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over the past few years there has been considerable work on the detection of two - dimensional galaxy overdensities in wide field imaging surveys for subsequent spectroscopic follow up
. however , these systematic searches for clusters , fuelled by the availability of large ccd camera mosaics , are in many cases biased towards high density peaks of the galaxy distribution .    with the next generation of large multi - object spectrographs in their final construction phase , deep redshift surveys ( keck / deep2 and vlt / virmos ) are well within the horizon .
it is now worthwhile to address the more ambitious task of constructing a three dimensional , statistically complete sample of high redshift galaxy clusters .
here we describe a fully automated and objective algorithm for identifying and reconstructing galaxy systems , based upon 3d voronoi polyhedra and delaunay triangulation .
a voronoi partition of the space into minimally sized convex polytopes is a natural way to measure packing .
a voronoi polyhedron is the uniquely defined region of space around a galaxy ( seed ) , within which each point is closer to the seed than to any other galaxy .
the faces of the voronoi cell are formed by planes perpendicular to the vectors between a galaxy and its neighbors .
the volume inside the polyhedron is inversely proportional to the packing efficiency of its central galaxy ; a large cell volume is indicative of an isolated galaxy .
the delaunay triangulation is the geometrical dual of the voronoi partition and is defined by the tetrahedron whose vertices are the 4 galaxies with the property that the uniquely determined circumscribing sphere does not contain any other galaxy .
we thus may identify high - density peaks by selecting as `` cluster seeds '' the centers of all the voronoi cells with volume smaller than some threshold , and then use the scale length of the delaunay mesh to infer the strength of the physical aggregation .
we then can define an adaptive cylindrical window in redshift space ( elongated in the @xmath0 direction ) with dimensions determined by the local scale factor and process all the delaunay connected galaxies with a rapidly converging `` inclusion - exclusion '' logic to identify cluster members .    in our algorithm ,
the parameters defining the adaptive search window are only weakly dependent on the galaxy density gradient which inevitably occurs in a flux limited survey , since our smoothing procedure immediately identifies regions of enhanced clustering where galaxies are generally more luminous .
moreover the number of cluster interlopers aggregated into a system in a redshift space analysis is minimized and we simultaneously obtain a non - parametric local estimate of the surrounding density environment for each galaxy and a quantitative measure of the distribution of cosmological voids in the survey volume .
we have tested our reconstruction scheme using mock catalogs modelling the deep2 redshift survey ( davis et al .
2000 ) derived from gif simulations ( coil et al .
this survey will obtain high quality spectra for @xmath1 60000 galaxies between z=0.7 - 1.5 using the keck 2 telescope and deimos spectrograph , with the twin goals of studying the evolution of the properties and the large - scale clustering of high redshift galaxies .
we used a lcdm model with @xmath2 , @xmath3 , @xmath4 and @xmath5 . in order to mimic the selection function of the magnitude limited survey we make a rest - frame b - band cut of 23.4 , which at z @xmath1 1 corresponds to an apparent i - band magnitude with the appropriate k - correction matching the deep2 photometric selection criteria ( @xmath6 ) . in fig .
2a we present a mock catalog for one deep2-like field collapsed along the smallest axis , covering the redshift range @xmath0=0.7 - 1.2 ( a depth which corresponds to selecting the 1200 l / mm grating in the deimos spectrograph ) and containing a volume of roughly @xmath7 mpc h@xmath8 ( @xmath1 15000 galaxies ) .
we have used six independent mock catalogs to determine how well the algorithm performs in identifying clusters of galaxies .    to determine the `` true '' distribution of clusters , we have applied in real space ( volume - limited simulation ) a standard percolation algorithm optimised for selecting virialized objects ( mean overdensity @xmath1 180 ) .
the resulting distribution of those clusters with more than 5 members is shown in fig 2b .
cluster candidates selected by the voronoi  delauney algorithm applied in redshift space ( flux - limited simulation ) are shown in fig 2c .
note how the large - scale pattern defined by galaxy systems and the associations of clusters in higher order structures reproduce in an unbiased way the underlying real space landscape .
the evolution of the comoving abundance of clusters as a function of their velocity dispersion @xmath9 and redshift @xmath0 is a sensitive function of cosmological parameters ( see newman et al .
the statistical significance and robustness of the cluster abundance test depends critically on an unbiased mapping of the distribution properties of the cluster observables between real to redshift space .
we have investigated the differential and integral distribution functions of the number of reconstructed clusters as a function of their richness , velocity dispersion and redshift .
the kolmogorov - smirnov statistical test confirms the similarity between the real and reconstructed distributions , and thus the reliability of our algorithm .
this can be seen visually in fig .
4 where we show the two - dimensional distributions of the real and reconstructed clusters in redshift z and projected velocity dispersion @xmath9 .
coil , a. , davis , m. , & szapudi , i. 2001 , pasp , in press , astro - ph/0107541 davis , m. , newman , j. a. , faber , s. a. , & phillips , a. c. , 2000 in proc . of the eso / ecf / stsci workshop on deep fields , ( garching , edition springer ) , astro - ph/0012189 newman , j. a. , marinoni , c. , davis , m. , & coil , a. 2001 , pasp , in press
we have developed a geometrical method based on 3d voronoi polyhedra and delaunay tessellation for identifying and reconstructing clusters of galaxies in the next generation of deep , flux - limited redshift surveys . 
 we here describe this algorithm and tests of it using mock catalogs that simulate the deep2/deimos redshift survey , which will begin observations in the spring of 2002 and will provide a detailed three dimensional map of the large scale structure up to redshift 1.5 .    
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