Best Of

While I’ve been driving around delivering parcels today, I’ve been thinking about Pythagorean triples and circles

I believe that every right angle triangle with all whole number length sides perfectly contains a circle with a whole number radius

I love geometry even more today

I imagine it'd be something more widely known if it was true. Heavy geometry really went entirely out of fashion, but as you've seen it's sort of cool.

And actually for integer sides you're right I guess because you can just write down the solution. Are there non-Integer sides for which this is true? Seems likely.

I presume that there are infinite numbers of right angle triangles that will fit perfectly around a circle with a radius of 1

I just love the fact that the most famous triangle with given side lengths is the 3,4,5; and that triangle perfectly contains a circle with the most famous area related number as its area - pi!

I didn't know about the radius of circles that you pointed out. Almost all of maths is stuff we don't know though. A Sunday evening doesn't find me at my mathematical best.

I'll try to report back tomorrow. Alcohol not featuring.

Discovering all sorts of cool and unexpected stuff in maths is what it's all about. Do look!

I'm not sure that anything will ever beat when I first understood Bayes's theorem

I am a great fan of Bayes's theorem and have used it tio solve all sorts of problems. However I do find it hard to apply to lost things, and wonder if any of you can advise on this. As I get older and more forgetful, and my wife buys more and more clutter, i spend an increasing proportion of my time looking for things that I have lost.
I am using it by listing all the places I could have lost it, estimating the probability of each. Then once I have looked in them all withiut success, I estimate the chance of not finding it each place I have looked if it had been there, and multiplying it by the probability of it having been there.
Then I look in each place in the order of the resulting relative probability. The problem comes when I lose the piece of paper with the calculations on it and then have to use Bayes's theorem to find it.

Ukrainian air strikes are certainly escalating. But so are Russian ones. Given Trump cut off all US aid to Ukraine it could certainly be worse but I'm not optimistic about things right now.

Trump is trying to force Zelensky into an awful deal because he wants to make money from business with Russia. Russia is happy to keep fighting, because Putin believes his army is winning, so is sticking to its maximalist war aims. The Europeans are running around like headless chickens with no direction or cohesion.

It's a bad situation.

Ukrainian successes with long-range strikes, or the counterattack around Kupiansk, are welcome, but they're not enough to turn the tide. And they're kinda bittersweet because they show what would be possible if Europe found the resolve and sense of purpose to fully back Ukraine.

Instead we're choosing to do enough to keep Ukraine fighting, but not enough to help them win. It's such a big mistake.

It clearly demonstrates why the EU can’t be the core of European defence structures. NATO and JEF have to be the way forward (with or without the US)

It’s been blindingly obvious since 2023 (but the signs were there in autumn 2022) that almost no one in the Western alliance wants Ukraine to “win”. Or rather they are too frightened of what might follow a decisive Russian loss - likely some
combo of wounded animal behaviour by Putin, disruption to global commodities markets and stray nukes ending up with regional Russian war lords.

This grates with me, but all of us have to accept that we have not seen the intelligence the decision makers have. It is lazy in the extreme to think the current US government’s main goal is to build a hotel in Moscow. There has been a remarkably consistent common position among the core counties even after domestic political transitions. Is what it is.

Given this seemingly immovable reality, the best thing for the Ukrainians really does now feel like getting done whatever deal will get the fighting to finish as soon as possible, followed by an influx of weapons and funding.

Is it ?
If Europe is so terrified that they can't defeat a Russian invasion, how do they deter a repeat in a few years' time ?

The sad and deeply cynical answer is what the west required was the exhaustion of the incredible quantities of kit that Russia had inherited from the Soviet Union which made them a threat. The brave resistance of Ukraine and the imbecility of the psychopath in the Kremlin mean that has been achieved. In addition the loss of over 1m men of fighting age (even on a broad definition) together with at least another million who fled has turned the already poor demographics of Russia into a catastrophe. Combine that with the profound economic damage and you are left with a country that would very probably struggle to take on Poland in a conventional war today and would have no chance whatsoever in 3 or 4 years time.

We owe Ukraine an incredible debt of gratitude for massively degrading a serious threat to our way of life. But countries, and certainly governments, are not sentimental. I hope we honour our debt and their sacrifice but I am not holding my breath.

As for the idea that an exhausted Russia is some threat to western Europe in any conventional sense? Please, don't be ridiculous.

"Conservatives to scrap plans to ban new petrol and diesel cars by 2030"

https://www.itv.com/news/2025-12-14/conservatives-to-scrap-plans-to-ban-new-petrol-and-diesel-cars-by-2030

So the Tories who won't get a say before late 2028 at the earliest and probably 2029, will try to reverse something that will be 99% by then...

It's the sort of stupid headline that looks like they are doing something when really it's a complete nothingburger.

I don't think that is the point

There is an increasing demand both here and in the EU to delay the ban and Badenoch's call is much in line with a changing mood on this policy

The irony is I believe Euro 7 is going to increase petrol engine costs to the point that BEV's will be cheaper.

Euro 7 has stricter rules on unburnt hydrocarbons. The only (cost effective) way to meet them will be to reduce cylinder count. Unburnt HC emissions vary with cylinder radius but engine output is proportional to displacement and hence the square of the cylinder radius. This raises a marketing problem, as Mercedes discovered, because people will not pay shitloads of money for 4 (and fewer) cylinder cars regardless of whether achieve or even exceed performance parity.

ICE engines will hang around in hybrid form for a while but KB is just railing against the inevitable to catch the attention of GB News watching morons who hate BEVs on principle.

Meanwhile BYD and similar are eating the legacy automakers market. The market for ICE cars is dying. Its like trying to sell C41 and E6 film in the digital age.

For anyone with a driveway or other off-road parking where charging at home is an option, it absolutely is.

For people who needs public charging on the other hand, which includes tens of millions of people, the market is not dying.

Even before the foolhardy introduction of an EV per mile tax, it was already cheaper to drive an efficient petrol vehicle over a publicly-charged electric one, despite the fact that the petrol tax is almost entirely taxation and the EV charging cost is not. With the per mile EV tax, that disparity has grown even worse.

Should private transportation only be the preserve of those with off-road parking?

It's quite possible to have charging with on-road parking, I have seen it in the Netherlands

just upgrade street lights surely and stick plugs in them

Yes - it's a good idea, but that would be limited to a small fraction of streetlights (perhaps 10%), without replacement of the entire infrastructure.

The current free for charging is the difference between the headroom created by the move to LED bulbs, and the max rating of the distribution cables.

On the upside that also means that turning one or two spaces per street into charging points means that they can be set aside without too much distruption.

The near future for road transport is fully autonomous and electric. People are going to look back on conversations about “bans” for new hydrocarbon road vehicles with a nostalgic chuckle. It’s not illegal to ride about in a horse and cart but the only people my way who do are the local scrap metal merchants.

Only if we can sort out charging issues.

When it is considerably cheaper to charge a petrol vehicle than an electric one, as it is currently for those without off-road parking, people will act rationally according to their incentives.

If you want the transition to electric to proceed, and I do, then we need to tackle this problem, not just stick our heads in the sand and pretend it does not exist.

4 years is not long to sort out this issue for the tens of millions of homes without off-road parking, that currently have and require a car.

Don’t worry about it. There will be little economic incentive for those without off street parking to own their own fully autonomous EV.

That's the point. 🤦‍♂️

The alternative to owning their own EV, is owning their own ICE vehicle. As much as some people would wish away the private ownership of vehicles, it has not happened and is not going to happen within 4 years.

Unless we can tackle the fact it is considerably cheaper to fuel an efficient vehicle by petrol (despite that being almost all tax already), than it is via public-charging, which tens of millions of homes require, the market alone is not going to magic away that problem.

We need serious investment in a solution for charging, or the transition is not going to be completed. Saying I'm alright as I have at-home charging is not a serious solution for those who don't.

The Government have just introduced a BEV per mile tax that makes it even more expensive to drive electrically, even if you don't charge at home, and is doing absolutely nothing I can see to sort out charging issues nationwide to have them be resolved within 4 years if the 2030 cessation of ICE sales is meant to take effect.

I know you don’t understand what I’m saying. That’s ok. You will do once the crushing certainty of economic gravity is measured. Not long now.

You're sounding like Doctor No.

A pity I am allergic to cats. Or was that the other one, I get the mixed up.

Could be worse, you could have been on Grauniad Blind Date this week.

https://www.theguardian.com/lifeandstyle/2025/dec/13/blind-date-tom-rita

I dunno. She seems well presented and a decent judge of character

Well, she could have been the cat owner!

Were I single, it would be an interesting experiment to go on that. See if there’s a guardian reader out there who would be willing to overlook my political outlook, which I assume they would find rather unsavoury. Leon managed it for a time so perhaps it’s not impossible. Is he coming back ever by the way?

Leon and you went on a date?!

The mind boggles…

We need serious investment in a solution for charging, or the transition is not going to be completed.

That's the nub of the issue, though you could say the same about a lot of issues in Britain today. And there's less than no money left.

It's not like the solutions for charging don't exist, it's just that it needs money to implement them.

Indeed.

And without implementation, then the status quo solution will be the continuation of millions of people buying, driving and refuelling ICE vehicles.

Well, that is a logical conclusion, but I wouldn't be surprised if the government simply drove their popularity off another cliff by sticking to the timetable and pushing another increase in the cost of living onto people by forcing them to rely on more expensive public charging.

Expect the price of second hand ICE cars to be high in the 2030s.

The problem with the roo-taxi vision is it's great for some use cases, and pretty rubbish for others. My work car is basically a self propelled portable toolbox with seating for 5. I don't want to be lugging three socket sets, a couple of work lamps, a roll of spanners, a tin of copper slip, a hammer, a couple of adjustable spanners, two tape measures, a battery drill, a box of torx drives etc into a robo-taxi every time I hail one, but I use various bits of this lot in various different places, and having it all in boot of the car means I never have to worry about taking it with me.

My wife's car has two child seats semi-permanently installed, plus a pram, baby change bag, some shopping bags etc in the boot. Just the childseat aspect alone sounds like a nightmare enough if we're hailing robo-taxis.

And there just isn't much money to be saved. Wife's car was £4k 8 years ago. It doesn't do many miles, although it needs a decent range as it does a couple of trips to the in-laws a year. We put ~£30 of petrol in it a month, insurance is about £300 pa, and I service it annually for £25 in consumables and 20 mins of my time. Even with a little MOT work, I doubt our total outgoings on it are £1k a year.

Sounds a reasonable theory of the case.

https://x.com/MikeLevin/status/1999625620549161290
For those, who still had ANY doubts or hopes, let me explain the simple truth of our reality.

1) The only interest the current US administration has is making sure everyone in Trumps circle gets their profit. People in that circle are extremely cynical, possess no moral compass or empathy, and they have employed aggressive business practices throughout their lives. Nothing personal, just business.

2) Whenever some “humanitarian” deeds happen, they happen ONLY as a cover for getting a new source of profit. For example, the release of Belarusian political prisoners was done in exchange for desanctioning the BelarusKaliy* ; I can bet €100 that somebody from the Trump’s circle is already in touch with the company, and has arranged some very profitable scheme.

3) Same goes with all negotiations with Russia. Putin, according to Trump, is a typical man of his circle. One you can do business with, disregarding all these boring ethical and national security concerns. Again, I am pretty sure, the main content of the private “negotiations” has nothing to do with ensuring peace; it is purely business talk.

4) Russia is an immensely rich country in terms of resources. Many American (and quite some European) businessmen are waiting impatiently for any solution that gives them access to the Russian resources and money associated with.

5) People tend to live in denial when something so earth-shuttering as the large scale, open corruption in the US happens. They start searching for a second layer, say there are deeper forces in play. No, it is what it looks like: the people who currently rule the most powerful country on Earth, try to make as much profit as possible. No matter how...

*At the same time the Administation desanctions Belarus potash, it's imposing tariffs on the Canadian supply.

"Conservatives to scrap plans to ban new petrol and diesel cars by 2030"

https://www.itv.com/news/2025-12-14/conservatives-to-scrap-plans-to-ban-new-petrol-and-diesel-cars-by-2030

So the Tories who won't get a say before late 2028 at the earliest and probably 2029, will try to reverse something that will be 99% by then...

It's the sort of stupid headline that looks like they are doing something when really it's a complete nothingburger.

I don't think that is the point

There is an increasing demand both here and in the EU to delay the ban and Badenoch's call is much in line with a changing mood on this policy

The irony is I believe Euro 7 is going to increase petrol engine costs to the point that BEV's will be cheaper.

Euro 7 has stricter rules on unburnt hydrocarbons. The only (cost effective) way to meet them will be to reduce cylinder count. Unburnt HC emissions vary with cylinder radius but engine output is proportional to displacement and hence the square of the cylinder radius. This raises a marketing problem, as Mercedes discovered, because people will not pay shitloads of money for 4 (and fewer) cylinder cars regardless of whether achieve or even exceed performance parity.

ICE engines will hang around in hybrid form for a while but KB is just railing against the inevitable to catch the attention of GB News watching morons who hate BEVs on principle.

Meanwhile BYD and similar are eating the legacy automakers market. The market for ICE cars is dying. Its like trying to sell C41 and E6 film in the digital age.

For anyone with a driveway or other off-road parking where charging at home is an option, it absolutely is.

For people who needs public charging on the other hand, which includes tens of millions of people, the market is not dying.

Even before the foolhardy introduction of an EV per mile tax, it was already cheaper to drive an efficient petrol vehicle over a publicly-charged electric one, despite the fact that the petrol tax is almost entirely taxation and the EV charging cost is not. With the per mile EV tax, that disparity has grown even worse.

Should private transportation only be the preserve of those with off-road parking?

It's quite possible to have charging with on-road parking, I have seen it in the Netherlands

just upgrade street lights surely and stick plugs in them

Yes - it's a good idea, but that would be limited to a small fraction of streetlights (perhaps 10%), without replacement of the entire infrastructure.

The current free for charging is the difference between the headroom created by the move to LED bulbs, and the max rating of the distribution cables.

On the upside that also means that turning one or two spaces per street into charging points means that they can be set aside without too much distruption.

The near future for road transport is fully autonomous and electric. People are going to look back on conversations about “bans” for new hydrocarbon road vehicles with a nostalgic chuckle. It’s not illegal to ride about in a horse and cart but the only people my way who do are the local scrap metal merchants.

Only if we can sort out charging issues.

When it is considerably cheaper to charge a petrol vehicle than an electric one, as it is currently for those without off-road parking, people will act rationally according to their incentives.

If you want the transition to electric to proceed, and I do, then we need to tackle this problem, not just stick our heads in the sand and pretend it does not exist.

4 years is not long to sort out this issue for the tens of millions of homes without off-road parking, that currently have and require a car.

Don’t worry about it. There will be little economic incentive for those without off street parking to own their own fully autonomous EV.

That's the point. 🤦‍♂️

The alternative to owning their own EV, is owning their own ICE vehicle. As much as some people would wish away the private ownership of vehicles, it has not happened and is not going to happen within 4 years.

Unless we can tackle the fact it is considerably cheaper to fuel an efficient vehicle by petrol (despite that being almost all tax already), than it is via public-charging, which tens of millions of homes require, the market alone is not going to magic away that problem.

We need serious investment in a solution for charging, or the transition is not going to be completed. Saying I'm alright as I have at-home charging is not a serious solution for those who don't.

The Government have just introduced a BEV per mile tax that makes it even more expensive to drive electrically, even if you don't charge at home, and is doing absolutely nothing I can see to sort out charging issues nationwide to have them be resolved within 4 years if the 2030 cessation of ICE sales is meant to take effect.

I know you don’t understand what I’m saying. That’s ok. You will do once the crushing certainty of economic gravity is measured. Not long now.

I know exactly what you're saying, that taxis are going to displace private ownership.

Taxis are not a new invention, and yet people want their own vehicles anyway. For a plethora of very good reasons.

The idea that within 4 years that is magically going to change, let alone ever going to change, is just insanity and wishful thinking by you.

Roadside charging would be easier if residents had a maximum of one car each. I know there are good reasons why one car per household is not always practical, but why does anyone need more than one personal car?

Sometimes people have more than one because the vehicles serve different purposes, eg a personal car for driving and a van for work.

Sometimes there's more than one person in the household and each of them might need their own vehicle.

If there's 2 parents working in a household they might both need a vehicle. If there's adult children who have not yet been able to afford a home of their own yet, they might too on top of their parents vehicles.

Either way though, roadside charging solutions have not been rolled out yet, and I don't see any concrete plans in place to do so by 2030, which makes a mockery of the 2030 deadline for ICE. Either tackle the issue, or be realistic.

If there’s not a generalised autonomous EV in mass production for the UK market then you are right. But if there is, the issue of road side charging will become redundant. Even for privately owned robotaxis, it will be far cheaper and quicker to build charging hubs that your car drives itself to park and charge at overnight.

It’s going to take a while to replace the whole fleet of course. But by 2040 (?) we will likely have see a transformative impact on our urban road architecture. Rows and rows of parked cars in narrow streets sitting there doing nothing is unlikely to be a thing any more.

Is the 2030 deadline the right one? I dunno. But it might surprise you how quickly demand for brand new ICEs gets crushed when this product is widely launched.

When you think about it, it's actually pretty strange that people are allowed to store their cars on the public highway, thus obstructing other users of said highway. If I had my way, I'd ban the storage of cars on the road completely. This would both free up our streets and eliminate the need for roadside charging. However, I appreciate this policy might be a little difficult to push though politically, and I will keep quiet about it during my paper candidacy for the upcoming council elections.