Kaktovik numerals

Kaktovik numerals (above) are a base-20 numeral system used by the Iñupiat people in northern Alaska. They were added to the Unicode standard last year.

Like a number of cultures around the world (e.g. ancient Maya and Dzongkha), the Iñupiat count in 20s. So, in Europe, did the ancient Celts, and their base-20 system survives in English in the phrase “four score and seven years ago” and in French in numbers like vingt = 20 and quatre-vingts = 80.

The Kaktovik numerals were invented in the 1990s by nine middle school students at Harold Kaveolook school in Kaktovik, Alaska. Like Maya numerals and Babylonian cuneiform numerals, they have internal structure, so that, for example, 17 is written as 3 × 5 + 2. This structure matches the spoken language, in that, for example, the number 17 is spoken as akimiaq malġuk (“fifteen-two”). Using extended European digits (e.g. 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F, G, H, I, J) would not match the language in the same way – and, as noted by the students, could cause confusion.

Wikipedia claims benefits for arithmetic if Kaktovik numerals are used, but I have seen no studies on that (improved test scores for the school in 1997 may simply reflect the talent of this particular cohort of nine students). It seems like the system is no longer used in mathematics classrooms in Alaska, but is used for teaching numbers in the Iñupiaq language (where it no doubt emphasises the fact that mathematics existed in their culture before European settlement). Detailed records of the origin of the system were sadly destroyed in a fire at the school in 2020. However, if the numerals are made available on mobile phones, we may see a resurgence of interest in the scheme.

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Mass Shootings in the US

Tragically, we have had another mass shooting in the US. The chart above summarises deaths from such events (data collated by Mother Jones). It is clear that the problem has been getting worse since about 2005 (statistically significant at p < 0.00001).

Social factors appear to be blame, since there has been no signficant change in the availability of weapons in that time. Those social factors might include mental health policy, education policy, social media, video games, drugs, the decline of religion, media coverage of past shootings, etc. It seems to me that serious study is urgently required. Some things we do know: psychiatrist Ragy Girgis suggests:

“With exceptions, many of these [perpetrators] tended to be younger males who were empty, angry, and nihilistic, felt rejected by society, were socially, occupationally and/or academically unsuccessful, and blamed society for their failures. These individuals tended to have very fragile egos and were highly narcissistic, feeling they were much more special than they actually were and deserving of fame and notoriety. They harbored a strong desire for this notoriety and infamy. Committing a mass shooting instantly produces these results in today’s culture.”

In the Mother Jones dataset (for 1982 onwards), 13 states have never had a mass shooting (Alaska, Delaware, Idaho, Maine, Montana, New Hampshire, New Mexico, North Dakota, Rhode Island, South Dakota, Vermont, West Virginia, and Wyoming), while in 13 other states, the chance of dying in a mass shooting exceeds 0.1 per million per year:

State	Total Fatalities	Annual Deaths per million
California	175	0.11
Nebraska	9	0.11
Oklahoma	19	0.11
Wisconsin	28	0.12
Washington	37	0.12
Hawaii	7	0.12
Texas	151	0.12
Florida	126	0.14
Virginia	53	0.15
Colorado	53	0.22
Connecticut	41	0.27
DC	12	0.43
Nevada	63	0.48

The relevant social factors are therefore not uniform across the United States. The map below shows the mean annual death rate per million for mass shootings in each state (for 1982 to 2023, excluding Alaska = 0 and Hawaii = 0.12):

Edit: Ragy Girgis, quoted above, notes that perpetrators tended to be “occupationally and/or academically unsuccessful.” Consequently, state unemployment rate is a statistically significant risk factor (p = 0.0148):

Even more significant (p = 0.0046) is the correlation with the Social Support Index from the US Joint Economic Committee Social Capital Project. Better social support helps to reduce the risk of mass shootings.

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Pi Day once more!

In honour of Pi Day (March 14), the chart shows six ways of randomly selecting a point in a unit disc. Four of the methods are bad, for various reasons.

A. Midpoint of random p, q on circumference

p = (cos(𝜃₁), sin(𝜃₁)) is a point on the circumference

q = (cos(𝜃₂), sin(𝜃₂)) is another point on the circumference

x = ½ cos(𝜃₁) + ½ cos(𝜃₂) and

y = ½ sin(𝜃₁) + ½ sin(𝜃₂), for random 𝜃₁ and 𝜃₂, define their midpoint.

B. Random polar coordinates

x = r cos(𝜃)

and y = r sin(𝜃), for random angle 𝜃 and radius r ≤ 1. This gives choices biased towards the centre.

C. Random y, then restricted x

Random y, followed by random x in the range −√(1−y²) to √(1−y²). This gives choices biased towards the top and bottom.

D. Random point on chord in A

Similar to A, but x = a cos(𝜃₁) + (1−a) cos(𝜃₂)

and y = a sin(𝜃₁) + (1−a) sin(𝜃₂), for random 𝜃₁ and 𝜃₂ on the circumference of the circle and random a between 0 and 1. This gives choices biased towards the periphery.

E. Random polar with sqrt(r)

Similar to B, but x = √r cos(𝜃)

and y = √r sin(𝜃), for random angle 𝜃 and radius r. The square root operation makes the selection uniform across the disc.

F. Random x, y within disc

Random x and y, repeating the choice until x² + y² ≤ 1. This is uniform, and the selection condition restricts the final choice to the disc.

Oh, and here are some Pi Day activities.

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The train crash in East Palestine, Ohio

A great deal has been written about the train derailment in East Palestine, Ohio. The preliminary NTSB report is one of the few solidly factual responses. One important question has been: what happens when vinyl chloride burns? Theoretically, in the presence of enough oxygen, you get this:

More realistically, based on the above reaction and this paper, you also get carbon monoxide, black soot (carbon), and traces of phosgene:

Why was it burned? Because vinyl chloride, if it gets too warm, can spontaneously polymerise into PVC plastic. That reaction is heat-producing and can lead to an explosion. Since the fire following the derailment had heated the rail cars containing vinyl chloride to dangerous levels, authorities believed an explosion was imminent. A controlled burn was probably a rational decision at the time.

Hydrogen chloride (HCl) in the smoke cloud was probably the immediate threat resulting from the controlled burn (and the likely cause of dead birds), although the HCl would soon have been safely diluted by rain. Unburned vinyl chloride in the subsoil is probably the longer-term threat, and (I understand) the focus of cleanup efforts.

Why did the train derail? This (somewhat fuzzy) map and chart is my best guess at a timeline. As the train travelled east, hotbox detectors (HBDs) noted increasing wheel bearing temperatures on car 23. Some media reports suggest flames of burning axle grease were seen in Columbiana, Ohio. The HBD at East Palestine noted a temperature of 253°F above ambient, higher than the railway company’s critical threshold. The crew immediately began to further slow the already slowing train, at which point the faulty wheel bearing on car 23 failed catastrophically, triggering the derailment and fire:

The map at database.defectdetector.net suggests that there used to be hotbox detector near Columbiana, Ohio (at MP 60.8), and this would presumably have caught the fault in time to avoid a derailment. One wonders what became of that HBD. It’s a pity that we need to wait 18 months for the NTSB’s final report.

Meanwhile, the EPA has put all their air, soil, and water testing results online.

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The Austin airport incident

Above is a chart of altitude data for the recent near-miss of two aircraft at Austin-Bergstrom International Airport in Texas, involving a Southwest 737 and FedEx 767, which the NTSB is investigating. Data for the chart is from Flightradar24 (at 25 foot resolution, and not totally accurate because it is calculated from air pressure). See that link also for the story, or this tweet and this other tweet.

In the chart, the emergency go-around by the FedEx aircraft is obvious (it began about 5 seconds before minimum separation, when horizontal separation was about 650 feet). The temporary rise of Southwest to 25 feet appears to be an artifact. I estimate minimum separation as 187 feet. FedEx was about 0.58 nautical miles (1.08 km) away, on course to land, when Southwest started rolling.

The landing approach was CAT III ILS due to heavy fog. It seems to me that the clear-thinking FedEx pilots saved lives that day. A recording of the radio communication is here. Transcript is as follows (accurate to-the-second timestamps are not available):

• FDX1432: Austin tower, FedEx 1432 heavy passing 5.4 for that CAT III ILS 18L.
• Tower: FedEx 1432 heavy, Austin Tower, 18L RVR [Runway Visual Range] touchdown 1400, midpoint 600, rollout 1800, 18L cleared to land.
• FDX1432: Cleared to land 18L, FedEx 1432 heavy.
• SWA708: Tower, Southwest 708, we’re short of 18L and we’re ready.
• Tower: Southwest 708, Austin Tower, runway 18L RVR 1200, midpoint 600, rollout 1600, fly heading 170, runway 18L, cleared for takeoff, traffic 3 mile final is a heavy 767.
• SWA708: Okay, 170 cleared for takeoff, 18L, copy the traffic, Southwest 708.
• FDX1432: Tower, confirm FedEx 1432 heavy is cleared to land on 18L [hearing the message to SWA708 and seeing a potential problem].
• Tower: FedEx 1432 heavy that is affirmative, 18L you are cleared to land, traffic departing prior to your arrival is a 737.
• FDX1432: Roger.
• Tower: Southwest 708 confirm on the roll.
• SWA708: Rolling now.
• FDX1432: Southwest abort [seeing SWA708 on the runway in front of him at the limit of visibility in the fog].
• FDX1432: FedEx is on the go [go-around].
• Tower: Southwest 708, roger [apparently believing “abort” came from SWA708], you can turn right when able.
• SWA708: Negative.
• Tower: FedEx 1432, climb and maintain 3000 [feet], when able you can turn left heading 080.
• FDX1432: Left turn to 080, 3000, FedEx 1432 heavy.
• Tower: Southwest 708, you can turn left heading 170.

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COVID-19 and Vitamin-D

The chart above shows national Covid mortality against latitude of national capitals (open circles are for the Southern Hemisphere, solid circles for the Northern). The trend line in blue has a correlation of 0.50 (with p < 10⁻¹³). Countries further away from the equator are definitely reporting more Covid deaths.

It is possible that these numbers reflect under-counting in the tropics (although this is unlikely for Singapore = SG) and over-counting in wealthier countries away from the tropics (e.g. by reporting deaths of patients with positive Covid tests as Covid deaths, even if the actual cause of death is unrelated). However, it seems unlikely that under-counting and over-counting can explain everything here.

This paper in The Lancet notes that “It has long been clear that groups that traditionally exhibit vitamin D deficiency or insufficiency, such as older adults and nursing home residents, and Black, Asian, and minority ethnic populations, are the same groups that have also been disproportionately impacted by COVID-19. Additionally, increased time spent indoors due to strict lockdowns and shielding triggered concerns that some people might not obtain the necessary physiological levels of vitamin D from sunlight.”

My chart above is consistent with this: decreased sunshine away from the equator appears to increase Covid mortality, presumably due to vitamin D deficiency. This study in QJM notes, “vitamin D supplementation is effective in reducing COVID-19 severity. Hence vitamin D should be recommended as an adjuvant therapy for COVID-19.” Personally, I have been taking this advice for quite some time.

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Human embryology again

Returning to the topic of human embryology, here is a human fetal timeline for the first 16 weeks post fertilisation (obstetricians count from the LMP = last menstrual period, which adds about 2 weeks). It is a little disturbing quite how much scientific misinformation is being circulated in regard to the topic. False information is not conducive to honest debate, and is highly corrosive of the trust people have in professionals such as scientists (it’s also unethical on both religious and Kantian grounds). In particular, contrary to what some have suggested:

• the fetal or embryonic heart begins beating at about 21 days after conception, as any textbook will point out
• as the table shows, after a few weeks, the fetus or embryo is not “too small to see with the naked eye”

Fetal length data in the table is mostly from here. Except where indicated, linked images are subsequent to miscarriage or to surgery to resolve ectopic pregnancy, so may be distressing to some readers.

Week post fertilisation	Week post LMP	Fetal length	Image
1	3	0.01 cm / 0.005 inches	8–cell image
2	4	0.02 cm / 0.008 inches	–
3	5	0.1 cm / 0.04 inches	heart begins beating at 21 days
4	6	0.5 cm / 0.2 inches	image on flickr
5	7	1 cm / 0.4 inches	image on wikimedia
6	8	1.6 cm / 0.6 inches	image on flickr
7	9	2.3 cm / 0.9 inches	image on flickr
8	10	3.2 cm / 1.3 inches	–
9	11	4.1 cm / 1.6 inches	–
10	12	5.4 cm / 2.1 inches	–
11	13	6.7 cm / 2.6 inches	ultrasound image
12	14	14.7 cm / 5.8 inches	–
13	15	16.7 cm / 6.6 inches	–
14	16	18.6 cm / 7.3 inches	–
15	17	20.4 cm / 8 inches	ultrasound image
16	18	22 cm / 8.7 inches	–

Below (from here) is a chart of heart development:

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Are there more hurricanes nowadays?

Hurricane Ian approaches Florida (NOAA image)

Florida is counting the cost of Hurricane Ian (above). While we empathise with the people affected by this terrible tragedy, and wish them well, it’s appropriate for a science blog to reflect on what’s happening. The paper “Trends in Global Tropical Cyclone Activity: 1990–2021” by Philip J. Klotzbach, Kimberly M. Wood, Carl J. Schreck III, Steven G. Bowen, Christina M. Patricola, and Michael M. Bell (Geophysical Research Letters, 14 March 2022) is a good guide.

Klotzbach et al. find that global hurricane counts have in fact decreased since 1990. We are not seeing more hurricanes. Restricting attention to Category 4–5 hurricanes (see below) shows no significant change. This overall decrease seems to be driven by more frequent La Niña years, which are associated with fewer hurricanes in the huge Pacific Ocean, and more in the smaller North Atlantic Ocean, giving a reduced total.

Global damage has increased significantly, however, due to having more people and more infrastructure in hurricane-prone coastal areas. It seems difficult to stop people from moving to Florida to live, so there is a clear need to make buildings and infrastructure more hurricane-proof. This document from the government of Queensland (Australia) offers some useful tips for home construction.

Figure 3c from Klotzbach et al. Category 4–5 hurricanes in six tropical cyclone basins during 1990–2021.

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Explaining Doppler ultrasound

Recently, I posted something about fetal heartbeats. This seems to be a hot political topic at the moment in the US. Many people don’t seems to understand that, as I noted in my earlier post, the human fetus has a functional (though not yet fully developed) heart from about 21 days after conception.

There seems to be an even greater confusion about the physics of Doppler ultrasound machines. These do not “detect electrical signals,” as has been suggested, but are essentially “speed cameras” for blood (although they use sound rather than radio waves). They detect movement, because movement causes a change in sound frequency, through the Doppler effect. When the heart contracts and the blood is moving fastest, the velocity signal is greater. The pulsing of the resulting velocity signal matches the pulsing of the heart, and can be made audible, although the timbre of the resulting heartbeat sound will not be identical to what a microphone would detect.

The graphic below summarises the operation of the device, at three moments in time (A, B, C). The sound signal sent and received is shown in blue. At the bottom of the graphic is the velocity signal, with the three moments A, B, C marked.

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Fetal heartbeat and political debate

Following the recent Dobbs v. Jackson Women’s Health Organization decision in the US, that nation is struggling with two moral/philosophical questions:

• Is the unborn human fetus a person, and if so from when?
• Does the unborn human fetus deserve legal protection, and if so from when?

These are independent questions – a puppy or kitten is not a person, but nevertheless has legal protection from animal cruelty. The now-repealed Roe v. Wade (1973) and Planned Parenthood v. Casey (1992) decisions essentially answered the second question as “yes, from viability.”

These questions are moral and philosophical, not scientific. However, scientific questions do arise in the debate. Is the fetus alive? Yes, obviously. Is it human? Well, it has different hemoglobin from adult humans, but the fetus is obviously Homo sapiens. Does the fetus have a heartbeat? Surprisingly, that seems to be controversial, although every textbook I have seen agrees that the heart is functional very early (how else would the developing fetus get oxygen and nutrients?). To quote some sources:

• “The heart is the first organ to develop. In the human embryo, the heart begins beating at about 21 days after conception [i.e. 5 weeks after LMP = last menstrual period].” (Anatomy and Physiology of the Circulatory and Ventilatory Systems, page 2)
• “The fetal–placental circulation begins at about 9 days postfertilization … A functional circulation is established by the end of the third developmental week [i.e. 5 weeks after LMP].” (Fetal MRI, page 405)
• “In a developing embryo, the heart has developed enough by day 21 post-fertilization to begin beating [i.e. 5 weeks after LMP]. Circulation patterns are clearly established by the fourth week of embryonic life. It is critical to the survival of the developing human that the circulatory system forms early to supply the growing tissue with nutrients and gases, and to remove waste products.” (Anatomy and Physiology 2e)
• “Circulation of fetal blood in the placental circulation begins approximately 21 days postfertilization in humans [i.e. 5 weeks after LMP].” (Handbook of Developmental Neurotoxicology, page 68)
• “1. Fetal heart development begins during the first month of gestation. At about 21 days of gestation, the fetal heart begins beating, and blood begins circulating [i.e. 5 weeks after LMP]. Between the second and seventh weeks of gestation [i.e. 4 to 9 weeks after LMP], the primitive fetal heart undergoes a series of changes that create the four-chambered heart and its great arteries. 3. During gestation, the lungs are nonfunctional, and fetal oxygenation occurs via the placenta.” (Pediatric Nursing, page 223)

In my view, accurate discussion of the scientific facts is a necessary preliminary to addressing the moral and philosophical questions.

Doppler ultrasound is routinely used to detect fetal heartbeat and the velocity of fetal blood flow. The scientific principle known as the Doppler effect allows the detection of motion. It is the principle behind speed cameras, and it allows bats to “hear” the fluttering wings of a distant insect. During first-trimester screening (at around 11 to 13 weeks after LMP), professional Doppler ultrasound devices are sensitive enough not only to detect fetal heartbeat, but to detect blood flow abnormalities in various parts of the fetal circulatory system.

Fetal heartbeat at 13 weeks (from here). S is the ventricular systolic wave, D the early diastolic, A the atrial contraction.

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