Journal of Advanced Forensic Sciences
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Research Article | Open Access
  • Available online freely | Peer Reviewed
  • Calculating the Number of Rifling on Highly Deformed Bullets: A Digital Measurement and Mathematical Formula Approach

    John Zheng Wang 1      

    1Forensic Studies Program, School of Criminology, Criminal Justice and Emergency Management


    One of the advanced research questions and practical challenges in firearms examinations is to determine the number of rifling on highly deformed bullets (ricocheted) at crime scenes, such as drive-by-shootings, police-suspect barricades, and mass shootings. In reality, when a bullet hits a hard surface from an angle and gets ricocheted off, e.g. car metal, concrete, brick walls, the bullet becomes a highly deformed bullet (HDB). While the number of rifling is one of the standard criteria for a bullet-weapon determination, such HDB usually has only one or two lands or grooves visible for examination, and thus it is unusable for an identification. The aim of the study proposes a new mathematical formula that allows calculating the number of rifling on highly deformed bullets fired from pistols and revolvers using a palm-sized digital device. With the number of rifling from highly deformed bullets, the new approach can provide a real-time method to improve crime scene investigations as well as later lab work in bullet-weapon identifications.

    Received 11 Oct 2018; Accepted 13 Nov 2018; Published 14 Nov 2018;

    Academic Editor:Yavuz Hekimoglu, Assistant Professor at Department of Forensic and Legal Medicine, Turkey.

    Checked for plagiarism: Yes

    Review by: Single-blind

    Copyright©  2018 John Zheng Wan

    Creative Commons License    This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

    Competing interests

    The authors have declared that no competing interests exist.


    John Zheng Wang (2018) Calculating the Number of Rifling on Highly Deformed Bullets: A Digital Measurement and Mathematical Formula Approach. Journal of Advanced Forensic Sciences - 1(1):1-14.
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    When a drive-by-shooting, barricade shooting, or mass shooting incident occurs, quite often badly deformed bullets are found at the crime scene due to hitting a hard surface, which is often termed “ricocheted trajectory.” The physical principle of the ricocheted trajectory is largely dependent on the ratio between the impact angle (incident) and the critical angle of the bullet 1. These bullets often turned out to be unusable because the bullet bears only one or two lands/grooves on the surface, rendering the total number of the rifling undetermined, which is one of the criteria for a bullet-weapon determination. Although the phenomena of ricocheted bullets have been investigated in details 2, 3, 4, 5, 6, there is little research found in the literature review on calculating the number of rifling on highly deformed bullets or ricocheted bullets. The scarce inquiry of the subject is due to two difficulties: lack of a portable device in the field that can measure the land/groove width of a highly deformed bullet and a calculation method for scientific reasoning. Using a palm-sized digital device, a quasi-experimental test was conducted in this study to calculate the number of rifling (numbers of lands/grooves) within minutes using a mathematical formula. The device can make a real-time and digital measurement of the number of rifling in the field for a bullet-weapon determination

    Materials and Methods

    Current Examination Criteria

    While there are different criteria or standards used by different firearms examiners, the following ten criteria are most commonly employed and recognized. The more criteria that are matched, the higher confidence level can be reached.

    a. Caliber Specific Rule: It refers to a normal measurement of the interior diameter of a muzzle/barrel, the interior diameter of a casing mouth, or the exterior diameter of a bullet in hundredth of an inch or millimeters. As a rule, one specific firearm can only fire the same diameter of the cartridge (casing/bullet). Therefore, a strong correlation of the diameter of the bullet should be found at the crime scene and the weapon involved. In other words, if the diameter of a bullet (.40) is found at a scene, the same diameter of a weapon should be sought.

    b. Type of Rifling: There are two types of rifling, the angled rifling and the polygonal rifling. While the former rifling has sharp edged lands and grooves that make a visible indentation on the bullet by the rifled barrel, the latter is curved-shape lands and grooves (hills and valleys) that produces less angled lands and grooves on the bullet, such as a Smith and Wesson .40. By looking at the rifling on the bullet, one can determine that the rifling was fired by a weapon with an angled or curved rifling.

    c. Direction of Rifling: The main purpose of rifling is to force a bullet to rotate as it passes through the barrel to increase its stability and accuracy in its flight. Again, two types of rifling exist inside the barrel: either clockwise (twisted to the right) or counter-clockwise (twisted to the left). Once placing a bullet found at the scene, the slanted rifling can indicate which direction; For example, if the rifling is slanted toward your right hand, it is a clockwise direction. Thus, the weapon that has fired the bullet must have the same direction of rifling.

    d. Number of Rifling: The lands and grooves are cut in a twisted (right vs. left) pattern inside a barrel. The number of lands and grooves are equal in number, and the common number is 4, 5, 6, 7 and 8. Therefore, if there are six lands and six grooves on the bullet, the weapon firing the bullet must have the same number of lands and grooves 7.

    e. Ratio between Land Width and Groove Width: The land width (LW) and the groove width (GW) are rifled in a ratio relationship: the LW is equal to the GW (1:1), the LW is two times wider than the GW (2:1), or the LW is three times wider than the GW (3:1), or vice versa.

    f. Land Width and Groove Width: The actual land width (LW) and the groove width (GW) can also be used to link a bullet to a weapon in an opposite manner, meaning the LW on the bullet must be identical to the GW inside the barrel where the LW is made by. By the same principle, the GW on the bullet must be identical to the LW of the barrel.

    g. Striations: When a bullet passes through a rifle barrel, the harder steel (the barrel) makes tiny lines, called striation marks, on the softer surface of the bullet (copper, brass or lead). This is the principle of rifling because the diameter of the barrel (the distance between the opposite land surface) is usually smaller than the diameter of the exterior of the bullet, which results in the bullet’s spinning. Under a comparison microscope, if there are six tiny lines matching in alignment between the bullet found at the scene and the other bullet from a test fire, it may suggest that the bullet was fired from the weapon producing the test-fired bullet regarding a striation comparison.

    h. Frontal Marks: When examining the front portion of a fired bullet, two types of marks can be used to distinguish if the bullet was fired by a pistol or a revolver. If there are some skid marks at the anterior part of the fired bullet, it strongly suggests that the bullet was fired from a revolver due to the bullet’s movement from the chamber (cylinder) to the barrel before its initial rotation. The second type of the marks is called shaving marks, which can indicate whether the bullet is also fired from a revolver due to the situation where the revolver barrel is poorly aligned to the chamber (cylinder).

    i. Rifling Pitch: A rifling pitch is a distance where the rifling makes a complete turn in a single revolution inside the barrel, e.g., one turn in 12 inches, also called rate of twist. The longer the barrel is, the slower the twist revolves. However, due to limited technology at present, the pitch is not being used in firearms examinations because it is very difficult to measure the pitch practically.

    j. Rifling Angle: Although the pitch has limited use in firearms examinations, the term does provide another useful indicator: the rifling angle at which the rifling is cut in the barrel.  The rifling angle can be used to determine whether two bullets are not fired from the same gun (exclusion). When two fired bullets are placed side by side in a vertical position, one can compare the degree of the slants of the rifling lines by the rifling angle. Therefore, expanding on the same principle, if the rifling angle inside a barrel can be measured and compared with the rifling angle on the bullet, then this may be a new method to determine if a bullet was fired from a particular weapon.

    Two Recent Reports

    In 2009, the National Research Council issued a report (often referred to as the NRC Report) and challenged that current forensic methods, except for nuclear DNA analysis, are less reliable and consistent to identify a specific individual or source due to a lack of quantifiable measurements 8. In 2016, The President’s Council of Advisors on Science and Technology (also known as the PCAST Report) recommended further actions to strengthen forensic science and promote its more rigorous use in the courtroom, again challenging that pattern-matching forensic procedures are less scientific due to its lack of standardization and computerization 9. While the two reports have received mixed feedback and responses, a quantifiable and computer standardization seems to be the future direction for the forensic science community.

    Research Design

    This quasi-experimental study was carried out to calculate the number of rifling from a highly deformed bullet (with only one intact land and one groove) with a mathematical formula using a palm-sized digital device. In essence, the research design is dependent on four related mathematical questions: (1)What is the correlation between the circumference and its equal units (rifling)? (2) What is the correlation between one unit of rifling and the number of rifling? 3) Is there a formula that can calculate the number of rifling using diameter and one unit of rifling? (4) Is there a device that can measure the one unit of rifling on a highly deformed bullet in the field?

    Mathematical Principles

    Based on the relationship among the diameter, the Pi (3.14), and the circumference, the following mathematical inference can be made and calculated:

    a. Given Mathematical Principle: Circumference = π (Pi=3.14) x Diameter

    b. One Unit of Rifling = Land Width + Groove Width (Equal Number of Both)

    c. If π (Pi) x Diameter ÷ One Unit of Rifling (One Land Width + One Groove Width) = Number of Rifling

    d. Then the Proposed Mathematical Formula for Calculation:

    Diameter (Caliber of the Bullet Measured) x π (Pi=3.14)


    One Unit of Rifling (One LW + One GW)

    ≈ Estimated Number of Rifling

    A Crime Scene Scenario

    At a drive-by shooting scene, a bullet was found that has hit a concrete driveway from an angle (ricocheted) and has become a highly deformed bullet (HDB). The HDB has only one visible intact land and groove due to the ricocheted trajectory. However, both the crime scene investigation teams and later the lab technicians would like to know the number of rifling for a bullet-weapon connection because the rifling number is one of the critical factors for an identification.

    Sampling Procedure

    To continue the study, the caliber variation for common pistols and revolvers should be the guide for the sampling procedure of the project. The author examined ten fired bullets of each of the following eleven calibers from the author’s firearm database. While the database of pistols consists of the following calibers: .45, .40, .30 (a foreign made), 9mm, and .25, and .22,the database of revolvers comprises the following: .44, .38, 357, and .22. From the preliminary screening, three observations were noted. First, the calibers of both pistols and revolvers can be divided into three categories. The large calibers (.45~.40) all have six lands and grooves without any variations. Second, the small calibers (.25~.22) all have six lands and groove without any exception. Finally, only the median calibers (.38~.30) showed some variations. Therefore, the median category of calibers becomes the focus of this study.

    Quasi-Experimental Test

    Based on the scenario above, a quasi-experimental test 10 was conducted and designed to simulate a real crime scene, in this case, in an outdoor shooting range. Two median caliber handguns were chosen due to their caliber variation observed from the prescreening examinations under a purposive sampling method 10. First, a revolver (.38) was used to fire five cartridges (lead bullets) from different angles onto a piece of car metal inside a big bucket with a cover and the most deformed bullet was selected (the HDB). The criterion for the selected HDBs required that at least one intact land and groove is visible at the bottom of the bullet. Next, the revolver was used to fire a cartridge into several telephone books that had been dampened with water to retrieve a control sample (an intact bullet). The same procedure was repeated for a pistol (9mm, fully metal jacketed) onto a piece of concrete inside a big bucket with cover. While the two highly deformed bullets from the hard surface were used for testing samples simulating spent bullets at crime scenes, the two intact bullets retrieved from the telephone books served as control samples or known samples for verification and a ground truth comparison. Due to the time constraints, availability, and the purposive sampling, one highly deformed bullet and one intact bullet from a pistol (.30, a foreign made pistol) were collected from a donation for the study.

    Results and Discussion

    Once the three pairs of bullets were selected and placed in order on the ground, the author picked up each bullet and examined, measured, and recorded each unit of rifling using a digital scope (palm-sized) on the spot, simulating a crime scene investigation. While Figure 1 shows the three highly deformed bullets (HDBs) before the measuring, Figure 2, Figure 4, and Figure 6 display the actual images of the measured rifling (land width and groove width) of the three HDBs. Figure 3, Figure 5 and Figure 7 portray the actual images of the measured rifling (land width and groove width) of the three control samples (intact) for a ground truth comparison. Each bullet was examined, measured, recorded, and calculated in approximately ten minutes. The digital device presented each image on a laptop (connecting via a USB cable), and the measurement was a real-time display, suggesting a practical implication for crime scene investigations. Then, three types of data were recorded and input into the formula for calculation: 1) the land width (LW), 2) the groove width (WG), and 3) the measured diameter at the bottom of the highly deformed bullets. The following Table 1 reports the comparison results:

    Figure 1. Three Highly Deformed Bullets: A Pistol Bullet (.30, left), A Revolver Bullet (.38, Middle), and a Pistol Bullet (9 mm, Right)
    Figure 1.

    Figure 2. The Highly Deformed Bullet Fired by a Foreign Made Pistol (.30) with the Measured Land Width (Right), the Measured Groove Width (Left), and the Measured Caliber by a Digital Caliper.
    Figure 2.

    Figure 3. The Bullet of the Control Sample (Intact) Fired by a Foreign Made Pistol (.30) with the Measured Land Width (Bottom) and the Measured Groove Width (Top).
    Figure 3.

    Figure 4. The Highly Deformed Bullet Fired by a Revolver (.38) with the Measured Land Width (Left) and the Measured Groove Width (Right).
    Figure 4.

    Figure 5. The Bullet of the Control Sample (Intact) Fired by a Revolver (.38) with the Measured Land Width (Left)) and the Measured Groove Width (Right).
    Figure 5.

    Figure 6. The Highly Deformed Bullet Fired by a Pistol (9 mm) with the Measured Land Width (Right) and the Measured Groove Width (Left).
    Figure 6.

    Figure 7. The Bullet of the Control Sample (Intact) Fired by a Pistol (9 mm) with the Measured Land Width (Right) and the Measured Groove Width (Left).
    Figure 7.

    Table 1. Comparison of the Measuring Results in the Field among the Three Pairs of Bullets
    Testing Samples (HDB Bullets) Estimated Number of Rifling from Testing Samples Estimated Number of Rifling from the Control Samples Control Samples (Intact Bullet) Error Margins between the two Samples Ground Truth of Number of Rifling
    .29 x 3.14/LW0.15+GW0.05 (inch)

     4.55  4.28 .30 x 3.14/=LW0.15+GW0.07 (inch)

    +0.27 4 (A foreign made pistol)
    .35 x 3.14/LW0.10+GW0.11(inch)

     5.23  5.68 .38 x 3.14/LW0.10+GW0.11 (inch)

    -0.45 5 (.38 Revolver)
    9.1 x 3.14/LW3.12+GW1.39(mm)

     6.34  6.53 9 mm x 3.14/LW2.91+1.42 (mm)

    -0.19 6 (9 mm Pistol)

    Following the proposed formula discussed earlier, the testing samples of the three bullets were calculated. The caliber (diameter) measured at the bottom of the HDB using a digital caliper multiplies the Pi, and then divides by the sum of the selected land width and the selected groove width. The result is the estimated number of the rifling for the bullet. Due to the non-definitive value of the Pi and the deformation of the bullet, the rifling number was calculated to the two places after the decimal point. However, if the result was taken as a whole number and the decimal number ignored, the whole number represents the number of rifling of the bullet under investigation. The same procedure was used for the control sample to calculate the estimated number of rifling from the intact bullet. For comparison purposes, a margin error, or the differences between the two estimated numbers of rifling, was provided for a degree of freedom of confidence. Finally, the actual number of rifling was given as ground truth information by counting the actual number of rifling from the revolver and the pistol as well as from the information from the donor of the foreign made pistol. Finally, the error margins between the estimated testing and the control samples are provided as +.27, +.45, and -.19, all of which are less than the normal confidence level of 0.5. As a result, the accuracy and reliability of the number of rifling from the three testing samples of the highly deformed bullets can be calculated and determined by this quasi-experimental study using the proposed formula by the palm-sized digital scope.

    Crime scene investigation of firearms-related evidence has seen several new methods introduced into the field 11, 12. Using a palm-sized digital device, this study provides a mathematical formula to calculate the number of rifling from highly deformed bullets of a revolver (.38) and two pistols (.30 and 9 mm). The device can display real time images and digital measurements in a few minutes. First, the highly deformed bullet from the foreign made pistol (.30, fully metal-jacked) indicates the same number of rifling (4) of the control sample with right twist lands and grooves and an equal land and groove width (4 R; L=3G) (see Figure 2). Second, the HDB from the revolver (.38, lead) displays the same number of rifling (5) of the control sample with right twist lands and grooves and an equal land and groove width (5 R; L=G) (see Figure 4). Finally, the HDB from the pistol (9 mm, fully metal jacketed) shows the same number of rifling (6) of the control sample with right twist lands and grooves and an equal land and groove width (6 R; L=2G) (see Figure 6). Noticeably, the proposed formula was not affected by the types of bullet metal (fully metal jacketed vs. lead) or the measuring units (inch vs. mm). While this study focuses on the phenomena of ricocheted bullets or highly deformed bullets from several previous studies 13, 14, 15, the new proposed mathematical formula examines the correlation between highly deformed bullets ricocheted from hard surfaces (car metal and concrete) and the number of rifling.

    This new proposed formula proposed by this quasi-experimental study may indicate other practical implications where a bullet has been fired into sand or onto a brick surface (jacked-pieces only). The strength of the proposed formula only requires one intact land and groove for a calculation. Finally, it is hoped that future studies will be expanded to more samples from more types of calibers and on more types of surfaces. The author strongly believes that many applications of the proposed method by this study may be carried out to test correlations between an incidental/ ricochet angle and a deflection angle at plain floating glass 16, wood grain 17, and laminated particle board 18 for crime scene shooting reconstructions.


    The author is grateful to the RSCA Grant of 2019~2020 from College of Health and Human Services at California State University-Long Beach.


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