The Plimpton 322 tablet re-examined

Several years back I blogged about the Plimpton 322 tablet – a Babylonian clay tablet from around 1,800 BC. It contains four columns of numbers, written in base 60 (with a small number of errors, as well as some numbers missing through damage – these are corrected below). For example, 1.59:00:15 = 1 + 59/60 + 0/3600 + 15/216000 = 1.983402777777778.

Column B of the tablet (with a label on the tablet containing the word “width”) is one of the sides of a Pythagorean triangle, and column C (with a label on the tablet containing the word “diagonal”) is the hypotenuse, such that C2 − B2 is always a perfect square (yellow in the diagram). Column A is exactly equal to C2 / (C2 − B2), the ratio of blue to yellow.

A B (“width”) C (“diagonal”) D
1.59:00:15 = 1.983402777777778 1:59 = 119 2:49 = 169 #1
1.56:56:58:14:50:06:15 = 1.949158552088692 56:07 = 3367 1:20:25 = 4825 #2
1.55:07:41:15:33:45 = 1.918802126736111 1:16:41 = 4601 1:50:49 = 6649 #3
1.53:10:29:32:52:16 = 1.886247906721536 3:31:49 = 12709 5:09:01 = 18541 #4
1.48:54:01:40 = 1.815007716049383 1:05 = 65 1:37 = 97 #5
1.47:06:41:40 = 1.785192901234568 5:19 = 319 8:01 = 481 #6
1.43:11:56:28:26:40 = 1.719983676268861 38:11 = 2291 59:01 = 3541 #7
1.41:33:45:14:03:45 = 1.692709418402778 13:19 = 799 20:49 = 1249 #8
1.38:33:36:36 = 1.642669444444444 8:01 = 481 12:49 = 769 #9
1.35:10:02:28:27:24:26:40 = 1.586122566110349 1:22:41 = 4961 2:16:01 = 8161 #10
1.33:45 = 1.5625 45 1:15 = 75 #11
1.29:21:54:02:15 = 1.489416840277778 27:59 = 1679 48:49 = 2929 #12
1.27:00:03:45 = 1.450017361111111 2:41 = 161 4:49 = 289 #13
1.25:48:51:35:06:40 = 1.430238820301783 29:31 = 1771 53:49 = 3229 #14
1.23:13:46:40 = 1.38716049382716 28 53 #15

What is this table all about? A good discussion is by Eleanor Robson [Words and pictures: new light on Plimpton 322,” American Mathematical Monthly, 109 (2): 105–120]. Robson sees Plimpton 322 as fitting into standard Babylonian mathematics, and interprets it as a teacher’s effort to produce a list of class problems.

Specifically, Robson believes that the table was generated by taking values of x (in descending order of x) from standard Babylonian reciprocal tables (specifically the values 2:24, 2:22:13:20, 2:20:37:30, 2:18:53:20, 2:15, 2:13:20, 2:09:36, 2:08, 2:05, 2:01:30, 2, 1:55:12, 1:52:30, 1:51:06:40, and 1:48) and then using the relationship (x − 1 / x)2 + 22 = (x + 1 / x)2 to generate Pythagorean triples. If we let y = (x − 1 / x) / 2 and z = (x + 1 / x) / 2, then B and C are multiples of y and z, and A = z2 / (z2 − y2).

Just recently, Daniel F. Mansfield and N. J. Wildberger [Plimpton 322 is Babylonian exact sexagesimal trigonometry,” Historia Mathematica, online 24 August 2017] interpret the table as proto-trigonometry. I find their explanation of the first column (“a related squared ratio which can be used as an index”) unconvincing, though. Why such a complex index? Robson calls such trigonometric interpretations “conceptually anachronistic,” and points out that there is no other evidence of the Babylonians doing trigonometry.

Mansfield and Wildberger also suggest that “the numbers on P322 are just too big to allow students to reasonably obtain the square roots of the quantities required.” However, I don’t think that’s true. The Babylonians loved to calculate. Using the standard square-root algorithm, even simplistic starting guesses for the square roots of the numbers in column A give convergence in 2 or 3 steps every time. For example, to get the square root of 1.59:00:15 (1.983402777777778), I start with 1.30:00:00 (1.5) as a guess. That gives 1.24:40:05 as the next iteration, then 1.24:30:01, and then 1.24:30:00 (1.408333333333333), which is the exact answer. That said, however, calculating those square roots was not actually necessary for the class problems envisaged by Robson.

Sadly, I do not think that Mansfield and Wildberger have made their case. Robson is, I believe, still correct on the meaning of this tablet.


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The Dakota Access Pipeline controversy

A recent letter to Science by Stephanie Januchowski-Hartley and three PhD students expresses strong opposition to the controversial Dakota Access Pipeline (DAPL). The letter cites (an old version of) the Environmental Assessment for the DAPL, though not the 1261-page Army Corps of Engineers (ACE) report on the project. The authors of the letter assert that “To date the potential impacts of DAPL construction, or any potential spills, on aquatic or terrestrial species has not been adequately assessed,” but unfortunately do not indicate which sections of the existing Environmental Assessment dealing with those subjects they consider to be inadequate.


A pallid sturgeon (Scaphirynchus albus) being released into the Yellowstone River by U.S. Fish and Wildlife Service personnel.

The long version of the letter also implies that the endangered pallid sturgeon (above) would be adversely affected by the proposed DAPL crossing of the Missouri River at Lake Oahe. However, as the ACE report notes, pallid sturgeon are in fact very scarce in Lake Oahe. This is because, ever since that lake was formed by the 1958 Oahe Dam, the waters have been unsuitable for reproduction of that species. The remaining pallid sturgeon are primarily found elsewhere. The ACE believes that the pallid sturgeon is unlikely to be adversely affected by the DAPL.

It is true that older oil pipelines can and do rupture with disturbing frequency. For example, the Poplar Pipeline in Montana, built in the 1950s using faulty welding techniques and laid in a very shallow trench under the Yellowstone River, spilled a substantial amount of oil in 2015. However, even that spill does not seem to have harmed the fish there (in contrast to the quite serious negative effects on fish typically seen for marine or wetland oil spills).


Map from the Army Corps of Engineers report, showing DAPL crossing point at Lake Oahe.

The authors of the letter also state that cultural impact assessments of the DAPL have been inadequate (although the court thus far disagrees, noting cultural surveys conducted by licensed archaeologists, and a consultation process that began in 2014). The proposed river crossing runs just north of the Standing Rock Indian Reservation (see map above). The Standing Rock Sioux Tribe claims that DAPL construction has destroyed cairns and sacred burial grounds near the crossing, although North Dakota’s chief archaeologist says that no burial sites or significant sites were destroyed (and it is a little difficult to see how DAPL construction in the disputed area could have damaged any significant sites, since the DAPL there closely follows the path of the 1982 Northern Border Pipeline, as indicated by a visible line on satellite imagery and by black and yellow “Caution: Gas Pipeline” signs visible in photographs taken at protest sites – i.e. the relevant land was already bulldozed and restored 34 years ago).

Early in the planning stage, a DAPL route further north was apparently considered. This would have not have been collocated with existing pipeline to the same extent, would have been 10.6 miles longer, would have crossed more agricultural land, wetlands, and floodplain, and would have cost the company behind the DAPL $22.6 million more. Still, the company may now be wishing that they had followed up that option.


DAPL construction, elsewhere along the route (photo: Tony Webster).

The whole topic is of course a political hot potato, being a major source of conflict between, on the one hand, mainstream US Democrats (including construction-worker unions and the Clinton campaign), and, on the other hand, followers of Bernie Sanders and the Greens. Further complicating matters is that some land in the Dakotas was assigned to the Sioux by the 1868 Treaty of Fort Laramie, and unjustly taken from them (over 1 billion dollars has been provided in compensation, following litigation by the tribe, but the tribe has nevertheless refused the money, wanting the land instead). At the same time, increasing tensions in the Dakotas are likely to damage the tribe’s casino business over the longer term.


Traffic churning out greenhouse gases.

Protests against the DAPL have also been linked to climate change, but the project in fact makes little or no difference to US fossil-fuel consumption. Oil can also be shipped within the US by rail (although this is less safe) and by ship from overseas oilfields. I think that activists would do better to campaign for e.g. public transport to replace inefficient individual automobiles, which produce copious greenhouse gases (solar cars would make a good alternative as well!). I must admit that I also struggle to understand activists who drive convoys of gasoline-powered vehicles to anti-fossil-fuel protests.


How many parts?

The diagram below shows the complexity, in terms of numbers of parts, of some human constructions. Interestingly, there is an approximate complexity plateau which starts at or before the Great Pyramid of Giza (constructed between about 2580 BC and 2560 BC, and composed of around 2.3 million stone blocks). The plateau continues through the dome of Florence Cathedral (brilliantly designed by Filippo Brunelleschi, made up of over 4 million bricks, and completed in 1436). A late member of the plateau is the Boeing 747 (first flown in 1969, and composed of around 6 million parts). The Great Pyramid required the resources of a nation, Brunelleschi’s dome those of a city-state, and the 747 those of a large company.

Somewhat less complex are the Antikythera mechanism and John Harrison’s H1 chronometer (a five-year effort by one man). The PDP-8/S (1966) and the original Apple Macintosh (1984) were widely popular low-cost computers. For those, I’ve interpreted “parts” as either transistors, individual bits of ferrite core memory, or bytes of semiconductor memory.

The recent iPhone 6s stands out from the simpler computers: the A9 processor has over 3 billion transistors, and the phone comes with at least 18 GB of memory. The iPhone 6s puts the power of a mid-80s Cray-2 supercomputer in a handheld device. Producing one requires the resources of an international network of specialised companies, with the processor and memory being fabricated in South Korea or Taiwan, the camera and display in Japan, and the accelerometer in Germany. The software is developed in the USA, and final assembly is mostly done in China. It seem unlikely that any one nation would be able to construct a device as complex as this.

Maybe I should get one.

The spectrum of history

I have previously blogged about carbon dating, a method which can be used to date organic items up to about 50,000 years old. Tests of carbon-dating have used tree-ring data back to around 10,000 BC.

The timeline above shows some highlights of the past 30,000 years, including one of the oldest cities, the oldest living tree, the oldest pyramid, and a Babylonian clay tablet I blogged about two years ago. The two caves in France are definitely on my “bucket list.”

Archaeology and Statistics

Statistics can be a useful tool in archaeology, as the 1996 book Statistics for Archaeologists: A Common Sense Approach by Robert Drennan points out. Quantifying Archaeology by Stephen Shennan is another book on the subject.

Elsewhere I have discussed the benefits of the R statistical toolkit. The image below uses R to plot some data from Drennan’s book. Specifically, it is a histogram of the lengths of stone scrapers found at two sites (from his Tables 1.9 and 1.10). It can be seen that there is no significant difference between the two archaeological sites involved (red vs blue), but a very clear difference between scrapers made from flint (light, mean length 42.9 mm) vs chert (dark, mean length 18.4 mm). The visual plot summarises the numbers better than the tables can, and R’s statistical tests for significance (which I used to confirm the visual impression) are critically important for testing hypotheses.

The R code for this plot is:

#Group means
m.pc <- mean(Scrapers$Length[Scrapers$Site == "Pine Ridge Cave" & Scrapers$Material == "Chert"])
m.pf <- mean(Scrapers$Length[Scrapers$Site == "Pine Ridge Cave" & Scrapers$Material == "Flint"])
m.wc <- mean(Scrapers$Length[Scrapers$Site == "Willow Flats" & Scrapers$Material == "Chert"])
m.wf <- mean(Scrapers$Length[Scrapers$Site == "Willow Flats" & Scrapers$Material == "Flint"])

#Histograms for each group
bks <- 2.5+5*(0:18)
h.pc <- hist(Scrapers$Length[Scrapers$Site == "Pine Ridge Cave" & Scrapers$Material == "Chert"], breaks=bks, plot=FALSE)
h.pf <- hist(Scrapers$Length[Scrapers$Site == "Pine Ridge Cave" & Scrapers$Material == "Flint"], breaks=bks, plot=FALSE)
h.wc <- hist(Scrapers$Length[Scrapers$Site == "Willow Flats" & Scrapers$Material == "Chert"], breaks=bks, plot=FALSE)
h.wf <- hist(Scrapers$Length[Scrapers$Site == "Willow Flats" & Scrapers$Material == "Flint"], breaks=bks, plot=FALSE)

#Matrix of histograms 
mat <- rbind (h.pc$counts, h.pf$counts, h.wc$counts, h.wf$counts)

#Plot matrix
legnd <- c(paste("Pine Ridge Cave, Chert (mean ", round(m.pc, digits=1), " mm)", sep=""),
           paste("Pine Ridge Cave, Flint (mean ", round(m.pf, digits=1), " mm)", sep=""),
           paste("Willow Flats, Chert (mean ", round(m.wc, digits=1), " mm)", sep=""),
           paste("Willow Flats, Flint (mean ", round(m.wf, digits=1), " mm)", sep=""))
barplot(mat, space=0, col=c("darkred", "pink", "navy", "skyblue"), legend.text=legnd, names.arg=5*(1:18), ylim=c(0,12),
        cex.names=0.7, ylab="Number of Scrapers", xlab="Scraper Length (mm)", cex.lab=1.3, args.legend=list(cex=0.8))

#Statistical tests
summary(lm(Scrapers$Length ~ Scrapers$Site + Scrapers$Material))

Ground-penetrating radar and archaeology

Ground-penetrating radar (shown in action above) is a useful application of science to archaeology. Exploring the underground with microwaves saves a lot of digging!

The image below (click for details) is of a “slice” though an historic cemetery. The vertical axis shows depth. Yellow arrows mark probable human burials, while dashed blue lines mark probable lines of bedrock. The upper half-metre is a tangle of tree-roots, which it would have been difficult to dig through (had that been permitted, which it was not).

You can imagine how useful this technique would be in searching for a lost and buried city!

A Street Through Time: a book review for Children’s Book Week


A Street Through Time, illlustrated by Steve Noon

In honour of Children’s Book Week, here is a look at a classic children’s book – A Street Through Time, from Dorling Kindersley. The strength of this book lies in the wonderful two-page spreads by Steve Noon. For example, this night-time plague scene:

Relationships between corresponding places at different times are particularly interesting in this book. For example, the high ground occupied by a Roman fort later holds a castle, which is then destroyed:

Several other structures also go through transformations over the centuries, as do the people, and it is these transformations that provide the young reader with a sense of change over time.

Although the genre here is history/archaeology, there are many great ideas here for children’s science books – an area in which Dorling Kindersley also has several excellent offerings.

* * * * *
A Street Through Time, illlustrated by Steve Noon: 5 stars