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# Second Woman Fields Medallist: E8 and Sphere Packing

[GitHub](https://github.com/HectorTablero/QED-Scripts/tree/main/Magazine%202/Second%20Woman%20Fields%20Medallist) (Open Source, MIT License) | [Live Demo](https://qed.mat.uam.es/revista/articulo/medallista-fields)

**Date:** July 2026
**Technologies:** JavaScript, Tailwind CSS

---

## Overview

This news bulletin covers Maryna Viazovska's proof that the $E_8$ lattice gives the densest sphere packing in 8 dimensions. Since neither an 8-dimensional lattice nor its packing density can be drawn directly, the two elements tackle the article's two hardest-to-picture claims separately: what the 240 roots of $E_8$ actually look like, via a 2D shadow that is not arbitrary but the genuine Coxeter-plane projection: and what "packing density" even means, via a fully interactive 2D analogue the reader can optimize by hand.

## E8RootProjection

### Description

Generates and projects the 240 roots of the $E_8$ root system, using the exact definition the article gives — $\pm e_i \pm e_j$ for all coordinate pairs, plus all $(\pm\tfrac12)^8$ vectors with an even number of minus signs.

### Implementation

```javascript
// Example element configuration
::: js E8RootProjection
8
e8root1
autoPlay, allowPause, autoPauseOnScroll, allowFullscreen, openControls, h=200
:::
```

The single parameter controls how many of the eight concentric "rings" of 30 roots are shown initially, letting the reader build up the full picture layer by layer or narrow in on the innermost shells.

### The Coxeter Plane

Rather than picking an arbitrary 2D projection, the element computes the genuine invariant plane on which the Coxeter element $w = s_1 s_2 \cdots s_8$ (the product of reflections in the eight simple roots, order $h=30$) acts as a pure rotation by $2\pi/30$. This plane is extracted with the Fourier projector $P = \tfrac{2}{h}\sum_k \cos(2\pi k/h)\, w^k$ applied to an arbitrary seed vector, which isolates the 2-dimensional isotypic component the Coxeter element rotates. Projected into this plane, the 240 roots collapse into exactly eight concentric rings of 30 points each — the structure the article calls hard to picture, made literal.

### Verification and Interaction

Every claim the article makes is checked live rather than merely asserted: a parity check confirms all 240 generated vectors satisfy the even-sum condition, and the minimum pairwise distance is computed by brute-force search and displayed alongside its expected value $\sqrt2$. Hovering a root displays its full 8-coordinate vector, and a "spin the plane" toggle rotates the projection continuously to emphasize that the ring structure is a genuine rotational symmetry, not a projection artifact.

## LatticePackingDensity

### Description

A fully interactive 2D sphere-packing sandbox: the reader drags two basis vectors to define a lattice, and the element draws the resulting circle packing (circles of radius half the shortest lattice vector) and computes its density live, letting them attempt to rediscover the hexagonal optimum $\pi/\sqrt{12}$ by hand.

### Implementation

```javascript
// Example element configuration
::: js LatticePackingDensity
1, 0, 0.25, 0.95
empaquetado1
autoPlay, allowFullscreen, openControls, h=190
:::
```

Parameters `[v1x, v1y, v2x, v2y]` set the initial basis vectors. Preset buttons snap directly to the square and hexagonal lattices, or randomize the second vector, for quick comparison against the reader's own attempts.

### Shortest-Vector Reduction

The packing radius depends on the lattice's shortest non-zero vector, which a naive fixed-window integer search gets wrong whenever the basis vectors are nearly parallel (the true shortest vector can then be a large combination of both, well outside any small search window — the source comments give the concrete failure case `v1=(1,0), v2=(0.1,0.01)`). Instead, `LatticePackingDensity.reduce()` performs an exact Lagrange–Gauss lattice reduction, repeatedly subtracting the nearest integer multiple of the shorter vector from the longer and swapping, terminating in $O(\log)$ steps with the true shortest vector in hand.

### Technical Considerations

A "best found" tracker records the highest density the reader has achieved across all their drags, turning the exercise into an implicit optimization the reader performs manually; once the density crosses $\pi/\sqrt{12} - \epsilon$ (the planar optimum), a congratulatory label and a color change on the density readout confirm they have reached it. A degenerate-basis guard (`det() < 0.05`) prevents the two vectors from being dragged into near-collinearity, which would otherwise blow up the reduction and the drawn circle radius.