Pythagorean tiling

The painting is Street Musicians at the Doorway of a House, by Jacob Ochtervelt (1665).  The pattern on the floor is an example of "Pythagorean tiling."

In geometry, the Pythagorean tiling or two squares tessellation is a tessellation of the plane by squares of two different sizes, in which each square touches four squares of the other size on its four sides. A tiling of this type may be formed by squares of any two different sizes.  It also is commonly used as a pattern for floor tiles; in this context it is also known as a hopscotch pattern...

This tiling is called the Pythagorean tiling because it has been used as the basis of proofs of the Pythagorean theorem by the ninth-century Arabic mathematicians Al-Nayrizi and Thābit ibn Qurra, and by the 19th-century British amateur mathematician Henry Perigal. If the sides of the two squares forming the tiling are the numbers a and b, then the closest distance between corresponding points on congruent squares is c, where c is the length of the hypotenuse of a right triangle having sides a and b. For instance, in the illustration the two squares in the Pythagorean tiling have side lengths 5 and 12 units long, and the side length of the tiles in the overlaying square tiling is 13, based on the Pythagorean triple... By overlaying a square grid of side length c onto the Pythagorean tiling, it may be used to generate a five-piece dissection of two unequal squares of sides a and b into a single square of side c, showing that the two smaller squares have the same area as the larger one.

Reposted from 2012 to accompany a related post. 

Tessellated pavement tiles in Granada

It just absolutely fascinates me that each of these hexagonal tiles has the same pattern, but that the resultant overall result can be so variable.

Tessellation longread in Wikipedia.

The price of eggs

I've been doing my own grocery shopping for the past 58 years.  I open the egg carton to peek inside and check the bottom for wet spots.  And I'm aware of the pricing.  Yesterday on a weekly visit I was somewhat startled by the low price, so I searched for a chart.  Found this one at a Federal Reserve website:

The gap in the curve several months ago reflects the absence of data that occurred during the government slowdown/shutdown, but the trend is clear.

The fallling price was a somewhat startling revelation since my focus (and most consumers' focus) has been on rising prices for gasoline, diesel, jet fuel, fertilizer etc etc since we started the war, and the anticipated roll-on effect on other commodities and goods.  I had frankly forgotten about the reason for the spike upward in egg prices in recent years, which is explained here.

Lots of other interesting data available at that Federal Reserve website for those interested.

U.S. Consumer Sentiment Index

Offered without comment from me.

World Central Kitchen in Gaza

Regular readers here may remember that World Central Kitchen is far and away my favorite charity.  I believe my last report on their work in Gaza was two years ago.  Because I'm a regular contributor, today I received an email update, which I'll share:

WCK is still cooking in Gaza—and we want to be direct with you about what is changing and why. Due to significant financial pressure, including rising food and fuel costs driven by regional conflict, WCK is making the difficult decision to reduce the scale of our meal distribution in Gaza. This decision reflects financial reality, not a reduction in need. Our teams remain on the ground, delivering hundreds of thousands of hot meals every day.

Since the start of the conflict in 2023, WCK has invested more than half a billion dollars feeding the people of Gaza—surging to one million hot meals a day. But no single NGO, funded primarily by small private donors, can sustain that level of output indefinitely. We specialize in emergency food relief, not long-term food security—and the long-term responsibility of feeding Gaza cannot rest on our shoulders alone. The people of Gaza have lost their homes and their economy. Governments, institutions, and international partners must commit the sustained, secure funding this crisis demands.

We know you have questions—here are answers to what we are being asked more frequently.

They note in the email that the upcoming wildfire season is expected to break records.  It's shameful to consider how much $ is currently being expended on weaponry and reparations to insurrectionists when basic human needs like this go unmet. 

U.S. treasuries being dumped

China has also publicly announced that they are lightening their reserves of U.S. treasuries.  

Here is a table of foreign holders of U.S. treasuries (latest data are from before we started the war).  The consequences of such shifts are complex and may be country-specific.  It's a bit over my head, so interested readers will need to do some searching or offer suggestions in the Comments.

The author of that Facebook post is a world-renowned economist and the former President of Queens College, Cambridge.

Addendum:  Here's an article about what happens if other countries stop buying U.S. debt.  I haven't read it yet.  It was written a year ago, so probably in response to tariff matters, but the principles outlined may be valid.

I don't know if all of this is true...

... but I have heard that parts of it are true, and if most or all of it is true, it's tremendously interesting and important.

Santa Marta, Colombia.  An activist demonstrating during a conference aimed at transitioning away from fossil fuels.  Photograph: Iván Valencia/AP, via The Guardian.

Neolithic trackway in Somerset

This interesting image was in a Guardian article describing and illustrating several of the most ancient pathways in England.  Shown above is the "Sweet Track", "built nearly 6,000 years ago (3806BC) by early farmers who needed access to an island, the collapsed boardwalk was preserved in peat."

It was interesting to me because of the technique used to secure the planks - suggestive of more sophisticated engineering than the most common corduroy roads.

The other ancient paths at the Guardian link are also interesting for different reasons; the article is worth a quick browse if you enjoy walking.

Top image cropped for size from the original, credit Craig Joiner Photography/Alamy.  Lower diagrams from Avalon Marshes.

Satire

Fortress Washington, drone-proofed and with friendly snipers

Holly Baxter of The Independent reported today that in the midst of all the chaos—including his war on Iran and rising fuel and food prices—Trump called a sudden, urgent press conference today as Blanche was testifying. But what was on his mind was not Iran, or prices, or his corrupt agreement with the Department of Justice. He wanted to talk about his ballroom.

Trump’s comments in that press conference have invited commentary suggesting he is turning the White House into a fortress. Describing the ballroom, he said: “Between the drone-proofing, the missile-proofing, we have ah, and the drone capacity upstairs, we can have all sorts of military—I hate to use the word snipers—but we have great sniper capacity. It’s built for our snipers, not enemy’s snipers, our snipers. And because of the height we get a very clear view of everything all over Washington.”

Via Letters from an American, May 19.  This is the report in The Independent.  I know some readers think this is normal behavior, not mental illness, but really... WTF?

Odd structure found "in the middle of nowhere"

I was delighted to discover what it is and was impressed to note how well-designed and constructed it is.  Read all about it at the whatisthisthing subreddit.

The economic cost of the Iran war

The embed is a screencap because I don't know how to embed the live graphic.  So by the time you read this the numbers are way out of date.  Go to this link to see the live numbers changing:

Energy Cost Tracker

Note that this does not include the cost of the military equipment being expended or destroyed (or replaced).  This is important, so please share the link freely.

A tip of the blogging cap to reader Kyle, who included the energy cost tracker link in a comment.

The antithesis of "diplomacy"

If anyone sees a way to a "negotiated settlement" based on this rant, please let me know.  This is a rant by a mentally ill man facing defeat and unable to accept such.  My only hope is that his hand-picked military brass who replaced the seasoned veterans can find a way to sweet-talk him out of putting more U.S. troops at risk.

Addendum:  Live coverage from Al Jazeera English at 1300 CDT (~10 pm in Qatar) indicates that Iran and the IRGC are expecting a resumption of active hostilities targeting Iranian government and military facilities very soon.  Lots of coverage there also of the Ebola outbreak in Africa; much less attention to that in Western news media.

Update:  A couple hours later Trump posted this message on Truth Social, indicating a postponement of aggressive action, which he describes as being in response to pleas from Gulf leaders (not because his military is advising him against it).

So perhaps less risk fot hot war (unless Netanyahu opts to intervene), but no obvious way forward to resolution, which means increasing pressure on world oil prices.

Why bus steering wheels are so big

I had never considered the question before, but thought I'd share an answer I discovered in the explainlikeImfive subreddit:

Back in the late Cretaceous when I was learning to drive, most cars and trucks did not have power steering. Larger/heavier vehicles had larger steering wheels because you actually had to muscle the front rolling wheels around to turn the vehicle, and the additional leverage from a larger steering wheel was important. (Incidentally, you could tell if one of your tires was low because it literally got harder to steer. Local truckers and other frequent drivers tended to build up their arm muscles from navigating corners.) My dad's little MG sports car had a 13" steering wheel; my VW van had a 16" steering wheel; pickup trucks' were more typically 17"; and buses were more typically 18-20".

Nowadays, practically every vehicle has power steering assist, but (CyberTruck aside) they're basically all designed so that if the power steering fails, you can still steer the car -- it's just harder to do so. So the big bus steering wheels are still around, as a safety measure.

Additional information at National Bus Sales:

A bus driver has to maneuver through lanes the same size as small cars but with a lot less clearance. With a smaller steering wheel, any adjustments could be too abrupt for safety. With a larger steering wheel, you can make a correction without changing the turning radius of the bus too dramatically. Smaller adjustments won’t cause any instability.

And this response to why the wheel is more horizontal:

This feature has changed over the years and varies in vehicles, but initially, the large steering wheels on buses sat almost horizontally. The driver sits directly above the tires, so for the steering column to correct the tires, the steering wheel needs to be positioned at a different angle. More recent bus models have options for the driver to adjust the position of the wheel.