 This history of science video will be on the question, did Witt discover the Leach lattice? There are obviously two further questions, who was Witt and what is the Leach lattice and why should anybody care about it? Well, first of all, Ernst Witt was a rather famous German mathematician. He's the guy who discovered things like Witt vectors and proved the Poincaré-Berkov-Witt theorem. He had a rather odd early life. He joined the SA, which was a sort of rather unpleasant street gang associated with Hitler's National Socialist Workers Party. On the other hand, he doesn't seem to have taken its ideology all that seriously because his supervisor was, in fact, Eminirta, who was Jewish and had to leave Germany because of the activities of this party. The rumor is that Witt used to turn up to Eminirta's house to listen to her lectures wearing a Nazi party uniform, which suggests he was somewhat clueless. Anyway, Leach discovered the Leach lattice in about 1965. So what is the Leach lattice? Well, the Leach lattice can be thought of as a sort of sphere packing. It's a sphere packing in 24 dimensions, which is a bit difficult to draw. So I will just draw it a sphere packing in two dimensions. So there's one obvious sphere packing in two dimensions. So a sphere in two dimensions is just a circle. So you could arrange spheres or other circles like this, and that would give you a packing in two dimensions. Well, obviously you can pack them a bit better if you arrange them like this. So instead of having a sort of rectangular pattern, I could arrange them in this sort of hexagonal pattern. And you can see this packs them a bit more densely. And this leads to the question, what is the densest way of packing spheres in n dimensions? And working at the answer, this is a really very difficult problem in anything more than two dimensions. And Witt was looking at, not Witt, Leach was looking at packings in 24 and 25 dimensions. And he knew a quite dense packing in 24 dimensions and was trying to use this to get a good packing in 25 dimensions. For example, if we want to get a good packing in three dimensions, we can take this two-dimensional packing and add extra spheres over these holes here, where you see that there are sort of holes in the packing. So what Witt did was he took this 24-dimensional packing that he knew about and tried to find the biggest holes in it that he could find on the grounds he would then put spheres on top of those holes in 25 dimensions and get a nice packing in 25 dimensions. And probably to his surprise, he found the holes in the 24-dimensional lattice packing was so big that there was room to put extra spheres in those holes in 24 dimensions. And this was how he found the Leach lattice, which was a very dense packing of spheres in 24 dimensions. So in two dimensions, you see this packing here, each sphere touches six other spheres. In 24 dimensions, the Leach lattice as the property of each sphere touches one, nine, six, five, six, oh, spheres. So it's really quite large. You can also ask what's the symmetry group of this lattice? So a symmetry group means all symmetries. For instance, if we take this packing in two dimensions, suppose I fix this sphere here, you can see there are 12 symmetries fixing that sphere because I can rotate these six spheres. And as well as those six rotations, I can also do reflections which doubles the number of symmetries. So there are 12 symmetries. So John Leach didn't actually work out the number of symmetries, but try to persuade other people to on grounds the answer might be interesting. And nobody bothered until he met John Conway and managed to persuade John Conway to find the symmetry group. And John Conway found the number of symmetry groups as given by this number here, eight, three, one, five, five, five, three, six, one, three, zero, eight, six, seven, two, zero, zero, zero, zero. So it's got absolutely huge symmetry group. And this generated a lot of excitement at the time because people are trying to classify things called finite simple groups, which are basically the building blocks that all other simple groups are built out of. They're the sort of analogs of elements in chemistry. Every chemical is built out of elements and every group is built out of simple groups, more or less. So Conway became quite famous because of this and went around the world giving lectures on this group. The simple group isn't quite this group. It turns out this group is sort of twice the size of the actual simple group that Conway discovered. The leach lattice also turns out to be the densest packing in 24 dimensions. And this was proved by Cohen, Kuma, Miller, Radchenko and Maria Viazowskaya using an amazing idea that Marina Viazowskaya had a few years before. And it's particularly remarkable because there are hardly any dimensions that we know for certain what the densest sphere packing is. We don't know what the answer is in four, five, six or seven dimensions, for example. But we do know the answer in 24 dimensions. So where does bit come in? Well, the leach lattice is one of 24 nemylattices. So the nemylattices are all in 24 dimensions and they have the property that they're what are called even and unimodular. So let me explain this. Well, if you take the centers of the sphere packing, they will sometimes form a nice regular lattice. For instance, it might look something like this. So a lattice is a kind of regular array of points. So here's a two-dimensional lattice. The nemylattice is a similar only in 24 dimensions. And the nemylattice has the following property. First of all, the distance between any two points is the square root of an even number. And secondly, there is one lattice point per unit volume. And lattices with this property are called even and unimodular. And nemy actually classified the nemylattices and found there are 24 nemylattices. So the number 24 turns up in two ways. There are 24 nemylattices and they live in 24 dimensions. And people have wondered if this is just a strange coincidence or if there's some significance, but it's probably just a coincidence, but who knows? Anyway, nemy classified the nemylattices and the list of them looks something like this. In fact, it looks exactly like this. So here's a list of them. I'll just magnify this so you can see it. So here is the list of the 24 nemylattices and down at the bottom is the leach lattice because the leach lattice is one of the 24 nemylattices. Nemy classified these a few years after leach found the leach lattice. You can also draw a picture of the nemylattices looking something like this. So this graph here has a circle for each nemylattice and the lines between show how the nemylattices are related to each other. As you can see, the nemylattices form a rather complicated sort of object. Anyway, let's go back to Witt. Witt has a very puzzling and cryptic note in one of his papers here. So again, let me magnify it so you can see it, which probably won't help unless you read German. So here he has a note in this paper. This was a paper he wrote in 1941. You can see up there. And in it he says that he found more than 10 different classes for Gamma 24. Well, Gamma 24 is his name for the nemylattices, except of course they weren't called nemylattices in those days. So he said he found more than 10 of the nemylattices and it's a bit of a puzzle, which ones he found. Let me give you an example of explicit construction of just one nemylattice. Instead of giving you a construction of a lattice in 24 dimensions, I'll give a construction in eight dimensions. So suppose you take the set of all vectors N1, N2 up to N8 with N8 integers, I guess I should zoom in again. So each Ni is an integer. So if you do this in two dimensions, you're just getting a sort of square lattice. And you can do the same thing in any other number of dimensions and you get a lattice. And this lattice is unimodular, meaning it has one point per unit volume. It's fairly obvious, but it's not even because the distance between two lattice points is of the form the square root of N1 squared plus N2 squared plus N8 squared. And this can be even or it can be odd. So it's not an even lattice. To turn it into an even lattice, you can do the following thing. You take all the vectors N1, N2 up to N8 with the N1 plus N2 and so on plus N8 is even. And all the Ni integers or all Ni integers plus a half. So by making the sum even, you're kind of throwing half of them away. And by allowing them to be half integers, you're kind of adding them back in again. So there's still one point per unit volume. And this has the nice property that this is now an even lattice. The distance between any two points is the square root of an even integer. For example, one of the vectors is a half a half up to a half. And the distance of this from the origin is the square root of a half squared plus a half squared plus a half squared. And you've got eight of these. So the distance is root two. And you can see whenever the dimension is divisible by eight, you can do a similar construction like this and you'll get an even lattice because the distance between any two lattice points will be the square root of some even integer. So VIT undoubtedly found the analog of this lattice in 24 dimensions. And there are a few other lattices that you can construct like that. Well, in 1970, VIT claimed that one of these 11 lattices he found in around 1940 was in fact, the leach lattice. He said he found nine lattices in 1938 and he found two more in January, 1940, one of which was the leach lattice. And he wrote a sort of note which is not very easy to read, but sort of looks like this. So this is his, it's handwritten German and not all that easy to read even in the version I have here. But what he does is he actually gives an explicit construction of his lattice. Here, if you look, there's a little matrix here and what he's describing here is a construction that does in fact give a lattice. And I sort of went through it and it does in fact give the leach lattice. However, it's a very peculiar construction because it's quite unlike any of the other constructions of the leach lattice that anyone else was giving in the 1960s. People have come up with about a dozen different construction and VITs is quite unlike all the others. However, it's very similar to a construction that O'Connor and Powell gave in 1944 of a lattice related to the leach lattice called the odd leach lattice. So it seems that VIT was using some ideas that were kind of floating around in the 1940s. His construction was closely related to something called the Steiner system. So let me explain what a Steiner system is. So a Steiner system, Steiner system is... Well, the example of the Steiner system that VIT was working with was called S5824. What does this mean? Well, the 24 means you take 24 points. One, two, three, four, five, six, one, two, three, four, five, six. And the five means you're interested in five element subsets of them. So here, for example, is a five element subset. And the eight means you're given various blocks, all of which have eight elements. So this Steiner system is 759 blocks and each block has eight points. And the point of the Steiner system is that every five element subset is contained in exactly one block. And where does the 759 come from? Well, the number of five element subsets of 24 points is 24 choose five. And the number of blocks is 759 and each eight element block contains eight choose five five element subsets. So every one of these 24 choose five five element subsets is in exactly one of these 759 blocks. And as each block contains this number of five element subsets, we get this equality here. So that's where the number 759 comes from. Anyway, Leach's construction of the Leach lattice made heavy use of the Steiner system. So knowing about the Steiner system would be very helpful in constructing the Leach lattice. And Vitt was actually an expert on this Steiner system. He wrote the fundamental paper on this Steiner system that was the basic paper everybody used for several decades. So let's summarize the evidence for Vitt discovering the Leach lattice. First of all, he said he constructed it. That's reasonably strong evidence. Secondly, he actually gave a construction of it which was unlike anybody else's, which suggests that. Thirdly, he was an expert in the technology you need to construct it. So it's very plausible he would have been able to. And fourthly, there were similar constructions of related lattices giving it about the same time. So that's quite strong evidence that he did construct the Leach lattice. The evidence against him constructing the Leach lattice is mainly that he never mentioned it. I mean, if the Leach lattice was so important, why did Vitt never say anything about it for nearly 30 years? I think the answer is he may just not have realized at the time it was important. He had found a handful of 10 or 11 lattices and there was no particular reason to think they were at all interesting and he hadn't managed to completely enumerate them and basically just forgot about it. I can, I mean, it's very plausible. I mean, I've enumerated collections of lattices, looked at them, decide they were boring and decided they weren't worth publishing. So I can well believe Vitt did the same. So to summarize, I think it very likely that Vitt did in fact discover the Leach lattice. You know, I put a probability of this of more than about 90%.