Biangular lines with angles arccos(1/5) and arccos(3/5)

Imagine you are standing in the center of a giant, invisible room. You hold a long, thin stick that points out from your hand to the wall. Now, imagine you have a whole bunch of these sticks, all starting from your hand, pointing in different directions. In the world of mathematics, this is called a "line system." Usually, mathematicians love it when everything is perfectly equal. If you have a set of sticks where every single pair makes the exact same angle, that's called "equiangular lines," and it's a bit like a perfectly balanced mobile hanging from the ceiling.

But life is rarely that perfect. Sometimes, you want to know what happens if you allow two different angles instead of just one. This is the world of "biangular lines." It's like a dance where partners can only step together at two specific angles: maybe a wide, lazy lean or a sharp, quick turn. The question mathematicians ask is: "How many dancers can we fit in this room before they start bumping into each other?" The bigger the room (the higher the dimension), the more dancers you might think you can fit, but the rules of geometry are strict. There's a limit to how many sticks you can pack in without breaking the "two-angle" rule.

This paper is a detective story about finding the absolute maximum number of these sticks in rooms of specific sizes, from dimension 7 up to dimension 10, and even peeking at dimension 15. The authors, Paul Tricot, are investigating a very specific type of dance: one where the angles between the sticks are fixed at two very precise values, determined by the numbers 1/5 and 3/5. They aren't just guessing; they are using a clever trick that connects these dancing sticks to "integral lattices." Think of a lattice as a giant, invisible grid of dots, like the points on a graph paper that extends into infinity. The authors discovered that if you arrange your sticks just right, they line up perfectly with the "roots" (special vectors) of these grids. By using this connection, they can prove exactly how many sticks fit in the room and show that the known arrangements are actually the best possible ones.

The Great Stick Packing Contest

The paper focuses on a specific puzzle: In a 7-dimensional space (which is hard to visualize, so imagine a room with 7 different directions you can move), what is the maximum number of lines you can draw through the center if every pair of lines must meet at an angle of either $\arccos(1/5)$ or $\arccos(3/5)$?

Before this paper, mathematicians knew some big numbers. For dimensions 7 through 20, the largest known groups of lines had sizes like 72, 126, 240, and so on. These were built using a clever method involving special grids called root lattices (specifically $E_6$, $E_7$, and $E_8$). But nobody knew for sure if these were the largest possible groups, or if someone could squeeze in a few more sticks by being even more clever.

Tricot's paper says: "Let's check if we can do better." And the answer, for dimensions 7, 8, and 9, is a resounding no.

Here is what the paper actually found, broken down by room size:

• Dimension 7: The paper proves that the maximum number of lines is exactly 72. If you try to fit 73, the geometry breaks. The only way to get 72 is to use the specific arrangement built from the $E_6$ lattice. It's the only solution.
• Dimension 8: The limit is 126 lines. Again, the paper proves you can't fit more. The only way to reach this number is by using the arrangement derived from the $E_7$ lattice.
• Dimension 9: The limit is 240 lines. The proof shows that the arrangement from the $E_8$ lattice is the unique winner. You cannot squeeze in a 241st line.
• Dimension 10: The paper sets out to classify the largest systems in this dimension as well, following the same rigorous process used for the lower dimensions. However, the provided text of the paper cuts off before the final proof for dimension 10 is completed, leaving that specific limit as a work in progress within this document.

The authors didn't just guess these numbers; they used a mix of logical deduction and computer searches to rule out every other possibility. They showed that if you assume there is a bigger group of lines, you eventually run into a mathematical contradiction—like trying to fit a square peg in a round hole, but the hole is made of pure math.

The "Special Triangle" Clue

How did they prove this? They used a detective tool called a "special triangle." Imagine three sticks in your hand. If two of them lean at the "wide" angle ($\arccos(3/5)$) and the third one leans at a "sharp" angle ($\arccos(-1/5)$, which is the opposite of the wide angle), they form a special triangle.

The paper proves that in any large group of these lines (72 or more), you must have these special triangles. Once you know they exist, the math gets very rigid. The authors showed that if you have these triangles, the entire group of lines is forced to align with a specific grid (a lattice). Once they forced the lines to align with the grid, they could use computer programs to check every possible way to arrange the sticks on that grid. The computers checked millions of combinations and found that in every case, you couldn't get more than the known numbers (72, 126, 240).

A New Discovery in Dimension 15

While the main job was proving the limits for dimensions 7 through 9 (and starting on 10), the paper also had a fun surprise. In dimension 15, the authors constructed a new system of 456 lines. This matches the size of the best-known system found by other mathematicians (Ganzhinov and Szöllősi), but this paper provides a fresh way to build it. It's like finding a new, slightly different recipe that makes the exact same delicious cake. This confirms that 456 is a very strong candidate for the maximum in dimension 15, though the paper doesn't claim to have proved it's the absolute limit (that's a harder job for higher dimensions).

The Bottom Line

This paper is a victory for certainty. For dimensions 7, 8, and 9, we now know the exact maximum number of lines that can dance to the tune of angles $\arccos(1/5)$ and $\arccos(3/5)$. The numbers are 72, 126, and 240. The paper proves that the arrangements we already knew about are not just good—they are the only ways to reach these maximums.

The authors used the connection between these lines and integral lattices (the invisible grids) as a superpower. By turning the problem of "fitting sticks" into a problem of "fitting points on a grid," they could use computers to check every possibility and prove that no one can do better. It's a reminder that in the vast, abstract world of high-dimensional geometry, sometimes the most beautiful patterns are also the only ones that fit.