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Double Trouble: When Identical Twins Run Into the Law

Last night, Law & Order: SVU did that “ripped from the headlines” thing where they borrow elements of real-world criminal cases or address current issues in law enforcement. The episode, “Double Strands,” revolved around a topic I've been reading a lot about lately: twins.

[Spoilers Ahead!]

The plot involves a serial rapist and his identical twin brother, who is falsely accused of the crimes. The SVU detectives eventually figured things out, but real-world police have had a lot of trouble with identical twins in several high-profile cases. DNA evidence — a tool that’s supposed to help convict the guilty and exonerate the innocent accurately and efficiently — has complicated cases where a twin or twins have been involved. This is because identical twins are the result of a single fertilized egg that formed one zygote, which then divided into two separate embryos. The siblings have nearly identical DNA, and we've yet to figure out how to discern one twin from the other using DNA analysis.

[End Spoilers]

The Headlines From Which The Story Was Ripped

One summer night in 2001, Darrin Fernandez attempted to break into an apartment in Boston’s Dorchester neighborhood. When he smashed a window to get inside, he not only alerted the person who lived in the apartment, but also cut himself deeply on the jagged glass. He fled, and police soon found him, still bleeding, as he tried to escape.

When the police analyzed DNA from the broken glass and from Fernandez, they found that he was a match for genetic material recovered in two unsolved sexual assaults cases, both of which were committed within a few blocks of the apartment Fernandez had tried to break into.

Fernandez was convicted of attempted breaking and entering at the apartment. He was also eventually convicted of one of the sexual assaults, for which there was plenty of evidence that implicated him, including the victim noticing a tattoo that his brother did not have (in the SVU episode, both twins have a similar tattoo in the same place). The police and prosecutors could not confidently put the second assault on him, though. The DNA match was the only substantive evidence that they had to go on, and it turns out that it also matched a second person: Fernandez’s identical twin brother, Damien.

There were no witnesses to the assault, no accomplices to roll, and no fingerprints at the scene. Darrin didn’t have an alibi to cover the time of the assault, but neither did Damien. The police couldn’t place either brother at the scene of the crime and the DNA that damned Darrin in another trial had only established reasonable doubt in this one. The case went to trial anyway and after four days of deliberation, there was a hung jury and a mistrial.  In a second trial several months later, the prosecution’s case rested heavily on the fact that Darrin worked as a painter near where the assault occurred, and he had the opportunity to case the neighborhood. Again, the jury was hung and a mistrial was declared.

In 2006 Fernandez went to trial a third time, and prosecutors were allowed to present for the first time evidence that he had committed four break-ins in the victim’s neighborhood within a year (and had been convicted of a similar sexual assault in one of those instances). The victim of that assault, who had not testified at the previous two trials, also took the stand this time around to highlight the similarities her attack shared with this case.

The jury returned a guilty verdict and, five years after his initial arrest, Darrin Fernandez was sentenced to 15 to 20 years on top of 10 to 15 year sentence he was already serving for the first assault.

Police and prosecutors in Grand Rapids, Michigan, may have had it even worse. In 1999, presented with DNA evidence in the rape of a college student, they couldn’t figure out which of their twin suspects to even charge with the crime.

Like the Fernandez cases, there were no witnesses and no fingerprints. To complicate matters, the suspects in this case, Tyrone and Jerome Cooper, both had records for sexual assault (Tyrone assaulted a 10-year-old girl in 1991 and Jerome a 12-year-old girl in 1998).

After hiring a biotechnology company to check some 100,000 DNA characteristics to match one twin or the other to the recovered evidence, the police came up empty. They could only tell both twins that the case would be not be forgotten and would get worked until the statute of limitations prevented prosecution.
* * *
Twins make the justice system work even harder when they’re attached to each other, literally. If a conjoined twin commits, and is convicted of, a crime, how do you punish them without also unjustly punishing their innocent sibling? Slate’s Daniel Engber and legal scholar Nick Kam have both looked at the available historic cases and suggested possible solutions to the problem.

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Animals
New Plankton Species Named After Sir David Attenborough Series Blue Planet
John Phillips, Getty Images for Tourism Australia
John Phillips, Getty Images for Tourism Australia

At least 19 creatures, both living and extinct, have been named after iconic British naturalist Sir David Attenborough. Now, for the first time, one of his documentary series will receive the same honor. As the BBC reports, a newly discovered phytoplankton shares its name with the award-winning BBC series Blue Planet.

The second half of the species' name, Syracosphaera azureaplaneta, is Latin for "blue planet," likely making it the first creature to derive its name from a television program. The single-cell organisms are just thousandths of a millimeter wide, thinner than a human hair, but their massive blooms on the ocean's surface can be seen from space. Called coccolithophores, the plankton serve as a food source for various marine life and are a vital marker scientists use to gauge the effects of climate change on the sea. The plankton's discovery, by researchers at University College London (UCL) and institutions in Spain and Japan, is detailed in a paper [PDF] published in the Journal of Nannoplankton Research.

"They are an essential element in the whole cycle of oxygen production and carbon dioxide and all the rest of it, and you mess about with this sort of thing, and the echoes and the reverberations and the consequences extend throughout the atmosphere," Attenborough said while accepting the honor at UCL.

The Blue Planet premiered in 2001 with eight episodes, each dedicated to a different part of the world's oceans. The series' success inspired a sequel series, Blue Planet II, that debuted on the BBC last year.

[h/t BBC]

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5 Ways You Do Complex Math in Your Head Without Realizing It
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The one thing that people who love math and people who hate math tend to agree on is this: You're only really doing math if you sit down and write formal equations. This idea is so widely embraced that to suggest otherwise is "to start a fight," says Maria Droujkova, math educator and founder of Natural Math, a site for kids and parents who want to incorporate math into their daily lives. Mathematicians cherish their formal proofs, considering them the best expression of their profession, while the anti-math don't believe that much of the math they studied in school applies to "real life."

But in reality, "we do an awful lot of things in our daily lives that are profoundly mathematical, but that may not look that way on the surface," Christopher Danielson, a Minnesota-based math educator and author of a number of books, including Common Core Math for Parents for Dummies, tells Mental Floss. Our mathematical thinking includes not just algebra or geometry, but trigonometry, calculus, probability, statistics, and any of the at least 60 types [PDF] of math out there. Here are five examples.

1. COOKING // ALGEBRA

Of all the maths, algebra seems to draw the most ire, with some people even writing entire books on why college students shouldn't have to endure it because, they claim, it holds the students back from graduating. But if you cook, you're likely doing algebra. When preparing a meal, you often have to think proportionally, and "reasoning with proportions is one of the cornerstones of algebraic thinking," Droujkova tells Mental Floss.

You're also thinking algebraically whenever you're adjusting a recipe, whether for a larger crowd or because you have to substitute or reduce ingredients. Say, for example, you want to make pancakes, but you only have two eggs left and the recipe calls for three. How much flour should you use when the original recipe calls for one cup? Since one cup is 8 ounces, you can figure this out using the following algebra equation: n/8 : 2/3.

algebraic equation illustrates adjustment of a recipe
Lucy Quintanilla

However, when thinking proportionally, you can just reason that since you have one-third less eggs, you should just use one-third less flour.

You're also doing that proportional thinking when you consider the cooking times of the various courses of your meal and plan accordingly so all the elements of your dinner are ready at the same time. For example, it will usually take three times as long to cook rice as it will a flattened chicken breast, so starting the rice first makes sense.

"People do mathematics in their own way," Droujkova says, "even if they cannot do it in a very formalized way."

2. LISTENING TO MUSIC // PATTERN THEORY AND SYMMETRY

woman enjoys listening to music in headphones
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The making of music involves many different types of math, from algebra and geometry to group theory and pattern theory and beyond, and a number of mathematicians (including Pythagoras and Galileo) and musicians have connected the two disciplines (Stravinsky claimed that music is "something like mathematical thinking").

But simply listening to music can make you think mathematically too. When you recognize a piece of music, you are identifying a pattern of sound. Patterns are a fundamental part of math; the branch known as pattern theory is applied to everything from statistics to machine learning.

Danielson, who teaches kids about patterns in his math classes, says figuring out the structure of a pattern is vital for understanding math at higher levels, so music is a great gateway: "If you're thinking about how two songs have similar beats, or time signatures, or you're creating harmonies, you're working on the structure of a pattern and doing some really important mathematical thinking along the way."

So maybe you weren't doing math on paper if you were debating with your friends about whether Tom Petty was right to sue Sam Smith in 2015 over "Stay With Me" sounding a lot like "I Won't Back Down," but you were still thinking mathematically when you compared the songs. And that earworm you can't get out of your head? It follows a pattern: intro, verse, chorus, bridge, end.

When you recognize these kinds of patterns, you're also recognizing symmetry (which in a pop song tends to involve the chorus and the hook, because both repeat). Symmetry [PDF] is the focus of group theory, but it's also key to geometry, algebra, and many other maths.

3. KNITTING AND CROCHETING // GEOMETRIC THINKING

six steps of crocheting a hyperbolic plane
Cheryl, Flickr // CC BY-SA 2.0

Droujkova, an avid crocheter, she says she is often intrigued by the very mathematical discussions fellow crafters have online about the best patterns for their projects, even if they will often insist they are awful at math or uninterested in it. And yet, such crafts cannot be done without geometric thinking: When you knit or crochet a hat, you're creating a half sphere, which follows a geometric formula.

Droujkova isn't the only math lover who has made the connection between geometry and crocheting. Cornell mathematician Daina Taimina found crocheting to be the perfect way to illustrate the geometry of a hyperbolic plane, or a surface that has a constant negative curvature, like a lettuce leaf. Hyperbolic geometry is also used in navigation apps, and explains why flat maps distort the size of landforms, making Greenland, for example, look far larger on most maps than it actually is.

4. PLAYING POOL // TRIGONOMETRY

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If you play billiards, pool, or snooker, it's very likely that you are using trigonometric reasoning. Sinking a ball into a pocket by using another ball involves understanding not just how to measure angles by sight but triangulation, which is the cornerstone of trigonometry. (Triangulation is a surprisingly accurate way to measure distance. Long before powered flight was possible, surveyors used triangulation to measure the heights of mountains from their bases and were off by only a matter of feet.)

In a 2010 paper [PDF], Louisiana mathematician Rick Mabry studied the trigonometry (and basic calculus) of pool, focusing on the straight-in shot. In a bar in Shreveport, Louisiana, he scribbled equations on napkins for each shot, and he calculated the most difficult straight-in shot of all. Most experienced pool players would say it’s one where the target ball is halfway between the pocket and the cue ball. But that, according to Mabry’s equations, turned out not to be true. The hardest shot of all had a surprising feature: The distance from the cue ball to the pocket was exactly 1.618 times the distance from the target ball to the pocket. That number is the golden ratio, which is found everywhere in nature—and, apparently, on pool tables.

Do you need to consider the golden ratio when deciding where to place the cue ball? Nope, unless you want to prove a point, or set someone else up to lose. You're doing the trig automatically. The pool sharks at the bar must have known this, because someone threw away Mabry's math napkins.

5. RE-TILING THE BATHROOM // CALCULUS

tiled bathroom with shower stall
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Many students don't get to calculus in high school, or even in college, but a cornerstone of that branch of math is optimization—or figuring out how to get the most precise use of a space or chunk of time.

Consider a home improvement project where you're confronted with tiling around something whose shape doesn't fit a geometric formula like a circle or rectangle, such as the asymmetric base of a toilet or freestanding sink. This is where the fundamental theorem of calculus—which can be used to calculate the precise area of an irregular object—comes in handy. When thinking about how those tiles will best fit around the curve of that sink or toilet, and how much of each tile needs to be cut off or added, you're employing the kind of reasoning done in a Riemann sum.

Riemann sums (named after a 19th-century German mathematician) are crucial to explaining integration in calculus, as tangible introductions to the more precise fundamental theorem. A graph of a Riemann sum shows how the area of a curve can be found by building rectangles along the x, or horizontal axis, first up to the curve, and then over it, and then averaging the distance between the over- and underlap to get a more precise measurement. 

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