Wednesday, 12 September 2018

the right angle

It is often said that there are no right angles in nature, but this is not strictly true. Of course, there are no animals or plants with square corners, but the human visual field is determined by two lines that intersect at 90 degrees: the horizontal, as defined by the horizon (if you live on the coast or are aboard a ship at sea); and the vertical, as defined by gravity. The need to define these lines starts when humans began to build structures in brick and stone.

Defining the vertical is easy—a plumb line does that job—but in order to define what is horizontal, it is necessary, somehow, to produce a right angle. The first civilization to do this was probably Sumer in the third millennium BC, but both the Babylonians and the Egyptians were able to construct right-angle triangles around 4,000 years ago. While the Egyptians used only the 3:4:5 triangle, they may have been aware of other number combinations, but the Babylonians certainly knew other ratios, which they calculated using a sexagesimal (base 60) system of arithmetic. This strikes me as being a seriously unwieldy way to perform calculations, but it’s why there are sixty seconds in a minute and sixty minutes in an hour, so it must have had some benefits.

Most people will be familiar with the theorem of Pythagoras—possibly the most famous theorem in the whole of mathematics—which states that in a right-angle triangle, the square on the hypotenuse is equal to the sum of the squares on the other two sides, as in this diagram, which illustrates the simplest of the ratios (3:4:5) that produce right-angle triangles:

However, I wondered whether I could work out any other ‘Pythagorean triples’. I already knew that the ratio 5:12:13 produced a right-angle triangle, but were there any more? That question turned out to be surprisingly easy to answer. First, it is not possible for the smallest value to be 2 or 4, and if the smallest value is 6, then the only combination that works is 6:8:10, in which the three numbers are not coprime (they have a common factor, 2). In other words, 6:8:10 is simply a multiple of 3:4:5 and therefore doesn’t count as a separate Pythagorean triple.

On the other hand, if I set the value of the smallest term at 7, then it’s straightforward to find two consecutive numbers, the squares of which have a difference of 49. At this point, I noticed that while the increases in the value of the smallest term were linear (3, 5, 7), the other terms increased more rapidly. Thus, 4, 12, 24. Could the middle term in the next triple be 40? It is. The left-hand column in the following table illustrates how far I went in identifying further triples, first calculating the appropriate values using the method I’ve just outlined, then confirming that the calculated values fitted the formula using the time-honoured method of long multiplication on the back of an envelope.


It seems to me that this is a series that will continue to infinity, and I could say the same about the right-hand column in the table, in which the difference between the two larger numbers is 2. This one works only when the smallest number is divisible by 4, because where the smallest number is divisible only by 2, the result is a multiple of one of the ratios in the left-hand column.

This is where things get more difficult. When I tried to find a ratio where the difference between the two largest numbers is 3, all I could come up with was multiples of other ratios that I’d previously discovered. At this point, I remembered that Jacob Bronowski had quoted, incredulously, the ratio 3367:3456:4825 in his landmark BBC TV series on the history of science, The Ascent of Man, as an indicator of the arithmetical prowess of the Babylonians 4,000 years ago. My first reaction was that this must be a multiple of a primitive Pythagorean triple, but I was amazed to find that the three terms are coprime (one of the prime factors of 4825 is 193).
This must have taken some calculating! It is clearly a primitive Pythagorean triple, but it immediately occurs to me that it may be part of a series like the simpler ratios described above. It was time to see what the internet had to say on the subject. The first page of my search included the statement that there are 16 Pythagorean triples in which the value representing the hypotenuse is less than 100. I’d already identified nine of these in the table above; here are the other seven:


I spotted immediately that the difference between the two larger numbers in the second and fourth ratios is 8, so I wondered whether they were part of the same sequence. They are, and this is another sequence that I’m assuming continues to infinity.


Notice that the difference between the two larger numbers in the first of these two tables is also 8, but the difference between the two smaller numbers is just 1, meaning that it is not part of the same sequence. However, it is difficult to identify a sequence from just one triple, especially when that triple is as large as the Babylonian example cited above.

I’ve concentrated so far on triangles where the values representing all three sides are integers, but there is what I assume to be an even larger category of Pythagorean triples involving irrational numbers. For example, if the lengths of the two sides enclosing the right angle are 2 and 3, respectively, then the length of the hypotenuse is the square root of 13, which is irrational (meaning that it cannot be represented by a fraction).

The general conclusion that I draw from my investigations is that there are an infinite number of ratios that meet the Pythagorean criteria, but I’m unable to explain why this particular juxtaposition of squares produces a right angle, or even what is special about 90 degrees that makes it the right angle.

Sunday, 2 September 2018

carved in stone

When we say that something is ‘carved in stone’, we are suggesting that it’s unchanging, that it’s the accepted way of doing things. However, I’d like to suggest that in the real world, with real stone, things that get carved into stone are eventually forgotten, and to emphasize the point, I’ve compiled a short quiz involving examples of stone carving in my home town, where the ubiquitous building material until the end of the nineteenth century was a red desert sandstone (Penrith sandstone).

If you are a casual visitor to the town, you may think that trying to find all of these examples would be an interesting way to spend an afternoon, but I should warn you that I suspect that not many Penrithians would know where they are all located, and not all are in the town centre. Most of the images are small, which should tell you that you will need to look up to see most of these carvings.

I’ll start with five simple plaques. Where is Castle View?


Where is the British School, built in 1847?


Where is the Primitive Methodist Chapel, built in 1837?


Where is the Infant School, built in 1833?


And where is Wigan Terrace?


Another apparent street name is Inglewood Terrace, although you won’t find either on any street map or directory. This name has been carved directly into the building:


The next image is a coat of arms:


…while the next carving is also vaguely heraldic:


And now for some dates:







You may guess that the building identifying itself as a bank in the next photo no longer functions in that capacity:


Here are two more conventionally sculptural carvings:



Finally, where is this elaborate decorative carving above a doorway?


There are no prizes for locating these eighteen examples, although if you think you know them, do leave a comment.

Monday, 27 August 2018

manchester miscellany

Although I’ve already posted five collections of graffiti that I photographed back in June in Manchester, these were all in locations where large numbers of graffiti had been painted in a relatively small area, and there were other places where perhaps there was only a single graffito. Some of these were well worth recording too, especially the first image:


I can’t help but see this as a row of semi-reclining creatures, possibly human, although that probably wasn’t the artist’s intention. The figure third from the left reminds me of an amorphous snowman wearing a bobble hat, despite the colour. Unfortunately, this was another graffito that I couldn’t fit into a single shot.

I spotted the next work on its own a short distance before I reached the hoarding that I described in The Writing on the Fence. There is nothing exceptional about the writing style, although I do wonder why the black outlining on the letters adjoining the mask is broken. Malign influence? Although I can actually read what’s written, it’s meaningless to me.


I came across an undeveloped area west of Oxford Street that was surrounded by walls and fences bearing graffiti, but I didn’t consider most of these worth recording. However, the next three artworks were a resounding exception. The first appears to be completely abstract, but the more I look at it, I wonder if I’m seeing, or imagining, concrete images.


The object obscuring part of the image in the bottom left of the picture is a lightweight tent. I saw several in similar locations around the city, and I did wonder whether the occupant was a homeless Mancunian, or merely a backpacker looking for a cheap option.

Around the corner to the right from the previous image, there are two more artworks side by side:


It’s worth looking closely at each work in more detail. Both are mostly abstract, but there are faces that were obviously intentional. Of course, once you see faces in paintings like these, you see them where they weren’t intended. And both feature the glinting light motif, which I’ve commented on before:



The red face in the next image doesn’t appear to have been painted by the artist who is responsible for the blue faces that I’ve commented on previously, although they are similar. The central graffito is what I now describe as ‘routine’, but the right-hand tag is much more nuanced—and colourful!


Finally, I’ve included this image precisely because all the lines are crude. Yet despite this apparent limitation, the lettering stands out extremely well:


This is my sixth and final report on graffiti in Manchester. I didn’t go looking for any of it, but in simply wandering around the city, this is what I saw.

more graffiti from manchester
The Writing on the Fence
Lost Horizon
Some Consolation: Part 1
Some Consolation: Part 2
A Graffiti Mystery

Wednesday, 22 August 2018

dom domination

Although there are many other things to see and do in Cologne, the chance to see the city’s cathedral (German: ‘Kölner Dom’) is the principal draw and the main reason we went to Cologne earlier this year. One thing I specifically wanted to check out was the reason for the Dom’s exclusion from the shortlist compiled a few years ago as part of the process of defining a new ‘seven wonders of the world’.

In fact, there were no European Gothic cathedrals on the shortlist of twenty, and only one European site—the Colosseum in Rome—in the final selection of seven. As I wrote at the time (Wonderful), this is a flawed list, because that final selection was determined by popular vote, so nationalistic rather than æsthetic sentiments were the deciding factor. What else would privilege the statue of Christ the Redeemer in Rio de Janeiro, which made the final cut, over the Statue of Liberty in New York, which merely made the shortlist. And while Machu Picchu seems an obvious choice, I wonder why Chichen Itzá was chosen over other Mesoamerican sites such as Palenque or Teotihuacán.

Paula had already been impressed by some of the churches that we visited in Brussels, especially that city’s cathedral, but when we emerged from the main railway station in Cologne to be confronted by the Dom on the other side of the road, she was absolutely gobsmacked. And with good reason. Cologne Cathedral is a veritable mountain of stone (the twin west towers are 157 metres high)—no reinforced concrete here. Here are two views of the west façade, the second of which was taken by Paula (notice the scaffolding on the left-hand tower—the cathedral employs a full-time maintenance staff of 100):



It was impossible to get far enough away to capture everything in a single shot, as this view from the south also demonstrates:


…although the view from the east is more compact. This is the oldest part of the cathedral, construction of which started in 1248 and continued for more than two centuries. Mainly due to a lack of both interest and money, building work ceased in the early sixteenth century, and the huge shell was left unfinished for more than 300 years. However, interest was revived in the nineteenth century, and construction was completed in 1880 in line with the original mediæval plan.


To demonstrate how the Dom dominates the city, I’ve selected the following five photos. The first is of the road that separates the railway station from the Dom. The scaffolding on the left-hand tower is clearly visible:


The next, taken by Paula, is the view from the opposite bank of the Rhine, with the Hohenzollern Bridge on the right of the picture. I’ve no idea what the exotic building to the left of the Dom is, because we couldn’t find any open doors.


Ancient and modern (also taken by Paula):


The following photo was taken downstream from the Hohenzollern Bridge, while the one after that was taken from the bridge itself. I cannot positively identify the rider on the horse, but given that Hohenzollern was the dynastic name of the Prussian imperial family, it’s likely to be one of the kaisers.



We signed up for a guided tour of the cathedral, although you could spend a week here and still be seeing something new. However, we were treated to some very fine snippets of information that you would never know about unless told. For example, all the mediæval stained glass was removed upon the outbreak of war in 1939 and stored in deep basements all around Germany, but all the nineteenth-century glass was destroyed during the war. Here are two of those mediæval windows, first a view of the entire window, then an enlargement of a portion:





I will leave it to you, the reader, to determine what is being depicted here, although the second window has a martial rather than a religious theme.

All the following interior shots were taken by Paula. Apart from the stained glass, I couldn’t take a photo without activating flash, which I didn’t think was appropriate in such a place. I hadn’t been paying full attention to our guide, but my ears pricked up when she pointed out the gold reliquary behind the high altar. It contained the bones of the three wise men!


Call me a cynic, but how were they ever identified?

They appear only in Matthew’s gospel, and then for only the first twelve verses of Chapter 2, at the end of which:
12 And being warned of God in a dream that they should not return to Herod, they departed into their own country another way.
In other words, they disappeared from history, if they were ever part of history in the first place. They are far more likely to be a story made up by Matthew to fulfill some Old Testament prophecy or other. Apparently, though, the relics were a major focus of pilgrimage during the Middle Ages (they were installed here in 1164, in an earlier version of the cathedral). In other words, they were a money-making scheme.

There were some interesting mediæval mosaics on the floor surrounding the reliquary, which weighs 600 kilograms, although I don’t think it can be solid gold, despite the vast wealth of the mediæval Church. Unfortunately, the light made it difficult to take a good picture. This one is the best (the wavy lines represent streams and rivers):


The choir stalls are also mediæval originals:


The main entrance is extremely grand:


…but the side doors are scarcely less elaborate:


…while this cast iron bas-relief is one of several along the side walls:


The most important architectural innovation of the Gothic period was the flying buttress, good examples of which can be seen in the next photo:


When a flying buttress cast a shadow across a window during our cathedral tour, our guide referred to it as a ‘bridge’, but this feature is better described as a ‘half-arch’. The effect of flying buttresses can be seen in the next photo. Because a flying buttress takes part of the weight of the roof, it is possible to pierce the walls with much bigger windows than were possible in earlier Romanesque churches, which relied on the round arch.


The only one of the new seven wonders of the world that I’ve seen is the Great Wall of China, and I think the Dom is more impressive. It is certainly a better representative of the Christian contribution to world civilization than Christ the Redeemer. I haven’t tried to include everything that I saw in this collection of photos, because that would be an impossibility. Would I go back to Cologne? Certainly. Would I take another look at the cathedral? Yes, but only if I could be allowed to explore the gallery that you can see in the previous photo just below the windows.