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X-shaped or peanut-shaped (X/P) bulges are observed in more than 40% of(nearly) edge-on disc galaxies, though to date a robust method to quantify themis lacking. Using Fourier harmonics to describe the deviation of galaxyisophotes from ellipses, we demonstrate with a sample of 11 such galaxies(including NGC 128) that the sixth Fourier component ($B_6$) carries physicalmeaning by tracing this X/P structure. We introduce five quantitativediagnostics based on the radial $B_6$ profile, namely: its `peak' amplitude($\Pi_{\rm max}$); the (projected major-axis) `length' where this peak occurs($R_{\Pi,max}$); its vertical `height' above the disc plane ($z_{\Pi,max}$);the $B_6$ profile's integrated `strength' ($S_{\Pi}$); and the $B_6$ peak`width' ($W_{\Pi}$). We also introduce different `classes' of $B_6$ profileshape. Furthermore, we convincingly detect and measure the properties ofmultiple (nested) X/P structures in individual galaxies which additionallydisplay the signatures of multiple bars in their surface brightness profiles,thus consolidating further the scenario in which peanuts are associated withbars. We reveal that the peanut parameter space (`length', `strength' and`height') for real galaxies is not randomly populated, but the 3 metrics areinter-correlated (both in kpc and disc scale-length $h$). Additionally, the X/P`length' and `strength' appear to correlate with $v_{rot}/\sigma_*$, lendingfurther support to the notion that peanuts `know' about the galactic disc inwhich they reside. Such constraints are important for simulations, as theyprovide a direct link between peanuts and their host disc. Our diagnosticsreveal a spectrum of X/P properties and could provide a means of distinguishingbetween different peanut formation scenarios. Moreover, nested peanuts, asremnants of bar buckling events, can provide insights into the disc and barinstability history.

Quantifying the (X/Peanut)-Shaped Structure in Edge-on Disc Galaxies: Length, Strength, and Nested Peanuts