B MaCrime and Forensics Student 1 CRIME: Criminal Investigation through Mathematical Examination Overview Welcome to our unit on the mathematical examination of fingerprints! This

BioMath

Crime: Criminal Investigation

through Mathematical Examination

Student Edition

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DIMACS

Crime and Forensics Student 1

CRIME: Criminal Investigation through Mathematical Examination

Overview Welcome to our unit on the mathematical examination of fingerprints! This unit is provides a possible method to identify individuals in a species. Tracking individuals can be an important step in learning more about the species as a whole. This unit concentrates on human fingerprints. To accomplish this, we use a fictitious crime case to explore a mathematical procedure to examine and compare fingerprints. The unit consists of multiple activities to investigate the characteristics of a fingerprint and the biology behind why everyone’s fingerprints differ.

Goals and Objectives

Goal: Students will understand the importance of identifying individuals within a species. Objectives:

Explain the importance of being able to identify individuals within a species. State the types of fingerprints that can be left at a scene. Explain the general sequence of investigation from crime scene to identification.

Goal: Students will understand how genetics and environment can interact to ‘individualize’ traits such as fingerprints. Objectives:

List and identify the three basic layers of the skin. State that fingerprints arise from the dermal papillae. Evaluate the importance of fingerprints arising from the dermis rather than the epidermis. Define epigenetic, penetrance and expressivity. Describe ways to alter a gene’s expression either genetically or environmentally. Explain why even identical twins do not have identical fingerprints.

Goal: Students will understand how to approach a problem by first looking at general patterns and then looking in more and more detail to observe similarities and differences. Objectives:

Identify the fingerprint pattern of a given print. State the relative frequency of the three basic fingerprint patterns. Define and identify the ‘delta’ and ‘core’ of a fingerprint. Analyze a fingerprint and apply the primary and secondary classification values of the

POP system. Goal: Students will understand the importance of applying scientifically and mathematically rigorous solutions to problem-solving.

Explore connections between the mathematical and biological sciences. Use graphs and matrices to match fingerprint characteristics.


Lesson 1 Fingerprinting Basics Broadly speaking, forensic science is the analysis of traces of evidence such as body fluids, glass fragments, footprints, and drugs left at the scene of a crime by the criminal, victim or others. The evidence may be used subsequently to either implicate or exonerate a person suspected of committing that crime, or just to gain further insight into the incident. Forensic science involves more than just identifying traces of evidence. Sometimes it isn’t obvious just what is a piece of evidence. Important questions that need to be answered are: How did the evidence come to be at the crime scene? From where did the evidence originally come? Who left it there? This suggests a natural role for statistics and mathematics, as these questions can only be answered in terms of probabilities. So it is not surprising that the primary task of forensic statisticians/mathematicians is to evaluate any evidence found at a crime scene, so that this evidence can be appropriately presented to a jury in court. The following are some of the fields associated with forensic science. Biology Entomology Botany Zoology Psychology Anthropology Chemistry Economics Mathematics Statistics Computer Science Criminal Justice Physics Sociology Accounting Medicine Crime Case Part 1: The Store Many times when police receive the report of a possible crime, they call forensic investigators to the scene. These investigators determine, through examination of forensic evidence, whether the incident is suspicious and requires further investigation, or if it is an accident. They determine whether there are any victims or suspects involved in the occurrence. The following is a conversation between two investigators concerning an incident that has occurred at a local store.

Frank: Why do I always seem to be the one on duty during full moons? Clearly seniority means nothing in this town. And this rain and darkness is not helping. I sent Frank a text ten minutes ago. He should be here soon. Barney: Hey Frank, I got your text, what’s up? Oh yeah, I heard a good one today. What can run but never walks, has a mouth but never talks, has a head but never weeps, and has a bed but never sleeps? Frank: Really Barney? Maybe now isn’t the best time for riddles. Barney: Whoa, look at that. There is a car in the middle of the store. How did that get there? Frank: Perhaps someone drove it there Barney, but that’s just a guess.


Barney: But there is no one in the car? Frank: Nice call, Captain Obvious, they seem to have fled. Barney: Yikes! Someone must’ve been a little angry at the store, eh? Frank: It would certainly seem that way, or angry with someone in the store. Barney: Any ideas on who should be in our suspect list? Frank: I have a few ideas on where to start, but we’ll need more info on who was driving the car. Barney: Any evidence to suggest who that might be? Frank: Not yet, but we are searching for evidence in the car. We’ve got a CSI team on it now. They should let us know by tomorrow. Barney: What are they looking for?

The investigation begins with a general assessment of what constitutes and what does not constitute forensic evidence. After that assessment, the team begins collecting that evidence. Questions for Discussion 1. What kinds of information might the investigators obtain from the car that could be used to identify the driver? 2. If investigators do find some evidence in the car, whom should they compare it to? In other words, who do you think should be on the initial suspect list? 3. Fold your hands and place them in your lap or under the table (do not look at your hands).

a. Starting from your fingertip, how far down your finger do you think your prints go? (e.g. first knuckle, second knuckle, etc.)

b. Do you have prints on the back on your fingers?

c. Why do you have fingerprints? What is their purpose? d. Where else would you find “prints” on your body? Crime Case Part 2: The Evidence All areas of friction skin on our bodies (fingerprints, palms, toes and soles of the feet) are permanent and unique. So, they do not change over time and they allow for individualization. Although television shows have popularized the method of “dusting” a print (using a fine brush along with a special powder), in fact this method is no longer the method of choice. Modern


technologies have seen the use of digital imaging. Nevertheless, older methods are used when nothing else is available. For instance, it is possible to “lift” a print with clear tape. It is important to preserve the details of a print as clearly as possible so that a good match can be made. The better the impression, the better it will be to clearly identify distinguishing characteristics. Digital imaging has vastly improved the ability to record fingerprint details clearly. Collection methods also depend on the type of fingerprint. Some prints are latent, meaning they are not visible to the naked eye. A patent print is visible, being left in some medium such as blood, ink, grease, etc. Impressed prints (also called plastic prints) are indentations left in soft pliable surfaces such as wax, clay, paint, etc. As you might imagine, each of these types of prints has its challenges and advantages. When fingerprint evidence is collected at a crime scene, investigators’ first step is to search a national database to see if there is a copy of that print on file, thus identifying the person associated with the print. The evidence may belong to the person responsible for committing the crime. The print may just belong to an innocent victim who was in the wrong place at the wrong time. Or the print may not have anything to do with the crime. Additional evidence may be needed to corroborate the investigators’ conclusions. Back to the crime…

Barney: Hi Frank, did they find any evidence in the car? Frank: Yes, they found some fingerprints. Barney: Great, who do they belong to? Frank: I’d like to be able to tell you but the lab says they are very backed up and our case isn’t exactly high on their to-do list. Barney: Well it is high on our to-do list so what are we going to do? Frank: I had a friend send over one of the partial prints they found so we can take a look. Barney: I hope you have ‘brushed up’ on your fingerprint skills, ha, get it? Frank: Very funny Barney, very funny. I know some people at a private lab who might be able to help us. I’ll send the print over to them. I’ve also put together a suspect list.


Figure 1.1: Print Sent to the Lab Much of the mathematics in future lessons of this unit deals with methods for searching a database for a fingerprint match. However, before you examine the search algorithms used by law enforcement you need to study the process of taking and recording fingerprints. Activity 1-1 Fingerprinting 101 Objective: Complete a fingerprint card. Materials:

Fingerprint recording sheets Number two pencils and sharpener (if needed) 5 by 7 index cards Clear wide scotch tape Moist sanitary towelettes Handout CF-H1: Fingerprinting 101 Activity Worksheet

1. Each member of the group fills out a fingerprint recording sheet. Partners should sign each other’s fingerprint sheets as the official taking the fingerprints. Follow the directions in 2-7 below for each group member. 2. Take the number 2 pencil and rub it on the index card until there is a solid smudge of lead large enough to cover a fingertip. 3. Rub your finger on the lead smudge firmly, but not too firmly, until the fingertip is completely covered. You must cover your finger from the tip to the first knuckle and from one side to the other side, so be sure to roll your finger in the pencil lead from side to side (nail edge to nail edge). Start with the right thumb. 4. Once your finger is sufficiently covered, you are ready to place your fingerprint in the appropriate box on the card. Locate the spot for your right thumb on the fingerprint card. Place one edge of your thumb in the box, press firmly and roll your finger to the opposite edge (nail to nail). Do NOT roll it back again or you will smudge the print. Be sure to keep your finger parallel to the table to get as much of your print as possible onto the card. 5. You may wish to place a piece of scotch tape over the print to protect it from damage (unless you used fingerprint ink rather than pencil lead).


6. Repeat this process for each finger until you have a complete set of fingerprint impressions. 7. At the bottom of the card there is a spot to press all four fingers at the same time as well as two spots for each thumb. This is a safety check to be sure that each finger was put in the correct box above.

Practice 1. Use your fingerprint cards to look for patterns. How many different ‘patterns’ do you see on your prints? For example, is the pattern the same on all 10 fingers for a particular individual? How about on just one hand? Both index fingers? Come to class tomorrow ready to discuss your findings. 2. Check the Internet for information on fingerprinting. Find one example of when fingerprinting can be helpful in solving a crime and one example when fingerprinting can be problematic in solving a crime. 3. The investigators were able to put together a list of possible suspects as to who may have driven the car into the store. Consider the following suspects and make your prediction of the top three suspects.

a. John McDonnell – the storeowner who is struggling to keep up with the rent. b. Christine Smith – a recently fired employee for smoking in the store with Frank Cooper. c. Frank Cooper – also recently fired along with Christine Smith. d. Lee Howe – an employee who was recently docked in pay for falling asleep in the

dressing room. e. Steven Riale – an employee recently accused of stealing candy from the store but they

could not prove it. So now he is being watched ‘like a hawk’. f. George Munson – a competing store owner who has recently threatened John McDonnell

to stop telling people lies about George’s store, “or else!” g. Caroline Fletcher – John McDonnell’s ex-wife who is angry with him being behind in

alimony payments. h. Kathy Sullivan – a recent customer who tried to return an item she just purchased the day

before but lost her receipt. The store did not accept the return and she was overheard saying, “You’ll pay for this!”

i. Rick Dawson – the car owner who says his friend borrowed the car that night. j. Darryl Wright – the friend who borrowed the car. He was in a bar that night and says

when he came out the car had been stolen. k. Jeremy Dawson – the 16-year-old son of Rick who frequently sneaks out with the car. l. Lin Huang – the employee of the month, for 14 straight months. m. Linda Pierce – the runner up for employee of the month for 14 straight months. n. Andy King – the 41 year old and longest employee (18 years) who was promised he

would be store manager one day (16 years ago) and is still a stock ‘boy’. o. Susan Tilly – a female employee whose health insurance was cut just one week before

the birth of her triplets. p. Ivan Carlson – the ex-boyfriend of Susan who recently got out of jail.


q. Donald French – an employee, the current boyfriend of Susan and probable father of the triplets.

r. Sadie Wiler – a local criminal who has a history of stealing cars. s. Teddy “Thunder” Chow – a local criminal who has a history of burglarizing stores. t. Thomas Calvin – the high school jock who went out ‘partying’ last night and no one can

seem to find. u. Sarah Spencer – the local jock’s girlfriend, who is also currently missing.


Source: FBI is a Government Agency. Image is in the Public Domain. Figure 1.2: Fingerprint Card


Lesson 2 Fingerprint Pattern Recognition Taking a closer look at fingerprints, we can clearly define the patterns found in fingerprint impressions. Fingerprint patterns contain certain features that allow us to classify fingerprints easily into three primary categories. Fingerprint Features and Patterns Let’s begin by defining the two features that are used to distinguish one major pattern type from another: deltas and cores. Deltas and cores are formed by ridge patterns. A delta is triangular-shaped ridge pattern that appears at the center of three intersecting ridge flows. A core is the termination of the innermost ridge that appears in the interior of a concentric set of curved ridges. A fingerprint ridge is a slightly raised, long, narrow line or curve that appears on the surface of the skin.

Figure 2.1: Examples of Deltas

Figure 2.2: Examples of Cores There are three main patterns within fingerprints: arches, loops, and whorls. The main categories of patterns found on fingerprints are classified according to the number of cores and deltas present in the pattern.


The following figures show examples of the three patterns of fingerprints. Examine the figures and compare their differences.

Figure 2.3: Example of Plain Arch (left) and Tented Arch (right)

Figure 2.4: Example of Loop (left) and Whorl (right)

Sfinge finger print graphics from University of Bologna reprint permission is granted from Professor Maio


Practice 1. What pattern is shown by each of the fingerprints below?

a. b. c. 2. What pattern is the following?

Sfinge finger print graphics from University of Bologna reprint permission is granted from Professor Maio

Activity 2-1 Classifying Fingerprints Objective: Identify distinguishing patterns in fingerprints and classify fingerprints. Material: Handout CF-H2: Classifying Fingerprints Activity Worksheet Handout CF-H3: Database of Suspect Fingerprints Part I: Examine Fingerprint Samples 1. Examine the three columns of fingerprint samples in Figure 2.5. Count the number of cores and deltas in each print and record the number in the corresponding column. 2. Compare the numbers you counted in part 1 with your other group members. Are your counts the same or different? If different, why? 3. Discuss with your group and annotate any observations about whorls, loops and arches. 4. What can you conclude from your numbers in relation to the fingerprint patterns you see?


Part II: The Pattern Ordered Pair (POP) Classification System For the purposes of this activity we will create our own system of categorizing fingerprints.1 We will use the following classification system, referred to here as the Pattern Ordered Pairs (POP). The POP system will use several features of prints in the classification process. In fact, the POP system is a multi-stage process, as are systems actually in use by law enforcement agencies. The first step, called the primary classification, is to write the ordered pair (C, D), in which

represents the number of cores present in the pattern, and represents the number of deltas present in the pattern.

5. Calculate the primary classification ordered pair, (C, D), for each of your ten fingers on your own fingerprint card. 6. Calculate the primary classification ordered pair, (C, D), for the subset of database suspects as assigned by your teacher. Questions for Discussion 1. How common is each type of print within our own class? How many of your prints are loops? Whorls? Arches? 2. How does our class data compare to larger population studies?

1 The reason for this is the limited database that we are using does not require the complete complexity of the Henry System, the extension of the Henry system, or recently employed systems by federal agencies such as ridge distribution systems. Although our system is simpler, it still reflects the process by which fingerprint experts would determine likely suspects through an algorithmic process of elimination.


Figure 2.5: Fingerprint Samples

Fingerprint # of

Cores # of

Deltas Fingerprint

# of Cores

# of Deltas

Fingerprint # of Cores # of

Deltas


Pattern Refinement The three main patterns are subdivided into subcategories. In Figure 2.3 you saw that there are two types of arches: plain (no cores) and tented (one core). There are also two types of loops: ulnar (the loop slants, top to bottom, toward the ulnar side of the hand) and radial (the loop slants, top to bottom, toward the radial side of the hand). There are four types of whorls: plain (one core, the line segment joining the deltas intersects some of the curled ridges), central pocket (one core, the line segment joining the deltas falls below the curled ridges), double loop (two cores), and accidental (two cores and three deltas). Loops are classified by the fact that they have one core, one delta, and the ridge count. In a loop pattern one or more of the ridges enters on either side of the impression, recurves, touches or crosses the line segment joining the core to the delta. Loops are of two types: Radial & Ulnar. Radial loops slant toward the radial (thumb) side of the hand. Ulnar loops slant toward the ulna (pinky finger) side of the hand. What type of loop is shown in the following figure?

Figure 2.6: ___?____ Loop

Whorls are characterized by having two or more deltas and the ridge count. A plain whorl and a central pocket whorl consist of one or more ridges that make a complete circuit, with two deltas.

If one were to draw a line segment connecting the two deltas: In a Plain Whorl the line segment joining the deltas intersects some of the ridge circuits. In a Central Pocket whorl the line segment joining the deltas falls below the ridge circuits

A double loop whorl consists of two separate loop formations, with two separate and distinct sets of “shoulders”, two cores, and two deltas. An accidental whorl has two cores and three deltas.

Core

Delta


Figure 2.7: Whorl Fingerprints

Practice 3. We’d like to narrow down the list of suspects. The fingerprint database contains all the prints on file for the people from the community, including the possible suspects for the crime (but not limited to just those suspects). Calculate the primary classification ordered pair (C, D) for five of the remaining database prints beyond the ones your were assigned in Activity 2-1. 4. Look again at your own fingerprints. Further classify your prints in terms of any types of arches, loops and whorls you can identify.


Lesson 3 The Process of Elimination The Pattern Ordered Pair System of Classification Your activity and practice in the previous lesson included calculating the primary classification for fingerprints in the crime database. We can now calculate the primary classification for each of the crime scene fingerprints. Identifying a likely suspect from a database of fingerprints uses a process of elimination. The first step in the process of elimination is to compare the primary classification of fingerprint obtained at the scene with the primary classifications of the fingerprints in the database. Any suspect whose primary classification differs from the primary classification of the fingerprint obtained at the scene can be eliminated as a possible suspect.

Did You Know? The United States Federal Bureau of Investigation categorizes complete sets of fingerprints using several component parts of the Henry classification system.[1,2] 1. The Primary. 2. The Secondary. 3. The Sub-Secondary. 4. The Major. 5. The Final. 6. The Key. This system of classification is still in use in many parts of the world. However, with technologies developed in the 1990’s this system has begun to be replaced by more technologically advanced systems such as Ridge Distribution Systems.

Activity 3-1 The Primary Classification Objective: Use the POP system to classify and eliminate suspects. Material: Handout CF-H4: The Primary Classification Activity Worksheet Handout CF-H3: Database of Suspect Fingerprints 1. As a class, agree upon and record the classifications of each fingerprint in the suspect database. 2. Calculate the primary classification using our POP system for the fingerprint obtained at the scene using the POP method.


3. Compare this primary classification with the primary classifications for each print in the database. Any fingerprint in the database that does not match the primary classification of the fingerprint found at the scene can be removed from consideration. How many prints from the database have the same primary classification as the fingerprint found at the scene? Which ones are they? 4. Which of the remaining fingerprints belong to individuals in our suspect list? Which suspects do not have the same primary classification as the fingerprint found at the scene? Can they now be eliminated as suspects? Questions for Discussion 1. What are some of the aspects of the fingerprint found at the scene that may help or hurt in finding a unique match from the database? 2. Is the primary classification enough to identify a single suspect? Explain. Fingerprint Uniqueness

Aren’t You Glad Dr. Faulds is Not Your Teacher? In the late 1800s, Dr. Henry Faulds conducted an experiment with his medical students. In this experiment he had them rub their fingertips with pumice stones (like sandpaper) until they rubbed off their fingerprints. Then they waited to see if… They would grow back? If they grew back – were they the same?

What do you think about Dr. Faulds experiment? What do you think happened? There are 3 basic layers of skin – the epidermis (outermost), dermis, and hypodermis (innermost layer that is mostly fat). The epidermis is subdivided into 5 layers, which is why you may have heard we have 7 layers of skin. But for this module we only need to break the skin into 3 layers.

Figure 3.1: How Do Prints Grow Back the Same?


The dermal papillae are the ‘bumps’ at the top of your DERMIS. These bumps are then covered by your epidermis and show through as your fingerprints. Therefore, if you only damage the epidermis, your fingerprints will grow back exactly the same because the new epidermis will lay back over the same bumps. It is possible to damage the skin deeply enough that you destroy the dermal layer and lose the fingerprint in that region. Deep cuts and bad burns may do this. There will be a scar at that portion of your print, but the rest of the print will still be the same as before. Potentially, you could damage or burn off your entire print. The skin will then probably be shiny and tight and more slippery when you pick up things. Questions for Discussion 3. What determines the pattern of dermal papillae and therefore the pattern of fingerprints? 4. Do identical twins have the same DNA? Do identical twins have the same fingerprint patterns? Penetrance refers to the fact that the same gene (genotype) may, or may not, be expressed in the phenotype of different individuals. In other words, one individual may express a trait while a second individual, with the same gene/allele, may not. This is likely due to either an epigenetic interaction or an interaction with the environment. Epigenetics refers to modifications in a gene’s expression (rather than modifying the gene itself). These modifications can include DNA methylation, which effectively turns off the gene, and post-transcriptional modifications that can affect the production of the protein for which the gene encodes.

Source: Biology MOOC by Robert Maxwell is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported

License.http://biomoocnews.blogspot.com/2012_03_01_archive.html

Figure 3.2: Penetrance vs. Expressivity Figure 3.2 shows the effects of penetrance and expressivity through a hypothetical character “pigment intensity.” In each row, all individuals have the same allele—say, P—giving them the same “potential to produce pigment.” However, effects deriving from the rest of the genome and from the environment may suppress or modify pigment production in an individual.


Questions for Discussion 5. Take a closer look at the 28 remaining prints. Although all of these prints have the same primary classification, and thus the same loop pattern, there are differences among the loops on each of the prints. Examine the loops from the 28 prints that remain from the database more closely. What are some differences that you notice among the loops? 6. At this point are you confident that those people who have been eliminated as suspects are indeed innocent? Secondary Classification It is clear that the primary classification is not enough to identify a suspect. Ideally a final comparison should be made between fingerprints found at a crime scene and as few prints from the database as possible. The next step is to reduce the database even more so that just a few prints remain as likely suspects. To do this it will be necessary to calculate the secondary classification. The secondary classification can be thought of as a way of measuring the difference in size of the general pattern and the direction (left or right) of the general pattern. As you have seen, even though two prints may both be (1, 1) loops, they may still be visibly different. For example, the number of ridges that fall between the core and the delta on a loop could be different for distinct prints. That is, the primary classification process of the POP system is not enough to distinguish among similar-looking prints. Therefore we introduce the secondary classification process of the POP system. To carry out the secondary classification process, begin by drawing a straight line from the center of each core to the center of each delta. For a (1, 1) print, that is just one line, but for other primary classifications you may need to draw several lines. The center of a core is the center ridge ending at the heart of the core. The center of a delta is the ridge ending or the “fork” of a ridge that occurs within the delta. Figure 3.3 provides an example.


Figure 3.3: Line Connecting Core and Delta

For each line you drew, count the number of ridges that it crosses (DO NOT count the core itself or the delta itself). If it happens to cross a point where a ridge splits into two ridges, count that as two crossings. Record your count as positive if the slope of the line is positive, and record your count as negative if the slope of the line is negative. The secondary classification is now defined as the ordered pair (M, m), where M represents the largest (in absolute value terms) count among your lines and m represents the smallest (in absolute value terms) count among your lines. Note that when calculating M and m for arches and loops, the maximum and minimum counts are the same, M = m, because there is at most one line segment. For example, for a loop if the line segment crosses 4 ridges, then (M, m) = (4, 4). For an arch, there is no delta. Therefore, the answer is always (M, m) = (0, 0) for both plain and tented arches. Whorls are the only pattern where M and m could be of different values. Note that for the fingerprint in Figure 3.3, M and m are both the same number because there is only one line segment. Both M and m are negative to indicate that the line segment has a negative slope and the number of ridges is counted as 9 instead of 8 because the line segment crosses a bifurcation at its point of intersection. Therefore, the fingerprint in Figure 3.3 has a secondary classification of (M, m) = (-9, -9). In general it is possible for one of M and m to be positive and the other to be negative. For the fingerprint in Figure 3.4, however, the maximum and minimum measurements are both positive slopes (the two line segments on the left-hand side). This is because the smallest number in terms of absolute value (out of 8, 9, 13, and 19) is 8 and the largest number in terms of absolute value is 19. Therefore, the fingerprint in Figure 3.4 has a secondary classification of (M, m) = (19, 8).

Core

Delta


Figure 3.4: Secondary Classification Measures

Practice Back to the crime… 1. Calculate the secondary classification for the fingerprint found at the crime scene. A larger version of the crime scene print is provided below.

Figure 3.5: Fingerprint From Crime Scene

2. Calculate the secondary classification for each of the 28 remaining fingerprints from the database with the (1,1) primary classification. That is, calculate the second ordered pair (M, m) from the POP Classification System outlined above. 3. Which prints from the 28 fingerprints from the database have the same secondary classification as the fingerprint from the crime scene?

9־ 19

13־8


Lesson 4 Zeroing In Your practice from the previous lesson was to calculate the secondary classification, the second POP ordered pair (M, m). The secondary classification identifies the “size” of the pattern (that is, the number of ridges between the core and the delta) and the “direction” of the pattern (that is, whether the delta is placed to the right or the left of the core). What did you discover in your practice from Lesson 3? Did everyone calculate the same secondary classifications? Was it difficult to determine the counts? Ridge Features and Clarity Now that the number of prints from the database has been narrowed down to a few choices, the final stage is to determine an exact match between the fingerprint found at the crime scene and the remaining suspects. To do this a more refined examination of the finer ridge characteristics of fingerprints will be examined. ACTIVITY 4-1 Ridge Characteristics Objective: Identify fingerprint ridge characteristics. Materials: Handout CF-H5: Ridge Characteristics Activity Worksheet A number two pencil, 5 by 7 index card, small to medium size balloon, moist sanitary

towelette 1. To examine the ridge details of your thumb more closely, you will create a print of your thumb on the balloon, and then fill the balloon with air.

a. Take the number 2 pencil and rub it on the index card until there is a solid smudge of lead large enough to cover a fingertip.

b. Rub your thumb on the lead smudge firmly, but not too firmly, until the thumb tip is completely covered.

c. Firmly place your thumb on the side of the balloon so that the entire print from tip to knuckle appears on the balloon.

d. Once your thumbprint is on the balloon blow up the balloon so that it is about ½ to ¾ full or until the ridge details of your thumb are clearly visible.


2. Look closely at your thumbprint on the balloon. You will notice many characteristics among the ridges. A list of some of the more common ones is given below:

Ridge Ending

Lake (Enclosure)

Trifurcation

Bifurcation Ridge

Ending

Opposed Bifurcation

Dot Bridge

Ridge Crossing

Island (Short Ridge)

Double Bifurcation

Opposed Bifurcation/

Ridge Ending

Figure 4.1: Ridge Characteristics

Study the ridge details on your balloon thumbprint and answer the following questions:

a. Which of the ridge characteristics listed above are evident in your thumbprint? How frequent is each of these characteristics on your thumbprint? Are most of the characteristics present in your thumbprint? Do you think that is the case for everyone?

b. Compare the characteristics of your thumbprint with the characteristics of your classmates’ thumbprints. Do these characteristics appear to occur in any pattern? That is, are they concentrated in a particular area of the thumbprint? Is one characteristic more common than any others?

c. Do you think the frequency and placement of ridge characteristics for your other nine fingers will be any different?

The level of clarity or quality of detail will determine the type of features available for comparison and the amount of detail needed to identify. There is not a set predetermined standard as to the amount of detail needed for an identification of a print.


Level One Detail – General Ridge Flow Class characteristics only – No individualizing detail Fingerprint patterns and ridge flow (shape) Level Two Detail – Individual Ridge Path Major ridge features Allows some individualization

Ridge endings, bifurcations, their positions and relationship to other features

Level Three Detail – Individual Ridge Appearance Highest level of detail The most individualizing detail

Smallest features (pore & ridge structure) are visible for comparison.

With the 12 different characteristics and the fact that any one of them could be located anywhere on a fingerprint, it is easier to see how everyone’s fingerprints could be unique. Extension Try to estimate the number of different possible prints, or even the order of magnitude of that number. Or, put another way, how many people could be born before you would have to start repeating prints? Keep in mind that each person will have 10 different fingerprints. Of course, you will have to create a set of rules and assumptions, but it is a good problem to practice mathematical and scientific thinking. Introduction to Graph Theory The final classification to determine a match between the fingerprint found at the scene and the remaining suspects is to compare ridge characteristics among the prints. At this point you may have already “guessed” who the suspect is. This most certainly would not be the case when searching a much larger database that may contain millions of fingerprint impressions. Even though you may have figured out the suspect in this example, it is important to know what the “next step” would be in a “real” crime case, one that compares a suspect print against a database of millions. One such method would be to apply the principles of graph theory to the suspect print in order to further reduce the remaining possible matches from the database.


The Königsberg Bridge Problem As a distinct mathematical field Graph Theory traditionally traces its roots back more than 350 years to the Königsberg Bridge Problem. In the sixteenth century the city of Königsberg was located along the river Pregel in what was then Prussia; its prominence due to the fact that it was the seat of the Prussian dukes. Today the city has been renamed Kaliningrad and is a major commercial and industrial center located in western Russia. The Königsberg Bridge Problem refers to the sixteenth century city which had seven bridges that connected the shores of the river Pregel (Pregolya or Pregola in Russian) with two islands that divide the town center into four sections (see diagram below). Some of the people of Königsberg wondered if it was possible to traverse the seven bridges exactly once in a continuous uninterrupted walk. The problem was posed to Leonard Euler in a letter written by a fellow mathematician. In 1735 Euler provided a solution that showed it was not possible.

By Merian-Erben [Public domain], via Wikimedia Commons

We can use graphs to further refine the search for a suspect. A graph is a commonly used diagram that represents problems in mathematics, biology, and other sciences. Diagrams that use dots and arcs (or curved line segments) can represent graphs. In studying the properties of a graph scientists can understand the nature of the problem it represents giving them a better opportunity to find a solution. A simple example of a graph is given by:


Figure 4.2: Example of a Graph

In order to study the properties of a graph, a more formal definition will be needed.

A graph (or network) G is a set of at least one vertex, , , … , , commonly represented as points, and a set of edges , , … , , commonly represented as a line or curve having a vertex at each end. The edge connecting vertices u and v is written uv. Therefore, ∈ .In that case, we say that vertices u and v are adjacent, u and v are called the endpoints of uv, and u and v are said to be incident to uv. Vertices are also known as nodes.

Because the vertex-edge connections do not depend on position, a graph has many visually different representations. For example, in the figure below all three diagrams represent equivalent graphs since connections in the one graph correspond to connections in the other graphs. (‘a’ is connected to ‘b’ and ‘e’, ‘d’ is connected to ‘c’ and ‘e’ etc.…)

Figure 4.3: Graph Examples

Graphs can be used to identify and connect ridge point characteristics. The resulting graph could then be used to match the suspect’s fingerprint with the fingerprint from the crime scene. Connecting the Dots The nodes or vertices of the graph will be ridge-point characteristics (dots, the tip of a ridge ending, the vertex of a bifurcation, etc.) in the fingerprint. To distinguish one type of point characteristic from another the nodes will be color-coded. This is referred to as vertex coloring.

Diagram G1 Diagram G2 Diagram G3

a a a

b d d

c b b d e e

e c c

a


For simplicity, this activity will concentrate on identifying only three point characteristics: bifurcations (red), ridge endings (blue), and dots or delta centers (green). The Figure 4.4 below shows the fingerprint from the crime scene with several nodes already identified. It is usually suggested to identify at least 8 to 12 point characteristics for better match identification. In general, it is best to choose point characteristics that are spread across the fingerprint. Look for nodes that are not concentrated in one area.

Figure 4.4: Crime Scene Fingerprint Vertex Coloring The next step is to add edges joining the vertices. Not all vertices will be joined by an edge. The rule for joining two vertices with an edge is this: A pair of vertices will be joined (be adjacent) if the edge drawn will “cross” the ridges in a nearly perpendicular fashion. If the edge appears to be almost “parallel” (that is, follows the general flow of the ridges), then it will not be added to the graph. The reason for this is that you will need to count the number of ridges that cross the edge. The edge will then be “labeled” with this number (count). This number is referred to as the weight of the edge. Enough edges should also be added so that the resulting graph is connected. Weighted edges have been added to the vertices of the fingerprint from the crime scene to obtain:

a

b

c

d

e

f


Figure 4.5: Crime Scene Fingerprint Edges Note that the graph added to the above fingerprint is just one example of a possible graph on this print. There are many other ridge characteristics that could have been used to create a graph associated with the fingerprint. As practice, try creating a different graph for the fingerprint above. Here are some suggestions when creating your graph:

1. Try to select point characteristics that are clearly visible so there is no ambiguity as to their classification.

2. Try to choose point characteristics that cover as much of the print as possible. 3. Try to choose point characteristics that can be joined to at least one other node by an edge

so that a connected graph can be constructed.

a

b

c

d

e

f

g

i

h 1

3 7

2 4

6


ACTIVITY 4-2 Matching With Graph Theory Objective: Construct a graph of point characteristic vertices and edges on a given fingerprint to match an existing fingerprint. Materials: Handout CF-H6: Matching With Graph Theory Activity Worksheet Reprinted below are the two suspect prints from the database. On one of the prints it will be possible to construct a graph that has identical connections, with corresponding colors and weights as the graph on the fingerprint from the crime scene above. When two graphs can be matched in such a way that all connections, colors, and weights are identical, the two graphs are said to be isomorphic. 1. Label each print with the print letter, POP primary classification and secondary classification. 2. Construct a graph on each of the suspect prints attempting to construct a graph that is isomorphic to the graph on the print from the crime scene. The one on which an isomorphic graph can be constructed will be the matching print. 3. Which print is the matching print?


Numerical Representation of Graphs As humans we can compare the characteristics of two prints with the benefit of our eyesight. However, comparing graph diagrams by visually studying the vertices and edges is not the way experts determine if two graphs are isomorphic. This sort of comparison is generally done algorithmically using computers. When working with computers and algorithms, it is important to find a way to represent graphs numerically. Representing the graphs numerically will allow the process to be more precise and to be able to encode the information into a computer so that the computer can make a comparison without the bias of human interference. One way to represent graphs numerically is with an adjacency matrix. An adjacency matrix for a graph compares which vertices are adjacent to each other and which are not adjacent to each other. The vertices are labeled and placed at the left of each row as well as at the top of each column. The entries in each row and column represent the graph’s edges with entries of 1 and 0. A 1 is placed in a matrix entry if the vertices represented by that row and column are joined by an edge (i.e., are adjacent). A 0 is placed in a matrix entry if the vertices represented by that row and column are not joined by an edge (i.e., are not adjacent). The adjacency matrix below represents the graph constructed for the fingerprint found at the crime scene. Note, for example, the 1 at row c and column a means that vertices a and c are adjacent.

a b c d e f g h i a 0 0 1 0 0 0 0 0 0b 0 0 1 0 0 0 0 0 1c 1 1 0 1 0 0 0 0 0d 0 0 1 0 1 0 0 0 0e 0 0 0 1 0 1 1 0 0f 0 0 0 0 1 0 1 0 0g 0 0 0 0 1 1 0 0 0h 0 0 0 0 0 0 0 0 1i 0 1 0 0 0 0 0 1 0

This matrix can be altered to include some additional information. For example, the 1’s in the matrix can be replaced with the edge weights for the corresponding edges to produce this new matrix, indicating that vertices a and c have 18 ridges between them. Instead of placing a 1 where two vertices are adjacent, the following matrix puts the ridge count (that is, the edge weight) in the row and column of two adjacent vertices:

a b c d e f g h i a 0 0 18 0 0 0 0 0 0b 0 0 12 0 0 0 0 0 3c 18 12 0 6 0 0 0 0 0d 0 0 6 0 4 0 0 0 0e 0 0 0 4 0 2 7 0 0f 0 0 0 0 2 0 3 0 0g 0 0 0 0 7 3 0 0 0h 0 0 0 0 0 0 0 0 3i 0 3 0 0 0 0 0 3 0


It is even possible to represent the vertex colorings numerically in the matrix. Let the color red be represented by 1, blue represented by 2, and green represented by 3. Each vertex will be labeled with a 1, 2, or 3 to identify its color. Each matrix entry will be an ordered triple with the first coordinate the edge weight, the second coordinate the row vertex color and the third coordinate the column vertex color. This new matrix will be called the expanded adjacency matrix for the graph. The new matrix shows that vertices a and c have 18 ridges between them, and that vertex c is red and vertex a is green:

a b c d e f g h i a 0 0 (18, 3, 1) 0 0 0 0 0 0 b 0 0 (12, 2, 1) 0 0 0 0 0 (3, 2, 2) c (18, 1, 3) (12, 1, 2) 0 (6, 1, 1) 0 0 0 0 0 d 0 0 (6, 1, 1) 0 (4, 1, 1) 0 0 0 0 e 0 0 0 (4, 1, 1) 0 (2, 1, 1) (7, 1, 1) 0 0 f 0 0 0 0 (2, 1, 1) 0 (3, 1, 1) 0 0 g 0 0 0 0 (7, 1, 1) (3, 1, 1) 0 0 0 h 0 0 0 0 0 0 0 0 (3, 2, 2) i 0 (3, 2, 2) 0 0 0 0 0 (3, 2, 2) 0

It is now possible to compare two “expanded” adjacency matrices for two fingerprint graphs numerically. If the matrices have identical numerical values in their rows and columns, the corresponding graphs can be said to be isomorphic. Note that if corresponding vertices are not labeled the same on both prints, then row/column operations (interchanging rows/columns with one another) may need to be performed in order for corresponding vertices to be “lined up” and the matrices to be identical. Even though the problem has been solved and the suspect identified, this last activity revisits the problem in a way that that turns the fingerprint graph into a numerical representation. Suppose for a moment that there are still two suspects and the fingerprint graphs are to be compared numerically to determine which of the two matches the print left at the scene of the crime. ACTIVITY 4-3 Using Matrices to Match Prints Objective: Construct adjacency matrices on given fingerprints to match the corresponding adjacency matrix of an existing fingerprint. Materials: Handout CF-H7: Using Matrices to Match Prints Activity Worksheet 1. Construct the expanded adjacency matrix for each of the two graphs you produced for the suspects’ fingerprints. 2. Using row operations if necessary compare the expanded adjacency matrices to determine which suspect print matches the fingerprint at the crime scene. 3. What is your conclusion?


Questions for Discussion 1. What are some of the shortcomings of the methods we have discussed in this unit for positively identifying suspects using fingerprinting? 2. How can technology assist investigators in identifying suspects? If you were developing a computer program to use in analyzing fingerprints, what algorithms would you program and what output would you want?


Glossary

Adjacent – two vertices in a graph are adjacent if they are joined by an edge. Adjacency matrix – a matrix representing a graph that compares which vertices are adjacent to each other and which are not adjacent. Arch – a ridge pattern having no deltas. Bifurcation – a characteristic of ridge classification where a single ridge splits into two ridges. Connected – a graph is connected if there is a path from each vertex in the graph to any other vertex in the graph. Core - the termination of the innermost ridge that appears in the interior of a concentric set of curved ridges. Delta - a triangular-shaped ridge pattern that appears at the center of three intersecting ridge flows. Dot – a ridge characteristic with a point within two ridges. Edge – a line segment or curve having a vertex at each end. Edge weight – another name for ridge count. Expanded adjacency matrix – an adjacency matrix with ordered triple entries representing the edge weight, the row vertex color and the column vertex color. Epigenetics – the study of the chemical modification of specific genes or gene-associated proteins and how these modifications produce their effect on the phenotype of that organism. Expressivity – the degree to which a particular gene produces its effect in an organism. Graph (or Network) – a set of at least one vertex, commonly represented as points, and a set of edges, commonly represented as a line or curve having a vertex at each end. Impressed print – a print left as an indentation in a soft pliable surface such as wax, clay, or paint (also know as a plastic print). Incident – a vertex and an edge are incident if the vertex is an endpoint of the edge. Island (or short) ridge – a characteristic of ridge classification with a short ridge section between two longer ridges.


Isomorphic – a classification of two graphs that can be matched in such a way that all connections, colors and weights are all identical. Lake (or enclosure) – a ridge characteristic with a ridge leading to a circle formation and then continuing back to a ridge. Latent print – a print that is not visible to the unaided eye. Loop – a ridge pattern having exactly one delta. Matrix – a rectangular array of numbers. Node – another name for a vertex. Patent print – a print that is visible to the unaided eye, made in some medium such as blood, ink or paint. Pattern ordered pairs (POP) – a system of fingerprint classification that identifies an ordered pair (C, D) in which C represents the number of cores present and D represents the number of deltas present in a given fingerprint. Penetrance – the frequency, under given environmental conditions, with which individuals possessing a specific genotype express that genotype. Usually given as a percentage. Plastic print – a print left as an indentation in a soft pliable surface such as wax, clay, paint, etc. (also know as an impressed print). Primary classification – the first level of classification in analyzing a fingerprint. Ridge – the lines in a fingerprint. Ridge ending – a characteristic of ridge classification in which the ridge(s) end. Secondary classification – a second level of classification in analyzing a fingerprint that includes measuring the difference in size of the general pattern and the direction (left or right) of the general pattern of the print. Trifurcation – a characteristic of ridge classification in which a single ridge splits into three distinct ridges. Vertex – a point on a graph often representing the endpoint of an edge (also known as a node). Vertex coloring – a method to distinguish one type of point characteristic from another by coloring the node/vertex. Whorl – a ridge pattern having two or more deltas.


References [1] Department of Justice Federal Bureau of Investigation. Classification Codes. Found at www.slideshare.net/guest8cbcb02/classification-4000506. [2] Fingerprint identification Chapter 10 fingerprint classification systems. (2005). San Clemente, CA: Law Tech Custom Publishing, Inc. Found at www.slideshare.net/KUL2700/ch-10-fingerprint-classification-systems.

Documents

B MaCrime and Forensics Student 1 CRIME: Criminal Investigation through Mathematical Examination Overview Welcome to our unit on the mathematical examination of fingerprints! This