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).
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.
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.
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):
- ★★★★ Out of the Silent Planet by C. S. Lewis (1938).
- ★★★★ The Lonely Planet by Murray Leinster (1949).
- ★★★★ Solaris by Stanislaw Lem (1961).
- ★★★★ “Vaster than Empires and More Slow” by Ursula Le Guin (1971).
- ★★★★ Powers that Be by Anne McCaffrey and Elizabeth Ann Scarborough (1993) and its sequels.
- ★★★★ The film Avatar by James Cameron (2009) and its upcoming sequels.
Readers, how do you feel the various books and films compare?
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.
The Cascades Raptor Center, Eugene, Oregon (image credit)
I am currently travelling in the US, and recently dropped in at the Cascades Raptor Center in Eugene, Oregon (see the Raptor Center website and Raptor Center instagram). This is primarily a hospital and rehabilitation centre for injured raptors (eagles, hawks, falcons, ospreys, owls, vultures, etc.). By my count, 37 birds are on public display in wire mesh aviaries (mostly birds too seriously injured to be released).
Admission is US$10 per adult, and there is a small gift shop. Viewing the birds takes about an hour, and I really enjoyed my visit (the Center would also appeal to children of all ages). Tripadvisor gives the Center 4½ stars, which is probably a little more than I would give, given the Center’s size (but the money supports the Center’s work). There are also more interactive night-time viewing events, which I did not experience.
Ferruginous hawk (Buteo regalis) and Gyrfalcon (Falco rusticolus) at the Raptor Center (photographs by Anthony Dekker)
Pi Day is coming up again (3/14 as a US date). The number π is, of course, 3.14159265… Here are some possible activities for children:
- Search for your birthday (or any other number) in the digits of π
- Follow in the footsteps of Archimedes, showing that π is between 22/7 = 3.1429 and 223/71 = 3.1408.
- Calculate 333/106 = 3.1415 and 355/113 = 3.1415929, which are better approximations than 22/7.
- Measure the circumference and diameter of a round plate and divide. Use a ruler to measure the diameter and a strip of paper (afterwards measured with a ruler) for the circumference. For children who cannot yet divide, try to find a plate with diameter 7, 106, or 113.
- Calculate π by measuring the area of a circle (most simply, with radius 10 or 100), using A = πr2. An easy way is to draw an appropriate circle on a sheet of graph paper.
You can also try estimating π using Buffon’s needle. You will need some toothpicks (or similar) of length k and some parallel lines (such as floorboards) a distance d apart (greater than or equal to k). Then the fraction of dropped toothpicks that touch or cross a line will be 2 k / (π d), or 2 / π if k = d. There is an explanation and simulator here (see also the picture below). And, of course, you can bake a celebratory pie and listen to Kate Bush singing π, mostly correctly!
This picture by McZusatz has 11 of 17 matches touching a line, suggesting the value of 2×17/11 = 3.1 for π (since k = d).
Actually, of course, π = 3.1415926535 8979323846 2643383279 5028841971 6939937510 5820974944 5923078164 0628620899 8628034825 3421170679 8214808651 3282306647 0938446095 5058223172 5359408128 … (digits in red are sung by Kate Bush, accurately, although some have said otherwise).
The 1968 film 2001: A Space Odyssey suggested that we would have extensive space flight in 2001. That turned out not to be the case. What we did get was the September 11 attacks on the USA and the military conflicts which followed. Nevertheless, NASA commemorated the film with the 2001 Mars Odyssey orbiter.
Films of 2000 included the superb The Lord of the Rings: The Fellowship of the Ring, several good animated films (including Monsters, Inc., Shrek, and Hayao Miyazaki’s Spirited Away), the wonderful French film Amélie, some war movies (Enemy at the Gates was good, but Black Hawk Down distorted the book too much for my taste), the first Harry Potter movie, and an award-winning biographical film about the mathematician John Nash.
Saul Kripke (belatedly) received the Rolf Schock Prize in Logic and Philosophy for his work on Kripke semantics, while Ole-Johan Dahl and Kristen Nygaard (also belatedly) received the Turing Award for their work on object-oriented programming languages (both these pioneers of computing died the following year).
The year 2001 also saw the completion of the Cathedral of Saint Gregory the Illuminator in Armenia, which I have sadly never visited.