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.

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.

Do gun laws save lives?

Do gun laws save lives? The chart above shows homicide rates for U.S. states (data from here) together with an A to F ranking of state gun laws from the Giffords organisation. As with my post from 2017, there is actually no statistically significant correlation (this is particular noticeable among the F’s, which include both the seven states with the highest murder rate and the two states with the lowest). In other words, the answer seems to be no.

Rather, it seems that guns don’t kill people, people kill people. The murder rate in the U.S. is driven by social factors which differ from state to state – factors which make New Hampshire and Maine pretty safe, but which produce a murder rate ranging from 14 per 100,000 to 20.5 per 100,000 in Missouri, Alabama, Louisiana, and Mississippi. For comparison, New Hampshire has a murder rate similar to that of Australia, but Louisiana and Mississippi, if they were countries, would rank among the most murderous 20 countries in the world.

There is some evidence that keeping guns out of the hands of criminals would reduce the murder rate in the U.S., but this is extremely difficult to do. The U.S. has a lengthy, porous southern border, across which there is a free flow of people, guns, and illegal drugs.

In addition, a concept from catastrophe theory is useful here. In the cusp pictured below, it is possible to “drop” from the top of the fold to the bottom, but a long roundabout journey would be required to get back up. Similarly, it is very easy to introduce guns into a society, but very difficult to remove them. Although such removal has been done elsewhere, laws forbidding gun ownership are likely to be ignored by precisely those violent criminals that one would not wish to carry them. And, of course, there is the 2nd Amendment.

Timeline of mathematical notation

Following up on my earlier timelines about zero and about Hindu-Arabic numerals, here is a timeline for some other mathematical notation, starting with the square root symbol (click to zoom).

0123456789 in Europe: an infographic

Following up on my earlier post about 0 and 1 in Greek mathematics and my timeline of zero in Europe, here is a timeline for the use of Hindu-Arabic numerals in Europe up to René Descartes (click to zoom).

The history of zero: an infographic

Following up on an earlier post about Zero in Greek mathematics, here is a timeline for the use of zero in Europe (click to zoom). I have used images of, or quotes from, primary sources where possible (reliably dated Indian primary sources are much harder to find than Greek ones, unfortunately).

Chinese uses of zero are probably also derived from the Greeks, but Mayan uses are clearly independent.

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.

Dante’s Heaven

In previous posts (Inferno, Purgatorio, Paradiso), I have mentioned the scientific content of Dante’s incredible theological poem, the Divine Comedy. Above, just for fun, is a chart of Heaven (the Solar System) in his Paradiso. Notice the sphere of fire which was believed to surround the Earth.

Planetary Intelligences

In a book review of Out of the Silent Planet, I mentioned last year that C. S. Lewis had pioneered the science fiction sub-genre of a planetary intelligence or sentient planet which resists outsiders. A planetary intelligence provides a way of exploring colonisation and other issues, while still having a positive ending to the story.

The chart above (click to zoom) shows a timeline of the concept. Although there are many other stories based on the idea, these six seemed particularly noteworthy (star ratings out of 5 are from GoodReads and RottenTomatoes):

Solaris was filmed in 1968, 1972 (★★★★☆), and 2002 (★★★☆). Here are trailers for the last two films:

Readers, how do you feel the various books and films compare?

Brouwer and his fixed point theorem


The Brouwer fixed-point theorem is one of my favourite mathematical theorems. It is named after the Dutch mathematician Luitzen Egbertus Jan Brouwer (above right). Brouwer is also known for his work in Intuitionism. I have mentioned the Brouwer fixed-point theorem before.

The theorem states that any continuous function f on a compact convex set (and specifically, on a disc in the plane) will have at least one fixed point – that is, there will be at least one point p such that f(p) = p. The picture below is intended to illustrate the theorem; it is explained further down.

In the case of a disc, the theorem can be proved by contradiction. Assume that f(p) ≠ p for every point p. Then the pair of f(p) and p always defines a continuous mapping g from p to the boundary of the disc, as illustrated above (left). However, such a continuous mapping is impossible (for complex reasons, but in simple terms, because it creates a hole, which continuous mappings cannot do).

So what about that picture? It shows a continuous function f from the disc to itself, combining an irregular rotation about the centre (rotating least towards the east of the disc) with a “folding” operation that leaves the centre and boundary untouched. The picture below shows a cross-section of the folding in action. The shades of blue in the picture above show how far each point p is from f(p), with lighter colours representing smaller values. Arrows show the action of the function on 6 randomly chosen points. There are two fixed points, marked with black dots: the centre and one other point where the folding and the irregular rotation cancel each other out.

The three-dimensional version of the theorem tells us that, when I stir my morning cup of coffee, at least one speck of liquid will wind up exactly where it started.