Euclidean construction paintings

I've enjoyed straightedge-and-compass constructions since high school—I remember that high school geometry was my favorite math class up until to a point in college after I had already decided to be a math major. And I also remember feeling cheated in high school geometry because our textbook used a construction system based on folding paper instead of a straightedge and compass!

Lately I've been painting a lot, and what I've been painting is the constructions in Book 1 of Euclid's Elements, a weirdly enduring piece of the mathematics canon. Written around 300 BCE, I think this is the oldest piece of mathematical writing I read any substantial amount of during my mathematics education by around 2000 years! (Most people could probably add another ~100-300 years on there... but at one point I assisted a professor with a project of translating an early calculus textbook from French to English.)

Gouache paintings of all 14 constructions from Book 1

All the paintings here use gouache on 5x7 Yupo heavy watercolor paper. This blog does pretty badly with photos; they might look nicer on instagram, where I'm planning to post one construction at a time. They look best in person, though, and I would happily show them off to you.

Individual Constructions

Numbering below follows the original: in Elements the constructions are interspersed with other propositions.

1. Equilateral Triangle

I find this surprising as a first proposition... the construction is so simple, and recurs so many times throughout what follows, that it's obvious in a practical sense why Euclid starts here. But my intuition insists that a line or an angle is a more fundamental building block of geometry than an equilateral triangle. 

2.  A straight line of given length

Another construction that will recur frequently. 

I wasn't satisfied with how close in hue the given point and line here are to the constructed ones. In later paintings I tried to create more separation to make the construction easier to follow. 

3. Cutting a given length from a given line

This construction is almost identical to the previous one; only one more circle is constructed.

9. Angle bisection

Between constructions 3 and 9 Euclid proves several propositions relating mainly to congruent triangles. These are used in verifying that the next set of constructions are correct.

This is one of my favorite paintings from the series; I loved the way the darker blue paint behaved.

10.  Segment bisection

Another surprise for my intuition... intuitively, I find it much easier to approximate bisecting a line segment than an angle. But this construction relies on the previous one. 

11. A perpendicular to a line from a point on the line

The second of three very similar constructions.

This painting bothers me... something about the eye I can see in it. 

12. A perpendicular to a line from a point not on the line

The triangles here are not needed for the construction, but for the proof that the lines really are perpendicular.

22. Given three appropriate lines, to construct a triangle

This construction was so much more complex than the preceding ones that it broke parts of the system I'd been using. 

The first few times I tried it, I kept losing track of which lines, points, and circles were which; I started color coding as I went instead of, as I had been doing, at the end. Then I started omitting some of the full circles used in the construction for transferring lengths, instead making just the hatches where they intersect (and only the intersection that is used moving forward).  This simplification also made the appearance a lot nicer, so I kept it for the final painting; it's also used in some of the other constructions that follow. 

This was also the only construction for which I didn't fit Euclid's full specifications on the page. The whole statement is: "To construct a triangle out of three straight lines which equal three given straight lines: thus it is necessary that the sum of any two of the straight lines should be greater than the remaining one."

23. To copy an angle at a given point on a line

This is really another construction of a triangle, this time in a specific place... but since you can choose two sides to be the same length, it is a little less complicated, both to carry out and to look at. 

31. A parallel line running through a given point

This is the first construction after Euclid starts using the "parallel postulate." If you'd like to explore what might happen without that postulate, one place you can start is this spherical geometry model: spherical geometry shares the first four of Euclid's postulates, but instead of some lines intersecting as specified in the parallel postulate, any two lines intersect.

42. A parallelogram with area equal to that of a triangle, on a given angle

Euclid states a number of theorems as pertaining to parallelograms where, practically, one would most likely want to use the special case parallelogram of a rectangle. If I'd painted a rectangle, though, I'd expect people to feel unsure whether the same construction works for other parallelograms. A rectangle just feels very special.

This is another construction that was a breaking point for my system. Up to this point, I'd used the method from construction 2 every time I needed to transfer a length. With a typical modern compass, this is not necessary—it stays fixed at the same length until it is adjusted. I found this construction unbearably messy, both to do and to look at afterwards, when performed using all the steps from construction 2. So I started using my compass to transfer lengths in a single step. I didn't want to—I did drafts both on paper and with geometry software that let me play with the arrangement more efficiently, trying to find a position or choice of angles that would clean things up. I couldn't do it, though.

44. A parallelogram with area equal to a triangle, on a given line and angle

This construction allows you to specify the length of one side of the parallelogram constructed. For example, you could use a side one unit long. Or, as in the next construction...

45. A parallelogram equal in area to a polygon

... you could chain multiple parallelogram constructions together, to construct a parallelogram with the same area as a more complicated shape. Here I've used a quadrilateral for simplicity, but any shape with straight sides would work, as it can be divided into triangles and then construction 44 can be applied to each in sequence. 

Here I've used a combination of transferring lengths with my compass and only marking portions of some circles. By this point in the book there are also multiple ways to perform the constructions: for example, I've used a different method for constructing the parallel lines around the original figure here than in construction 44.

46. A square

 A return to simplicity... this was a relief after the last several parallelogram constructions of increasing complexity. It was nice to have relatively large areas to play with paint in again.

Some thoughts on the ending of Roe vs. Wade

 I was going to write you something well-organized about abortion. This is not that. Perhaps I will write something well organized in the future.

***

I heard the news on the TV at work while I was trying to get my patient to sit on the edge of his bed without help. (Explained badly, my job is to force elderly people to get up when they were perfectly comfortable lying down.) I was lucky in the timing of the decision, in that it was a reasonable time to take a lunch break. After finishing the session, of course.

***

Last summer I had an abortion at seventeen weeks. I wanted to be pregnant—I still want to be pregnant. Perhaps because of this, the experience is still fresh, something I think about many times a month, sometimes many times a day. 

***

After lunch, I went to get another patient for our session. My patient, an 81 year old white man, was sitting in the unit common area, watching Elizabeth Warren speak over a backdrop of protest footage. I asked him to come to therapy, and he told me that he couldn't leave. He needed to witness the protest. It was an important day, too important to leave. He cried when he looked back at the television. I asked if he needed anything, and he said "to watch this." I left to get another patient. 

I spent my time with that next patient trying to redirect my attention; I wanted to cry too, if not actually to watch the protest. It felt unfair that my patient was able to put other things on hold to attend to the political moment, while I had to move on, to act like I was fine. I considered calling out sick for the rest of the day, but I knew I could stay, and I knew no one else would see my patients if I didn't. We're understaffed already.

***

I'm glad I had the abortion—I want a child with a good chance at life. My doctor told me—and I believe her—that even if I carried my pregnancy to term, it would not end that way. That the ultrasound showed I would have a baby whose brain hadn't separated into two hemispheres, such a basic fact of our functioning as humans that it is shared with all vertebrates, from fish to birds to mammals. That outside my body, the baby might live for hours or days. Genetic testing performed after the abortion suggested that this view was optimistic if anything. 

***

Before, I thought about the retort that "the pro-life movement only cares about life before birth" as relating to a wide variety of policies, but nothing specific. To paid parental leave, or the lack thereof. To expanded access to healthcare, or the lack thereof. To concern or lack of concern for maternal health, for educational access for all children, for gun control, for policies that keep children and adults fed and housed and safe. 

I still think all that resonates with the actions of many politicians, pundits, and voters. But it misses something crucial.

***

I think about the life my baby would have had. It's plausible they were healthy and thriving—in the environment of the womb. We can't know what, if anything, they felt. But their life before birth was, almost literally, the only life they ever could have had. How can we consider a few hours of failing to adapt to an alien environment a life worth living? 

The only way my baby could be fed, housed, and safe was in my womb. This made theirs the perfect life to care about, for some people. 

***

By the time I got home, I was grateful for my patient who insisted on sitting in front of the television crying. I appreciated the compassion for others in his emotional response to a decision that will likely have no personal impact upon him. I remembered that he was doing all he could do to be in solidarity with the protesters—that a man who can't operate his television remote isn't going to be able to call his congressional representatives, to show up in person at a protest, to donate to abortion funds. 

If every 81 year old white man in this country cared as much for other people, for people unlike them, as this one does, we would not be in this mess. 

***

I don't have a nice ending to this piece. I want to say that I will write more, that I would be happy to discuss more one on one. But the truth is, I may not write more. I would be happy to talk, if this topic feels as personal to you as it does to me. If it doesn't, I might want to talk, or I might want to just make it through the day. I don't know.

some experiences with one way masking

 Right now, local and national guidance on wearing masks to prevent covid transmission is changing quickly. The omicron spike has mostly passed in my local area, and there's a mix of people trying to get back to "normal" and people feeling concerned that a rush to remove restrictions will lead to a new spike in caseloads or in their personal risk exposure.

Meanwhile I spent the omicron spike working in healthcare, in a setting where our covid patients were mostly not wearing masks, and at times made up about 50% of our caseload. 

To the best of my understanding, neither I, nor any of my closest colleagues, contracted covid at work this winter. I know I didn't contract it at work, because I didn't contract it at all—I've been PCR tested twice weekly with consistent negative tests, as part of our surveillance testing at work. (A PCR test is so sensitive that it can stay positive for some time following an infection—patients and staff with a positive test are exempt from our surveillance testing for 90 days, because it is presumed their positive tests will reflect the known infection. For this reason, the specific day of testing is not as important with PCR as with antigen (rapid) tests.)

I work in a rehab department in a skilled nursing facility. In my building, the rehab team has tended to have better compliance with personal protective equipment guidelines than other teams, and in some cases our masking behavior has been more conservative than required by guidelines. For example, several of us on the rehab team were wearing N95s for all patient contact prior to our facility implementing a requirement for all staff to wear N95s for all patient contact this January. I would guess there are two main factors here: rehab staff are likely more educated than facility staff on average due to licensing requirements, which correlates to beliefs around the importance of mask wearing, and rehab staff have more separation of times when we are and are not around patients compared to nursing staff in particular, making it easier to wear good masks properly whenever we're around patients. 

The variety of N95 most people use at my workplace.

For myself and my closest colleagues, this means that the majority of our at-work covid exposure took place when we were wearing fit-tested N95s, eye protection, gown, and gloves. However, the other person in the scenario was usually not taking any transmission prevention steps—we were treating covid positive patients in their rooms, and most were not wearing masks. We were in close contact with covid patients, helping them eat, drink, or move their bodies. Patients might cough or sneeze without covering their mouths. It is not possible to socially distance from someone if your job is to help them stand up safely.

I do have some colleagues from the rehab department (2 out of 6 team members) who contracted covid this winter. In both cases, they had other family members who were sick and tested positive for covid before they did, with known exposures from their own daily activities before developing symptoms. Neither was trying to isolate from the symptomatic family member(s) at home, while both were being especially careful at work just prior to testing positive, because of their home exposure. It is possible that these colleagues actually contracted covid from their masked exposures to patients at work, but I think it's much more likely that unmasked close contact at home was responsible. 

I also know, of course, many staff and patients who presumably (or certainly, in the case of some patients) did contract covid at our facility. Because covid was so prevalent this winter, it's hard to know where all these infections came from and how they were passed around. Some seem very likely to have arisen from patient-visitor or patient-patient contact, which is usually unmasked on both sides. (Regardless of guidance issued to visitors, they tend to remove masks once in a patient room—and no one wears a mask all the time around their roommate!) Some seem very likely to have passed from staff to and/or from patients, with staff who were wearing a surgical mask or an N95 and patients who were mostly unmasked. 

Besides mask-related behavior, it is worth noting that vaccination rates are very high at my workplace—staff have been required to be fully vaccinated for covid since August, and everyone in the rehab department was boosted before the omicron wave arrived at our facility.

Of course, my anecdata should not substitute for real data in assessing the risks and benefits of various policies around masking. But stories are helpful for calibrating our feelings and risk tolerances in ways that numbers usually are not. And that's my story, for now. 

Understanding the Transition of Polio from an Endemic to an Epidemic Disease

One of the diseases covered in physical therapy education to a disproportionate extent from its current prevalence is polio. This is due to its role in the history of the profession—which itself has dubious clinical relevance. However, the story of polio is itself quite interesting, as we have documentation of its transition from an endemic to an epidemic disease, to one which no longer circulates in much of the world. There is a well-known story here about the triumphs of vaccination, and a less well-known story about how public health experts began to appreciate the burden of disease polio presented only as a society with less polio, or without polio, became imaginable. The following is a paper I wrote in 2019, as part of a writing portfolio requirement.  

Polio is now a rare disease, nearing global eradication in the wild as a result of the work of the Global Polio Eradication Initiative. As of mid-2018, the Initiative reported that wild poliovirus was still endemic to only three countries, Nigeria, Afghanistan, and Pakistan, with Afghanistan and Pakistan treated as “a single epidemiological block” due to repeated transmission of the virus across their shared border (7). This is especially remarkable given that poliovirus is commonly believed to have been endemic in most human populations up until the late 19th and early 20th century, with most people being infected within the first few years of life (De Jesus; Nathanson and Kew 1214; Horstmann 80). Perhaps equally remarkable is the fact that polio became a prominent subject of medical concern only as it ceased to reliably infect members of the population at early ages, transitioning from an endemic to an epidemic disease (Nathanson and Kew 1214; Horstmann 80). How is it that a disease which reliably infected nearly an entire population early in life could be described as “not uncommon” (Horstmann 80), while that same disease, transmitted slightly less reliably, led to large-scale public concern and public health responses including vaccine trials involving over a million school-age children (Meldrum 1233)?

Polio’s unique pathophysiology played a key role in this transformation. One crucial aspect of the disease is that while its characteristic symptoms are neurological, most typically weakness or paralysis in one or more limbs, it is now known that around 96% of non-immune children infected with poliovirus exhibit either no manifestations of disease, or vague symptoms that could be attributed to any number of viral infections, and fewer than 1% experience lasting paralysis (U.S. Centers for Disease Control and Prevention, “Poliomyelitis”). Prior to the development of modern medical technology and modern models of illness, it was not possible to track the spread of a disease when so few infected individuals exhibited identifiable symptoms. Even now, the infrequency with which infection with poliovirus produces clinical symptoms leads to challenges in monitoring and tracking the spread of the disease—one case of characteristic paralysis may indicate hundreds of undetected infections, and workers for the Global Polio Eradication Initiative track the presence of the wild virus through environmental surveillance in addition to through individual reports of infection (Global Polio Eradication Initiative 7–12).

Another aspect of polio’s pathophysiology that decreased its visibility during the endemic period is that a small number of exposures can confer long-term immunity. Unlike some other viral illnesses such as influenza, polio only has a small number of antigenic types: exposure to a strain of polio from each of three groups is sufficient to produce immunity to all polioviruses (Horstmann 83). The duration of immunity produced either by exposure to wild poliovirus or by immunization is unclear, particularly with regard to effects on subclinical infection and transmission (Duintjer Tebbins et. al. 596). However, the pattern of disease leading to polio having been known as “infantile paralysis” prior to the epidemic period, with few or no cases being reported in older children and adults, suggests that typical levels of immunity in those populations under endemic conditions are enough to prevent development of characteristic symptoms (Nathanson and Kew 1214; Horstmann 80). Additionally, a study of an isolated Inuit population where polio was not endemic found that antibodies to specific strains persisted over spans of 40 or more years (Horstmann 83). Thus, an individual’s chances of developing characteristic symptoms of polio where it is endemic do not increase with their frequency of exposure to the disease, but are concentrated in the first exposure to each immunologically distinct strain.

The lasting immunity conferred by exposure to polio viruses means that as polio shifted from an endemic to epidemic disease, it began to affect more age groups. In addition to any corresponding differences in the perceived social impact of the disease, the immediate, physical effects of the disease while it is endemic (and thus primarily affects infants) are potentially less severe for two reasons. Firstly, infants who are breastfed, as was typical prior to the 20th century, receive some immune benefits. Therefore, if a first polio infection occurs while a child is mostly or exclusively breastfed, it may allow the child to develop their own lasting immunity, while still being partially controlled by an immune response mediated by antibodies received from the mother (Nathanson and Kew 1214). Secondly, while the evidence for age-specific effects of polio is limited, some evidence suggests that among individuals with paralytic polio, older individuals more frequently experience more severe or life-threatening forms of paralysis (Nathanson and Kew 1217-1218, Nielsen 181). For example, one study of case reports from Sweden found that 23.5% of cases in individuals over the age of 25 led to death, compared to only 4.5% of cases in individuals under 3 years of age (Nathanson and Kew 1217). Another study, looking at cases of polio leading to hospitalization in Denmark, found that for each analyzed age group with patients 30 years and older, over 10% of cases led to death, while no age group with patients under 25 years of age had over 5% of cases leading to fatality (Nielsen 183).  

None of this, however, should be taken to mean that polio presented a low burden of disease in areas and times when it was endemic. Rather, its effects, although significant, were continual and expected, allowing them to exist in plain sight even while they were rarely noted as a coherent and potentially avoidable pattern. The effects of polio were continual in the sense that polio infections occurred relatively continuously over time, although possibly with some seasonality in temperate climates (Nathanson and Kew 1214, 1218; Horstmann 80). They were also continual in that paralysis induced by polio is frequently permanent, so that affected individuals continued to display the signs of the disease throughout their lives; as a result others in their community might come to expect that permanent paralysis was one potential outcome of a childhood illness. In the growing tradition of Western medicine, this came to be called infantile paralysis (Horstmann 80). We don’t have records of how common infantile paralysis was in Europe or North America prior to polio’s emergence as an epidemic disease, but during the 1970s and 1980s many surveys were conducted to try to establish the prevalence of lingering paralysis characteristic of polio in areas where polio was still endemic, in part in order to understand the benefits of vaccination in those areas (Bernier S371). These surveys looked specifically for gait abnormalities in children, therefore potentially missing other presentations of polio-related paralysis, such as flaccid paralysis of an upper extremity, or cases of polio that ended in death (Bernier S371). Still, they found prevalence rates ranging from under 1 child in 1000 affected, up to 25 children in 1000 affected; rates around 5-10 children per 1000 were common (Bernier S371-S373). For comparison, modern studies find that cerebral palsy, the current most common cause of childhood motor disability, affects under 5 children per 1000 (U.S. Centers for Disease Control and Prevention, “Data and Statistics for Cerebral Palsy”). 

Despite these continual infections, and continual reminders of past infections, it is clear from the scientific record that researchers and policy-makers in the late 1970s and early 1980s were surprised by mounting formal evidence of the high burden of disease polio presented in areas where it was still endemic. (By this time, polio was no longer circulating as a wild virus in the United States, or in an increasing number of other countries (Nathanson and Kew 1220). ) Writing in 1980, LaForce et. al remarked that “Until recently poliomyelitis has not been considered as a major public health problem in developing countries,” then went on to note that recent studies had found incidence rates for polio in those countries comparable to those seen in the United States prior to the development of polio vaccines (609).  Indeed, only two years earlier, and with access to some of the studies reported on by these authors, a World Health Organization publication on the benefits of polio vaccination discussed vaccination in areas where polio was endemic almost exclusively as a public health program seeking to “anticipate and forestall the epidemic phase of poliomyelitis” (Melnick 25). The publication included recommendations to test populations for prior exposure to polio viruses in order to determine the correct time to begin vaccination, rather than waiting for epidemics to occur, underscoring the perception that polio began to present an unacceptably high burden of disease only when it began to appear as an epidemic disease (Melnick 25-26). 

The policy transition in response to the mounting evidence, however, was swift; by 1984, Bernier noted that results of surveys on infantile paralysis had influenced several countries’ decisions to begin vaccination programs (S371). And in 1988 the World Health Assembly announced a goal of eradicating wild poliovirus by the year 2000 (Nathanson and Kew 1220; DeJesus). This policy required a coordinated effort to address polio not only where it was (or, increasingly, had been) epidemic, but also in the endemic areas where it had previously been seen as a low priority. Indeed, work in these areas has been extensive, and as polio is eliminated in these areas, the Global Polio Eradication Initiative plans to transition the infrastructure built by responses to polio to address other public health concerns in “priority countries” where polio remained endemic longest and polio eradication infrastructure is most developed (18).

Due to its unique pathophysiology, polio was a challenging disease to understand before and during the early part of the 20th century. It became visible as a serious public health issue primarily as it began to operate as an epidemic disease, affecting larger numbers of individuals in a short span of time and affecting older individuals. This rise in visibility did not correspond neatly to an increasing burden of disease. However, it took until late in the 20th century, once polio had already been eradicated in many of the areas where it first rose to prominence, for scientists and policy makers to fully understand the burdens that polio causes in areas where it is endemic. This delay resulted from incomplete reporting, along with the challenges of tracking a disease in which the majority of cases do not include the symptoms thought of as characteristic of that disease. That careful studies eventually exposed the burden of polio on communities where it is endemic may be in part responsible for the current state of near-global eradication of wild poliovirus, a state made possible by intensive worldwide vaccination campaigns.

Works Cited

Bernier, Roger H. "Some Observations on Poliomyelitis Lameness Surveys." Reviews of Infectious Diseases, vol. 6, no. Supplement 2, 1984, pp. S371-S375. doi.org/10.1093/clinids/6.Supplement_2.S371.

De Jesus, Nidia H. “Epidemics to Eradication: The Modern History of Poliomyelitis.” Virology Journal, vol. 4, no. 70, 2007. doi.org/10.1186/1743-422X-4-70.

Duintjer Tebbens, Radboud J., et al. "Expert Review on Poliovirus Immunity and Transmission." Risk Analysis, vol. 33, no. 4, 2013, pp. 544-605. doi.org/10.1111/j.1539-6924.2012.01864.x. 

Global Polio Eradication Initiative. “Semi-Annual Status Report January to June 2018.” World Health Organization, 2018. polioeradication.org/wp-content/uploads/2018/11/2018_January-June_SemiAnnualReport_POL_highresolution.pdf. 

Horstmann, Dorothy M. "The Poliomyelitis Story: A Scientific Hegira." The Yale Journal of Biology and Medicine, vol. 58, no. 2, 1985, pp. 79-90. PubMed Central, www.ncbi.nlm.nih.gov/pmc/articles/PMC2589894/.

LaForce, F. M., et al. "Clinical Survey Techniques to Estimate Prevalence and Annual Incidence of Poliomyelitis in Developing Countries." Bulletin of the World Health Organization, vol. 58, no. 4, 1980, pp. 609-620. PubMed Central, www.ncbi.nlm.nih.gov/pmc/articles/PMC2395925/.

Meldrum, Marcia. "“A Calculated Risk”: The Salk Polio Vaccine Field Trials of 1954." BMJ, vol. 317, no. 7167, 1998, pp. 1233-1236. PubMed Central, www.ncbi.nlm.nih.gov/pmc/articles/PMC1114166/.

Melnick, Joseph L. "Advantages and Disadvantages of Killed and Live Poliomyelitis Vaccines." Bulletin of the World Health Organization, vol. 56, no.1, 1978, pp. 21-38. PubMed Central, www.ncbi.nlm.nih.gov/pmc/articles/PMC2395534/.

Nathanson, Neal, and Olen M. Kew. "From Emergence to Eradication: The Epidemiology of Poliomyelitis Deconstructed." American Journal of Epidemiology, vol. 172, no. 11, 2010, pp. 1213-1229. doi.org/10.1093/aje/kwq320. 

Nielsen, Nete Munk, et al. "The Polio Model. Does it Apply to Polio?." International Journal of Epidemiology, vol. 31, no. 1, 2002, pp. 181-186. doi.org/10.1093/ije/31.1.181. 

United States, Centers for Disease Control and Prevention. “Poliomyelitis.” Epidemiology and Prevention of Vaccine-Preventable Diseases. 13th ed. Edited by Jennifer Hamborsky, Andrew Kroger, and Charles (Skip) Wolfe. Washington D.C. Public Health Foundation, 2015. www.cdc.gov/vaccines/pubs/pinkbook/polio.html. 

---. “Data and Statistics for Cerebral Palsy.” Centers For Disease Control and Prevention.  9 Mar. 2018. www.cdc.gov/ncbddd/cp/data.html. Accessed 15 July 2019.

Geometry in Motion

On a visit to the Exploratorium this weekend, we were hanging out near the harmonograph, watching kids make really cool pictures, when the Cinema Arts Saturday Cinema program started just across the way. Since the program for this week was titled Geometry in Motion, I told the people I was with that I was going to watch. We all enjoyed the program, and each had a different favorite film.  Since most of the five films shown are available to watch for free online, I thought I'd share them with you. 

The first was Symmetry from the 1961 IBM Mathematics Peep Shows by Ray and Charles Eames. This one is available in a free iPad app from IBM, but unfortunately I haven't been able to find it elsewhere. It is a very short introduction to formal definitions of symmetry, defining the degree of symmetries of an object as the number of different ways it can be placed in a box that fits it exactly. This made me really want a dodecahedron box, and a dodecahedron to fit in it exactly! I'm not sure I have the construction skills and equipment to make that happen, though. I was also entertained by the cameo appearance of group multiplication tables at the end of this very short film; the reference without explanation felt similar to the jokes in kids' movies that are really aimed at parents. 

The second film, Aesthetic Species Maps, is available to watch online from its creator, David C. Montgomery. This film set the tone for the rest of the program, which consisted of basically wordless art films with strong ties to geometry. Each segment was assembled from still images taken of many specimens of the same type of plant of animal, in approximately the same orientation. For me the first segment was the most interesting, because you could clearly see variation of the degree of symmetry between specimens. 

The next film we saw I can't find anywhere online. It was Eights by Seth Olitzky, and it used computer animation to play with symmetrical figures in a way reminiscent of kaleidoscopes or screensavers. I stared at a lot of screensavers around the time the film was made, though, and it was more interesting than any of them. 

After that we watched a more modern take on computer animated geometry shorts, Nature by Numbers. Etérea Estudios, the creator of this film, has a webpage for it with not just the movie, but also still images from it and a page explaining the mathematics behind it. I'd seen a lot of the material towards the beginning of the video before, whether as the countless fibonacci spirals that had filled my notebooks since we learned to make them in 6th grade, or more recently in this series of videos about being a plant by Vi Hart. At the end, though, I had a big surprise, as a Voronoi tessellation turned a grid of sunflower seeds into a dragonfly's wing. This was my favorite film, so I'll embed it here, too, though if it makes you curious about things, Etérea's website is a great place to start.

The final film, Rectangle & Rectangles by René Jodoin, is available from the National Film Board of Canada. This one didn't directly explore a mathematical concept as much as some of the other, but it was one of the more fascinating films to watch and is even better to watch online.  It uses strobe effects and a variety of growing and shrinking rectangles to create an interesting and somewhat confusing visual experience. The Exploratorium loves to create a "Why is this happening?" moment with its exhibits, so I can definitely see why they would show this film! In the theater, I did a lot of fast blinking to try to understand the technique behind the visual effects. I'd love to have a frame-by-frame replay option for this, but the freedom to pause and rewind at will makes the streaming version nearly as good. 

Thus ended the film series, and thus ends this post, although we saw plenty of other fascinating things at the museum. I hope you enjoyed these films, and I hope you get a chance to visit the Exploratorium and see some of the other stuff for yourself.

More on statistical significance and small samples

In yesterday's post, I pointed out that for small sample sizes and cases where success is unlikely, standard tests of statistical significance can use even one observation of "success" to reject the null hypothesis. This is jarring, since statistical significance sounds important and official, like it should be more rigorous than our intuitions about what the data says. So what's going on here, and when do we have to worry about it?

We'll work through this with an example.  Suppose we conduct a poll of five wizards and five muggles, and all the wizards and four of the muggles eat at least one piece of chocolate per day, while one of the muggles is on a diet and eats no chocolate at all.

Some Definitions, Applied 

The null hypothesis is the hypothesis that every observation is being drawn from the same distribution, or that the treatment group has the same distribution as the control group. In this case, the null hypothesis might say "Wizards and muggles eat the same amount of chocolate,"  or perhaps "The same proportion of wizards and muggles eat (or don't eat) chocolate." It's good to be precise about what you are measuring; we'll use the second formulation this time.

In the case of our survey about chocolate, we might be tempted to conclude that the null hypothesis is wrong, because more of the wizards eat chocolate.  But first we should decide whether we've just come to the conclusion by chance-- after all, we've only talked to ten people total.  This is where statistical significance comes in.

To determine whether our results are statistically significant, we first must decide how willing we are to reject the null hypothesis when it is actually true. That is, suppose that the Actual Real Truth is that the same percentage of wizards and muggles abstain entirely from chocolate. How willing are we to conclude that they don't? It's common to accept a 5% or 1% chance of rejecting the null hypothesis wrongly, though some disciplines are okay with even a 10% chance. Whatever chance we accept, we'll be looking for statistical significance "at that level", for instance, statistical significance at the 5% level, otherwise known as statistical significance with p < .05.

Checking for Statistical Significance (Through Simulation)

To show that our results are statistically significant at the 5% level, we have to show that if the null hypothesis is true, we would expect to get results as extreme as ours less than 5% of the time, if we repeated our experiment many times.  To do this, we don't actually repeat the experiment many times-- remember, we don't know if the null hypothesis is true in the real world. Instead, we can simulate repeating it many times in a computer world where the null hypothesis is true, or we can use analytical methods to find out what would be the results of carrying out such simulations.

We use what we know in order to make the null hypothesis more precise so that we can carry out our simulations. The null hypothesis that we started with in our example just said that wizards and muggles are equally likely to eat chocolate, but now we have data that says that 9 of 10 people surveyed consume chocolate. So the best we can do is to assume that the null hypothesis is true, and 90% of people eat at least one piece of chocolate per day and 10% eat no chocolate at all. Since the null hypothesis says there is no consumption difference between wizards and muggles, in our simulation these figures are the same for both groups.

One way to simulate this situation is simply to erase the wizard/muggle data from our observations and replace it randomly, so that a random five observations are from "wizards" and  the other five are from "muggles". If we simulate this way (also known as sampling without replacement) then every time either the wizards or the muggles appear to eat more chocolate, to an extent exactly as extreme as was seen in our original sample, because the one abstainer is always either a wizard or a muggle.  We would conclude that our findings are not statistically significant.

Another way to simulate it is to sample with replacement. This time, we will make five "wizard" observations and five "muggle" observations, each with a 90% chance of eating chocolate and a 10% chance of abstaining.  We will get some cases when exactly one of our observations is an abstainer, like in the original sample, and others where none are, or where several people abstain.  The situation is more complicated than the previous case, and much of the time we do get samples showing exactly the same consumption of chocolate for wizards and muggles, but over half the time, we don't. Again, our single observation difference is not considered statistically significant. This should agree with our intuition that we need a larger sample size if we want to find a difference between the populations that was not immediately obvious.

Randomly sampling with replacement 10,000 times, the most common situation is for wizards and muggles to have the same number of abstainers in our sample, but this case makes up fewer than half of our samples. 

Tweaking the Example

Things get more complicated if we have much more data about one population than the other. For instance, suppose the five wizards we surveyed were all the wizards in the world. Then it is inappropriate to suppose that we could have sampled from a population that includes wizards who do not eat chocolate; all the wizards do eat chocolate. And one of the muggles doesn't. There is no way the null hypothesis could be true now.  This is not just statistical significance, which says that if the null hypothesis is true our results are unlikely. Our study actually disproved the null hypothesis, which is much stronger. (And, practically speaking, almost never the case.)

The example above is a special case of a more general situation in which much more is known about the distribution of one population than about the other. Another example of such a situation is the case in which our study was carried out by wizard researchers who knew very few muggles, so that they hadn't surveyed 5 wizards and 5 muggles, but 5000 wizards and 5 muggles. Let's say they found 50 wizards who did not eat chocolate and, as before, 1 muggle who did not eat chocolate. They can do either of our tests above; let's see what happens.

If they sample without replacement, they're looking at 5005 total observations, of which 51 are chocolate-abstainers. Randomly assigning five of these observations to be "muggles" 10,000 times, in 9,511 cases I got no muggles who were chocolate abstainers. This is (just) over 95% of my samples, so in fewer than 5% of cases, I got a result as extreme as the wizard researchers' original result.[1] From one muggle chocolate-abstainer in their sample, they could conclude statistical significance at the 5% level.

Sampling with replacement is technically easier and gives similar results. In 9,526 of my 10,000 repetitions, no muggles were chocolate abstainers. Once again, this would allow the researchers to conclude a statistically significant difference at the 5% level based on the one muggle chocolate abstainer in their original sample.  This doesn't agree so well with our intuitions; although the researchers now have a sample of wizards that seems large enough to determine their chocolate consumption habits with some precision, the sample of muggles still feels awfully small for most purposes.  

A Broader View

By having a lot of information about the proportion of wizards who eat chocolate, the researchers in the last example are able to use very little information about the proportion of muggles who do to conclude that the two populations are different. This isn't all bad, and sometimes such techniques are necessary. But their results, even though statistically significant by both methods described, aren't as solid as that makes them sound. The small size of their sample of muggles makes it especially easy for them to accidentally get a sample that is not representative of the total muggle population by bad sampling methodology or for some other reason.  Imagine polling the next five people you see about some issue, and then imagine polling the next 1000 people you see. Both these polls use the same method to get a non-random sample of the population of your area, but the first poll will likely have respondents that are much less diverse, and a much less representative sample, simply because there are so few of them.  

This example of researchers who have an easier time finding subjects in one of the groups they're studying than another is not purely fictional; there are many reasons it may occur in reality. An intervention might be much more expensive or difficult than the appropriate control procedure and than the follow-up data collection, so that it is easier to fund or conduct a study with a large control group and small treatment group than with two groups of equal sizes. Researchers might be trying to compare a group that is a small fraction of the population with a group that is a much larger fraction of the population.  In the case of the study I was working on in yesterday's post, with our original study design we expected to have trouble finding the targets of our intervention in order to follow up with them and therefore expected to survey a much larger number of people in the control group (in that case, the population from which we had originally drawn targets for our intervention) than from the treatment group.  

In any case, if a study design allows for statistical inferences to be drawn from a small number of surprising observations in the treatment group, caution is warranted.  Use common sense as a back-up check on the meaningfulness of statistical significance, just as you use statistical significance as a back-up check on the meaningfulness of your intuitive reaction to the data.

Code used in this post is available on github.

[1]: In this case, any result but the most likely result. Normally we'd need to check both for cases where the muggle group has too many abstainers and for cases where it has too few, but since zero is the most likely number for it to have under the null hypothesis, in this case we do not have to be concerned with there being too few.

Statistical power at small sample sizes

Recently I was working on a team to design a study where we expected to find relatively few examples of the phenomenon we were studying. While it wasn't out of the question that we'd find an example in the control population, we expected to find very few examples there, and only a few more in the treatment group. I made a chart to help us understand how large a sample size we'd need to decide our intervention was making a difference, and it looked something like this:

Each line on this chart shows the probability that we would find our treatment had an effect significant at a given level, depending on the size of our treatment group. But why are there sudden drops in the probability that we'd find a statistically significant effect? And why, for so many sample sizes, does it not appear to matter whether we're seeking an effect significant at the 5% or 10% level? 

The answer is that this is the wrong chart to draw, in this situation. If we expected only 1 in 2000 observations from the control group to be a success, and 1 in 500 observations from the treatment group to be a success, a simple application of the definition of significance levels gives the graph above. But besides looking weird, it is misleading. Notice where it implies that we'd have about a 30% chance of concluding our treatment works, with a sample size of about 225? Well, that's the probability we'd get at least one success in 225 trials.

This chart, where each line shows the probability of getting at minimum a given number of successes, depending on the size of the treatment group, is much easier to read, and much more illuminating. Even if a single success in the treatment group would technically be statistically significant, our study would be much more persuasive if we chose a sample size that allowed us to expect to find two or three successes, at a minimum. 

Overlaid, the charts are even more useful. They show the probability of getting a certain number of successes at a given sample size, together with the level of statistical significance that would imply. And they answer our questions about the first chart-- the drops come when a particular level of statistical significance requires us to find an additional success, and the times when two levels of significance are equally likely are times when they are requiring us to find equal numbers of successes.

Code used in this post is available on github.