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The statistical analysis of cosmic large-scale structure is most often based on simple two-point summary statistics, like the power spectrum or the two-point correlation function of a sample of galaxies or other types of tracers. In contrast, topological measures of clustering are also sensitive to higher-order correlations, and thus offer the prospect to access additional information that may harbor important constraining power. We here revisit one such geometric measure of the cosmic web in the form of the so-called percolation analysis, using the recent MillenniumTNG simulation suite of the LCDM paradigm. We analyze continuum percolation statistics both for high resolution dark matter particle distributions, as well as for galaxy mock catalogues from a semi-analytic galaxy formation model within a periodic simulation volume of 3000 Mpc on a side. For comparison, we also investigate the percolation statistics of random particle sets and neutrino distributions with two different summed particle masses. We find that the percolation statistics of the dark matter distribution evolves strongly with redshift and thus clustering strength, yielding progressively lower percolation threshold towards later times. However, there is a sizable residual dependence on numerical resolution which we interpret as a residual influence of different levels of shot noise. This is corroborated by our analysis of galaxy mock catalogues whose results depend on sampling density more strongly than on galaxy selection criteria. While this limits the discriminative power of percolation statistics, our results suggest that it still remains useful as a complementary cosmological test when controlled for sampling density., Comment: 14 pages, 12 figures, accepted by ApJ, updated to match published version

Planets are often covered with thin cracked shells. From mud films to lithospheres of rock or ice, fracture networks form two-dimensional (2D) tessellations of convex polygons whose geometry encodes their genesis. Here we chart the geometry of 2D fracture mosaics across the solar system, and decode their formative conditions using a new dynamical crack model. We show that mosaics can be projected onto a Symbolic Ternary Diagram, where the relative proportions of ``T'', ``X'' and ``Y'' junctions are uniquely related to contributions from distinct modes of fracture. Most planetary mosaics cluster in a region associated with hierarchical fracture networks, where sequential cracking favors formation of T junctions. Exceptions to this rule may betray the presence of water. Europa's fracture networks stand apart due to the predominance of X junctions; this is a special feature of ice, where healing of cracks by refreezing of water allows new fractures to overprint older ones. Several fracture networks on Mars appear as outliers due the high proportion of Y junctions. These patterns -- previously interpreted as ancient mudcracks and frozen polar terrain, based on geological evidence -- are consistent with the twisting of crack junctions by cyclic volume change. Our findings suggest that counting cracks could aid in the identification of other water-influenced planetary environments.

A central problem of geometry is the tiling of space with simple structures. The classical solutions, such as triangles, squares, and hexagons in the plane and cubes and other polyhedra in three-dimensional space are built with sharp corners and flat faces. However, many tilings in Nature are characterized by shapes with curved edges, non-flat faces, and few, if any, sharp corners. An important question is then to relate prototypical sharp tilings to softer natural shapes. Here, we solve this problem by introducing a new class of shapes, the \textit{soft cells}, minimizing the number of sharp corners and filling space as \emph{soft tilings}. We prove that an infinite class of polyhedral tilings can be smoothly deformed into soft tilings and we construct the soft versions of all Dirichlet-Voronoi cells associated with point lattices in two and three dimensions. Remarkably, these ideal soft shapes, born out of geometry, are found abundantly in nature, from cells to shells., Comment: 18 pages, 7 figures

We prove that every homogeneous convex polyhedron with only one unstable equilibrium (known as a mono-unstable convex polyhedron) has at least $7$ vertices. Although it has been long known that no mono-unstable tetrahedra exist, and mono-unstable polyhedra with as few as $18$ vertices and faces have been constructed, this is the first nontrivial lower bound on the number of vertices for a mono-unstable polyhedron. There are two main ingredients in the proof. We first establish two types of relationships, both expressible as (non-convex) quadratic inequalities, that the coordinates of the vertices of a mono-unstable convex polyhedron must satisfy, taking into account the combinatorial structure of the polyhedron. Then we use numerical semidefinite optimization algorithms to compute easily and independently verifiable, rigorous certificates that the resulting systems of quadratic inequalities (5943 in total) are indeed inconsistent in each case., Comment: Submitted for publication

Type-I Superluminous Supernovae (SLSNe) are an exotic class of core-collapse SN (CCSN) that can be up to 100 times brighter and more slowly-evolving than normal CCSNe. SLSNe represent the end-stages of the most massive stripped stars, and are thought to be powered by the spin-down energy of a millisecond magnetar. Studying them and measuring their physical parameters can help us to better understand stellar mass-loss, evolution, and explosions. Moreover, thanks to their high luminosities, SLSNe can be seen up to greater distances, allowing us to explore how stellar physics evolves as a function of redshift. The High Latitude Time Domain Survey (HLTDS) will provide us with an exquisite dataset that will discover 100s of SLSNe. Here, we focus on the question of which sets of filters and cadences will allow us to best characterize the physical parameters of these SLSNe. We simulate a set of SLSNe at redshifts ranging from z = 0.1 to z = 5.0, using six different sets of filters, and cadences ranging from 5 to 100 days. We then fit these simulated light curves to attempt to recover the input parameter values for their ejecta mass, ejecta velocity, magnetic field strength, and magnetar spin period. We find that four filters are sufficient to accurately characterize SLSNe at redshifts below $z = 3$, and that cadences faster than 20 days are required to obtain measurements with an uncertainty below 10\%, although a cadence of 70 days is still acceptable under certain conditions. Finally, we find that the nominal survey strategy will not be able to properly characterize the most distant SLSNe at $z = 5$. We find that the addition of 60-day cadence observations for 4 years to the nominal HLTDS survey can greatly improve the prospect of characterizing these most extreme and distant SNe, with only an 8\% increase to the time commitment of the survey., Comment: Submitted as part of the Nancy Grace Roman Space Telescope's Core Community Surveys call for white papers

Pair-instability supernovae (PISNe) are explosions of very massive stars that may have played a critical role in the chemical evolution and reionization of the early Universe. In order to quantify their roles, it is required to know the PISN event rate at z > 6. Although Roman Space Telescope has a capability to discover PISNe at z > 6, identifying rare high-redshift PISN candidates among many other transients is challenging. In order to efficiently identify PISN candidates at z > 6, we propose to add sparse F213 observations reaching 26.5 mag (or deeper) every half year in the High Latitude Time Domain Survey. By adding the F213 information, PISNe at z > 6 can be efficiently identified in the color-magnitude diagram., Comment: 5 pages, 2 figures, submitted as a Roman Core Community Survey White Paper

The monostatic property of polyhedra (i.e. the property of having just one stable or unstable static equilibrium point) has been in a focus of research ever since Conway and Guy \cite{Conway} published the proof of the existence of the first such object. In the same article they also proved that a homogeneous tetrahedron has at least two stable equilibrium points. By using polar duality, the same idea has been used \cite{balancing} to prove that a homogeneous tetrahedron has at least two unstable equilibria. Conway \cite{Dawson} also claimed that among inhomogeneous tetrahedra one can find monostable ones. Here we not only give a formal proof of this statement and show that monostatic tetrahedra have exactly 4 equilibria, but also demonstrate a startling new aspect of this problem: being monostatic implies certain \emph{visible} features of the shape and vice versa. Our results also imply that mono-monostatic tetrahedra (having just one stable and just one unstable equilibrium point) do not exist. In contrast, we show that for any other legal number of faces, edges, and vertices there is a mono-monostatic polyhedron with that face vector., Comment: 12 pages, 7 figures

In the study of monostatic polyhedra, initiated by John H. Conway in 1966, the main question is to construct such an object with the minimal number of faces and vertices. By distinguishing between various material distributions and stability types, this expands into a small family of related questions. While many upper and lower bounds on the necessary numbers of faces and vertices have been established, none of these questions has been so far resolved. Adapting an algorithm presented in (Boz\'{o}ki et al., 2022), here we offer the first complete answer to a question from this family: by using the toolbox of semidefinite optimization to efficiently generate the hundreds of thousands of infeasibility certificates, we provide the first-ever proof for the existence of a monostatic polyhedron with point masses, having minimal number (V=11) of vertices (Theorem 3) and a minimal number (F=8) of faces. We also show that V=11 is the smallest number of vertices that a mono-unstable polyhedron can have in all dimensions greater than 1. (Corollary 6.), Comment: Accepted for publication

We introduce and study a general framework for modeling the evolution of crack networks. The evolution steps are triggered by exponential clocks corresponding to local micro-events, and thus reflect the state of the pattern. In an appropriate simultaneous limit of pattern domain tending to infinity and time step tending to zero, a continuous time model, specifically a system of ODE is derived that describes the dynamics of averaged quantities. In comparison with the previous, discrete time model, studied recently by two of the present three authors, this approach has several advantages. In particular, the emergence of non-physical solutions characteristic to the discrete time model is ruled out in the relevant nonlinear version of the new model. We also comment on the possibilities of studying further types of pattern formation phenomena based on the introduced general framework., Comment: 30 pages, 7 figures

We have extracted 636 spectra taken at the positions of 583 transient sources from the third Data Release of the Hobby-Eberly Telescope Dark Energy eXperiment (HETDEX). The transients were discovered by the Zwicky Transient Facility (ZTF) during 2018 - 2022. The HETDEX spectra are useful to classify a large number of objects found by photometric surveys for free. We attempt to explore and classify the spectra by utilizing machine learning (ML) and template matching techniques. We have identified two transient sources, ZTF20aatpoos = AT2020fiz and ZTF19abdkelq as supernova candidates. We classify AT2020fiz as a Type IIP supernova observed ~10 days after explosion, and we propose ZTF19abdkelq as a likely Type Ia SN caught ~40 days after maximum light. ZTF photometry of these two sources are consistent with their classification as supernovae. Beside these two objects, we have confirmed several ZTF transients as variable AGNs based on their spectral appearance, and also determined the host galaxy types for several other ZTF transients., Comment: submitted to ApJ

The 2-m aperture Chinese Space Station Telescope (CSST), which observes at wavelengths ranging from 255 to 1000 nm, is expected to start science operations in 2024. An ultra-deep field observation program covering approximately 10 square degrees is proposed with supernovae (SNe) and other transients as one of its primary science drivers. This paper presents the simulated detection results of type Ia supernovae (SNe Ia) and explores the impact of new datasets on the determinations of cosmological parameters. The simulated observations are conducted with an exposure time of 150 s and cadences of 10, 20, and 30 days. The survey mode covering a total of 80 observations but with a random cadence in the range of 4 to 14 days is also explored. Our simulation results indicate that the CSST can detect up to $\sim 1800$ SNe Ia at z $<$ 1.3. The simulated SNe Ia are then used to constrain the cosmological parameters. The constraint on $\Omega_m$ can be improved by 37.5% using the 10-day cadence sample in comparison with the Pantheon sample. A deeper measurement simulation with a 300 s exposure time together with the Pantheon sample improves the current constraints on $\Omega_m$ by 58.3% and $\omega$ by 47.7%. Taking future ground-based SNe Ia surveys into consideration, the constraints on $\omega$ can be improved by 59.1%. The CSST ultra-deep field observation program is expected to discover large amounts of SNe Ia over a broad redshift span and enhance our understanding of the nature of dark energy., Comment: 10 pages, 6 figures, 2 tables, accepted for publication in SCIENCE CHINA Physics, Mechanics & Astronomy

We investigate the potential of using a sample of very high-redshift ($2\lesssim z \lesssim6$) (VHZ) Type Ia supernovae (SNe~Ia) attainable by the James Webb Space Telescope (JWST) on constraining cosmological parameters. At such high redshifts, the age of the universe is young enough that the VHZ SNIa sample comprises the very first SNe~Ia of the universe, with progenitors among the very first generation of low mass stars that the universe has made. We show that the VHZ SNe~Ia can be used to disentangle systematic effects due to the luminosity distance evolution with redshifts intrinsic to SNIa standardization. Assuming that the systematic evolution can be described by a linear or logarithmic formula, we found that the coefficients of this dependence can be determined accurately and decoupled from cosmological models. Systematic evolution as large as 0.15 mag and 0.45 mag out to $z=5$ can be robustly separated from popular cosmological models for the linear and logarithmic evolution, respectively. The VHZ SNe~Ia will lay the foundation for quantifying the systematic redshift evolution of SNIa luminosity distance scales. When combined with SNIa surveys at comparatively lower redshifts, the VHZ SNe~Ia allow for a precise measurement of the history of the expansion of the universe from $z\sim 0$ to the epoch approaching reionization., Comment: Accepted for publication in ApJ. Typos corrected. Fig.14 updated

Molecular self-assembly plays a very important role in various aspects of technology as well as in biological systems. Governed by the covalent, hydrogen or van der Waals interactions - self-assembly of alike molecules results in a large variety of complex patterns even in two dimensions (2D). Prediction of pattern formation for 2D molecular networks is extremely important, though very challenging, and so far, relied on computationally involved approaches such as density functional theory, classical molecular dynamics, Monte Carlo, or machine learning. Such methods, however, do not guarantee that all possible patterns will be considered and often rely on intuition. Here we introduce a much simpler, though rigorous, hierarchical geometric model founded on the mean-field theory of 2D polygonal tessellations to predict extended network patterns based on molecular-level information. Based on graph theory, this approach yields pattern classification and pattern prediction within well-defined ranges. When applied to existing experimental data, our model provides an entirely new view of self-assembled molecular patterns, leading to interesting predictions on admissible patterns and potential additional phases. While developed for hydrogen-bonded systems, an extension to covalently bonded graphene-derived materials or 3D structures such as fullerenes is possible, significantly opening the range of potential future applications.

We examine geophysical crack patterns using the mean field theory of convex mosaics. We assign the pair $(\bar n^*,\bar v^*)$ of average corner degrees to each crack pattern and we define two local, random evolutionary steps $R_0$ and $R_1$, corresponding to secondary fracture and rearrangement of cracks, respectively. Random sequences of these steps result in trajectories on the $(\bar n^*,\bar v^*)$ plane. We prove the existence of limit points for several types of trajectories. Also, we prove that cell density $\rho = \bar v^*/\bar n^*$ increases monotonically under any admissible trajectory., Comment: 15 pages, 4 figures

We regard a smooth, $d=2$-dimensional manifold $\mathcal{M}$ and its normal tiling $M$, the cells of which may have non-smooth or smooth vertices (at the latter, two edges meet at 180 degrees.) We denote the average number (per cell) of non-smooth vertices by $\bar v^{\star}$ and we prove that if $M$ is periodic then $v^{\star} \geq 2$ and we show the same result for the monohedral case by an entirely different argument. Our theory also makes a closely related prediction for non-periodic tilings. In 3 dimensions we show a monohedral construction with $\bar v^{\star}=0$., Comment: 11 pages, 4 figures

The monostatic property of convex polyhedra (i.e. the property of having just one stable or unstable static equilibrium point) has been in the focus of research ever since Conway and Guy published the proof of the existence of the first such object, followed by the constructions of Bezdek and Reshetov. These examples establish $F\leq 14, V\leq 18$ as the respective \emph{upper bounds} for the minimal number of faces and vertices for a homogeneous mono-stable polyhedron. By proving that no mono-stable homogeneous tetrahedron existed, Conway and Guy established for the same problem the lower bounds for the number of faces and vertices as $F, V \geq 5$ and the same lower bounds were also established for the mono-unstable case. It is also clear that the $F,V \geq 5$ bounds also apply for convex, homogeneous point sets with unit masses at each point (also called polyhedral 0-skeletons) and they are also valid for mono-monostatic polyhedra with exactly on stable and one unstable equilibrium point (both homogeneous and 0-skeletons). Here we present an algorithm by which we improve the lower bound to $V\geq 8$ vertices (implying $f \geq 6$ faces) on mono-unstable and mono-monostable 0-skeletons. Our algorithm appears to be less well suited to compute the lower bounds for mono-stability. We point out these difficulties in connection with the work of Dawson and Finbow who explored the monostatic property of simplices in higher dimensions., Comment: 45 pages, 3 figures

This is the third sequel in a series discussing the discovery of various types of extragalactic transients with the {\it James Webb Space Telescope} in a narrow-field ($\sim 0.1$ deg$^2$), moderately deep ($m_{AB} \sim 27$ mag) survey. In this part we focus on the detectability and observational characteristics of Direct Collapse Black Holes (DCBH) and Tidal Disruption Events (TDE) around them. We use existing models for DCBH accretion luminosities and spectra as well as for TDE light curves, and find that accreting DCBH seeds may be bright enough for detection up to $z \sim 7$ with JWST NIRCam imaging, TDEs of massive ($M \gtrsim 50$ Msol) stars around them can enhance the chance for discovering them as transient objects, although the rates of such events is low, a few per survey time. TDEs around non-accreting black holes of $M \sim 10^6$ Msol may also be detected at $z < 7$ redshifts in the redder NIRCam bands between 3 and 5 microns. It is also shown that accreting DCBHs appear separate from supernovae (SNe) on the NIRCam color-color plot, but TDEs from quiescent black holes fall in nearly the same color range as Superluminous Supernovae (SLSNe), which makes them more difficult to identify., Comment: 17 pages, 11 figures, ApJ in press

Pair-instability supernovae (PISNe) are the ultimate cosmic lighthouses, capable of being observed at z<25 and revealing the properties of primordial stars at cosmic dawn. But it is now understood that the spectra and light curves of these events evolved with redshift as the universe became polluted with heavy elements because chemically enriched stars in this mass range typically lose most of their hydrogen envelopes and explode as bare helium cores. The light curves of such transients can be considerably dimmer in the near infrared (NIR) today than those of primordial PISNe of equal energy and progenitor mass. Here, we calculate detection rates for PISNe whose progenitors lost their outer layers to either line-driven winds or rotation at z<10, their detection limit in redshift for the James Webb Space Telescope (JWST). We find that JWST may be able to detect only Pop II (metal-poor) PISNe over the redshift range of z<4, but not their Pop III (metal-free) counterparts., Comment: Accepted for ApJ, in press

Future time-domain surveys for transient events in the near- and mid-infrared bands will significantly extend our understanding about the physics of the early Universe. In this paper we study the implications of a deep (~27 mag), long-term (~3 years), observationally inexpensive survey with the James Webb Space Telescope (JWST) within its Continuous Viewing Zone, aimed at discovering luminous supernovae beyond z~2 redshift. We explore the possibilities for detecting Superluminous Supernovae (SLSNe) as well as Type Ia supernovae at such high redshifts and estimate their expected numbers within a relatively small (~0.1 deg^2) survey area. It is found that we can expect ~10 new SLSNe and ~50 SNe Ia discovered in the 1 < z < 4 redshift range. We show that it is possible to get relatively accurate (sigma_z < 0.25) photometric redshifts for Type Ia SNe by fitting their Spectral Energy Distributions (SED), redshifted into the observed near-IR bands, with SN templates. We propose that Type Ia SNe occupy a relatively narrow range on the JWST F220W-F440W vs F150W-F356W color-color diagram between +/- 7 rest-frame days around maximum light, which could be a useful classification tool for such type of transients. We also study the possibility of extending the Hubble-diagram of Type Ia SNe beyond redshift 2 up to z~4. Such high-z SNe Ia may provide new observational constraints for their progenitor scenario., Comment: accepted for publication in ApJ

We define the mechanical complexity $C(P)$ of a convex polyhedron $P,$ interpreted as a homogeneous solid, as the difference between the total number of its faces, edges and vertices and the number of its static equilibria, and the mechanical complexity $C(S,U)$ of primary equilibrium classes $(S,U)^E$ with $S$ stable and $U$ unstable equilibria as the infimum of the mechanical complexity of all polyhedra in that class. We prove that the mechanical complexity of a class $(S,U)^E$ with $S, U > 1$ is the minimum of $2(f+v-S-U)$ over all polyhedral pairs $(f,v )$, where a pair of integers is called a polyhedral pair if there is a convex polyhedron with $f$ faces and $v$ vertices. In particular, we prove that the mechanical complexity of a class $(S,U)^E$ is zero if, and only if there exists a convex polyhedron with $S$ faces and $U$ vertices. We also give asymptotically sharp bounds for the mechanical complexity of the monostatic classes $(1,U)^E$ and $(S,1)^E$, and offer a complexity-dependent prize for the complexity of the G\"omb\"oc-class $(1,1)^E$., Comment: 29 pages, 13 figures

JWST was conceived and built to answer one of the most fundamental questions that humans can address empirically: "How did the Universe make its first stars?". Our First Lights At REionization (FLARE) project transforms the quest for the epoch of reionization from the static to the time domain. It targets the complementary question: "What happened to those first stars?". It will be answered by observations of the most luminous events: supernovae and accretion on to black holes formed by direct collapse from the primordial gas clouds. These transients provide direct constraints on star-formation rates (SFRs) and the truly initial Initial Mass Function (IMF), and they may identify possible stellar seeds of supermassive black holes (SMBHs). Furthermore, our knowledge of the physics of these events at ultra-low metallicity will be much expanded. JWST's unique capabilities will detect these most luminous and earliest cosmic messengers easily in fairly shallow observations. However, these events are very rare at the dawn of cosmic structure formation and so require large area coverage. Time domain astronomy can be advanced to an unprecedented depth by means of a shallow field of JWST reaching 27 mag AB in 2 and 4.4 microns over a field as large as 0.1 square degree visited multiple times each year. Such a survey may set strong constraints or detect massive Pop III SNe at redshifts beyond 10, pinpointing the redshift of the first stars, or at least their death. Based on our current knowledge of superluminous supernovae (SLSNe), such a survey will find one or more SLSNe at redshifts above 6 in five years and possibly several direct collapse black holes. Although JWST is not designed as a wide field survey telescope, we show that such a wide field survey is possible with JWST and is critical in addressing several of its key scientific goals.

In the context of the ESA M5 (medium mission) call we proposed a new satellite mission, Theia, based on relative astrometry and extreme precision to study the motion of very faint objects in the Universe. Theia is primarily designed to study the local dark matter properties, the existence of Earth-like exoplanets in our nearest star systems and the physics of compact objects. Furthermore, about 15 $\%$ of the mission time was dedicated to an open observatory for the wider community to propose complementary science cases. With its unique metrology system and "point and stare" strategy, Theia's precision would have reached the sub micro-arcsecond level. This is about 1000 times better than ESA/Gaia's accuracy for the brightest objects and represents a factor 10-30 improvement for the faintest stars (depending on the exact observational program). In the version submitted to ESA, we proposed an optical (350-1000nm) on-axis TMA telescope. Due to ESA Technology readiness level, the camera's focal plane would have been made of CCD detectors but we anticipated an upgrade with CMOS detectors. Photometric measurements would have been performed during slew time and stabilisation phases needed for reaching the required astrometric precision., Comment: http://theia.phyip3.dur.ac.uk

Context. The study of the galaxy stellar mass function (SMF) in relation to the galaxy environment and the stellar mass density profile, rho(r), is a powerful tool to constrain models of galaxy evolution. Aims. We determine the SMF of the z=0.44 cluster of galaxies MACS J1206.2-0847 separately for passive and star-forming (SF) galaxies, in different regions of the cluster, from the center out to approximately 2 virial radii. We also determine rho(r) to compare it to the number density and total mass density profiles. Methods. We use the dataset from the CLASH-VLT survey. Stellar masses are obtained by SED fitting on 5-band photometric data obtained at the Subaru telescope. We identify 1363 cluster members down to a stellar mass of 10^9.5 Msolar. Results. The whole cluster SMF is well fitted by a double Schechter function. The SMFs of cluster SF and passive galaxies are statistically different. The SMF of the SF cluster galaxies does not depend on the environment. The SMF of the passive population has a significantly smaller slope (in absolute value) in the innermost (<0.50 Mpc), highest density cluster region, than in more external, lower density regions. The number ratio of giant/subgiant galaxies is maximum in this innermost region and minimum in the adjacent region, but then gently increases again toward the cluster outskirts. This is also reflected in a decreasing radial trend of the average stellar mass per cluster galaxy. On the other hand, the stellar mass fraction, i.e., the ratio of stellar to total cluster mass, does not show any significant radial trend. Conclusions. Our results appear consistent with a scenario in which SF galaxies evolve into passive galaxies due to density-dependent environmental processes, and eventually get destroyed very near the cluster center to become part of a diffuse intracluster medium., Comment: A&A accepted, 15 pages, 13 figures

We aim at constraining the assembly history of clusters by studying the intra cluster light (ICL) properties, estimating its contribution to the fraction of baryons in stars, f*, and understanding possible systematics/bias using different ICL detection techniques. We developed an automated method, GALtoICL, based on the software GALAPAGOS to obtain a refined version of typical BCG+ICL maps. We applied this method to our test case MACS J1206.2-0847, a massive cluster located at z=0.44, that is part of the CLASH sample. Using deep multi-band SUBARU images, we extracted the surface brightness (SB) profile of the BCG+ICL and we studied the ICL morphology, color, and contribution to f* out to R500. We repeated the same analysis using a different definition of the ICL, SBlimit method, i.e. a SB cut-off level, to compare the results. The most peculiar feature of the ICL in MACS1206 is its asymmetric radial distribution, with an excess in the SE direction and extending towards the 2nd brightest cluster galaxy which is a Post Starburst galaxy. This suggests an interaction between the BCG and this galaxy that dates back to t <= 1.5 Gyr. The BCG+ICL stellar content is 8% of M_(*,500) and the (de-) projected baryon fraction in stars is f*=0.0177 (0.0116), in excellent agreement with recent results. The SBlimit method provides systematically higher ICL fractions and this effect is larger at lower SB limits. This is due to the light from the outer envelopes of member galaxies that contaminate the ICL. Though more time consuming, the GALtoICL method provides safer ICL detections that are almost free of this contamination. This is one of the few ICL study at redshift z > 0.3. At completion, the CLASH/VLT program will allow us to extend this analysis to a statistically significant cluster sample spanning a wide redshift range: 0.2