Diminishing areas in fingerprints are funnel or cone-shaped ridge patterns that force ridges to end or merge in localized areas and as a result display same minutiae orientations.  These funnel or cone-shaped patterns typically bear clusters of ending or merging (bifurcating) ridges that are directed towards the apex of the funnel (Figure 1).  Fingerprint diminishing areas in the shape of funnels were initially examined in terms of ridge count and relative size with respect to the “Golden Ratio” and friction ridge skin elasticity thresholds.  It has been said that the Golden Ratio is a universal law which permeates all forms and proportions, and finds its fullest realization in the human form, that it is found not only in the human body, hands and fingers, but also art, architecture, biology, nature and mathematics, and that in nature it is displayed in the arrangement of branches along the stems of plants, of veins in leaves, and within spirals found in sea shells, horns of rams, and arms of galaxies [14] [Video].  

Golden Ratio in the spiral of a shell 

Golden Ratio in horns of a ram 

 Golden Ratio in the arms of a galaxy 

In geometry the Golden Ratio is reflected in the isosceles triangle with the property that bisecting a particular angle produces a new triangle which is a similar triangle to the original [15].  In terms of mathematics, two quantities are in the golden ratio if the ratio between the sum of those quantities and the larger one is the same as the ratio between the larger one and the smaller. The golden ratio is universally defined as 1.618 [16]. 

Figure 1

Diminishing area pattern force causes ending ridge tails to point in the same direction and makes them less random [17].

"Nature does, it seem, favor the golden ratio."

Keith Devlin, Stanford University Mathematician [83] 

The diminishing area funnel was deemed to display similar combinative spiral and triangular shapes characteristic of the golden triangle and equiangular spiral (Figure 2).  As a result, the ratio of 1.618 was used to initially define relative boundaries for diminishing areas in fingerprints with consideration for friction ridge skin elasticity thresholds (see Friction Ridge Skin Elasticity Thresholds).  Diminishing area funnels, or “Golden Funnels”, were initially defined as clusters of directional ridge formations oriented towards the apex and having a funnel top to funnel length ratio (in millimeters) of approximately 1 - 1.618mm.  However, based on a 20% maximum friction ridge skin elasticity thresholds for compression and stretch, these dimensions were redefined as follows:  The tolerance range of smallest to largest funnel top and smallest to largest funnel length ratios were defined as 0.80 – 1.20 and 1.29 – 1.94 respectively.  A smallest top to largest funnel length ratio of 0.80 – 1.94, or approximately 1 - 2, was used to define the boundary for the narrowest and longest diminishing area funnel.  For purposes of simplicity, this threshold was used to identify diminishing area funnels in which the length of the top of the funnel (X) should be greater than or equal to half the length of the longest side of the funnel (Y).  The formula used to define diminishing area funnels is as follows:

X ≥ Y/2

where,

X = length of funnel’s top                                                                                                   

Y = length of funnel’s longest side

3 ending ridges in a funnel

A similar 2 to 1 ratio was applied to numbers of ridges located at the funnel top and apex in which the ridge count at the funnel top should be at least twice the ridge count at the apex as follows: 

Ridge count of funnel top ≥ 2 x ridge count of funnel apex

For example, the below image contains 5 ridges at the top of the funnel and 2 at the apex of the funnel.  Also the distance of the top of the funnel is roughly greater than or equal to half the distance of the longest side of the funnel.  As a result the configuration qualifies as a 3 ending ridges in a "funnel".   

3 ending ridges in a funnel

Note:  Borders for funnel tops and apexes were defined by the beginning and end of directional ridge units.  Borders for funnel sides were defined by either continuous ridges or the outside ridge of bifurcations.  No boundary for shortest funnels was applied. 

Conservatively, directional ridge formations found in areas displaying ambiguous diminishing area funnels should default to the lower bound value.  In any case, objective observation and conservative interpretation should be used to identify diminishing area funnels whereby its presence or absence should speak for itself and allow for demonstration, even to the layman.

The below image illustrates how directional ridge features, e.g., ending ridges and bifurcations, located in a funnel and within delta areas (see Pattern Force 1/2) can be identified in a fingerprint.

Ending ridges and bifurcations in funnels and delta areas impacted by pattern force are marked in red. 

Expanded Weights for Directional Ridge Formations Not Impacted by Pattern Force

Directional ridge formations, i.e. ending and bifurcating ridges located in areas that as a result of pattern force makes orientation predictable, have reduced quantitative weight.  However, ridge formations not impacted by pattern force are comparatively less predictable  should have expanded weight.  The theory of ridge formations found outside pattern forced areas having expanded quantitative weights corresponds with position statements that minutiae found in areas not impacted by pattern force are judged to be more random events regarding the aspects of direction and location and by nature are deemed to be of a higher value [17]. 

In order to define expanded quantitative weights for directional minutiae located in areas not impacted by pattern force, the following study was performed:  A random sample of 39 flat fingerprints was examined to determine the number and type of ridge formations present in diminishing area funnels.  The number of diminishing area funnels found was 27 with a total of 172 ending ridges and 65 bifurcations (the tail of each ending ridge and bifurcation was oriented toward the apex of each funnel).

Based on a study of 234 flat fingerprints from ten-print files, 139 ending ridges and 94 bifurcations were found between the delta and diverging type lines all with tails oriented toward the diverging types lines.  A total of 311 ending ridges and 159 bifurcations were found in these pattern force areas.  Based on reduction factors applied to quantitative weights for ridge formations found inside these areas, quantitative weights for directional minutiae found outside these areas were expanded such that the aggregate amounts for each balanced.  The following equations were established in which values for X and Y, defined expanded quantitative weights for ending and bifurcating ridge formations:

(13.34) ^ 1734 ≈ (X ^ 1423) x (10 ^ 311)

(25.01) ^ 925   ≈ (Y ^ 766) x (18.75 ^ 159)

The values for X and Y were solved and the resulting expanded quantitative weights for ending and bifurcating ridges with orientations not impacted by pattern force were rounded to the nearest .25 and defined as 14.25 and 26.75 respectively.

The match probability values for these ridge formation types were subsequently defined as 1/14.25 and 1/26.75 respectively.

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Example

The funnel in Figure 1 displays a diminishing area funnel in which the ridge count at the funnel apex is 9 and the ridge count at the funnel top is 18.  The longer side of the funnel (bottom side) is not greater than twice the length of the top.  As a result, the quantitative weight for each of the 13 ending ridges that are directed towards the apex, are subject to reduction.  Although a 14^th ending ridge is located inside the diminishing area, it is directed away from the apex, and therefore its quantitative weight is not subject to reduction (see Figure 3).

Figure 3

Diminishing Area Funnel

Quantitative weights for directional minutiae influenced by diminishing area pattern force were reduced based on similar reduction factors applied to ending and bifurcating ridge formations found between delta areas and inside diverging type lines.