Key to Palaearctic Cardiocondyla
Key to the species of Cardiocondyla with outdoor occurrence in the Palaearctic, based on Seifert (2023).
This key almost thoroughly used dichotomous decision schedules for cases requiring combinations of multiple characters and in the same time it aspired to achieve a low error rate inevitably resulting in tedious procedures. Lamenting does not help here and those who think that simple answers can be given for difficult problems bark up the wrong tree. The finally best approach is here using a software providing linear discriminant functions (LDA), measuring the full set of morphometric characters and running the investigated samples as wildcards within the data space of the hypothesis-providing supplementary file SI2. It is convenient to run the analysis with primary (raw) measurements instead with ratios. A good check of the result is then comparing with the images that are given for any species in four aspects and to consider geographic distribution. The low error rates reported in the key cannot be achieved for such tiny ants without high-resolution optical systems, a good measuring stage and a careful realization of character definitions.
See Character Definitions below for characters used in this key.
1
- Anterior postpetiolar sternite laterally more prominent than in median part—as result an imagined cross section shows a deeply concave ventral margin of the sternite. The lateral protrusions appear in frontolateral view as dents (Fig. 113). Cardiocondyla wroughtonii group => 2
- Anterior postpetiolar sternite in median part more prominent or at equal level as the lateral surface—as result an imagined cross section shows a convex or straight ventral margin of the sternite => 3
2
return to couplet #1
- With all measurements in mm, discriminant 34.21*CW-59.33*PoOc-177.83*EYE+72.15*SL-113.56*FRS-106.46*SP+140.69*PpW-14.15 < 0 [error 0% in 37 nest sample means of two workers measured]. 1st gaster segment in a majority of samples with absent or weakly developed blackish pigmentation. Worldwide tramp species. Outdoor occurrence confirmed for the Middle East. Figs. 105–108 => Cardiocondyla wroughtonii
- Discriminant > 0 [0% error in 65 nest sample means of two workers]. 1st gaster segment in a majority of samples with strongly developed blackish pigmentation. Worldwide tramp species. Outdoor occurrence confirmed for southern Spain and the Middle East. Figs.109–113 => Cardiocondyla obscurior
3
return to couplet #1
- True foveolae on vertex dorsal of the eyes completely absent. Instead the bases of pubescence hairs are placed in the center of flat tubercles or flat pits of only 4–10 µm diameter, giving a finely punctate surface appearance at lower magnifications (Figs. 89, 97). Postocular distance large, PoOc/CL 0.416–0.467. Frons usually very wide, FRS/CS > 0.288 (except [[Cardiocondyla tibetana, here 0.280 ± 0.006). Propodeal spines reduced to blunt dents. Metanotal depression deep. Cardiocondyla stambuloffii group => 4
- Bases of pubescence hairs on vertex dorsal of the eyes placed in the center of foveolae or cuticular meshes of > 10 µm diameter (Figs. 9, 25, 117)—if foveolae reduced (Cardiocondyla verdensis), than PoOc/CL < 0.394. Frons narrower, FRS/CS < 0.288. Metanotal depression varying from nearly absent to deep. Propodeal spines variable => 8
4
return to couplet #3
- Eye larger, EYE/CS >0.235. Frons narrower, FRS/CS <0.295. Waist segments lower, PeH/CS <0.343, PpH/CS < 0.283. Southern and northern margin of Tarim Basin. Figs. 101–104 => Cardiocondyla tibetana
- EYE/CS <0.235, FRS/CS > 0.295, PeH/CS >0.343, PpH/CS < 0.283 => 5
5
return to couplet #4
- Posterior 40% of vertex surface completely smooth except for tiny pits of 4–5 µm around the bases of pubescence hairs just visible with high-resolution stereomicroscopy. Pubescence longer. With all measurements in mm, discriminant 334.68*PLg-45.456*CW+88.43*PoOc-66.63*EYE-3.60 > 2.3. South Kazakhstan. Figs. 98–100 => Cardiocondyla gibbosa
- Posterior part of vertex not completely smooth, at least with weak rugulae and small tubercles around the bases of pubescence hairs. Discriminant < 2.3 => 6
6
return to couplet #5
- Westpalaearctic, known range extending east to the western shores of Caspian Sea. With all measurements in mm, discriminant 44.95*CW+54.68*PoOc+31.14*SL-130.25*SPBA-79.81*PpW +40.84*PeH+90.89*PpH -28.33 < 0.4 [0% error in 82 individuals]. Figs. 86–89 => Cardiocondyla stambuloffii
- Central Palaearctic, known range extending from western coast of Caspian Sea east to Mongolia. Discriminant > 0.4 [0% error in 59 individuals] => 7
7
return to couplet #6
- With all measurements in mm, discriminant 120.6*PoOc-29.53*SL+109.54*PeW-109.42*PeH-157.97*PLg-7.70 < 0.7 [0% error in 37 individuals]. Figs. 94–97 => Cardiocondyla rolandi
- Discriminant > 0.7 [0% error in 22 individuals]. Figs. 90–93 => Cardiocondyla koshewnikovi
8
return to couplet #3
- Propodeal spines reduced to obtusely angled corners (Fig. 115). Postpetiole rather narrow, PpW/CS < 0.46, in dorsal view never suggestively hexagonal. With all measurements in mm, discriminant 48.06*PoOc+27.66*SL-56.96*PpW+119.07*Mgr-13.79 > 0 [error 0% in 17 individuals] => 9
- Propodeal spines not reduced to obtusely angled corners. If spines occasionally reduced to blunt corners, then EYE/CS > 0.221 and PpW/CS >0.450. Discriminant < 0 [error 0% in 1140 individuals] => 10
9
return to couplet #8
- Head elongated CL/CW > 1.16 (Fig. 114). Dorsum of promesonotum and lateral pronotum with more developed microsculpture (Figs. 115, 116). Head sculpture stronger, foveolae with regular margin (Fig. 117). Egypt and Saudi Arabia => Cardiocondyla fajumensis
- Head very short CL/CW < 1.16 (Fig. 118). Dorsum of promesonotum and lateral pronotum nearly glabrous (Figs. 119, 120). Head sculpture weaker, foveolae sometimes with irregular margin (Fig. 121). Iran => Cardiocondyla unicalis
10
return to couplet #8
- Frons very narrow, FRS/CS 0.196—0.228 (Fig. 122), postpetiole with a bulging sternite and high (Fig.123.), PpH/CS 0.306–0.358. With all measurements in mm, discriminant 128.52*FRS+42.56*PeW-143.94*PpH+1.616 < 0 [error 0% in 145 specimens]. Confirmed outdoor occurrence in S Iberia, N Africa and Middle East. Figs. 122–125 => Cardiocondyla emeryi
- Frons wider and postpetiole without a strongly bulging sternite. Discriminant > 0 [error 0% in 863 specimens] => 11
11
return to couplet #10
- Small, CW < 420 mm. Metanotal depression very weak or absent, spines well developed, postpetiole much lower than petiole and with a completely flat sternite (Fig. 127). vertex with deeply impressed foveolae (Fig.126). All surfaces of head, mesosoma and waist matt (Fig. 128). Outdoor occurrence confirmed for Egypt and Saudi Arabia => Cardiocondyla minutior, Cardiocondyla tjibodana & Cardiocondyla goa
- Note: These species are inseparable by single or few characters. It is recommeded to measure CL, CW, SL, EYE, FRS, SP, PpW, PeH, PpH and PLg and then running the test specimens as wild-cards in a LDA against the data given in supplementary information SI2. The total classification error in 210 worker individuals in SI2 is 2.4%. Users who refrain from doing this time consuming and challenging separation are recommended to name this collective cluster the Cardiocondyla minutior aggr.
- Larger, CW > 420 mm. Minute specimens of Cardiocondyla mauritanica may match the character combination of 11a but differ by shorter spines and a thinner petiolar peduncle (compare Figs. 127 and 130) => 12
12
return to couplet #11
- Outlines of postpetiole in dorsal view approximately hexagonal (Fig. 131). Metanotal depression shallow (Mgr/CS 2.14±0.52). Propodeal spines developed as short dents (Fig. 130). gastral pubescence dense (sqPDg 3.72 ± 0.28). Postocular distance large (PoOc/CL 0.447 ± 0.008, Fig. 129). Microsculpture on vertex strongly reticulate (Fig. 132). With Mgr, PoOc and SP in mm and sqPDg in µm, discriminant 136.0*Mgr-25.0*PoOc*73.99*SP+1.81*sqPDg-7.2322 < 0 [error 0% in 151 specimens]. Mediterranean and South Temperate zone of the Palaearctic => Cardiocondyla mauritanica
- Outlines of postpetiole in dorsal view not approximately hexagonal. Metanotal depression usually deeper, propodeal spines more developed, gastral pubescence often more dilute, postocular distance usually smaller. Microsculpture on vertex different. Discriminant > 0 [error 0.3% in 619 specimens] => 13
13
return to couplet #12
- Postocular distance extremely small (PoOc/CL <0.340). Petiole height and postpetiole width very low. With all measurements in mm, discriminant 30.96*PoOc-38.98*CL+17.12*PeH+26.77*PpW+ 7.196 <0 [error in 4 specimens 0%]. Kuweit. Figs. 62–65 => Cardiocondyla opistopsis
- Postocular distance larger (PoOc/CL 0.342–0.469]. Petiole height and postpetiole width higher. Discriminant >0 [error 0% in 612 specimens] => 14
14
return to couplet #13
- Head surface between inner eye margin and paramedian vertex strongly microrugulose (Fig. 69). Dorsal promesonotum longitudinally microrugulose (Fig. 68). All surfaces of head, mesosoma and petiole matt due to more developed microsculpture. Dorsal profile of mesonotum straight, contrasting the clearly convex dorsal profile of propodeum, metanotal depression feebly suggested (Fig. 67). With all measurements in mm, discriminant 95.88*EYE-164.99*Mgr-12.035 > 0. Yemen. Figs. 66–69 => Cardiocondyla rugulosa
- Body surfaces less microrugulose, often shiny. Dorsal profile of mesosoma different. Discriminant < 0 => 15
15
return to couplet #14
- Microsculpture of head with irregular foveolar margins (Fig. 73). Additionally differing from members of the elegans and ulianini group by larger eye (EYE/CS 0.265–0.274] and from members of the batesii group by longer tergite pubescence (PLg/CS 6.84–6.94). Afghanistan. Figs. 70–73 => Cardiocondyla kushanica
- Foveolae on vertex not with irregular margins. Character combination different => 16
16
return to couplet #15
- Foveolae on vertex extremely small (dFOv 7.0–9.1 µm). Tergite pubescence very dilute (sqPDg 5.88–5.92). Scape and head much elongated (SL/CS 0.854–0.867, CL/CW 1.238–1.256). All body surfaces very shiny. Island endemic of Cape Verde, Figs. 82–85 => Cardiocondyla verdensis
- Foveolae much larger (dFOv 12–21 µm). Tergite pubescence often more dense. Scape and head usually shorter => 17
17
return to couplet #16
- With sqPDg in µm and all other measurements in mm, discriminant 201.57*PLg-0.32*sqPDg-40.86*CL-10.32*PoOc+70.74 *SL+23.72*SPBA-24.27*SP+26.48*PeW+12.96*PpW-73.42*PeH-8.61 > 0 [error 0% in 177 individuals].Cardiocondyla elegans group => 18
- Discriminant < 0 [error 0% in 322 individuals]. Cardiocondyla batesii & Cardiocondyla ulianini groups => 21
18
return to couplet #17
- Occurring from Iberia accross S France to Italy. Contact zone with dalmatica in the border region of NE Italy and Slovenia. With all measurements in mm, discriminant 56.7*SP+156*PeW-61.9*PoOc-126.2*PeH-48.2*PpW+126.6*Mgr+17.86 >0 [error 0% in 81 two-specimen nest samples]. Figs. 6–9 => Cardiocondyla elegans
- Occurring farther east. If found in NE Italy or Slovenia, discriminant < 0 => 19
19
return to couplet #18
- Combination of larger foveloar diameter on vertex, lower postpetiolar height, longer propodeal spines and higher pubescence distance on gaster tergites. With dFOv, sqPDg in µm and SP, PpH in mm, discriminant 1.623*sqPDg+0.718*dFOv+96.14*SP-62.63*PpH-15.233 > 2 [error 0% in 13 specimens]. S georgia, NE Turkey, Iran, Afghanistan. Figs. 18–21 => Cardiocondyla brachyceps
- Foveolar diameter smaller, postpetiolar height larger, spines shorter, pubescence distance smaller. Discriminant < 2 [error 1.4% in 72 specimens] => 20
20
return to couplet #19
- Bases of propodeal spines more approached, petiole narrower and higher, surface of propodeum glabrous, propodeal spines very acute, petiole high with a very steep anterior profile of the node. With all measurements in mm, discriminant 163.8*PeH-29.1*PeW-134.63*SPBA-5.68 >2 [error 0% in 6 specimens]. Asia Minor. Figs. 14–17 => Cardiocondyla dalmaticoides
- Bases of propodeal spines more distant, petiole wider and lower, surface of propodeum less glabrous, propodeal spines less acute, petiole lower with a less steep anterior profile of the node. Discriminant <2 [error 0% in 59 specimens]. From N Italy across the whole Balkans, Cyprus and Asia Minor east to the Iran. Northern range border in Austria, Slovakia and Hungary at 48°N. Figs. 10–13 => Cardiocondyla dalmatica
21
return to couplet #17
- With all measurements in mm, discriminant 32.44*CW-103.36*CL+134.03*PoOc-44.05*EYE+23.08*SL-41.77*SPBA+48.21*PpW-0.135 < 0 [error 0% in 98 individuals]. Species of the Cardiocondyla batesii group => 22
- Discriminant > 0 [error 0.4% in 241 individuals]. Species of the Cardiocondyla ulianini group => 25
22
return to couplet #21
- Frontal carinae notably diverging frontad, FL/FR >1.109. Postocular distance large, PoOc/CL >0.408. Metanotal depression very shallow, Mgr < 2.5 %. Jordan. Figs. 74–77 => Cardiocondyla tenuifrons
- Frontal carinae less diverging, FL/FR <1.110. Postocular distance smaller, PoOc/CL <0.409. Metanotal depression deeper, Mgr usually > 2.5 % => 23
23
return to couplet #22
- Petiole massive, ratio PeW /PpW > 0.57. With all measurements in mm, discriminant 152.578*PeW-69.53*PpW+2.092*SL+30.63*PoOc-10.57 > 2.5 [error 0% in 4 individuals]. Only known from two sites in Asia Minor. Figs. 58–61 => Cardiocondyla semirubra
- Petiole less massive, ratio PeW/PpW in 97% of individuals <0.57. Discriminant < 2.5 [error 0% in 132 individuals. Figs. 54–57 => 24
24
return to couplet #23
- With all measurements in mm, discriminant 63.22*CL-63.18*PoOc+55.89*SL+64.64*PeW-184.93*PeH-93.82*PpH-11.19 < 0 [error 0% in 37 specimens]. Morocco, Tunisia, Algeria, Spain. Figs. 54–57 => Cardiocondyla batesii
- Discriminant > 0 [error 0% in 95 specimens]. Portugal, Morocco, Algeria, Tunisia, Egypt, Cyprus, Turkey, Middle East, Iran eastwards to 56°E. Color polymorphic. Figs. 78–81 => Cardiocondyla nigra
25
return to couplet #21
- Frons very narrow, FRS/CS 0.225; propodeal spines acute but very short, SP/CS 0.071; postocular distance small, PoOc/CL 0.403; postpetiole wide PpW/CS 0.552. With all measurements in mm, discriminant 160.18*FRS+41.17*PoOc-45.99*PpW-15.23 <0. Israel. Figs. 34–37 => Cardiocondyla gallilaeica
- Frons wider; propodeal spines longer; postocular distance larger; postpetiole often narrower. Discriminant > 0 [error 0% in 240 specimens] => 26
26
return to couplet #25
- Combination of large propodeal spine base distance, small eye, large postocular distance, broad and high postpetiole. With all measurements in mm, discriminant 44.74*PoOc-167.14*EYE+161.96*SPBA-56.07*PpW+31.77*PpH+0.543 >0 [error 0% in 179 specimens] => 27
- Propodeal spine base distance smaller, eye larger, postocular distance smaller, postpetiole narrower and lower. Discriminant < 0 [error 0% in 61 specimens] => 29
27
return to couplet #26
- With sqPDg in µm and all other measurements in mm, discriminant 163.54*PoOc+0.755*sqPDg-57.89*CL-58.26*CW+197. 78*EYE+47.39*FRS-54.85*SP+55.08*PeW-109.15*PeH+53.45*PpH-178.59*PLg-4.19 > 0.5 [error 0% in 84 specimens]. S Balkans and Asia Minor. Figs. 42–45 => Cardiocondyla bulgarica
- Discriminant < 0.5 [error 0% in 95 specimens] => 28
28
return to couplet #27
- Microsetae on clypeus shorter (Fig. 46), pubescence on all body parts shorter, foveolae on vertex less densely packed (Fig.49), postpetiole wider, petiole node in lateral view more symmetric (Fig.47), dorsum of mesosoma and waist shiny and mesosoma frequently darker. With dFOv in µm and all other measurements in mm, discriminant 0.965*dFOv+164.79*EYE-38.64*PpW+123.49*PLg +77.67*Mgr-32.49 < 0 [error 2.9% in 69 specimens]. SE Balkans, Asia Minor, Israel, Caucasus, Caspian region, Iran, Afghanistan, Kyrgistan. Figs, 46–49 => Cardiocondyla sahlbergi
- Microsetae on clypeus longer (Fig. 50), pubescence on all body parts longer, foveolae on vertex more densely packed (Fig. 53), postpetiole narrower, petiole node in lateral view more asymmetric (Fig. 51), dorsum of mesosoma and waist less shiny and mesosoma frequently lighter. Discriminant > 0 [error 7.7% in 26 specimens]. georgia, Iran. Figs 50–53 => Cardiocondyla persiana
29
return to couplet #26
- Vertex with rather large, well-demarcated, bicoronate and densely-packed foveolae and in anteromedian part with well-developed longitudinal sculpture(Figs. 38, 41). Pubescence on gaster tergites more dense. With dFOv, sqPDg in µm and SPBA in mm, discriminant 2.559*sqPDg-0.472*dFov+15.117*SPBA-4.857 < 0 [error 0% in 10 specimens]. Egypt, Israel. Figs. 38–41 => Cardiocondyla israelica
- Vertex with smaller, less demarcated and more spaced foveolae and with very weak or absent longitudinal microsculpture. Pubescence on gaster tergites less dense. Discriminant > 0 [error 0% in 51 specimens] => 30
30
return to couplet #29
- Postpetiole and frons wider, pubescence hairs on gaster tergites shorter. With all measurements in mm, discriminant 93.21*FRS+27.62*PpW-406.34*PLg-6.246 > 0 [error 0% in 44 specimens]. Continuously distributed from the S Ukraine (32°E) eastwards to N Xinjang (88°E). Figs. 22–25 => Cardiocondyla ulianini
- Postpetiole and frons narrower, pubescence hairs on gaster tergites longer. Discriminant 93.21*FRS+27.62*PpW-406.34*PLg-6.246 < 0 [error 0% in 7 specimens] => 31
31
return to couplet #30
- Postocular distance and postpetiole height much larger (PoOc/CL > 0.423, PpH/CS > 0.260). Foveolae on vertex mediad of eye with diffuse margins (Fig. 29). Anterior face of petiole node moderately sloping down (Fig. 27). SE Kazakhstan. Figs. 26–29 => Cardiocondyla littoralis
- Postocular distance and postpetiole height smaller (PoOc/CL < 0.423, PpH/CS < 0.260). Foveolae on vertex mediad of eye more strongly demarcated and bicoronate (Fig. 33). Anterior face of petiole node steeper (Fig. 31). Iranian Caspi region. Figs. 30–33 => Cardiocondyla caspiensis
Character Definitions
- CL
- maximum cephalic length in median line; the head must be carefully tilted to the position yielding the true maximum; excavations of hind vertex and/or clypeus reduce CL.
- CW
- maximum cephalic width; the maximum is usually found across and including the eyes, exceptionally posterior of the eyes.
- CS
- cephalic size; the arithmetic mean of CL and CW, used as a less variable indicator of body size.
- dFOV
- mean inner diameter of foveolae or of meshes of a reticulum on vertex at about half way between the median line of head and the inner eye margin. These structures usually have the base of a pubescence hairs in their center. At least seven measurements at magnifications of 360x are averaged.
- EYE
- eye-size: the arithmetic mean of the large (EL) and small diameter (EW).
- Foveolae types
- either simple cup-shaped, simple flat-bottomed, bicoronate or compound depressions of cuticular surface which are usually most strongly developed on dorsal head and have the base of a pubescence hair in the center. Bicoronate foveolae show in perpendicular view additionally to the outer margin ring a second smaller ring just around the hair base. Compound foveolae show in perpendicular view inner cuticular ridges reminiscent of the outlines of a three- or four-leafed clover leaf.
- FRS
- distance of the frontal carinae immediately caudal of the posterior intersection points between frontal carinae and the lamellae dorsal of the torulus. If these dorsal lamellae do not laterally surpass the frontal carinae, the deepest point of scape corner pits may be taken as reference line. These pits take up the inner corner of scape base when the scape is fully switched caudad and produce a dark triangular shadow in the lateral frontal lobes immediately posterior of the dorsal lamellae of scape joint capsule (Fig. 1).
- Longitudinal mesosomal axis
- in lateral view is defined as a straight line from the center of propodeal lobe (center of circle in Fig.4) to the border point between anterior pronotal shield and propleuron.
- MGr
- Depth of metanotal groove or depression, measured from the tangent connecting the dorsalmost points of promesonotum and propodeum.
- ML
- mesosoma length in the alates; measured in lateral view from the caudalmost portion of propodeum to the frontalmost point of the anterior pronotal slope (i.e. not to the frontalmost point of the whole pronotum that is usually concealed by the occiput).
- MW
- maximum mesosoma width of alates before the tegulae
- PeH
- maximum petiole height. The straight section of ventral petiolar profile at node level is the reference line perpendicular to which the maximum height of petiole node is measured at node level.
- PeW
- maximum width of petiole.
- PigCap
- pigmentation score of dorsal head.
- PigCon
- pigmentation contrast between dorsal head and mesosoma; PigCon = PigCap / PigMes.
- Pigmentation score
- variation of pigmentation from light yellowish or reddish (score 4) to blackish brown (score 12) assessed by subjective comparison of a standard color table (Fig. 5) with the stereomicroscopic image seen at a magnification of 150x and reflected-light with a color temperature of 2800 K.
- PigMes
- pigmentation score of dorsal mesosoma.
- PLG
- mean length of pubescence hairs on dorsum of first gaster tergite as arithmetic mean of at least seven measurements taken at magnifications of 360x.
- PpH
- maximum postpetiole height; the lateral suture of dorsal and ventral sclerites is the reference line perpendicular to which the maximum height of postpetiole is measured.
- PpW
- maximum width of postpetiole.
- PoOc
- postocular distance. Use a cross-scaled ocular micrometer and adjust the head to the measuring position of CL.
- Caudal measuring point
- median occipital margin; frontal measuring point: median head at level of posterior eye margin. Note that many heads are asymmetric; therefore average the left and right postocular distance (Fig. 2).
- SL
- maximum straight line length of scape excluding the articular condyle given as the arithmetic mean of both scapes.
- SP
- maximum length of propodeal spines; measured in dorsofrontal view along the long axis of the spine, from spine tip to a line, orthogonal to the long axis that touches the bottom of the interspinal meniscus (Fig. 3). Left and right SP are averaged. This measuring mode is less ambiguous than other methods but yields higher spine length values in species with reduced spines. This is the case in the dentiform spines found in the C. nuda group where it is difficult to correctly define the long axis. In such cases, the deviation of the assumed spine axes from longitudinal mesosomal axis should not exceed 30°.
- SPBA
- the smallest distance of the lateral margins of the spines at their base. This should be measured in dorsofrontal view, since the wider parts of the ventral propodeum do not disturb the measurement in this position. If the lateral margins of spines diverge continuously from the tip to the base, a smallest distance at base is not defined. In this case SPBA is measured at the level of the bottom of the interspinal meniscus.
- sqPDG
- square root of pubescence distance on dorsum of first gaster tergite.
The number of pubescence hairs in crossing a transverse measuring line of length L is counted; hairs just touching the line are counted as 0.5. The pubescence distance PDg is then given by L/n. In order to normalize the positively skewed distributions, the square root of PDg is calculated. Exact counts are promoted by clean surfaces and flat, reflection-reduced illumination directed slightly skew to the axis of the pubescence hairs. Counting is performed at a magnification of 360x. Tergite pubescence is easily torn-off in Cardiocondyla. An effort should be made to evaluate undamaged surface spots. In specimens with mostly removed pubescence, PDg can be calculated from the mean distance of hair base pits (BD) and PLg using the formula PDg = BD2 /PLg.