results presented here are insensitive to the exact values of the ΛCDM parameters, in particular to the differences between the WMAP and Planck parameters.

## Cosmic V-web

The cosmic web is defined here by the means of the V-web model^{
17
}. The normalized velocity shear tensor at a given grid cell is defined by:
$$\begin{array}{}\text{(1)}& {\Sigma}_{\alpha \beta}=-\frac{1}{2{H}_{0}}({\partial}_{\alpha}{v}_{\beta}+{\partial}_{\beta}{v}_{\alpha})\end{array}$$

The standard definition of the velocity shear tensor is modified here by the Hubble constant (*H*
_{0}) normalization, which makes it dimensionless. The minus sign is introduced so that a positive eigenvalue corresponds to a contraction. Eigenvalues are ordered by decreasing value, hence
${\stackrel{\mathbf{\u02c6}}{\mathbf{e}}}_{i}$
points in the direction of maximum collapse and
${\stackrel{\mathbf{\u02c6}}{\mathbf{e}}}_{3}$
points toward maximum expansion.

The V-web model starts with the continuous velocity field and its associated velocity shear tensor (Equation (1)). Consider a given point in space at which the shear tensor is evaluated, and thereby its eigenvalues and eigenvectors. The V-web is defined by a threshold value (*λ*
_{th}) — a free parameter that defines the web. The number of eigenvalues above *λ*
_{th} defines the web classification at that point: 0, 1, 2 or 3 corresponds to the point being a void, sheet, filament or knot.

The V-web is defined by the effective resolution of the velocity field and by the value of the threshold. Here a Gaussian smoothing of *R*
_{s} = 250 km s^{−1} and *λ*
_{th} = 0.04 are assumed.

The sources and sinks of the velocity field, namely the repellers and attractors, are closely associated with the voids and knots of the V-web. The voids (knots) are regions of diverging (converging) flow, namely regions where the Hessian of the velocity potential is negative (positive) definite, yet these regions are in general moving with respect to the CMB frame of reference. The repellers and attractors are defined here as stationary voids and knots respectively, and hence correspond to local extrema of the gravitational potential. Note that the Great Attractor, which moves towards the Shapley attractor^{
14
}, is not an attractor as defined here.

## Multipole expansion of the flow

A first-order expansion of a potential (that is, irrotational) velocity field, **v**(**r**), around a point labeled by 0 yields:
$$\begin{array}{}\text{(2)}& {v}_{\alpha}(\mathbf{r})\approx {v}_{\mathrm{0,}\alpha}+({\partial}_{\beta}\phantom{\rule{.5em}{0ex}}{v}_{\alpha})\phantom{\rule{.5em}{0ex}}{r}_{\beta}={v}_{\mathrm{0,}\alpha}-{H}_{0}\phantom{\rule{.5em}{0ex}}{\Sigma}_{\alpha \beta}\phantom{\rule{.5em}{0ex}}{r}_{\beta}\end{array}$$
where *v*
_{0,α
} and Σ_{
αβ
} are evaluated at the point 0. This expansion is equivalent to a dipole and quadrupole expansion of the (velocity) potential. The flow in a sphere of radius *R* is modelled here as the sum of a bulk flow, *V*
_{bulk}(*R*), and a shear term, Σ_{
αβ
}(*R*), in the manner of Equation (2). The parameters of the model (that is, the bulk velocity vector and the symmetric tensor) are found by minimizing the quadratic residual between the model and the actual velocity field, with a spherical top-hat window function weighting.

## Streamlines

In the linear regime the flow is irrotational —that is, it is a potential flow — and hence the velocity field can be written as a gradient of a scalar (velocity) potential,
$\mathbf{v}=\nabla {\varphi}_{v}(\mathbf{r})$
. In the linear regime the peculiar velocity (**v**) and gravitational field (**g**) are simply related by
$\mathbf{v}=\frac{2}{3}\frac{f({\Omega}_{\mathrm{m}},{\Omega}_{\Lambda})}{H{\Omega}_{\mathrm{m}}}\mathbf{g}$
, where Ω_{m}, Ω_{Λ} and *H* are, respectively, the time-dependent cosmological density parameters for matter and for dark energy, and Hubble’s constant. The velocity and gravitational potential are similarly related. Inspired by the similarity between the gravitational potential in linear theory, hence also the velocity potential, to the electrical potential in electrostatics, we present the flow field by field lines which we call here stream or flow lines. The line equation of a streamline, **l**(*s*), where *s* is the line parameter, is calculated by integrating
$d\mathbf{l}(s)=\mathbf{v}(\mathbf{l}(s))ds$
. The numerical calculation of a streamline involves the determination of the seeds of the streamlines and the number of integration steps. For a small number of integration steps and a regular grid of seeds, the streamlines resemble velocity arrows. For a large number of steps, the flow and anti-flow lines are either trapped by attractors or repellers or leave the box. It should be emphasized here that the streamlines are a graphical means for the presentation of a vector field and do not represent trajectories of objects. Mass elements and galaxies move only a few megaparsecs in a Hubble time along streamlines en route from a repeller to an attractor.

## Quantitative comparison of repeller versus attractor

The contribution (in km s^{−1}) of the dipole repeller and the Shapley attractor to the motion of the Local Group is compared by means of setting spheres of radius *R* around both objects, and calculating the velocity induced by the density inside these spheres at the Local Group. The linear-theory density–velocity relation is used in such a calculation. A word of caution is needed before proceeding to the comparison. The attractor region is much better sampled by the Cosmicflows-2 data than that of the dipole repeller; hence the Wiener-filter suppression of the signal is stronger at the dipole repeller, and the constrained variance of the residual around the mean is larger there (Supplementary Fig. 1). The figure presents the contribution (in km s^{−1}) to the Local Group velocity projected onto the direction of the CMB dipole. Out to *R* ≈ 8,000 km s^{−1}, the contributions of the attractor and repeller are almost identical. At 8,000 km s^{−1}, the repeller and the attractor contribute 59 ± 26 and 67 ± 27 km s^{−1}, respectively, to the CMB dipole. The slight deficiency of the dipole repeller is fully consistent with the enhanced suppression of the Wiener filter. The Perseus–Pisces supercluster, at a distance of *R* > 8,000 km s^{−1} away from the repeller, leads to the decrease in the velocity contributed by the dipole-repeller-centric spheres. The dip is the reflection of the tug-of-war between the Perseus–Pisces supercluster and the Great Attractor^{
15
}. Very similar results are obtained when considering the contribution to the amplitude of the full 3D velocity of the Local Group. The conclusion that follows here is that out to *R* ≈ 8,000 km s^{−1}, the Shapley attractor’s basin of attraction and the dipole repeller’s basin of repulsion contribute equally to the Local Group motion.

## Uncertainties assessment

The probability distribution of the alignment of the bulk velocity and the eigenvectors of the shear tensor is sampled by means of an ensemble of 20 constrained realizations, constrained by the Cosmicflows-2 data and evaluated within the WMAP parameters of the ΛCDM model. The constrained realizations are evaluated on a grid of size 256^{3} spanning a box of 256,000 km s^{−1} on its side. The bulk velocity and the velocity shear tensor are obtained by a convolution of the velocity field with a spherical top-hat window of radius *R* and are evaluated at the centre of the box, in other words the location of the Local Group. Supplementary Fig. 2 presents the alignment of the bulk velocity and the third eigenvector of the shear tensor with the dipole repeller and the Shapley attractor, respectively, over a range of radii of *R* = (20, 30,..., 300) × 100 km s^{−1}. The uncertainty in *μ*
_{bulk}(*R*) is used to assess the uncertainty in the position of the dipole repeller. At *R* ≈ 6,000 km s^{−1}, we find *μ*
_{bulk}(*R*) = −0.96 ± 0.04. Assuming the dipole repeller to be responsible for the direction of the bulk velocity, the uncertainty in *μ*
_{bulk} is translated to a projected distance (at a radial distance of 16,000 km s^{−1}) of Δ*R*
_{DR} ≈ 4,500 km s^{−1}. The uncertainty in *μ*
_{bulk} changes significantly over the range *R* = 11,000 to 20,000 km s^{−1}. This again translates to an uncertainty in the radial position of roughly 4,500 km s^{−1}.

The robustness of the dipole repeller against subsampling of the data is considered. First, the Cosmicflows-2 data is subsampled by type: the ‘singles’ subsample consists of 4,264 data entries based on a single galaxy only, and the ‘TF-singles’ subsample is made of 3,943 single data with TF-only distances. The ungrouped Cosmicflows-2 data consists of 8,399 galaxies. The relative fractional distance error of the TF distances is 0.20, and for all the singles is 0.186 ± 0.046. The majority of the non-‘TF-singles’ are data with supernova type-Ia distances for which the relative distance errors are 0.088 ± 0.007. In assessing the constraining power of the various subsamples, one should consider both the number of data points and the magnitude of the errors. For comparison, the relative error for the grouped entries (that is, data entries based on more than one galaxy) is 0.098 ± 0.037. The anti-flow streamlines of the two subsamples are shown in Supplementary Fig. 3, and the details of their repellers are given in Supplementary Table 1. For both subsamples, a repeller (TF-only) or two (all singles) have been found at the back side of the CMB dipole; they both ‘push’, and the dipole repeller is located within an underdense region.

Next, a possible ‘edge of the data’ systematic effect is considered. The analysis is conducted by trimming the full Cosmicflows-2 dataset in spheres of radii 6,000, 8,000 and 10,000 km s^{−1}, which contain 49%, 67% and 82% of the full data, respectively. The Wiener filter has been applied to these subsets of data, and the resulting anti-flow field has been compared with that of the full data (Supplementary Fig. 3). The overall structure of the flow fields of the subsamples follows that of the full Cosmicflows-2 data. The anti-flow converges into two repellers in the 6,000 km s^{−1} case and into single repellers for the 8,000 and 10,000 km s^{−1} cases. Supplementary Table 1 provides the location and distances from the Local Group and the dipole repeller for each case. It also provides the fraction of data contained in each subsample. It is remarkable that with less than half the data (6,000 km s^{−1} case), the Wiener filter recovers the general anti-flow towards the general position of the dipole repeller. Taking the mean position of the two repellers, it is found to be a mere 2,500 km s^{−1} from the dipole repeller. Single repellers are found for the larger subsamples that consist of 67% and 82% of the data at distances of 4,300 and 500 km s^{−1} from the dipole repeller, respectively.

We conclude that the dipole repeller is not a fictitious structure induced by an ‘edge of the data’ effect, and that subsets of the data, chosen either by distance or galaxy type, uncover a basin of repulsion that ‘pushes’ the Local Group in the direction pointed by the CMB dipole.

## Data availability

The data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request. They are also available freely at the Extragalactic Distance Database (http://edd.ifa.hawaii.edu/) and through the NED interface. Use of the data and flow model must cite this article.

## Additional information

**How to cite this article:** Hoffman, Y., Pomarède, D., Tully, R. B & Courtois, H. M. The dipole repeller. *Nat. Astron.* **1,** 0036 (2017).

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## Acknowledgements

We thank J. Sorce and S. Gottloeber for discussions and A. Dupuy for her help in preparing Fig. 3. We thank K. Bowles and S. Thompson for the narration in the Supplementary Video. Support has been provided by the Israel Science Foundation (1013/12), the Institut Universitaire de France, the US National Science Foundation, Space Telescope Science Institute for observations with Hubble Space Telescope, the Jet Propulsion Lab for observations with Spitzer Space Telescope and NASA for analysis of data from the Wide-field Infrared Survey Explorer.

## Author information

## Affiliations

### Racah Institute of Physics, Hebrew University, Jerusalem 91904, Israel

- Yehuda Hoffman

### Institut de Recherche sur les Lois Fondamentales de l'Univers, CEA, Université Paris-Saclay, 91191 Gif-sur-Yvette, France

- Daniel Pomarède

### Institute for Astronomy (IFA), University of Hawaii, 2680 Woodlawn Drive, Honolulu, Hawaii 96822, USA

- R. Brent Tully

### IPN Lyon, UCB Lyon 1/CNRS/IN2P3, University of Lyon, 69622 Villeurbanne, France

- Hélène M. Courtois

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## Contributions

R.B.T. and H.M.C. carried out the observations and data analysis; D.P. contributed graphics and visualization; Y.H. carried out the numerical and theoretical analysis. All co-authors contributed to the writing of the paper, led by Y.H.

## Competing interests

The authors declare no competing financial interests.

## Corresponding author

Correspondence to Yehuda Hoffman.

## Supplementary information

## PDF files

- 1.
### Supplementary Information

Supplementary Figures 1–3, Supplementary Table 1.

## Videos

- 1.
### Supplementary Video

The dipole repeller.