These 10 Maths Facts Will Blow Your Mind (I Promise)

Here are my top 10 favourite mind blowing maths facts for all the maths lovers out there. And for the haters too, just so you know what you’re missing out on.

Breakthrough on 125 Year-Old Physics Problem

David Hilbert’s Sixth Problem is 125 years old and asks for an axiomatic foundation of physics. A good place to start with this, said Hilbert, would be fluid dynamics – physicists should be able to prove that our fluid dynamics equations are rooted in how we understand atoms behave when they bump into each other. This would also explain the origin of irreversibility in our lives, or the “arrow of time” as physicists like to say. Over a century later, mathematicians have made a major breakthrough in this arena. Let’s take a look.

Einstein-Tile Discovered in Nature

Einstein-tiles, discovered in 2023, are shapes that can be used to cover a surface without gaps with a pattern that never repeats. A group of researchers claim they’ve found one of these rare shapes in a group of molecules that spontaneously configured themselves into a surface cover that behaves like an Einstein-tile. Let’s take a look.

This Physicist Says We’re Using Maths Entirely Wrong

Intuitionist mathematics is the idea that the entire discipline of maths is a mental construct based on human thought rather than a platonic realm of eternal truths. According to physicist Nicolas Gisin, one of this idea’s biggest proponents, that maths is based on human intuition is the reason why quantum mechanics seems so strange. Let’s take a look.

This Physics Problem Can’t Be Solved

According to physicists, everything in the universe is physics. Chemistry, biology, why you crave cake at 11:00 pm – you name it, it can eventually be explained by physics. But what if some physics problems require calculations that are just impossible? The authors of a new paper claim they’ve found a measurable quantity in a physical system that just can’t be computed. Let’s take a look.

Scientists Discover a Law of Natural Laws

People often talk about mathematics as a sort of language. Languages are known to have near-universal regularities, which brings up the question of whether the mathematics that we use in physics has similar features. According to a new study which just appeared, this is indeed the case: The maths of natural laws has a hidden law itself — but it’s not the same law that governs other languages. What might this mean? Let’s take a look.

A New Pattern in Nature

Mathematicians are obsessed with patterns. They've been obsessed for so long you'd think they've got it figured all out. But just recently they discovered a new type of pattern that is everywhere in nature. Once you have seen it, you can't help but notice that it's really everywhere.

String Theorists Have Calculated the Value of Pi

String theorists have calculated the value of pi. Didn’t we already know the value of pi? At least the first one hundred trillion digits or so. Yes, but this is an interesting story about the relation between maths and physics. Let’s have a look.

This video comes with a quiz which you can take here:

This New Idea Could Explain Complexity

The universe creates complexity out of simplicity, but despite many attempts at understanding how, scientists still have not figured it out. We do know that complexity relies on the emergence of new features and laws, but then again we don't understand emergence either. The first step must be to clearly define what we are talking about and to measure it. A group of scientists now put forward a way to do exactly this. Let’s have a look.

2+2 doesn't always equal 4

[This is a transcript of the video embedded below.]



2 plus 2 makes 5 is the paradigmatic example of an obvious falsehood, a falsehood that everybody knows to be false. Because 2 plus 2 is equal to 1. Right? At the end of this video, you’ll know what I am talking about.

George Orwell famously used two plus two equals five in his novel nineteen eighty-four as an example for an obviously false statement that you can nevertheless make people believe in.

The same example was used already in seventeen eighty-nine by the French priest and writer Emmanuel Sieyès in his essay “what is the third estate”. At this time the third estate – the “bourgeoisie” – made up the big bulk of the population in France, but wasn’t allowed to vote. Sieyes wrote
“[If] it be claimed that under the French constitution two hundred thousand individuals out of twenty-six million citizens constitute two-thirds of the common will, only one comment is possible: it is a claim that two and two make five.” This was briefly before the French revolution.

So you can see there’s a heavy legacy to using two plus two is five as an example of an obvious untruth. And if you claim otherwise that can understandably upset some people. For example, the mathematician Kareem Carr recently got fire on twitter for pointing out that 2+2 isn’t always equal four.

He was accused of being “woke” because he supposedly excused obviously wrong math as okay. Even he was surprised at how upset some people got about it, because his point is of course entirely correct. 2+2 isn’t always equal to four. And I don’t just mean that you could change the symbol “4” with the symbol “5”. You can do that of course, but that’s not the point. The point is that two plus two is a symbolic representation for the properties of elements of a group. And the result depends on what the 2s refer to and how the mathematical operation “+” is defined.

Strictly speaking, without those definitions 2+2 can be pretty much anything. That’s where the joke comes from that you shouldn’t let mathematicians sort out the restaurant bill, because they haven’t yet agreed on how to define addition.

To see why it’s important to know what you are adding and how, let’s back up for a moment to see where the “normal” addition law comes from. If I have two apples and I add two apples, then that makes four apples. Right? Right.

Ok, but how about this. If I have a glass of water with a temperature of twenty degrees and I pour it together with another glass of water at 20 degrees, then together the water will have a temperature of 40 degrees. Erm. No, certainly not.

If both glasses contained the same amount of water, the final temperature will be one half the sum of the temperatures, so that’d still be 20 degrees, which makes much more sense. Temperatures don’t add by the rule two plus two equals four. And why is that?

It’s because temperature is a measure for the average energy of particles and averages don’t add the same way as apples. The average height of women in the United States is 5 ft 4 inches, and that of men 5 ft 9 inches, but that doesn’t mean that the average American has a height of 11 ft 1. You have to know what you’re adding to know how to add it.

Another example. Suppose you switch on a flashlight. The light moves at, well, the speed of light. And as you know the speed of light is the same for all observers. We learned that from Albert Einstein. Yes, that guy again. Now suppose I switch on the flashlight while you come running at me at, say, ten kilometers per hour. At what velocity is the light coming at *you. Well, that’s the speed of light plus ten kilometers per hour. Right? Erm, no. Because that’d be faster than the speed of light. What’s going on?

What’s going on is that velocities don’t add like apples either. They merely approximately do this if all the velocities involved are much smaller than the speed of light. But strictly speaking they have to be added using this formula.

Here u and v are the two velocities that you want to add and w is the result. C is the speed of light. You see immediately if one of the velocities, say u, is also the speed of light, then the resulting velocity stays the speed of light.

So, if you add something to the speed of light, the speed of light doesn’t change. If you come running at me, the light from my flashlight still comes at you with the speed of light.

Indeed, if you add the speed of light to the speed of light because maybe you want to know the velocity at which two light beams approach each other head on, you get c plus c equals c. So, in units of the speed of light, according to Einstein, 1+1 is 1.

That’s some examples from physics for quantities that just have different addition laws. Here is another one from mathematics. Suppose you want to add two numbers that are elements of a finite group, to keep things simple, say one with only three elements. We can give these elements the numbers zero, one, and two.

We can then define an addition rule on this group, which I’ll write as a plus with a circle around it, to make clear it’s not the usual addition. This new addition rule works like this. Take the usual sum of two number, then divide the result by three and take the rest.

So, for example 1+2 = 3, divide by three, the rest is 0. This addition law is defined so that it keeps us within the group. And with this addition law, you have 1 plus 2 equals 0. By the same rule 2 plus 2 equals one.

I know this looks odd, but it’s completely standard mathematics, and it’s not even very advanced mathematics, it just isn’t commonly taught in school. This remainder after division is usually called the modulus. So this addition law can be written as the plus with the circle equals the normal plus mod 3. A set of numbers with this addition law is called a cyclic group.

You can’t only do this with 4, but with any integer number. For example if you take the number 12, that just means if you add numbers to something larger than 12 you start over from zero again. That’s how clocks work, basically, 8+7=3, add another 12 and that gives 3 again. We’re fairly used to this.

Clocks are a nice visual example for how to add numbers in a cyclic group, but time-keeping itself is not an example for cyclic addition. That’s because the “real” physical time of course does not go in a circle. It’s just that on a simple clock we might not have an indicator for the time switching from am to pm or to the next day.

So in summary, if you add numbers you need to know what it is that you are adding and take the right addition law to describe what you are interested in. If you take two integers and use the standard addition law, then, yes, two plus two equals four. But there are many other things those numbers could stand for and many other addition laws, and depending on your definition, two plus two might be two or one or five or really anything at all. That’s not “woke” that’s math.