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The Brachistochrone and Snell’s Law: A Surprising Connection

May 3, 2023

Brachistochrone – A word originating from Greece that means “shortest time.” The brachistochrone problem is a well known problem in the world of physics that involves finding a curve such that, assuming there is no friction, a ball traveling along this curve would reach its destination in the shortest possible time. Initially discovered by Johann Bernoulli, a famous Swiss mathematician, the brachistochrone curve has a variety of interesting properties, such as the fact that it is a cycloid, the path mapped out by a point on a circle as it rolls on a flat surface. Additionally, a subsection of the curve also acts as a tautochrone curve, a curve that causes a ball to reach its destination in the same amount of time regardless of where it begins its journey. However, perhaps the most interesting fact about the brachistochrone is that it was initially solved by Bernoulli using Snell’s law, a law that dictates the path traveled by light when moving between different mediums.

A cycloid is mapped out in red. When inverted, this curve matches the shape of the brachistochrone.

Initially, it may not be readily obvious why Snell’s law would approximate a brachistochrone curve, but with a bit of physical intuition, it makes perfect sense. First, the reason why Snell’s law works must first be understood. Snell’s law states that the ratio of the sines of the angles formed by the refraction of light when passing through a medium is equal to the ratio of the speed within the two media.

The index of refraction dictates how much light “slows down” in a medium.

With a bit of calculus not shown here (perhaps in a future article), it can be shown that this equation comes as a consequence of minimizing the time taken for light to travel from one point to another between the two media. This is because light naturally always takes the path of shortest time, and this is what connects Snell’s law to the brachistochrone. When a ball is dropped, it is constantly changing velocity due to gravity, so you can envision a ball as traveling through a continuous set of layers of different velocities. If the ball wanted to travel the path of shortest time, it would follow Snell’s law at every point in its path, so the brachistochrone curve should have the property of having the ratio of instantaneous change in the velocity of the ball match the ratio of the angles formed by a vertical line at that point of the curve.

Here, the line travels through several “layers” with different speeds due to gravity and diffracts according to Snell’s law each time.

Of course, it’s one thing to be told that Snell’s law maps out the brachistochrone, but it’s not as satisfying as seeing it for yourself, which is why I coded a simulation in Python to try and match up a curve constructed from Snell’s law with a brachistochrone curve.

The top two sliders change the initial conditions of the actual brachistochrone in blue, while the bottom two sliders change the initial angle and the number of layers for the Snell’s law approximation in orange.

For more information on the brachistochrone curve, check out Vsauce or 3Blue1Brown’s videos on the topic below:

 

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  Timothy Durham was convicted of rape when a test showed his DNA genotype matched the DNA recovered from the crime scene. Eye witnesses testified that Durham was in a different state at the time of the incident, but he wasn’t exonerated from his 3000 year sentence until follow up DNA analysis showed that the preliminary forensic analysis used in the trial was misinterpreted (Thomas et al.). DNA testing serves an invaluable part in our criminal justice system, but it comes with complications and bias that can lead to wrongful convictions and misinformed juries.  

Background:

Forensic genetics aims to identify the origin of a biological sample. It can be traced back to Karl Landsteiner who discovered human ABO blood groups. With blood groups, suspects could be eliminated based on their blood type. For example, if a suspect had type A blood and there was type B blood left at the scene by the perpetrator, the investigators could rule out the suspect. Unfortunately, the reverse didn’t work. If the suspect and perpetrator both had the same blood type, it can’t be concluded that the suspect was guilty. Blood identifiers evolved into 8 distinct categories which improved the ability to eliminate suspects, but could only conclude that (if the system matched) it might be the perpetrator or this person might be related to the perpetrator (Jobling and Gill). The field evolved in the 1980s with the discovery of Minisatellites, lines of repetitive DNA that don’t encode proteins. Minisatellite patterns in DNA create what we know today as the DNA fingerprint, and it has a much higher accuracy rate than previous methods (Augustyn et al.). Around the same time, a breakthrough for victims of sexual assault was discovered. In the case of assault, bodily fluids of both the victim and the assaulter can get mixed. A process called differential lysis was created which could selectively increase the concentration of sperm in a fluid sample thereby allowing for accurate DNA testing (Jobling and Gill). Forensics evolved to use single-locus probe testing after several years because of its ease of interpretation. It looked at specific minisatellites that had high variability between subjects. This was the first DNA-based method to be used in a criminal investigation. Two murder homicides from several years apart were connected using DNA evidence. The evidence was then used to exonerate an innocent man, and the guilty party was found when he tried to evade mandatory DNA testing for all men in the small town. His guilt was then reinforced when the DNA profiles matched, and he gave a confession (Jobling and Gill). Finding DNA at a crime scene doesn’t guarantee that you’ll find a murderer. There are issues with non-pure samples and limited samples that make DNA work difficult, and accreditation of facilities and new techniques and take time (Jobling and Gill).  

Subjectivity of DNA Interpretation:

In a 2009 publication, the National Academy of Science discusses the possibility for subjective influence in lab based forensics. The interpretation bias happens in complex cases where, for example, there could be a mixture of DNA. A study published in 2011 looked specifically at this issue. A gang rape happened in Georgia, and the mixture in DNA from the case was used to prosecute one suspect after another suspect took a plea bargain to testify against him. With DNA mixtures, there are three possible conclusions: suspect excluded, suspect cannot be excluded, and inconclusive. If the suspect is excluded, it means that there’s little to no possibility that his DNA is part of the sample. If the suspect cannot be excluded, it means that there’s a possibility that his DNA is part of the sample. In the trial, the DNA examiners who knew the context of the case concluded that the suspect being prosecuted could not be excluded. The 2011 study gave the DNA evidence from this case and the DNA profiles of the suspects to 17 forensic examiners. These examiners had no context of the case, and they all worked independently. Of the 17, only 1 came to the same conclusion (suspect cannot be excluded) as the examiners who knew the context of the case. Twelve of the 17 concluded that the suspect who was prosecuted and convicted had no relation to the DNA found in the sample. Not only does this show that subjectivity occurs due to the context of the case, but the 17 examiners came up with different findings based on their past experiences meaning that not all DNA analysis is objective. Some organizations like the Scientific Working Group on DNA Analysis Methods say that quantitative statistical evidence should be presented with every categorical conclusion, but the US doesn’t require this (Dror and Hampikian).

False Positives:

There are two categories that cause wrongful identification in a DNA report: coincidental matches and false positives. Coincidental matches are DNA matches between two unrelated people. The probability of these matches is about 1 to 500,000 for white individuals and about 1 to 350,000 for black individuals. This small probability of coincidental matches makes people feel confident in DNA evidence. False positives, on the other hand, are matches between DNA and test DNA that result from issues in the forensics lab. There have been multiple cases of forensics labs accidentally switching victim and suspected rapist DNA which lead to convictions that were later overturned when the lab realized the error. There are also cases of misreading or misunderstanding a sample. In a case from 1993, a man was convicted of rape because someone interpreted a mixed sample of victim and perpetrator DNA as an individual sample. The conviction was again overturned. To prevent false convictions, retesting DNA is allowed, but it’s not an efficient solution. Many people are pushing for statistical information on the probability of false positives, so that the jurors can make accurate judgements based on DNA evidence (Thomas et al.).  

Conclusion:

DNA analysis is a useful tool for the criminal justice system, but subjectivity can be an issue in forensic testing. To combat wrongful convictions, there are organizations like The Innocence Project who use DNA testing to free the innocent, but reformed forensic testing standards should be put in place to prevent subjectivity and to ensure juries are correctly informed during trials. https://www.innocenceproject.org/    

Citation:

Augustyn , Adam, et al. “DNA Finerprinting.” Encyclopedia Britannica , 26 Apr. 2018, www.britannica.com/science/DNA-fingerprinting#ref153186. Dror, Itiel E., and Greg Hampikian. “Subjectivity and Bias in Forensic DNA Mixture Interpretation.” Science & Justice, vol. 51, no. 4, 2011, pp. 204–208., doi:10.1016/j.scijus.2011.08.004. Jobling, Mark A., and Peter Gill. “Encoded Evidence: DNA in Forensic Analysis.” Nature Reviews: Genetics, vol. 5, no. 10, Oct. 2004, pp. 739–751., doi:10.1038/nrg1455. Thompson, William C, et al. “How the Probability of a False Positive Affects the Value of DNA Evidence.” Journal of Forensic Sciences, vol. 48, no. 1, Jan. 2003, pp. 1–8., doi:10.1520/jfs2001171.