The process of creating the HexMyTD cartogram is a two-stage process. Starting with the geographically true representation of the constituencies, the first job is to create a cartogram that distributes population evenly. To accomplish this, I use a diffusion based cartogram code available from here based on an algorithm described in Gastner and Newman (2004). You might have seen the global population cartograms doing the same previously.
The act of this on Irish constituencies is, yes—you guessed it, to expand the Dublin constituencies. Cork and Limerick city constituencies are also expanded, as the only predominantly city based constituencies outside of Dublin. The skewing is probably worst in the constituencies adjacent to Dublin: Wicklow is unrecognisable, Meath and Kildare distorted. Some of the rural constituencies suffer from squeezing. Notably, Mayo takes a big hit—maybe these cartograms won't be a big hit with an Taoiseach.
The final and most subtle of the steps is going from this cartogram to the final hexagonal grid. The areas are approximately right so that the number of hexagons corresponding to the number of TDs but there is a lot of shuffling at this stage. The aim is to lay out the hexagons so that they bear a semblance to the geographical shape of the actual constituencies. Donegal is good; Kerry isn't bad. Poor old Wexford is badly deformed and Dublin's a complete nightmare! Especially to a bogger like me.
Michael T. Gastner and M. E. J. Newman
Diffusion-based method for producing density-equalizing maps
Proc. Nat. Acad. Sci. USA, 101, 7499-7504 (2004).